1 Introduction↩︎

1.1 Smoothing fibre bundles↩︎

The study of smooth and topological fibre bundles with a compact smooth \(d\)-manifold \(M\) as fibre lies at the heart of geometric topology. Through the lens of homotopy theory, it becomes the study of the homotopy types of the classifying spaces \(B\mathrm{Diff}_\partial(M)\) and \(B\mathrm{Homeo}_\partial(M)\) of the topological groups of diffeomorphisms and homeomorphisms of \(M\), fixing the boundary. In this work, we are mostly concerned with the case \(M=D^d\) of a closed disc, which plays a distinguished role in the theory as it, informally speaking, measures the “difference” between smooth and topological fibre bundles. Let us explain why.

Formally, this “difference” is the homotopy type of the fibre \(\mathrm{Homeo}_\partial(M)/\mathrm{Diff}_\partial(M)=\mathrm{hofib}(B\mathrm{Diff}_\partial(M)\rightarrowB\mathrm{Homeo}_\partial(M))\) since this homotopy type classifies smooth fibre bundles together with a topological trivialisation. It is the source of a scanning map \[\label{equ:intro-smoothing-theory}\mathrm{Homeo}_\partial(M)/\mathrm{Diff}_\partial(M)\longrightarrow\Gamma_\partial\big(\mathrm{Fr}(M)\times_{\mathrm{O}(d)}\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow M\big)\tag{1}\] whose target is the space of sections, standard near the boundary, of the bundle associated to the frame bundle \(\mathrm{Fr}(M)\) using the left \(\mathrm{O}(d)\)-action on the fibre \(\mathrm{Top}(d)/\mathrm{O}(d)=\mathrm{hofib}(B\mathrm{O}(d)\rightarrow B\mathrm{Top}(d))\) where \(\mathrm{Top}(d)=\mathrm{Homeo}(\mathbf{R}^d)\) is the topological group of homeomorphisms of \(\mathbf{R}^d\). Smoothing theory asserts that the scanning map 1 is a weak equivalence onto the path-components it hits as long as \(d\neq 4\) [1] so, by obstruction theory, the main limiting factor in understanding the difference between smooth and topological fibre bundles is sufficient knowledge of the homotopy groups of \(\mathrm{Top}(d)/\mathrm{O}(d)\). In degrees up to \(d\), these groups are classically well understood, via surgery theory and work of Kirby–Siebenmann [1]. Beyond this degree these groups are less understood, but admit the following alternative description: specialising 1 to \(M=D^d\), the Alexander trick \(B\mathrm{Homeo}_\partial(D^d)\simeq*\) results in Morlet’s equivalence \[\label{equ:intro-Morlet}B\mathrm{Diff}_\partial(D^d)\simeq \Omega^d_0\mathrm{Top}(d)/\mathrm{O}(d),\tag{2}\] so we have \(\pi_{d+k}(\mathrm{Top}(d)/\mathrm{O}(d))\cong \pi_k(B\mathrm{Diff}_\partial(D^d))\), leaving us wishing to understand the latter.

1.2 Rational homotopy groups of \(B\mathrm{Diff}_\partial(D^d)\)↩︎

As a first approximation, one might attempt to determine \(\pi_k(B\mathrm{Diff}_\partial(D^d))\) after rationalisation (indicated by a \((-)_\mathbf{Q}\)-subscript). There are three known sources of nontrivial classes in these rational homotopy groups:

Algebraic \(K\)-theory↩︎

Based on Waldhausen’s work on concordance and algebraic \(K\)-theory [2], Farrell–Hsiang [3] expressed \(\pi_*(B\mathrm{Diff}_\partial(D^d))_\mathbf{Q}\) partially in terms of \(K_*(\mathbf{Z})\): \[\label{equ:into-farrell-hsiang} \pi_k(B\mathrm{Diff}_\partial(D^d))_\mathbf{Q}\cong\begin{cases} K_{k+1}(\mathbf{Z})_\mathbf{Q}&d\text{ odd}\\0&d\text{ even}\end{cases}\quad\text{for }k\le \phi(D^d).\tag{3}\] Here \(\phi(D^d)\) is the concordance stable range, so \(d/3 \lesssim\phi(D^d)\) by work of Igusa [4] (but see 1.3). From Borel’s work [5] it is known (and used by Farell–Hsiang) that \(K_{k+1}(\mathbf{Z})\cong\mathbf{Q}\) for positive \(k\equiv0\ (\mathrm{mod}\ 4)\), and \(K_{k+1}(\mathbf{Z})=0\) otherwise. Via different methods, the isomorphism 3 was later improved to \(k\le d-5\) for even \(d\) and to \(k\le d-6\) for odd \(d\), by Randal-Williams [6] and Krannich [7] respectively.

Watanabe’s graph classes↩︎

In [8], [8]–[10], Watanabe constructed classes in the homotopy groups \(\pi_{k(d-3)}(B\mathrm{Diff}_\partial(D^d))_\mathbf{Q}\) for \(k\ge1\) and \(d\ge4\), indexed by trivalent graphs, and showed that many of them are nontrivial by evaluating them against characteristic classes which had been described by Kontsevich, in terms of configuration space integrals. The classes of lowest degree Watanabe finds show that one has \[\label{equ:intro-Watanabe-first-class}\pi_{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1)}))_\mathbf{Q}\neq 0 \text{ for ``many'' }n\quad \text{and}\quad\pi_{4n-6}(B\mathrm{Diff}_\partial(D^{2n}))_\mathbf{Q}\neq0\text{ for all }n\ge2;\tag{4}\] “many” refers to an numerical condition on \(n\ge2\) that is often satisfied. The Kontsevich classes responsible for 4 are indexed by the “theta” graph and the complete graph with four vertices.

Pontryagin–Weiss classes↩︎

By work of Sullivan [11] and Kirby–Siebenmann [1], the map \(B\mathrm{O}\rightarrow B\mathrm{Top}\), obtained by stabilising with \((-)\times\mathbf{R}\) the maps \(B\mathrm{O}(d)\rightarrow B\mathrm{Top}(d)\) that classify the underlying topological \(\mathbf{R}^d\)-bundle of a vector bundle, is a rational equivalence, so we have \(\mathrm{H}^*(B\mathrm{Top};\mathbf{Q})=\mathrm{H}^*(B\mathrm{O};\mathbf{Q})=\mathbf{Q}[p_1,p_2,\ldots]\). Pulling back \(p_i\) along the map \(B\mathrm{Top}(d)\rightarrow B\mathrm{Top}\), one obtains Pontryagin classes \(p_i\in\mathrm{H}^{4i}(B\mathrm{Top}(d);\mathbf{Q})\) for topological \(\mathbf{R}^d\)-bundles which extend the usual classes for vector bundles, so one cannot help but wonder whether the relations

1. \(p_i=0\) in \(\mathrm{H}^{4i}(B\mathrm{O}(d);\mathbf{Q})\) for \(i> \lfloor \tfrac{d}{2}\rfloor\)

2. \(p_n=e^2\) in \(\mathrm{H}^{4n}(B\mathrm{SO}(2n);\mathbf{Q})\)

one is used to for vector bundles are also satisfied in \(\mathrm{H}^*(B\mathrm{STop}(d);\mathbf{Q})\); here \(e\in\mathrm{H}^{2n}(B\mathrm{STop}(2n);\mathbf{Q})\) denotes the Euler class. Remarkably the answer is no: Weiss [12] showed that [P1] and [P2] do not hold in \(\mathrm{H}^*(B\mathrm{STop}(d);\mathbf{Q})\) for “many” values of \(d\) (see loc.cit. for precise statements). Moreover, using [P1] and [P2], one sees that there are unique classes \[p_i^\tau\in\mathrm{H}^{4i-1}(\mathrm{O}(d)/\mathrm{Top}(d);\mathbf{Q})\text{ for }i>\lfloor \tfrac{d}{2}\rfloor\quad\text{and}\quad (p_n-e^2)^\tau\in\mathrm{H}^{4n-1}(\mathrm{O}(2n)/\mathrm{Top}(2n);\mathbf{Q})\] which transgress to \(p_i\) and \(p_n-e^2\) in the Serre spectral sequence of the fibration \(\mathrm{O}(d)/\mathrm{Top}(d)\rightarrow B\mathrm{SO}(d)\rightarrow B\mathrm{STop}(d)\). Looping these \(d\) times and using 2 thus yields classes

1. \(\Omega^{d}p_i^\tau\) \(\in \mathrm{H}^{4i-d-1}(B\mathrm{Diff}_\partial(D^{d});\mathbf{Q})\) for \(i>\lfloor\tfrac{d}{2}\rfloor\)

2. \(\Omega^{2n}(p_n-e^2)^\tau\) \(\in \mathrm{H}^{2n-1}(B\mathrm{Diff}_\partial(D^{2n});\mathbf{Q})\).

Weiss’ work also shows that [PW1] and [PW2] “often” evaluate nontrivially against classes in the image of the rational Hurewicz map, which in particular gives (see Proposition 6.2.1. loc.cit.) \[\pi_{4i-d-1}(B\mathrm{Diff}_\partial(D^{d}))_\mathbf{Q}\neq 0\quad\text{for }d>c_1\text{ and }\lfloor \tfrac{d}{2}\rfloor<i<\tfrac{9}{8}d-c_2,\] for certain positive constants \(c_1\) and \(c_2\). We will refer to classes in \(\pi_*(B\mathrm{Diff}_\partial(D^d))\) which evaluate nontrivially against [PW1] or [PW2] as Pontryagin–Weiss classes.

1.2.1 First main result↩︎

The first result we state determines the groups \(\pi_k(B\mathrm{Diff}_\partial(D^d))_\mathbf{Q}\) completely for \(k\lesssim \tfrac{3}{2}d\), and it in particular shows that the three sources of classes \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}\)–\(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}\) exhaust these groups in this range. We also discover that Watanabe’s first graph class from \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}\) is in fact a Pontryagin–Weiss class, in a sense to be explained (see 3).

1.2.2 The class \(E\)↩︎

In the statement of 1, the first summand of \(\pi_k(B\mathrm{Diff}_\partial(D^{2n+1}))_\mathbf{Q}\) and all of \(\pi_k(B\mathrm{Diff}_\partial(D^{2n}))_\mathbf{Q}\) consists of Pontryagin–Weiss classes in the sense of \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}\). Note however, that for \(d=2n+1\) the first class in [PW1] has degree \(2n+2\), but the first summand in 1 contains a class of degree \(2n-2\). This is due to an odd-dimensional analogue of [P2], which seems to be less well known. Namely, writing \(\mathrm{G}(d)=\mathrm{hAut}(S^{d-1})\) and \(\mathrm{SG}(d)\subset \mathrm{G}(d)\) for its orientation-preserving variant, there is a unique class \[\label{intro-E} E\in \mathrm{H}^{4n}(B\mathrm{SG}(2n+1);\mathbf{Q})\tag{5}\] which pulls back to the square of the Euler class \(e^2\in \mathrm{H}^{4n}(B\mathrm{SG}(2n);\mathbf{Q})\) along the stabilisation map \(B\mathrm{SG}(2n)\rightarrow B\mathrm{SG}(2n+1)\) induced by suspension. Pulled back along the composition \(B\mathrm{SO}(2n+1)\rightarrow B\mathrm{STop}(2n+1)\rightarrow B\mathrm{SG}(2n+1)\) involving the map classifying the spherical fibration of an \(\mathbf{R}^d\)-bundle, the class \(E\) agrees with \(p_n\), so we can add to [P1] and [P2] the relation

1. \(p_n=E\) in \(\mathrm{H}^{4n}(B\mathrm{SO}(2n+1);\mathbf{Q})\)

which, as with [PW1] and [PW2], gives rise to a characteristic class

1. \(\Omega^{2n+1}(p_n-E)^\tau\in \mathrm{H}^{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1});\mathbf{Q})\).

This invariant detects the class of degree \(2n-2\) in the first summand in 1.

We proceed this introduction by explaining two applications of 1, one to the concordance stable range as featuring in \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}\), and one to the rational homotopy type of \(B\mathrm{Top}(d)\).

1.3 The rational concordance stable range↩︎

Closely related to the diffeomorphism group \(\mathrm{Diff}_\partial(M)\) of a compact smooth manifold \(M\) is its space of concordances \[\mathrm{C}(M)=\{\phi\colon M\times [0,1]\xrightarrow{\cong}M\times [0,1]\mid \phi_{M\times \{0\}\cup \partial M\times [0,1]}=\mathrm{id}\},\] which in turn relates to the concordance space of \(M\times [0,1]\) via the Hatcher stabilisation map \[\label{equ:stabilsation}s_M\colon\mathrm{C}(M)\longrightarrow\mathrm{C}(M\times [0,1]),\tag{6}\] given by taking products with \([0,1]\) and “bending” the resulting diffeomorphism appropriately to make it satisfy the boundary condition (see e.g. [4]). The concordance stable range of \(M\) is defined in terms of these stabilisation maps as the quantity \[\phi(M)\coloneq \min\{k\in \mathbf{Z}\mid s_{M\times [0,1]^m}\text{ is }k\text{-connected for all }m\ge0\},\] and the rational concordance stable range \(\phi(M)_\mathbf{Q}\) is defined analogously in terms of the rational connectivity of the stabilisation maps. This range plays a crucial role in the study of \(B\mathrm{Diff}_\partial(M)\) and \(B\mathrm{Homeo}_\partial(M)\) in that it is the main limitation when trying to access these homotopy types via the classical route in terms of unstable homotopy theory and \(K\)- and \(L\)-theory (see e.g. [14] for a survey). As briefly mentioned in \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}\), the main result on \(\phi(M)\) is due to Igusa [4] who established a lower bound \(\phi(M)\gtrsim\dim(M)/3\) (more precisely \(\phi(M)\ge \min(\tfrac{1}{3}(\dim(M)-4),\tfrac{1}{2}(\dim(M)-7))\)), but the precise concordance stable range is the subject of speculation (see e.g. [4], [15], [16]) and is still not known for a single manifold \(M\). Based on a strengthening of 1, we determine the rational concordance stable range for high-dimensional discs and compute the first rational obstruction to stability.

1.4 The rational homotopy type of \(B\mathrm{STop}(d)\)↩︎

Combining 2 and the known rational homotopy type of \(B\mathrm{O}(d)\) with a variant of 1, one can easily calculate \(\pi_*(B\mathrm{Top}(d))_\mathbf{Q}\) in a range. But unlike \(B\mathrm{Diff}_\partial(D^d)\), the space \(B\mathrm{Top}(d)\) is not a loop space, so its rational homotopy type need not be determined by its rational homotopy groups. Our second main result describes this rational homotopy type in a \((\approx\tfrac{5}{2}d)\)-range, using a strong form of 1 (see 2). We state the answer in terms of the orientation-preserving variant \(B\mathrm{STop}(d)\), and it involves (i) the Euler class \(e\in \mathrm{H}^{2n}(B\mathrm{STop}(2n);\mathbf{Q})\), (ii) the odd-dimensional analogue of its square \(E\in \mathrm{H}^{4n}(B\mathrm{STop}(2n+1);\mathbf{Q})\) from 1.2.2, (iii) the stabilisation map \(s\colon B\mathrm{STop}(d)\rightarrow B\mathrm{STop}\simeq_\mathbf{Q}\prod_{i\ge1}K(\mathbf{Q},4i)\), and (v) a certain map \(\tilde{k}\colon B\mathrm{STop}(2n+1)\rightarrow\Omega^{\infty-2n-1}(K(\mathbf{Z})/\mathbf{S})_\mathbf{Q}\) that factors a canonical map \(k\colon \mathrm{Top}(2n+2)/\mathrm{Top}(2n+1)\rightarrow\Omega^{\infty-2n-1}K(\mathbf{Z})\) (more about this map in 1.5.4) followed by quotienting out the unit \(\mathbf{S}\rightarrow K(\mathbf{Z})\) and rationalising.

1.5 Orthogonal calculus and the second derivative of \(B\mathrm{Top}(-)\)↩︎

All of the above can be recast (and strengthened) in terms of Weiss’ orthogonal calculus, which provides a unifying perspective that greatly conceptualises our results. This part of the introduction serves to explain this viewpoint, and what we prove about it.

1.5.1 The Taylor tower↩︎

The spaces \(B\mathrm{O}(d)\), \(B\mathrm{Top}(d)\), and \(\mathrm{Top}(d)/\mathrm{O}(d)\) may be viewed as the values at \(\mathbf{R}^d\) of continuous functors \(\mathsf{Bo}\coloneq B\mathrm{O}(-)\), \(\mathsf{Bt}\coloneq B\mathrm{Top}(-)\), and \(\mathsf{t}\mskip-2mu/\mskip-2mu\mathsf{o}\mskip 1mu\coloneq \mathrm{Top}(-)/\mathrm{O}(-)\) from the category of finite-dimensional inner product spaces and isometries to the category of pointed spaces. Functors of this type are called orthogonal, and they are the input for Weiss’ apparatus of orthogonal calculus [17]. To an orthogonal functor \(\mathsf{E}\) this calculus associates for each \(k\ge1\) a (naïve) \(\mathrm{O}(k)\)-spectrum \(\Theta \mathsf{E}^{(k)}\)—the \(k\)th derivative—and a tower \[\begin{tikzcd}[column sep=2cm,row sep=0.3cm,/tikz/column 3/.append style={anchor=base west}] &\smash{\vdots} \dar\\ &T_2\mathsf{E}(V) \dar &[-1.5cm]\lar\Omega^\infty\big((S^{2\cdot V}\wedge \Theta \mathsf{E}^{(2)})_{h\mathrm{O}(2)}\big) \\ &T_1\mathsf{E}(V) \dar &\lar\Omega^\infty\big((S^{1\cdot V}\wedge \Theta \mathsf{E}^{(1)})_{h\mathrm{O}(1)}\big)\\ \mathsf{E}(V) \arrow[r,swap, pos=0.7 ] \arrow[ru,swap, pos=0.69] \arrow[ruu, swap,pos=0.67]\arrow[ruuu,]& T_0\mathsf{E}(V)&\arrow[l,equal]\mathsf{E}(\mathbf{R}^\infty) \end{tikzcd}\] of orthogonal functors—the Taylor tower. Some explanation is in order: the right-hand horizontal maps—the layers—indicate the homotopy fibre inclusions of the vertical maps between the stages \(T_k\mathsf{E}\), the zeroth stage \(T_0\mathsf{E}\) (also called the zeroth layer) admits a preferred equivalence to the constant functor on the value at infinity \(\mathsf{E}(\mathbf{R}^\infty)\coloneq \mathop{\mathrm{hocolim}}_d\mathsf{E}(\mathbf{R}^d)\), the space \(S^{k\cdot V}\) is the one-point compactification of \(k\cdot V\coloneq \mathbf{R}^k\otimes V\), and \(\mathrm{O}(k)\) acts on \(S^{k\cdot V}\wedge \Theta E^{(k)}\) diagonally.

1.5.2 Derivatives↩︎

Forgetting the \(\mathrm{O}(k)\)-actions for a moment, let us outline how the derivatives \(\Theta\mathsf{E}^{(k)}\) arise: setting \(\mathsf{E}^{(1)}(V)\coloneq \mathrm{hofib}(\mathsf{E}(V)\rightarrow\mathsf{E}(\mathbf{R}\oplus V))\), rotating in a \(2\)-plane induces maps \[\label{intro-first-unstable-derivative}\mathsf{E}^{(1)}(V)\longrightarrow\Omega \mathsf{E}^{(1)}(\mathbf{R}\oplus V),\tag{7}\] which define a sequential spectrum with \(d\)th space \(\mathsf{E}^{(1)}(\mathbf{R}^d)\); this is the first derivative \(\Theta \mathsf{E}^{(1)}\) as constructed in [17] (see also [18] for an explicit formula for 7 involving the aforementioned rotation in a plane). Moreover, setting \(\mathsf{E}^{(2)}(V)\coloneq \mathrm{hofib}(\mathsf{E}^{(1)}(V)\rightarrow\Omega \mathsf{E}^{(1)}(\mathbf{R}\oplus V))\), there are further natural maps \[\mathsf{E}^{(2)}(V)\longrightarrow\Omega^2 \mathsf{E}^{(2)}(\mathbf{R}\oplus V)\] which give rise to a sequential spectrum with \(2d\)th space \(\mathsf{E}^{(2)}(\mathbf{R}^d)\); this is \(\Theta \mathsf{E}^{(2)}\). Continuing in this manner by taking iterated homotopy fibres gives rise to the higher derivatives \(\Theta \mathsf{E}^{(k)}\) as spectra, though this iterative description obscures the \(\mathrm{O}(k)\)-action (see [17] for details).

1.5.3 An example: the tower for \(B\mathrm{O}(-)\)↩︎

In the example \(\mathsf{Bo}=B\mathrm{O}(-)\), the derivatives \(\Theta \mathsf{Bo}^{(k)}\) are well understood. Here the map 7 giving rise to \(\Theta\mathsf{Bo}^{(1)}\) agrees with the loop-suspension map \[\label{intro-suspension-map} S^d=\mathrm{O}(d+1)/\mathrm{O}(d)\longrightarrow\Omega\mathrm{O}(d+2)/\mathrm{O}(d+1)=\Omega S^{d+1},\tag{8}\] so \(\Theta\mathsf{Bo}^{(1)}\) is the sphere spectrum \(\mathbf{S}\). Moreover, from the metastable EHP-description of the fibre of 8 one sees \(\Theta \mathsf{Bo}^{(2)}\simeq \mathbf{S}^{-1}\). Moreover, the \(\mathrm{O}(1)\)-action on \(\mathbf{S}\) can be seen to be trivial, and the \(\mathrm{O}(2)\)-action on \(\mathbf{S}^{-1}\) to be rationally trivial. Based on this, it is an instructive exercise to compute the layers of the tower of \(\mathsf{Bo}\) evaluated at \(\mathbf{R}^d\) after rationalisation (fibrewise over \(B\pi_1(-)\), indicated by a \((-)_\mathbf{Q}\)-subscript) as: \[\label{equ:tower-o} \begin{tikzcd}[ampersand replacement=\&, column sep=2cm, row sep=0.5cm, /tikz/column 3/.append style={anchor=base west}] \&T_2B\mathrm{O}(d)_\mathbf{Q}\dar\&[-1.5cm]\lar \prod_{i=\lceil d/2\rceil}^{ \infty}K(\mathbf{Q},4i-1)\\ \&T_1B\mathrm{O}(d)_\mathbf{Q}\dar \&\lar \smash{\begin{cases}K(\mathbf{Q},d)&d\text{ even}\\ *&d\text{ odd}\end{cases}}\\ B\mathrm{O}(d)_\mathbf{Q}\arrow[r ] \arrow[ru,swap, pos=0.69] \arrow[ruu,] \& B\mathrm{O}_\mathbf{Q}.\& \end{tikzcd}\tag{9}\] Vaguely speaking, the zeroth layer creates the Pontryagin classes, the first layer creates the Euler class in even dimensions, the second layer imposes the relations [P1] and [P2] (the relation [P3] need not be imposed as \(E\) is never created), and the higher layers are rationally trivial. In particular, the map to the second stage \(B\mathrm{O}(d)\rightarrow T_2B\mathrm{O}(d)\) is a rational equivalence.

We now turn from the tower of \(\mathsf{Bo}=B\mathrm{O}(-)\) to that of its topological cousin \(\mathsf{Bt}=B\mathrm{Top}(-)\).

1.5.4 The first derivative of \(B\mathrm{Top}(-)\)↩︎

As mentioned in , the map between the zeroth stages \(B\mathrm{O}\rightarrow B\mathrm{Top}\) of the towers of \(\mathsf{Bo}\) and \(\mathsf{Bt}\) is a rational equivalence. Waldhausen’s work on concordance theory [20] shows that the spectrum \(\Theta\mathsf{Bt}^{(1)}\) defined by the topological incarnation of the loop-suspension map 8 \[\label{intro-topological-loop-suspension}\mathrm{Top}(d+1)/\mathrm{Top}(d)\longrightarrow\Omega \mathrm{Top}(d+2)/\mathrm{Top}(d+1)\tag{10}\] agrees with his \(\mathrm{A}(*)=K(\mathbf{S})\). Combined with the linearisation \(K(\mathbf{S})\rightarrow K(\mathbf{Z})\) this yields the map \(k : \mathrm{Top}(d+1)/\mathrm{Top}(d)\rightarrow\Omega^\infty(S^d\wedge K(\mathbf{Z}))\) mentioned before the statement of 3. Moreover, the map \(\mathbf{S}\simeq\Theta \mathsf{Bo}^{(1)}\rightarrow\Theta \mathsf{Bt}^{(1)}\simeq K(\mathbf{S})\) induced by the inclusion \(\mathrm{O}(-)\subset \mathrm{Top}(-)\) is given by the unit, so \(\Theta \mathsf{t}\mskip-2mu/\mskip-2mu\mathsf{o}\mskip 1mu^{(1)}\simeq \Omega K(\mathbf{S})/\mathbf{S}=\Omega\mathrm{Wh}^{\mathrm{Diff}}(*)\) since taking derivatives preserves fibre sequences. Furthermore, using smoothing theory and a bit more of the theory of orthogonal calculus, one sees that the first stage \(\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow T_1(\mathrm{Top}(d)/\mathrm{O}(d))\) of the tower for \(\mathsf{t}\mskip-2mu/\mskip-2mu\mathsf{o}\mskip 1mu=\mathrm{Top}(-)/\mathrm{O}(-)\) is \((\phi(D^d)+d+1)\)-connected where \(\phi(D^d)\) is the stable range from 1.3. Taking fibres over the zeroth stage \(\mathrm{Top}/\mathrm{O}\), using \(\mathrm{\widetilde{Diff}}_\partial(D^d)/\mathrm{Diff}_\partial(D^d)\simeq \Omega^d\mathrm{hofib}\big(\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow\mathrm{Top}/\mathrm{O}\big)\), and looping \(d\) times, this yields the \((\phi(D^d)+1)\)-connected map \[B\mathrm{Diff}_\partial(D^d)\simeq_\mathbf{Q}\mathrm{\widetilde{Diff}}_\partial(D^d)/\mathrm{Diff}_\partial(D^d) \longrightarrow\Omega^{\infty+d}((S^{1\cdot \mathbf{R}^d})\wedge \Omega\mathrm{Wh}^{\mathrm{Diff}}(*))_{h\mathrm{O}(1)}),\] featuring in the work of Weiss–Williams [21]. Using that the \(\mathrm{O}(1)\)-action on \(\pi_*(\Omega\mathrm{Wh}^{\mathrm{Diff}}(*))_\mathbf{Q}\) is by multiplication with \(-1\), this recovers Farrell–Hsiang’s result from .

1.5.5 The second derivative of \(B\mathrm{Top}(-)\)↩︎

Unlike for orthogonal groups, the second derivative \(\Theta\mathsf{Bt}^{(2)}\) of \(\mathsf{Bt}\) has proved to be difficult to access (see e.g. [17], [23], [24]). Prior to this work, the one piece of nontrivial information on \(\Theta\mathsf{Bt}^{(2)}\) available resulted from Weiss’ work explained in which in a sense is all about \(\Theta\mathsf{Bt}^{(2)}\). This slogan is already indicated by the above discussion of the rational Taylor tower for \(\mathsf{Bo}\) which suggests that the (non)validity of [P1] and [P2] in the cohomology of \(B\mathrm{Top}(d)\) ought to be closely related to \(\Theta\mathsf{Bt}^{(2)}\). Reis–Weiss made this precise in [25] by showing that the map of \(\mathrm{O}(2)\)-spectra \(\mathbf{S}^{-1}\simeq \Theta\mathsf{Bo}^{(2)}\rightarrow\Theta \mathsf{Bt}^{(2)}\) induced by the inclusion has a rational left homotopy inverse if and only if [P2] holds in \(B\mathrm{STop}(2n)\) for all \(n\), which we know is not the case by Weiss’ theorem.

In the final part of this work, we completely determine the second derivative \(\Theta\mathsf{Bt}^{(2)}\) rationally.

Combining 4 with the equivalence \(\Theta\mathsf{Bt}^{(1)}\simeq \mathrm{A}(*) \simeq_\mathbf{Q}K(\mathbf{Z})\) explained in 1.5.4, the first two layers of the rationalised tower for \(B\mathrm{Top}(d)\) can be computed as follows:

\[\label{equ:tower-top} \begin{tikzcd}[ampersand replacement=\&, column sep=2cm, row sep=0.6cm,, /tikz/column 3/.append style={anchor=base west}] \&T_2B\mathrm{Top}(d)_\mathbf{Q}\dar\&[-1.5cm]\lar \smash{\begin{cases}*&d\text{ even}\\ K(\mathbf{Q},2d-2)&d\text{ odd}\end{cases}}\\ \&T_1B\mathrm{Top}(d)_\mathbf{Q}\dar \&\lar \smash{\begin{cases}K(\mathbf{Q},d)&d\text{ even}\\ \Omega^{\infty-d}(K(\mathbf{Z})/\mathbf{S})_\mathbf{Q}&d\text{ odd}\end{cases}}\\ B\mathrm{Top}(d)_\mathbf{Q}\arrow[r] \arrow[ru] \arrow[ruu] \& B\mathrm{Top}_\mathbf{Q}.\& \end{tikzcd}\tag{11}\] Vaguely speaking—as for \(B\mathrm{O}(d)\)—the zeroth layer creates the Pontryagin classes and the first layer creates the Euler class but—unlike for \(B\mathrm{O}(d)\)—the first layer also creates \(K\)-theory classes and instead of imposing [P1] and [P2] the second layer creates the class \(E\) described in 1.2.2; this difference in behaviour of the second layer explains Weiss’ theorem from .

From the point of view of the tower 11 , 3 says equivalently that the second Taylor approximation \(B\mathrm{Top}(d)\rightarrow T_2B\mathrm{Top}(d)\) is rationally \((5\lfloor\tfrac{d}{2}\rfloor-5)\)-connected for \(d\ge6\), and that the second stage splits rationally into its layers when passing to the orientation preserving variant \(B\mathrm{STop}(d)\). To fit 1 into this picture, one uses that with respect to the equivalence in 4, the natural map \(\mathbf{S}^{-1}\simeq \Theta\mathsf{Bo}^{(2)}\rightarrow\Theta\mathsf{Bt}^{(2)}\simeq \mathrm{Map}(S^1_+,\mathbf{S}^{-1})\) is given by the inclusion of the constant loops, which results in an equivalence of rational \(\mathrm{O}(2)\)-spectra \[\Theta\mathsf{t}\mskip-2mu/\mskip-2mu\mathsf{o}\mskip 1mu^{(2)}\simeq_\mathbf{Q}\mathbf{S}^{-3},\] where the \(\mathrm{O}(2)\)-action on \(\mathbf{S}^{-3}\) is via the determinant. Together with \(\Theta\mathsf{Bt}^{(1)}\simeq_\mathbf{Q}\Omega K(\mathbf{S})/\mathbf{S}\) this allows for a computation of the first two layers of the Taylor tower for \(\mathsf{t}\mskip-2mu/\mskip-2mu\mathsf{o}\mskip 1mu=\mathrm{Top}(-)/\mathrm{O}(-)\) which reproduces 1 via Morlet’s equivalence 2 : the \(K\)-theory classes are created by the first layer, the Pontryagin–Weiss classes by the second, and the rational connectivity of \(B\mathrm{Diff}_\partial(D^d)\simeq \Omega^d_0\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow\Omega_0^dT_2\mathrm{Top}(d)/\mathrm{O}(d)\) follows from that of \(B\mathrm{O}(d)\rightarrow T_2B\mathrm{O}(d)\) and \(B\mathrm{Top}(d)\rightarrow T_2B\mathrm{Top}(d)\) discussed above.

1.6 Structure of the proof of 1↩︎

To provide some guidance, we give an overview of our strategy to prove the first main result, 1, on which all other results rely.

Our approach is not to directly calculate the rational homotopy groups of \(B\mathrm{Diff}_\partial(D^{2n+1})\) and \(B\mathrm{Diff}_\partial(D^{2n})\), but rather calculate those of \(B\mathrm{C}(D^{2n})\) and \(B\mathrm{Diff}_\partial(D^{2n+1})\) and then use the fibration relating these three spaces. To do this we use certain “delooped Weiss fibre sequences” \[\label{intro-weiss-fs} \def\arraystretch{1.5} \begin{array}{l@{\hskip 0.1cm} c@{\hskip 0.1cm} c@{\hskip 0.1cm} c@{\hskip 0.1cm} l@{\hskip 0.1cm}} B\mathrm{Diff}^{\mathrm{fr}}_{D^{2n}}(V_{g})_{\ell_V} &\longrightarrow&B\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_{\ell_V}&\longrightarrow&B(B\mathrm{C}(D^{2n}))\\ B\mathrm{Diff}^{\mathrm{fr}}_\partial(W_{g,1})_{\ell_W} &\longrightarrow&B\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(W_{g,1})_{\ell_W}&\longrightarrow&B(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n})_B) \end{array}\tag{12}\] Without explaining these in full detail (see 2.3), we have \[V_g \coloneq \natural^g (D^{n+1}\times S^n)\quad\text{ and }\quad W_{g,1} \coloneq \partial V_g \backslash \mathrm{int}(D^{2n}),\] the fibres in 12 are certain path-components of classifying spaces for framed smooth manifold bundles subject to certain boundary conditions, \(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n})_B\) differs from \(B\mathrm{Diff}_\partial(D^{2n})\) (up to issues with path-components) by \(\Omega^{2n}\mathrm{O}(2n)\) whose rational homotopy groups are known, and the total spaces are classifying spaces of certain group-like monoids of framed self-embeddings of the manifolds \(W_{g,1}\) and \(V_g\), again subject to certain boundary conditions. Results of Botvinnik–Perlmutter and Galatius–Randal-Williams on parametrised surgery show that for \(g \gg 0\) the fibres are rationally acyclic (see 2.4), so the right-hand maps in 12 become rational homology isomorphisms for \(g\gg 0\). As the base spaces are loop spaces, their rational homotopy groups can be extracted from the rational homology, so the problem becomes to determine the rational homology of the total spaces for \(g \gg 0\) in degrees \(*\lesssim 3n\). To do this, we first compute the rational homotopy groups of these spaces, as modules over the fundamental groups. In principle one may attempt this using embedding calculus, but this approach leads to various ambiguities that seem hard to resolve. We rather employ a different strategy, taking advantage of the simple handle-structures of the manifolds \(V_g\) and \(W_{g,1}\):

1. instead of applying the machinery of embedding calculus to the total spaces of 12 , we take a more “hands-on” approach to access such spaces of self-embeddings (see 4).

In a bit more detail: to compute the rational homotopy groups of the total spaces of 12 , we first replace framings by stable framings (which has a negligible effect) and replace the space \(B\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_{\ell_V}\) by a variant \(B\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2}\partial}(V_g)_{\ell_V}\) involving a different boundary condition; these are related (up to issues with path-components) by a fibre sequence \[B\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2}\partial}(V_g)_{\ell_V} \longrightarrowB\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_{\ell_V}\longrightarrowB\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2}\partial}(W_{g,1})_{\ell_W}.\] Then we calculate the rational homotopy groups of the homotopy fibres of certain maps \[\label{equ:intro-map-to-haut}B\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(W_{g,1}) \longrightarrowB\mathrm{hAut}_\partial(W_{g,1}) \quad\text{and}\quad B\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(V_{g}) \longrightarrowB\mathrm{hAut}_\partial(V_g)\tag{13}\] to the delooped spaces of homotopy automorphisms fixing the full boundary by an induction over handles. This step in particular relies on Goodwillie’s work on multiple disjunction and calculations of Arone, Fresse, Turchin, and Willwacher on the rational homotopy groups of spaces of long embeddings. It also involves

1. a general method to compute rational homotopy groups of spaces of block embeddings of a point into simply-connected high-dimensional manifolds (see 5).

The handle-induction breaks the symmetry of the manifolds: it gives the rational homotopy groups of the fibres in 13 only as vector spaces instead of as modules over (a variant of) the mapping class groups. In the case of \(W_{g,1}\) this is irrelevant as the answer turns out to be zero, but in the case of \(V_g\) it does matter. There is however an obvious guess of what the equivariant answer ought to be, and we confirm this guess based on

1. a new perspective on Watanabe’s parametrised form of the Goussarov–Habiro theory of claspers, offered as part of 6. In particular, we give an axiomatic description of the family of genus \(3\) handlebodies central to Watanabe’s work in odd dimensions, which he described explicitly in terms of a Borromean ring construction.

Combining the resulting equivariant computation with calculations of the rational homotopy groups of the targets of 13 due to Berglund–Madsen and Krannich eventually yields the required computation of the higher homotopy groups of the total spaces of 12 , as modules over their fundamental groups. To calculate the rational homology groups, in 7, we first determine the Lie algebra structure on the rational homotopy groups up to some ambiguities, then calculate the rational homology of the universal covers, and finally analyse the Serre spectral sequences of the universal covers and show that the ambiguities do not matter. This final step in particular involves a number of computations of stable cohomology groups of certain arithmetic groups with coefficients in various algebraic representations.

Acknowledgements↩︎

We would like to thank Søren Galatius, Erik Lindell, Samuel Muñoz-Echaniz, Arthur Soulié, and the referee for their comments on an earlier version of this paper. We were partially supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.), and by a Philip Leverhulme Prize from the Leverhulme Trust. MK was also supported by the European Union through an ERC grant (MaFC, 101221003).

2 Automorphisms, self-embeddings, and mapping class groups↩︎

As outlined in 1.6, the overall strategy of proof of 1 is to compare the homotopy type of \(\mathrm{Diff}_\partial(D^d)\) to that of different types of automorphism and embedding spaces of the auxiliary manifolds \[\label{equ:the-stars} V_g\coloneq \natural^g (D^{n+1}\times S^n),\quad W_g\coloneq \partial V_g\cong\sharp^g(S^n\times S^n),\quad\text{and}\quad W_{g,1}\coloneq W_g\backslash \mathrm{int}(D^{2n}).\tag{14}\] where \(D^{2n}\subset W_g\) is a fixed embedded disc. The discs fit into this as the case \(g=0\): we have \[V_0=D^{2n+1},\quad W_0= S^{2n},\quad\text{and}\quad W_{0,1}=D^{2n}.\] This section serves to define the various automorphism spaces we shall we use, and to discuss some of their properties. The most important result in this section is 17 which enables us to access the rational homotopy type of diffeomorphisms of discs by studying certain spaces of self-embeddings of \(W_{g,1}\) and \(V_g\) for large \(g\).

2.1 (Block) diffeomorphisms, the derivative map, and tangential structures↩︎

For a compact \(d\)-manifold \(M\) and subspaces \(K,L\subset M\), we denote by \(\mathrm{Diff}_{K}(M,L)\subset \mathrm{Diff}(M)\) the subgroup of the group of diffeomorphisms \(\mathrm{Diff}(M)\) of those diffeomorphisms that fix \(L\) setwise and a neighbourhood of \(K\subset M\) pointwise, equipped with the smooth topology. We use the following abbreviations.

1. If \(K\) or \(L\) is empty, we omit it from the notation.

2. If \(K=\partial M\), we use \(\partial\) as subscript.

3. If \(K=\partial M\backslash \mathrm{int}(D^{d-1})\) for an embedded disc \(D^{d-1}\subset \partial M\), we use \(\frac{1}{2}\partial\) as subscript.

Similarly to \(\mathrm{Diff}_K(M,L)\), we write \(\mathrm{hAut}_K(M,L)\subset \mathrm{hAut}(M)\) and \(\mathrm{\widetilde{Diff}}_{K}(M,L)\subset \mathrm{\widetilde{Diff}}(M)\) for the corresponding subspaces of the monoid of homotopy automorphisms in the compact open topology and of the realisation of the semi-simplicial group of block diffeomorphisms. Recall that these spaces are related by a preferred factorisation \[\label{equ:hAut-factorisation} \mathrm{Diff}_K(M,L)\rightarrow\mathrm{\widetilde{Diff}}_{K}(M,L) \rightarrow\mathrm{hAut}_{K}(M,L)\tag{15}\] of the map that assigns a diffeomorphism its underlying homotopy equivalence. By definition, the first map is surjective on path components. As explained in [26] (see [7] for an alternative point-set model), the second map factors further as \[\label{equ:block-bundle-factorisation} \mathrm{\widetilde{Diff}}_{K}(M,L) \rightarrow\mathrm{hAut}_{K}(\tau_M^s,L)\longrightarrow\mathrm{hAut}_{K}(M,L),\tag{16}\] where \(\mathrm{hAut}_{K}(\tau_M^s,L)\) is the topological monoid of bundle maps of the stable tangent bundle \(\tau^s_M\) of \(M\) that are the identity on \(\tau_M^s|_K\) and preserve \(\tau_M^s|_L\), equipped with the compact open topology. The first map in the composition is induced by taking derivatives and the second map by forgetting bundle data (see the above references for precise definitions). We denote by \[\mathrm{hAut}^\cong_{K}(\tau_M^s,L)\subset\mathrm{hAut}_{K}(\tau_M^s,L)\quad\text{and}\quad \mathrm{hAut}^\cong_{K}(M,L)\subset \mathrm{hAut}_{K}(M,L)\] the components in the image of the map from \(\mathrm{\widetilde{Diff}}_{K}(M,L)\), or equivalently from \(\mathrm{Diff}_{K}(M,L)\).

2.1.1 Tangential structures↩︎

A tangential structure is a map \(\theta\colon B\rightarrow B\mathrm{O}(d)\) from some space \(B\) to the classifying space \(B\mathrm{O}(d)\) of \(d\)-dimensional vector bundles. Given a bundle map \(\ell_K\colon \tau_M|_K\rightarrow\theta^*\gamma_d\) where \(\tau_M\) is the tangent bundle of \(M\) and \(\gamma_d\rightarrow B\mathrm{O}(d)\) the universal bundle, a \(\theta\)-structure on \(M\) is a bundle map \(\ell\colon \tau_M\rightarrow\theta^*\gamma_d\) extending \(\ell_K\). We call \(\ell_K\) a boundary condition, since \(K\) agrees with the boundary of \(M\) in many of the situations we have have in mind. The set of \(\theta\)-structures extending \(\ell_K\) forms a space \(\mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K)\) with the compact open topology and it is acted upon by the diffeomorphism group \(\mathrm{Diff}_{K}(M,L)\) via precomposition with the derivative. We denote the resulting homotopy quotient by \[B\mathrm{Diff}^{\theta}_K(M,L;\ell_K)\coloneq \mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K)/\mkern-6mu/\mathrm{Diff}_K(M,L).\] Recall from [7] that if \(\theta\) is stable, i.e. pulled back from a map \(\theta'\colon B'\rightarrow B\mathrm{O}\) along the canonical map \(B\mathrm{O}(d)\rightarrow B\mathrm{O}\), then there is an action of \(\mathrm{\widetilde{Diff}}_K(M,L)\) and \(\mathrm{hAut}^\cong_K(\tau_M^s,L)\) on \(\mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K)\) in the \(A_\infty\)-sense (i.e.an actual action on a space which comes equipped with a preferred equivalence to \(\mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K)\)) whose homotopy quotients \[\begin{align} B\mathrm{\widetilde{Diff}}_K^\theta(M,L;\ell_K)&\coloneq \mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K)/\mkern-6mu/\mathrm{\widetilde{Diff}}_K(M,L)\\ B\mathrm{hAut}_K^{\theta,\cong}(\tau_M^s,L;\ell_K)&\coloneq\mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K)/\mkern-6mu/\mathrm{hAut}^\cong_K(\tau_M^s,L) \end{align}\] fit into a diagram of homotopy cartesian squares \[\label{equ:comparison-tangential} \begin{tikzcd} B\mathrm{Diff}^\theta_K(M,L;\ell_K)\rar\dar&B\mathrm{\widetilde{Diff}}^\theta_K(M,L;\ell_K)\dar\rar&B\mathrm{hAut}_K^{\theta,\cong}(\tau_M^s,L;\ell_K)\dar\\ B\mathrm{Diff}_K(M,L)\rar&B\mathrm{\widetilde{Diff}}_K(M,L)\rar&B\mathrm{hAut}^\cong_K(\tau^s_M,L) \end{tikzcd}\tag{17}\] induced by the maps in 15 and 16 . From the long exact sequence in homotopy groups of the fibration associated to a homotopy quotient, we see that sets of path components of the three spaces forming the upper row all agree with the orbits of the \(\pi_0(\mathrm{Diff}_K(M,L))\)-action on \(\pi_0(\mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K))\). We denote the component corresponding to a fixed tangential structure \(\ell\in \mathrm{Bun}_K(\tau_M,\theta^*\gamma_d;\ell_K)\) by a subscript \((-)_\ell\), e.g. \(B\mathrm{Diff}^\theta_K(M,L;\ell_K)_\ell\subset B\mathrm{Diff}^\theta_K(M,L;\ell_K)\) in the case of diffeomorphisms. We also write \[\mathrm{hAut}^{\cong,\ell}_K(M,L)\subset \mathrm{hAut}^\cong_K(M,L)\] for the collection of components in the image of the map \[\pi_1(B\mathrm{\widetilde{Diff}}^{\theta}_{K}(M,L;\ell_K);\ell)\longrightarrow\pi_1(B\mathrm{hAut}^{\cong}_{K}(M,L))\cong \pi_0\mathrm{hAut}^{\cong}_{K}(M,L).\]

2.1.2 A rational model for block diffeomorphisms↩︎

In [26], a variant of the bottom right horizontal map in 17 was shown to be rational equivalence on universal covers for \(K=\partial M\cong S^{d-1}\) with \(d\ge5\) and simply-connected \(M\). We shall need a version of this result, established in [7], for triads satisfying a \(\pi\)-\(\pi\)-condition. To state it, recall that a manifold triad \((M;\partial_0M,\partial_1M)\) is a compact \(d\)-manifold \(M\) together with a decomposition of its boundary \(\partial M=\partial_0 M\cup \partial_1M\) into (possibly empty) submanifolds with \(\partial(\partial_0M)=\partial_0M\cap\partial_1M=\partial(\partial_1M)\). It satisfies the \(\pi\)-\(\pi\)-condition if \(\partial_1M\subset M\) induces an equivalence on fundamental groupoids. The following is part of Theorem 2.2 and Corollary 2.4 of [7].

2.2 Diffeomorphisms and self-embeddings of handlebodies↩︎

We now turn to the specific manifolds 14 . Restricting diffeomorphisms to the moving part of the boundary induces a fibre sequence of the form \[\label{equ:diff-restriction-sequence} \mathrm{Diff}_\partial(V_g)\longrightarrow\mathrm{Diff}_{D^{2n}}(V_g)\longrightarrow\mathrm{Diff}_\partial(W_{g,1}),\tag{18}\] which we shall compare to a certain fibre sequence of self-embedding spaces via a diagram \[\label{equ:diff-emb-comparison} \begin{tikzcd} \mathrm{Diff}_\partial(V_g)\rar\dar& \mathrm{Diff}_{D^{2n}}(V_g)\rar\dar& \mathrm{Diff}_\partial(W_{g,1})\dar\\ \mathrm{Emb}_{\frac{1}{2}\partial}(V_g)\rar&\mathrm{Emb}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\rar&\mathrm{Emb}_{\frac{1}{2}\partial}(W_{g,1}). \end{tikzcd}\tag{19}\] To define the bottom row, we decompose the disc \(D^{2n}=D^{2n}_+\cup D^{2n}_-\subset \partial V_g\) into two half-discs intersecting in a \((2n-1)\)-disc and define the space \(\mathrm{Emb}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\) as the space of self-embeddings of \(V_g\) that restrict to a self-embedding of \(W_{g,1}\subset \partial V_g\) and agree with the identity on a neighbourhood of \(D^{2n}_+\) (but may send \(\mathrm{int}(D^{2n}_-)\subset \partial V_g\) to the interior). Restricting such embeddings to \(W_{g,1}\) gives a map to the space \(\mathrm{Emb}_{\frac{1}{2}\partial}(W_{g,1})\) of self-embeddings of \(W_{g,1}\) that agree with the identity on a neighbourhood of \(W_{g,1}\cap D^{2n}_+\cong D^{2n-1}\) (but may send \(\mathrm{int}(W_{g,1}\cap D^{2n}_-)\cong\mathrm{int}(D^{2n-1})\subset \partial W_{g,1}\) to the interior). By isotopy extension, the homotopy fibre of this map agrees with the space \(\mathrm{Emb}_{\frac{1}{2}\partial}(V_g)\) of self-embeddings of \(V_g\) which agree with the identity on a neighbourhood of \(W_{g,1}\cup D^{2n}_+=\partial V_g\backslash \mathrm{int}(D^{2n}_-)\) (but may send \(\mathrm{int}(D^{2n}_-)\subset \partial V_g\) to the interior).

We denote the components in the image of the two right vertical maps in 19 by \[\mathrm{Emb}^\cong_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\subset \mathrm{Emb}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\quad\text{and}\quad \mathrm{Emb}^{\cong}_{\frac{1}{2}\partial}(W_{g,1})\subset \mathrm{Emb}_{\frac{1}{2}\partial}(W_{g,1}),\] denote by \[\label{equ:restricted-components-ext-1} \mathrm{Diff}^{\mathrm{ext}}_\partial(W_{g,1})\subset \mathrm{Diff}_\partial(W_{g,1})\quad\text{and} \quad \mathrm{Emb}^{\cong,\mathrm{ext}}_{\frac{1}{2}\partial}(W_{g,1})\subset \mathrm{Emb}^\cong_{\frac{1}{2}\partial}(W_{g,1})\tag{20}\] the union of those path components hit by the maps from \(\mathrm{Diff}_{D^{2n}}(V_g)\), and denote by \[\label{equ:restricted-components-ext-2} \mathrm{Emb}^\cong_{\frac{1}{2}\partial}(V_g)\subset \mathrm{Emb}_{\frac{1}{2}\partial}(V_g)\tag{21}\] those components that map to \(\mathrm{Emb}^\cong_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\subset \mathrm{Emb}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\).

Given a tangential structure \(\theta\colon B\rightarrow B\mathrm{O}(2n+1)\) and a \(\theta\)-structure \(\ell\colon \tau_{V_g}\rightarrow\theta^*\gamma_{2n+1}\) for \(V_g\), we fix an identification \(\tau_{V_g}|_{W_{g,1}}\cong \tau_{W_{g,1}}\oplus \varepsilon\) induced by a choice of an inwards pointing vector field where \(\varepsilon\) is the trivial line bundle. Using this, we consider the maps \[\begin{tikzcd}[column sep=0.2cm] \mathrm{Bun}_{\partial}(\tau_{V_g},\theta^*\gamma_{2n+1};\ell_{\partial V_g})\rar\dar&\dar \mathrm{Bun}_{D^{2n}}(\tau_{V_g},\theta^*\gamma_{2n+1};\ell_{D^{2n}})\rar\dar& \mathrm{Bun}_\partial(\tau_{W_{g,1}}\oplus\varepsilon,\theta^*\gamma_{2n+1};\ell_{\partial W_{g,1}})\dar\\ \mathrm{Bun}_{\frac{1}{2}\partial}(\tau_{V_g},\theta^*\gamma_{2n+1};\ell_{\frac{1}{2}\partial V_g})\rar& \mathrm{Bun}_{\frac{1}{2}D^{2n}}(\tau_{V_g},\theta^*\gamma_{2n+1};\ell_{\frac{1}{2}D^{2n}})\rar& \mathrm{Bun}_{\frac{1}{2}\partial}(\tau_{W_{g,1}}\oplus\varepsilon,\theta^*\gamma_{2n+1};\ell_{\frac{1}{2}\partial W_{g,1}}) \end{tikzcd}\] of spaces of \(\theta\)-structures, where the horizontal maps are induced by restriction and are fibre sequences, and the vertical maps are induced by relaxing the boundary conditions, all of which obtained by restriction from \(\ell\). Write \((1\text{-}\theta)\colon B'\rightarrow B\mathrm{O}(2n)\) for the tangential structure given by the pullback of \(\theta\) along \(B\mathrm{O}(2n)\rightarrow B\mathrm{O}(2n+1)\), so that a \((1\text{-}\theta)\)-structure on a \(2n\)-dimensional vector bundle is simply a \(\theta\)-structure on its one-fold stabilisation. Then the bases of these fibrations are equivalent to the space of \((1\text{-}\theta)\)-structures on \(W_{g,1}\) relative to \(\partial W_{g,1}\) and to \(\frac{1}{2}\partial W_{g,1}\) respectively. Both rows of 19 act compatibly on this fibre sequence by precomposition with the derivative, so after passing to the subsets of components specified in 20 and 21 , we may form homotopy quotients to obtain a diagram of horizontal fibre sequences \[\begin{tikzcd} B\mathrm{Diff}^{\theta}_{\partial}(V_g;\ell_{\partial})\rar\dar& B\mathrm{Diff}^{\theta}_{D^{2n}}(V_g;\ell_{\partial})\rar\dar& B\mathrm{Diff}^{1\text{-}\theta,\mathrm{ext}}_{\partial}(W_{g,1};\ell_{\frac{1}{2}\partial})\dar\\ B\mathrm{Emb}^{\theta,\cong}_{\frac{1}{2}\partial}(V_g;\ell_{\frac{1}{2}\partial})\rar& B\mathrm{Emb}^{\theta,\cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1};\ell_{\frac{1}{2}D^{2n}})\rar& B\mathrm{Emb}^{1\text{-}\theta,\cong,\mathrm{ext}}_{\frac{1}{2}\partial}(W_{g,1};\ell_{\frac{1}{2}\partial}), \end{tikzcd}\] with fibres taken over the points induced by \(\ell\).

2.3 Weiss fibre sequences↩︎

For a tangential structure \(\theta\colon B\rightarrow B\mathrm{O}(2n)\) and a \(\theta\)-structure \(\ell\) on \(W_{g,1}\), there is a fibre sequence of the form \[\label{equ:weiss-fs-Wg} B\mathrm{Diff}^\theta_\partial(D^{2n};\ell_{\partial_0})\longrightarrowB\mathrm{Diff}^\theta_\partial(W_{g,1};\ell_{\partial})\longrightarrowB\mathrm{Emb}^{\theta, \cong}_{\frac{1}{2}\partial}(W_{g,1};\ell_{\frac{1}{2}\partial}),\tag{22}\] where the boundary conditions are obtained by restricting \(\ell\). The first map is induced by extension by the identity along the inclusion \(D^{2n}\subset W_{g,1}\) and the second by the forgetful map \(\mathrm{Diff}_\partial(W_{g,1})\rightarrow\mathrm{Emb}^\cong_{\frac{1}{2}\partial}(W_{g,1})\). For \(\theta=\mathrm{id}\), this sequence appeared in [12] and was shown in [28] to deloop once to the right with respect to the \(E_{2n}\)-structure on the fibre induced by boundary connected sum. The version with tangential structures \(\theta\) is noted in [27] and deloops in the same way. The same arguments establish fibre sequences \[\label{equ:weiss-fs-Vg} \begin{gather} B\mathrm{Diff}^\Xi_\partial(D^{2n+1};\ell_{\partial_0})\longrightarrowB\mathrm{Diff}^\Xi_\partial(V_{g};\ell_{\partial})\longrightarrowB\mathrm{Emb}^{\Xi, \cong}_{\frac{1}{2}\partial}(V_{g};\ell_{\frac{1}{2}\partial})\quad\text{and}\\ B\mathrm{Diff}^\Xi_{D^{2n}}(D^{2n+1};\ell_{D^{2n}})\longrightarrowB\mathrm{Diff}^{\Xi}_{D^{2n}}(V_{g};\ell_{D^{2n}})\longrightarrowB\mathrm{Emb}^{\Xi,\cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1};\ell_{\frac{1}{2}D^{2n}}), \end{gather}\tag{23}\] for tangential structures \(\Xi\colon B'\rightarrow B\mathrm{O}(2n+1)\), which both deloop once to the right with respect to the \(E_{2n+1} / E_{2n}\)-structure on the fibre induced by boundary connected sum. Similar to 22 , the right-hand maps are induced by the comparison maps between diffeomorphisms and embeddings and the left-hand maps are induced by extension by the identity along a fixed embedding \(D^{2n+1}\subset V_g\) that restricts to the chosen embedding \(D^{2n}\subset W_{g,1}\) on a hemisphere. Note that forgetting tangential structures induces an equivalence \[\label{equ:no-tangential-structure-for-moving-boundary} B\mathrm{Diff}^\Xi_{D^{2n}}(D^{2n+1};\ell_{D^{2n}})\overset{\simeq}{\longrightarrow} B\mathrm{Diff}_{D^{2n}}(D^{2n+1}),\tag{24}\] since the homotopy fibre is equivalent to the space of maps \(D^{2n+1}\rightarrow B'\) together with a homotopy from its composition with \(\Xi\) to a fixed classifier \(D^{2n+1}\rightarrow B\mathrm{O}(2n+1)\) of the tangent bundle with fixed behaviour on the hemisphere \(D^{2n}\subset D^{2n+1}\). As the latter inclusion is an equivalence, this space is contractible.

In [28], Kupers explains how to set up similar sequences for the corresponding groups of homeomorphisms (his discussion is phrased without tangential structures), resulting in a commutative diagram of fibre sequences \[\begin{tikzcd} B\mathrm{Diff}_\partial(D^{2n})\rar\dar& B\mathrm{Diff}_\partial(W_{g,1})\rar\dar& B\mathrm{Emb}^{\cong}_{\frac{1}{2}\partial}(W_{g,1})\dar\\ B\mathrm{Homeo}_\partial(D^{2n})\rar& B\mathrm{Homeo}_\partial(W_{g,1})\rar& B\mathrm{TopEmb}^{\cong}_{\frac{1}{2}\partial}(W_{g,1}) \end{tikzcd}\] and similarly for the extensions of 23 with \(\Xi=\mathrm{id}\). Here the bottom row is defined in the same way as the top row, but using spaces of homeomorphisms and locally flat topological embeddings instead. As the spaces \[B\mathrm{Homeo}_\partial(D^{2n}),\quad B\mathrm{Homeo}_\partial(D^{2n+1}),\quad\text{and}\quadB\mathrm{Homeo}_{D^{2n}}(D^{2n+1})\] are contractible by the Alexander trick, we obtain maps \[\begin{gather} B\mathrm{Emb}^{\cong}_{\frac{1}{2}\partial}(V_{g})\longrightarrowB\mathrm{Homeo}_\partial(V_g)\quad\quad B\mathrm{Emb}^{\cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\longrightarrowB\mathrm{Homeo}_{D^{2n}}(V_{g})\\ B\mathrm{Emb}^{\cong}_{\frac{1}{2}\partial}(W_{g,1})\longrightarrowB\mathrm{Homeo}_\partial(W_{g,1}) \end{gather}\] up to contractible choices, which we may postcompose with the comparison maps to homotopy automorphisms to obtain maps of the form \[\label{equ:self-embs-to-haut} \begin{gather} B\mathrm{Emb}^{\cong}_{\frac{1}{2}\partial}(V_{g})\longrightarrowB\mathrm{hAut}_\partial(V_g)\quad\quad B\mathrm{Emb}^{\cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\longrightarrowB\mathrm{hAut}_{D^{2n}}(V_{g},W_{g,1})\\ B\mathrm{Emb}^{\cong}_{\frac{1}{2}\partial}(W_{g,1})\longrightarrowB\mathrm{hAut}_\partial(W_{g,1}). \end{gather}\tag{25}\]

2.4 Stable cohomology↩︎

For \(n \geq 3\) and large values of \(g\), the homology of \(B\mathrm{Diff}^\theta_\partial(W_{g,1};\ell_{\partial})\) admits a homotopy theoretical description by work of Galatius–Randal-Williams [29]. Botvinnik–Perlmutter [30] proved a similar result for \(B\mathrm{Diff}^{\Xi}_{D^{2n}}(V_{g};\ell_{D^{2n}})\) as long as the domain of \(\Xi\) is \(n\)-connected and \(n\ge4\). We will only use the case of framings, in which case these results read as follows. We denote the component of the constant map in a loop space by \(\Omega_0X\subset \Omega X\) and the sphere spectrum by \(\mathbf{S}\).

By specialising the sequence 22 and the second sequence of 23 to the case of framings, restricting to the components of given framings \(\ell_{V_g}\) and \(\ell_{W_{g,1}}\) of \(W_{g,1}\) and \(V_g\), and delooping we obtain for \(n\ge3\) fibre sequences of the form \[\label{equ:delooped-framed-Weiss-fs} \begin{gather} B\mathrm{Diff}^{\mathrm{fr}}_\partial(W_{g,1};\ell_{\partial})_{\ell_{W_{g,1}}}\longrightarrowB\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(W_{g,1};\ell_{\frac{1}{2}\partial})_{\ell_{W_{g,1}}} \longrightarrow B(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n};\ell_{\partial_0})_B)\\ B\mathrm{Diff}^{\mathrm{fr}}_{D^{2n}}(V_{g};\ell_{D^{2n}})_{\ell_{V_g}} \longrightarrowB\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1};\ell_{\frac{1}{2}D^{2n}})_{\ell_{V_g}} \longrightarrow B(B\mathrm{Diff}_{D^{2n}}(D^{2n+1})) \end{gather}\tag{26}\] where \(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n};\ell_{\partial_0})_B\subset B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n};\ell_{\partial_0})\) is a collection of components which is finite by [27]. To arrive at the second sequence, we used the equivalence 24 .

2.5 A standard model↩︎

We fix \(g\ge0\) and write \[\bar{H}_W\coloneq \mathrm{H}_n(W_{g,1};\mathbf{Q}),\quad \bar{H}_V\coloneq \mathrm{H}_n(V_{g}; \mathbf{Q}),\quad\text{and}\quad \bar{K}_W\mathrel{\vcenter{:}}=\ker(\iota_*\colon \bar{H}_W\rightarrow\bar{H}_V),\] where \(\iota\) is the inclusion \(W_{g,1}\subset V_g\). Their integral cousins are denoted by \[\bar{H}_W^\mathbf{Z}\coloneq H_n(W_{g,1};\mathbf{Z}),\quad \bar{H}_V^\mathbf{Z}\coloneq H_n(V_{g}; \mathbf{Z}),\quad\text{and}\quad \bar{K}_W^\mathbf{Z}\mathrel{\vcenter{:}}=\ker(\iota_*\colon \bar{H}_W^\mathbf{Z}\rightarrow\bar{H}_V^\mathbf{Z}).\] The bar-superscript indicates that we consider these as plain vector spaces or abelian groups, whereas later we shall mostly consider graded objects (see 3.1.1)

At several points in this work, it will be convenient to have fixed a standard model of \(V_g\) as a submanifold of \(\mathbf{R}^{2n+1}\) with fixed embedded cores \(e_i,f_i\colon S^n\hookrightarrow \partial V_g\) for \(1\le i\le g\) representing a basis of \(\bar{H}_W^\mathbf{Z}\cong\mathbf{Z}^{2g}\), denoted by the same symbols, such that

1. \((e_i,f_i)_{1\le i\le g}\) is a hyperbolic basis for the intersection form \[\lambda\colon \bar{H}_W^\mathbf{Z}\otimes \bar{H}_W^\mathbf{Z}\longrightarrow\mathbf{Z},\] i.e. \(\lambda(e_i,e_j)=\lambda(f_i,f_j)=0\) and \(\lambda(e_i,f_j)=\delta_{ij}\) for all \(1\le i,j\le g\),

2. the \(e_i\)’s form a basis of the subspace \(\bar{K}_W^\mathbf{Z}\subset \bar{H}_W^\mathbf{Z}\), and

3. the \(f_i\)’s map to a basis of \(\bar{H}_V^\mathbf{Z}\cong \mathbf{Z}^g\) under the projection \(\iota_*\colon \bar{H}_W^\mathbf{Z}\rightarrow\bar{H}_V^\mathbf{Z}\).

4. the \(V_g\)’s come with inclusions \(V_{h}\subset V_{g}\) for \(h\le g\) which preserve the \(e_i\)’s and \(f_i\)’s.

To this end, we use the standard embedding \(S^n\subset \mathbf{R}^{n+1}\) and abbreviate by \(\iota_{S^n}\colon S^n\hookrightarrow \mathbf{R}^{2n+1}\) the inclusion into the last \((n+1)\)-coordinates. We consider the normal bundle \(\nu(\iota_{S^n})\) of \(\iota_{S^n}\) as a subspace of \(S^n\times \mathbf{R}^{2n+1}\) using the standard metric and framing of \(\mathbf{R}^{2n+1}\). Writing \(\vec{e}_i\) for the \(i\)th standard unit vector, we view \(V_1\) as the image of the embedding \[\phi\colon\begin{array}{rcl} \{(x,y)\in \nu(\iota_{S^n})\subset S^n\times \mathbf{R}^{2n+1}\mid |y|\le 1/2\} & \lhook\joinrel\longrightarrow& \mathbf{R}^{2n}\times (0,2) \\[0.3em] (x,y) & \longmapsto & \tfrac{1}{2}(\iota_{S^n}(x)+y)+\vec{e}_{2n+1}, \end{array}\] equipped with the cores \(e_1\colon S^n\hookrightarrow \partial V_1\) and \(f_1\colon S^n\hookrightarrow \partial V_1\) given by precomposing \(\phi\) with \[S^n\ni x\mapsto (\vec{e}_{1},(\tfrac{1}{2}x,0))\in S^n\times \mathbf{R}^{2n+1} \quad\text{and}\quad S^n\ni x\mapsto (x,(0,-\tfrac{1}{2}x))\in S^n\times \mathbf{R}^{2n+1}\] respectively. The cores \(e_1\) and \(f_1\) intersect in the single point \(\phi(\vec{e}_{1},-\tfrac{1}{2}\vec{e}_{n+1})=\tfrac{1}{4}\vec{e}_{n+1}+\vec{e}_{2n+1}\). Now consider the embedding \(\psi\colon \sqcup^g V_1 \rightarrow\mathbf{R}^{2n}\times (0,2g)\) given by translating the \(i\)th disjoint summand by \(2(i-1)\cdot \vec{e}_{2n+1}\) so that it lies in \(\mathbf{R}^{2n}\times (2(i-1),2i)\). The translates of the cores \(e_1,f_1\) above give a pair of cores \((e_i,f_i)\) for each of the summands. We define \(V_g\subset \mathbf{R}^{2n+1}\) as the union of the image of \(\psi\) with suitable tubular neighborhoods of the arcs \([0,1]\ni s\mapsto (2s+i+\tfrac{1}{2})\cdot \vec{e}_{2n+1}\in \mathbf{R}^{2n+1}\) connecting the \(i\)th and \((i+1)\)th disjoint summand, disjoint from the cores and axially symmetric around the \(\vec{e}_{2n+1}\)-axis. This embedding \(V_g\subset\mathbf{R}^{2n+1}\) together with its cores \((e_i,f_i)_{1\le i\le g}\) satisfies the desired properties (i)–(iv). See 1 for a schematic figure.

In addition, we once and for all choose an embedded codimension \(0\) disc \(D^{2n+1}\subset V_g\cap \mathbf{R}^{2n}\times(2g-1,2g)\) which is disjoint from the cores and axially symmetric around the \(\vec{e}_{2n+1}\)-axis. Moreover, we choose the half-disc \(D^{2n}_+\subset D^{2n}\) used in the definition of \(\mathrm{Emb}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})\) and \(\mathrm{Emb}_{\frac{1}{2}\partial}(W_{g,1})\) to have the same axial symmetry.

2.5.1 Standard (stable) framings↩︎

The standard framing of \(\mathbf{R}^{2n+1}\) induces by restriction a framing \(\ell_{V_g}\) on \(V_g\subset \mathbf{R}^{2n+1}\), which we call the standard framing of \(V_g\). Similarly, we fix a standard framing \(\ell_{W_{g,1}}\) of \(W_{g,1}\) which we require to (i) induce the \(\pm\)-structure on \(W_{g,1}\) of [13] and (ii) have trivial Arf invariant (see [32] for an explanation of this condition). From now on, we shall always use the standard framings \(\ell_{V_g}\) and \(\ell_{W_{g,1}}\) and the induced boundary conditions on submanifolds on the boundaries on \(\ell_{W_{g,1}}\) and \(\ell_{W_{g,1}}\). When we consider stable framings of \(V_g\) and \(W_{g,1}\), we always use the ones induced by the standard framings. If there is no source of confusion we abbreviate the standard framings \(\ell_{V_g}\) and \(\ell_{W_{g,1}}\) as well as the induced (stable) framings on submanifolds of \(V_g\) and \(W_{g,1}\) by \(\ell\), and we omit the reference to the induces boundary conditions from the notation. For instance we abbreviate \[B\mathrm{Diff}^\mathrm{fr}_{D^{2n}}(V_g)=B\mathrm{Diff}^\mathrm{fr}_{D^{2n}}(V_g;\ell_{D^{2n}})\quad\text{and}\quadB\mathrm{Emb}_{\frac{1}{2}D^{2n}}^\mathrm{fr}(V_g,W_{g,1})=B\mathrm{Emb}_{\frac{1}{2}D^{2n}}^\mathrm{fr}(V_g,W_{g,1};\ell_{\frac{1}{2}D^{2n}}).\] Also, we consider the different variants of (stably) framed automorphism and self-embedding spaces of \(V_g\) and \(W_{g,1}\) as based spaces via the points induced by the standard framings. As in 2.1.1, a subscript \((-)_\ell\) indicates that we restrict to this basepoint component.

2.6 Framings and stable framings↩︎

At various points in later sections, we will need to transition between framings and stable framings. The following lemma allows this.

2.7 Mapping class groups↩︎

In view of 17, in order to prove 1, we are faced with the task of computing the rational cohomology of the spaces (see 2.5.1 for the shortened notation in use) \[\label{equ:self-embedding-spaces} B\mathrm{Emb}^{\mathrm{fr}}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_{\ell}\quad\text{and}\quadB\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_{\ell}.\tag{27}\] To achieve this we need some understanding of the fundamental groups \[\label{equ:fundamental-groups-embedding-spaces} \check{\Lambda}_{V_{g}}\coloneq \pi_1(B\mathrm{Emb}^{\mathrm{fr}}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_\ell)\quad\text{and}\quad \check{\Lambda}_{W_{g,1}}\coloneq \pi_1(B\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_\ell)\tag{28}\] which come with comparison maps to \(\mathrm{GL}(\bar{H}_W^\mathbf{Z})\) (see 2.5 for the notation) \[\label{equ:homology-action-mcg-emb-spaces} \check{\Lambda}_{V_{g}}\longrightarrow\mathrm{GL}(\bar{H}_W^\mathbf{Z})\quad\text{and}\quad\check{\Lambda}_{W_{g,1}}\longrightarrow\mathrm{GL}(\bar{H}_W^\mathbf{Z})\tag{29}\] induced by the action on the homology of \(W_{g,1}\subset \partial V_g\). These maps are almost injective:

Up to these finite kernels, the groups 28 thus agree with the subgroups \[\label{equ:linear-part-of-mcgs} G_{V_g}\mathrel{\vcenter{:}}= \mathrm{im}\Big(\check{\Lambda}_{V_{g}}\rightarrow\mathrm{GL}(\bar{H}_W^\mathbf{Z}) \Big)\quad \text{and}\quad G_{W_{g,1}}\coloneq \mathrm{im}\Big(\check{\Lambda}_{W_{g,1}}\rightarrow\mathrm{GL}(\bar{H}_W^\mathbf{Z})\Big)\tag{31}\] of \(\mathrm{GL}(\bar{H}_W^\mathbf{Z})\). If \(g\) is clear from the context, we abbreviate these and the groups in 28 as \[\check{\Lambda}_{V}\coloneq \check{\Lambda}_{V_{g}},\quad \check{\Lambda}_{W}\coloneq \check{\Lambda}_{W_{g,1}},\quad G_V\coloneq G_{V_g}, \quad\text{and}\quad G_W\coloneq G_{W_{g,1}}.\]

The group \(G_V\) is closely related to \(\mathrm{GL}(\bar{H}_V^\mathbf{Z})\). Indeed, any self-embedding of the pair \((V_g,W_{g,1})\) preserves the subspace \(\bar{K}^\mathbf{Z}_W\subset \bar{H}_W^\mathbf{Z}\), so we have an inclusion \[G_V\subset\{\Phi\in\mathrm{GL}(\bar{H}_W^\mathbf{Z})\mid\Phi(\bar{K}^\mathbf{Z}_W)\subset \bar{K}^\mathbf{Z}_W\}\] and thus—using the canonical isomorphism \(\bar{H}_W^\mathbf{Z}/\bar{K}_W^\mathbf{Z}\cong \bar{H}_V^\mathbf{Z}\)—a canonical map \[\label{equ:homology-action-Vg}G_V\longrightarrow\mathrm{GL}(\bar{H}_V^\mathbf{Z}).\tag{32}\] Precomposed with \(\check{\Lambda}_{V}\rightarrow G_V\), this agrees with the usual action on homology.

More explicitly, the subgroups \(G_V\) and \(G_W\) can be described in terms of the hyperbolic basis \(\{e_i,f_i\}_{1\le i\le g}\) of \(\bar{H}_W^\mathbf{Z}\cong\mathbf{Z}^{2g}\) fixed in 2.5, ordered by \((e_1,\ldots,e_g,f_1,\ldots,f_g)\). Namely, with respect to the induced identification \(\mathrm{GL}(\bar{H}_W^\mathbf{Z})\cong \mathrm{GL}_{2g}(\mathbf{Z})\), we have \[\begin{align} \label{equ:matrix-description-Gg} \begin{aligned} G_W&\cong \begin{cases} \mathrm{O}_{g,g}(\mathbf{Z})&\text{for }n\text{ even}\\ \mathrm{Sp}^q_{2g}(\mathbf{Z})&\text{for }n\text{ odd}, \end{cases}\le \mathrm{OSp}_{g}(\mathbf{Z})\quad\text{and}\\ G_V&\cong \{\Big(\begin{smallmatrix}A&AB\\0&(A^T)^{-1}\end{smallmatrix}\Big)\in \mathrm{GL}_{2g}(\mathbf{Z}),\mid A\in\mathrm{GL}_g(\mathbf{Z}), B\in M^{(-1)^{n+1}}_g\}, \end{aligned} \end{align}\tag{33}\] see [32] for the first identification (this uses our condition on the standard framing in 2.5.1) and [7] for the second¹. Here

1. \(\mathrm{OSp}_{g}(\mathbf{Z})\) is defined as the automorphism group \(\mathrm{O}_{g,g}(\mathbf{Z})\) of the symmetric form \(\left(\begin{smallmatrix}0&I\\I&0\end{smallmatrix}\right)\) for \(n\) even and the automorphism group \(\mathrm{Sp}_{2g}(\mathbf{Z})\) of the antisymmetric form \(\left(\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}\right)\) for \(n\) odd. We omit the parameter \(n\) from the notation; it will be clear from the context.

2. \(\mathrm{Sp}^q_{2g}(\mathbf{Z})\le \mathrm{Sp}_{2g}(\mathbf{Z})\) is the finite index subgroup of those symplectic matrices which preserve the \(\mathbf{Z}/2\)-valued quadratic refinement \[q\colon\begin{array}{rcl} \mathbf{Z}^{2g} & \longrightarrow & \mathbf{Z}/2 \\[0.3em] \sum_{i=1}^g(A_i e_i+B_i f_i) & \longmapsto & \sum_{i=1}^gA_iB_i \ (\mathrm{mod}\ 2). \end{array}\]

3. \(M^{\pm}_g\subset \mathbf{Z}^{g\times g}\) is the additive subgroup of integral \((g\times g)\)-matrices which are \((\pm1)\)-symmetric and whose diagonal entries are even.

2.8 The reflection involution↩︎

It will be convenient for us to keep track of the following piece of additional structure. The manifold \(V_g\) admits an orientation-reversing smooth involution \(\rho\)—the reflection involution—which preserves setwise the subspaces \(W_{g,1}\subset W_g\subset V_g\) and the discs \(D^{2n+1}\subset V_g\) and \(D^{2n}\subset W_g\), and agrees with a reflection on these discs. Using the standard model of \(V_g\subset \mathbf{R}^{2n+1}\) from 2.5, we choose \(\rho\) given by acting by \(-1\) on the first coordinate (we could use any of the first \(n\) coordinates). This fixes the cores \(f_i\colon S^n\hookrightarrow V_g\) pointwise, and on the cores \(e_i\colon S^n\hookrightarrow V_g\) restricts to a reflection of \(S^n\). It also restricts to a reflection on the discs \(D^{2n}\subset \partial V_g=W_g\), \(D^{2n+1}\subset V_g\), and \(D^{2n}_+\subset D^{2n}\) as chosen in 2.5.

2.8.1 Action on automorphism spaces↩︎

Conjugation by the reflection involution \(\rho\) induces compatible pointed involutions \(\rho\) on each of the spaces in the diagram \[\label{equ:big-diagram-with-the-players} \begin{tikzcd} B\mathrm{Diff}_\partial(V_g) \dar\rar& B\mathrm{Diff}_{D^{2n}}(V_g) \dar \rar& B\mathrm{Diff}_\partial(W_{g,1}) \dar \\ B\mathrm{Emb}^\cong_{\frac{1}{2}\partial}(V_g) \dar\rar&B\mathrm{Emb}^\cong_{\frac{1}{2}D^{2n}}(V_g, W_{g,1})\rar\dar &B\mathrm{Emb}^\cong_{\frac{1}{2}\partial}(W_{g,1}) \dar\\ B\mathrm{hAut}_\partial(V_g)\rar& B\mathrm{hAut}_{D^{2n}}(V_g,W_{g,1})\rar& B\mathrm{hAut}_\partial(W_{g,1}). \end{tikzcd}\tag{34}\] In [13], it is explained that these involutions lift along the forgetful maps \[B\mathrm{Diff}^{\mathrm{fr}}_\partial(W_{g,1})_{\ell}\rightarrowB\mathrm{Diff}_\partial(W_{g,1})\quad\text{and}\quad B\mathrm{Emb}_{\frac{1}{2}\partial}^{\cong,\mathrm{fr}}(W_{g,1})_{\ell}\rightarrowB\mathrm{Emb}_{\frac{1}{2}\partial}^\cong(W_{g,1})\] to involutions in the pointed \(A_\infty\)-sense on \(B\mathrm{Diff}^{\mathrm{fr}}_\partial(W_{g,1})_\ell\) and \(B\mathrm{Emb}_{\frac{1}{2}\partial}^{\cong,\ell}(W_{g,1})_\ell\), so in particular there are involutions on these spaces in the pointed homotopy category. It will be convenient for us to know that the same is true for the framed analogues of the spaces forming the upper left square of 34 . We will explain this for \(B\mathrm{Diff}_{D^{2n}}(V_g)\); the other cases are analogous. To this effect, we consider the map of fibre sequences \[\label{equ:framed-involution-lifts-diagram} \begin{tikzcd} B\mathrm{Diff}^\mathrm{fr}_{D^{2n}}(V_g)\rar\dar& B\mathrm{Diff}^\mathrm{fr}(V_g,D^{2n})\rar\dar& B\mathrm{Diff}^{\mathrm{1\text{-}fr}}(D^{2n})\dar\\ B\mathrm{Diff}_{D^{2n}}(V_g)\rar& B\mathrm{Diff}(V_g,D^{2n})\rar& B\mathrm{Diff}(D^{2n}), \end{tikzcd}\tag{35}\] induced by restriction, and view the spaces forming the upper row as based via the basepoints induced by the standard framing. As the loop space of the total space of any fibration of pointed spaces naturally acts (in the pointed \(A_\infty\)-sense) on the fibre, it suffices to prove:

2.8.2 Action on mapping class groups↩︎

In particular, the reflection involution \(\rho\) acts on the groups \(\check{\Lambda}_W\), \(\check{\Lambda}_V\) by conjugation, and also on the groups \(G_W\), \(G_V\), and \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\) (see 2.7). As we chose the cores \(f_i\) to form a basis of \(\bar{H}_{V}^\mathbf{Z}\) and the cores \(e_i\) to form a basis of \(\bar{K}_W^\mathbf{Z}\) (see 2.5 for the notation), the involution \(\rho\) acts trivially on \(\bar{H}_V^\mathbf{Z}\) and by multiplication by \(-1\) on \(\bar{K}_W^\mathbf{Z}\). From this we see that when considering \(G_W\) and \(G_V\) as subgroups of \(\mathrm{GL}_{2g}(\mathbf{Z})\) via the fixed basis the involution is given by conjugation by \(\left(\begin{smallmatrix}-I&0\\0&I\end{smallmatrix}\right)\). In particular, the involution preserves the semi-direct product decomposition \(G_V = M_V^\mathbf{Z}\rtimes \mathrm{GL}(\bar{H}_V^\mathbf{Z})\): it is given by the identity on \(\mathrm{GL}(\bar{H}_V^\mathbf{Z})\) and by multiplication by \(-1\) on \(M_V^\mathbf{Z}\).

2.8.3 Interaction with the intersection form↩︎

As \(\rho\) reverses the orientation, care is needed when it comes to its interaction with the intersection form \(\lambda\) of \(W_g\). Writing \(\mathbf{Z}^\pm\) for \(\mathbf{Z}\) acted upon by \(\rho\) as \(\pm1\), and similarly \(\mathbf{Q}^\pm\), the intersection form is \(\rho\)-equivariant when considered as a morphism \(\lambda\colon \bar{H}_W^\mathbf{Z}\otimes \bar{H}_W^\mathbf{Z}\rightarrow\mathbf{Z}^-,\) and similarly after rationalising, and it induces isomorphisms \(\bar{H}_W^\vee\cong \bar{H}_W\otimes \mathbf{Q}^-\) as \(G_W\rtimes\langle \rho\rangle\)-modules and \(\bar{H}_V^\vee\cong \bar{K}_W^\mathbf{Z}\otimes \mathbf{Q}^-\) as \(G_V\rtimes\langle\rho\rangle\)-modules. By dualising, the intersection form induces a map \(\omega\colon \mathbf{Q}^- \rightarrow\bar{H}_W \otimes \bar{H}_W\). Thinking of \(\omega\) as an element of \(\bar{H}_W \otimes \bar{H}_W\), the involution \(\rho\) acts on it by \(-1\). Moreover, as the \(f_i\)’s are fixed by \(\rho\), they induce a splitting of the \(G_V\rtimes\langle\rho\rangle\)-equivariant extension \[\label{equ:filtration-H95W}0\longrightarrow\bar{H}_V^\vee\otimes\mathbf{Q}^-\longrightarrow\bar{H}_W\longrightarrow\bar{H}_V\longrightarrow 0\tag{36}\] when restricted to the subgroup \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\times\langle \rho\rangle\le G_V\rtimes\langle\rho\rangle\), so we have an isomorphism \(\bar{H}_W\cong \bar{H}_V\oplus (\bar{H}_V^\vee\otimes\mathbf{Q}^-)\) as \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\times\langle \rho\rangle\)-modules.

3 Homotopy automorphisms and mapping spaces↩︎

Our strategy for computing the rational homology of the spaces of self-embeddings appearing in 17 is to compare them to the relevant spaces of homotopy automorphisms forming the bottom row in 34 . The rational homotopy Lie algebras of the universal covers of these spaces of homotopy automorphisms are known by work of Berglund–Madsen [26] and Krannich [7]. We explain their answers in this section, alongside with some preliminaries and further information on the rational homotopy type of related spaces needed in later sections.

3.1 Gradings and Schur functors↩︎

We begin by fixing some conventions on gradings and Schur functors used in the remainder of this work.

3.1.1 Grading conventions↩︎

Unless said otherwise, throughout this and the following sections, we work in the symmetric monoidal category of \(\mathbf{Z}\)-graded vector spaces over \(\mathbf{Q}\), where the symmetry includes the Koszul sign rule, i.e.  is induced by \(v\otimes w\mapsto (-1)^{\deg(v)\deg(w)} w\otimes v\). For a graded vector space \(A\), we write \(A_k\) for the subspace of elements of degree \(k\) and \(|x|\) for the degree of an homogenous element \(x\in A\). For an ungraded vector space \(B\), we write \(B[k]\) for the graded vector space concentrated in degree \(k\) with \(B[k]_k=B\). The graded dual of a graded vector space \(A\) is denoted \(A^\vee\coloneq \mathrm{Hom}(A,\mathbf{Q}[0])\), and the \(k\)-fold suspension by \(s^kA\). Total dimension is denoted \(\dim(A)\). A superscript \((-)^+\) indicates that we pass to the subspace of elements of positive degree. We write \((-)_\mathbf{Q}\coloneq (-)\otimes_\mathbf{Z}\mathbf{Q}\) for rationalisation. Given a space \(X\), we abbreviate its desuspended reduced homology by \[H_X\coloneq s^{-1}\widetilde{\mathrm{H}}_*(X;\mathbf{Q}).\] For \(X=V_g\) and \(X=W_{g,1}\), we often neglect the genus \(g\) from the notation and simply write \[H_V\coloneq H_{V_g}=s^{-1}\widetilde{\mathrm{H}}_*(V_g;\mathbf{Q})\quad\text{and}\quad H_W\coloneq H_{W_{g,1}}=s^{-1}\widetilde{\mathrm{H}}_*(W_{g,1};\mathbf{Q}).\] This relates to the ungraded vector spaces \(\bar{H}_V\) and \(\bar{H}_W\) introduced in 2.5 by \[H_V=\bar{H}_V[n-1]\quad\text{and}\quad H_W=\bar{H}_W[n-1].\] Similarly, we write \(K_W\coloneq \ker\big(\iota_*\colon H_W\rightarrow H_V\big)\), so that \(K_W=\bar{K}_W[n-1]\).

3.1.2 Schur functors↩︎

Given a partition \(\mu\vdash k\) of an integer \(k\ge0\) the graded Schur functor associated to \(\mu\) is the endofunctor on graded vector spaces given by \[S_\mu(A)\coloneq [(\mu)\otimes A^{\otimes k}]^{\Sigma_k}\] where \((\mu)\) is the irreducible \(\mathbf{Q}[\Sigma_k]\)-module associated to \(\mu\) and \([-]^{\Sigma_k}\) are the invariants with respect to the symmetric group action. The relation to usual Schur functors on ungraded vector spaces is as follows: because of the Koszul sign rule, given an ungraded vector space \(B\), we have \[S_{\mu}(B[m])= [(\mu)\otimes (1^k)^{\otimes m}\otimes B^{\otimes k}]^{\Sigma_k}[km]=s^{km}\begin{cases} S_{\mu}(B[0])&\text{for }m\text{ even}\\ S_{\mu'}(B[0])&\text{for }m\text{ odd} \end{cases}\] where \(\mu'\vdash k\) denotes the conjugate partition, so \((\mu)\otimes (1^k)=(\mu')\).

3.1.3 Symplectic Schur functors↩︎

Now suppose the graded vector space \(A\) comes with an antisymmetric nondegenerate pairing of degree \(k\), i.e.a map of graded \(\mathbf{Q}[\Sigma_2]\)-modules \[\lambda\colon A\otimes A\longrightarrow(1^2)\otimes \mathbf{Q}[k]\] such that the adjoint \(A\rightarrow\mathrm{Hom}(A,\mathbf{Q}[k])=s^{k}A^\vee\) is an isomorphism.

In this case, for \(r\ge0\), we write \[\textstyle{A^{[r]} \coloneq \ker\big(A^{\otimes r}\xrightarrow{\lambda_{i,j}}\bigoplus_{1\le i,j\le r}A^{\otimes r-2}\otimes \mathbf{Q}[k] \big)},\] where \(\lambda_{i,j}\) applies \(\lambda\) to the \(i\)th and \(j\)th coordinate, and define the graded symplectic Schur functor \[V_{\mu}(A) \coloneq [(\mu)\otimes A^{[k]}]^{\Sigma_k}\] for partitions \(\mu\vdash k\). Similarly to the non-symplectic case discussed above, if \(A=B[m]\) for an ungraded vector space \(B\) and \(m\ge0\), we have \[V_\mu(B[m])\cong s^{km}\begin{cases} V_{\mu}(B[0])&\text{for }m\text{ even}\\ V_{\mu'}(B[0])&\text{for }m\text{ odd}. \end{cases}\]

3.2 Lie algebras and their derivations↩︎

For a graded \(\mathbf{Q}\)-vector space \(A\), we write \(\mathbf{L}(A)\) for the free graded Lie algebra on \(A\) and \(\mathbf{L}^k(A)\subset \mathbf{L}(A)\) for the subspace spanned by \(k\)-fold brackets. For subspaces \(C,D\subset L\) of a graded Lie algebra \(L\), we write \([C,D]\subset L\) for the subspace spanned by elements \([c,d]\) with \(c\in C\) and \(d\in D\). For a morphism \(f\colon L\rightarrow L'\) of graded Lie algebras, we write \(\mathrm{Der}^f(L,L')\) for the graded vector space of \(f\)-derivations whose degree \(k\) component consists of linear maps \(\theta\colon L\rightarrow L'\) that increase the degree by \(k\) and satisfy the condition \(\theta([x,y])=[\theta(x),f(x)]+(-1)^{k|x|}[f(x),\theta(y)]\). Having fixed an element \(\omega\in L\), we write \(\mathrm{Der}_\omega^f(L,L')\subset \mathrm{Der}^f(L,L')\) for the subspace of derivations that vanish on \(\omega\). If \(f=\mathrm{id}\), we abbreviate \(\mathrm{Der}(L)=\mathrm{Der}^{\mathrm{id}}(L,L)\) and and consider this graded vector space as a graded Lie algebra via the bracket \([\theta,\psi] \coloneq \theta\circ \psi-(-1)^{|\psi|}\psi\circ\theta\) for homogeneous elements \(\theta,\psi\in \mathrm{Der}(L)\). Note that for \(\omega \in L\) the subspace \(\mathrm{Der}_\omega(L)\subset \mathrm{Der}(L)\) is a graded sub Lie algebra.

3.2.1 A dimension formula↩︎

In calculations within 5, we will use the following decomposition of the subspace of bracket lengths \(\le4\) of the free graded Lie algebra \(\mathbf{L}(A\oplus x)\) on a graded vector space \(A\) concentrated either in even or in odd degrees and an additional generator \(x\). We write \(A^{\mathrm{even}}\) and \(A^{\mathrm{odd}}\) for the subspaces of elements of even respectively odd degree.

3.3 Homotopy groups of mapping spaces↩︎

We will need some information on the rational homotopy groups of \(B\mathrm{hAut}_\partial(M)\) for various compact \(d\)-manifolds \(M\) and on \(\mathrm{Map}_*(M,X)\) for spaces \(X\). Most of what we need can be extracted from work of Berglund–Madsen [26]. Their results apply in larger generality, but to simplify the exposition, we restrict to

• a \(1\)-connected compact oriented \(d\)-manifold \(M\) equivalent to a wedge of spheres, together with an orientation-preserving equivalence \(\partial M\simeq S^{d-1}\) and based at a point in \(\partial M\),

• a \(1\)-connected based space \(X\) equivalent to a wedge of spheres.

Using the notation of 3.1.1, the Hurewicz homomorphism induces maps of graded vector spaces \(\pi_*(\Omega_0 M)\rightarrow H_M\) and \(\pi_*(\Omega_0 X)\rightarrow H_X\) that are epimorphisms as \(M\) and \(X\) are equivalent to wedges of spheres. A choice of splittings \[s_M\colon H_M\rightarrow\pi_*(\Omega_0 M)\quad\text{and}\quad s_X\colon H_X\rightarrow\pi_*(\Omega_0 X)\] induces isomorphisms of graded Lie algebras \(\mathbf{L}(H_M) \cong \pi_{*}(\Omega_0M)\) and \(\mathbf{L}(H_X) \cong \pi_{*}(\Omega_0X)\), the first being equivariant with respect to the subgroup \[\label{equ:haut-split}\pi_0\mathrm{hAut}^\mathrm{split}_*(M,\partial M)\subset \pi_0\mathrm{hAut}_*(M,\partial M)\tag{39}\] of homotopy automorphisms of \((M,\partial M)\) that preserve the splitting \(s_M\). Note that 39 is an equality whenever \(s_M\) is unique, e.g. if \(M\) is equivalent to a wedge of equidimensional spheres.

In the case \(X=M\) and \(f=\mathrm{id}\), the bottom map in the diagram of the previous lemma is surjective since \(s^{-(d-2)}\mathbf{L}(H_M)\) has no indecomposable elements in positive degrees, so combining 26 with the long exact sequence induced by the fibration \[\mathrm{hAut}_\partial(M)\longrightarrow\mathrm{hAut}_*(M)\overset{(-)\circ \iota_\partial}{\longrightarrow} \mathrm{Map}_*(\partial M,M),\] one arrives at a chain of \(\pi_0\mathrm{hAut}^\mathrm{split}(M,\partial M)\)-equivariant isomorphisms in positive degrees \[\pi_*(\Omega_0B\mathrm{hAut}_\partial(M))_\mathbf{Q}\cong \mathrm{Der}_\omega(\mathbf{L}(H_M))\cong \ker\big(s^{-(d-2)}\mathbf{L}(H_M)\otimes H_M\otimes\mathbf{Q}^-\overset{[-,-]}{\longrightarrow}s^{-(d-2)}H_M\big).\] Berglund–Madsen [26] moreover show that the first isomorphism is compatible with the Lie bracket on both sides (see Proposition 3.7 loc.cit.), and a minor extension of their Proposition 6.6 incorporating \(\mathbf{Q}^-\) identifies the right-hand kernel canonically with \(\mathcal{L}ie(\!(H_M,d)\!)\otimes \mathbf{Q}^-\) where we write more generally for a graded vector space \(A\) \[\label{eq:interpretation-cyclic-lie}\mathcal{L}ie(\!(A,d)\!)\coloneq\textstyle{ s^{-(d-2)}\bigoplus_{k \geq 2} Lie(\!(k)\!)\otimes_{\Sigma_k} A^{\otimes k}}\tag{40}\] with \(Lie(\!(k)\!)\) the arity \(k\) part of the cyclic Lie operad. We summarise:

3.3.1 Relative homotopy automorphisms of \(V_g\)↩︎

Setting \(M=W_{g,1}\), 27 describes the rational homotopy groups of the base space of the fibration \[\label{equ:fibre-sequence-haut} \mathrm{hAut}_\partial(V_g)\longrightarrow\mathrm{hAut}_{D^{2n}}(V_g,W_{g,1})\longrightarrow\mathrm{hAut}_\partial(W_{g,1}).\tag{41}\] induced by restriction. The homotopy groups of base and fibre were determined as part of [7]. The following is a minor extension of Theorem 4.6 loc.cit.to include the effect of the reflection involution \(\rho\colon V_g\rightarrow V_g\) from 2.8.

3.4 Some characteristic classes↩︎

Specialising Theorems 27 and 28 to degree \(n-1\) and using 29 gives isomorphisms for \(n\ge2\) of the form \[\pi_n(B\mathrm{hAut}_\partial(W_{g,1}))_\mathbf{Q}\cong S_{1^3}(H_W)_{3(n-1)} \otimes \mathbf{Q}^-\;\;\;\text{and}\;\;\; \pi_{n-1}(B\mathrm{hAut}_\partial(V_{g}))_\mathbf{Q}\cong S_{1^3}(H_V)_{3(n-1)} \otimes \mathbf{Q}^-\] but for some purposes we will need a more geometric description of these identifications. To this effect, we will interpret them as certain characteristic classes of \(W_{g,1}\)- or \(V_g\)-fibrations.

3.4.1 The case of \(W_{g,1}\)↩︎

This case has already been discussed in detail in [13], see Section 5.3.3. We briefly recall the essentials. A class \(\xi \in \pi_n(B\mathrm{hAut}_\partial(W_{g,1}))\) classifies a relative fibration \[\pi\colon (E, S^n \times \partial W_{g,1}) \longrightarrow S^n.\] with an identification of the fibre over the basepoint \(*\in S^n\) with \((W_{g,1}, \partial W_{g,1})\). Identifying \(\partial W_{g,1}=S^{2n-1}\) and gluing in \(S^n \times D^{2n}\) results in an oriented fibration \(\bar{\pi}\colon \bar{E} \to S^n\) with closed fibre \(W_g\), with \(\bar{E}\) a Poincaré duality space, which comes with a section \(s \colon S^n \to \bar{E}\) given by the centres of the discs we glued in. The Serre spectral sequence for \(\bar{\pi}\) gives an exact sequence \[0 \longrightarrow\mathrm{H}^n(S^n;\mathbf{Q}) \overset{\bar{\pi}^*}\longrightarrow\mathrm{H}^n(\bar{E};\mathbf{Q}) \longrightarrow\mathrm{H}^n(W_{g};\mathbf{Q}) \longrightarrow 0\] and the section \(s\) provides a preferred splitting, inducing a map \(\iota \colon \bar{H}_W^\vee = \mathrm{H}^n(W_{g};\mathbf{Q}) \rightarrow\mathrm{H}^n(\bar{E};\mathbf{Q})\). This can be used to define a map \[\kappa_W\colon\begin{array}{rcl} \pi_n(B\mathrm{hAut}_\partial(W_{g,1})) & \longrightarrow & \mathrm{Hom}((\bar{H}_W^\vee)^{\otimes 3}, \mathbf{Q}) = \bar{H}_W^{\otimes 3} \\[0.3em] \xi & \longmapsto & \left(v_1 \otimes v_2 \otimes v_3 \mapsto \int_{\bar{E}} \iota(v_1) \cdot \iota(v_2) \cdot \iota(v_3)\right) \end{array}\] As the \(\iota(v_i)\)’s have degree \(n\), on permuting the \(v_i\) the number \(\kappa_W(\xi)(v_1\otimes v_2\otimes v_3)\) transforms as the \(n\)th tensor power of the sign representation of \(\Sigma_3\), and hence taking the grading of \(H_W\) into account the image of \(\kappa_W\) lies in \([(1^3) \otimes {H}_W^{\otimes 3}]^{\Sigma_3}_{3(n-1)}= S_{1^3}(H_W)_{3(n-1)}\). Furthermore, the \(\pi_0\mathrm{hAut}_\partial(W_{g,1})\)-action on \(\pi_n(B\mathrm{hAut}_\partial(W_{g,1}))\) is given by changing the identification of the fibre over the basepoint, so \(\kappa_W\) is equivariant. Finally, the reflection \(\rho\) has the effect of changing the identification of the fibre over the basepoint and the orientation of \(\bar{E}\), so \(\kappa_W\) is \(\rho\)-equivariant when twisting the target by \(\mathbf{Q}^-\). Together with Proposition 5.13 (and the subsequent discussion) of [13], this gives the following.

3.4.2 The case of \(V_g\)↩︎

An element \(\zeta \in \pi_{n-1}(B\mathrm{hAut}_\partial(V_g))\) classifies a relative fibration \[(E', S^{n-1} \times W_{g}) \overset{\pi'}\longrightarrow S^{n-1}\] with an identification of the fibre over the basepoint \(*\in S^n\) with \((V_{g}, W_{g})\). Gluing to this the trivial \(\tilde{V}_g\)-bundle \(S^{n-1} \times \tilde{V}_g \to S^{n-1}\) using the \(\partial V_g=W_g\) yields an oriented fibration \(\bar{\pi}' \colon \bar{E}' \to S^{n-1}\) with closed fibre \(U_g \coloneq V_g\cup_{\partial V_g}\tilde{V}_g\cong \sharp^{g} S^n \times S^{n+1}\), and with section \(s \colon S^{n-1} \to \bar{E}'\) given by the centre \(*\in D^{2n}\subset W_g\). Here \(\tilde{(-)}\) indicates reversal of orientation. The Serre spectral sequence for \(\bar{\pi}'\) gives an isomorphism \[\mathrm{H}^n(\bar{E}';\mathbf{Q}) \overset{\cong}{\longrightarrow} \mathrm{H}^n(U_{g};\mathbf{Q})\cong \mathrm{H}^n(V_{g};\mathbf{Q})=\bar{H}_V^\vee\] with inverse \(\iota' \colon \bar{H}_V^\vee\to \mathrm{H}^n(\bar{E}';\mathbf{Q})\). This can be used to define a map \[\kappa_V\colon\begin{array}{rcl} \pi_{n-1}(B\mathrm{hAut}_\partial(V_{g})) & \longrightarrow & \mathrm{Hom}((\bar{H}_V^\vee)^{\otimes 3}, \mathbf{Q}) = \bar{H}_V^{\otimes 3} \\[0.3em] \zeta & \longmapsto & \left(v_1 \otimes v_2 \otimes v_3 \mapsto \int_{\bar{E}'} \iota'(v_1) \cdot \iota'(v_2) \cdot \iota'(v_3)\right), \end{array}\] and the same degree considerations as in the \(W_g\)-case show that the image of \(\kappa_V\) lands in the subspace \([(1^3) \otimes {H}_V^{\otimes 3}]^{\Sigma_3}_{3(n-1)}=S_{1^3}(H_V)_{3(n-1)}\).

33 allows us to understand the functoriality of the left-hand side with respect to codimension \(0\) embeddings:

3.4.3 A vanishing result↩︎

At a certain point in Section 7.2 we will need to know that the map \(\pi_n(B\mathrm{Emb}^{\cong,\mathrm{fr}}_{\frac{1}{2}\partial}(W_{g,1})_\ell)_\mathbf{Q}\rightarrow\pi_n(B\mathrm{hAut}_\partial(W_{g,1}))_\mathbf{Q}\) induced by the bottom map in 25 is not surjective. To analyse this problem, we consider the composition \[\chi \colon \pi_n(B\mathrm{hAut}_\partial(W_{g,1}))_\mathbf{Q}\overset{\kappa_W}\longrightarrow\bar{H}_W^{\otimes 3} \xrightarrow{(x_1, x_2, x_3) \mapsto \lambda(x_1, x_2) x_3} \bar{H}_W,\] which is nontrivial for \(g \geq 2\), as 30 shows that the image of \(\kappa_W\) are the \((-1)^n\)-symmetric tensors, and the second map sends the \((-1)^n\)-symmetric tensor \[e_1 \otimes f_1 \otimes e_2 + e_2 \otimes e_1 \otimes f_1 + f_1 \otimes e_2 \otimes e_1 + (-1)^n (f_1 \otimes e_1 \otimes e_2 + e_2 \otimes f_1 \otimes e_1 + e_1 \otimes e_2 \otimes f_1)\] to \(2 e_2 \neq 0\). This implies the second part of the following lemma, whose statement uses 25 and the framing conventions from 2.5.1.

4 The fibre to homotopy automorphisms↩︎

To compute the rational cohomology of the framed self-embedding spaces featuring in 17, we first consider the stably framed self-embedding spaces \[B\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(V_{g})_{\ell}\quad \text{and} \quad B\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_{\ell}.\] By forgetting stable framings and using 25 , they admit maps (see 2.1.1 for the notation) \[\label{sec:recall-maps-to-haut} B\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_\ell\rightarrowB\mathrm{hAut}_\partial^{\cong,\ell}(W_{g,1}) \quad\text{and}\quad B\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(V_{g})_\ell\rightarrowB\mathrm{hAut}^{\cong,\ell}_\partial(V_g)\tag{42}\] whose homotopy fibres we denote by \[\label{equ:hofibs-self-emb-to-haut} \mathrm{hAut}^{\cong,\ell}_\partial(V_g)/\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(V_g)_\ell\quad\text{and}\quad \mathrm{hAut}^{\cong,\ell}_\partial(W_{g,1})/\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_\ell.\tag{43}\] These fibres are connected, since the maps 42 are surjective on fundamental groups by construction. In this section, we compute their rational homotopy groups in a range.

4.1 Reduction to a disjunction problem and separating the handles↩︎

As a first step towards 37, we will express the homotopy fibres in consideration rationally in terms of the total homotopy fibres of the commutative squares \[\label{equ:diff-blockdiff-squares} \begin{tikzcd} B\mathrm{Diff}_\partial(D^{2n+1})\arrow[d, phantom, shift left=1.4cm, "\square_{V_g}"]\dar\rar & B\mathrm{Diff}_\partial(V_g)\dar\\ B\mathrm{\widetilde{Diff}}_\partial(D^{2n+1})\rar& B\mathrm{\widetilde{Diff}}_\partial(V_g) \end{tikzcd}\quad\text{and}\quad \begin{tikzcd} B\mathrm{Diff}_\partial(D^{2n})\dar\arrow[d, phantom, shift left=1.4cm, "\square_{W_g}"] \rar& B\mathrm{Diff}_\partial(W_{g,1})\dar\\ B\mathrm{\widetilde{Diff}}_\partial(D^{2n})\rar& B\mathrm{\widetilde{Diff}}_\partial(W_{g,1}) \end{tikzcd}\tag{44}\] induced by extending (block)-diffeomorphisms of an embedded disc by the identity.

Motivated by 39, we now focus on analysing the total homotopy fibres of the squares \(\square_{V_{g}}\) and \(\square_{W_g}\). It will be convenient to factor \(\square_{V_{g}}\) as \[\label{equ:factorisationVg} \begin{tikzcd} B\mathrm{Diff}_\partial(D^{2n+1})\dar\arrow[d, phantom, shift left=1.4cm, "\square_{V_{g-1}}"]\rar & B\mathrm{Diff}_\partial(V_{g-1})\dar\rar\arrow[d, phantom, shift left=1.4cm, "\square_{V_{g-1,g}}"] &B\mathrm{Diff}_\partial(V_{g})\dar\\ B\mathrm{\widetilde{Diff}}_\partial(D^{2n+1})\rar& B\mathrm{\widetilde{Diff}}_\partial(V_{g-1})\rar & B\mathrm{\widetilde{Diff}}_\partial(V_{g}) \end{tikzcd}\tag{46}\] by using the inclusion \(V_{g-1}\subset V_g\) and extending diffeomorphisms by the identity. Moreover, viewing \(W_{g,1}\) as \(W_{g-1,1} \natural (D^{n}\times S^n\cup_{D^n\times D^n}S^n\times D^n)\), we factor \(\square_{W_g}\) as \[\label{equ:factorisationWg} \begin{tikzcd}[column sep=0.4cm] B\mathrm{Diff}_\partial(D^{2n})\dar\rar\arrow[d, phantom, shift left=1.4cm, "\square_{W_{g-1}}"]& B\mathrm{Diff}_\partial(W_{g-1,1})\dar\rar \arrow[d, phantom, shift left=1.7cm, "\square_{W_{g-1,g-1/2}}"] &B\mathrm{Diff}_\partial(W_{g-1,1}\natural D^n\times S^n)\dar\rar\arrow[d, phantom, shift left=2cm, "\square_{W_{g-1/2,g}}"] &B\mathrm{Diff}_\partial(W_{g,1})\dar\\ B\mathrm{\widetilde{Diff}}_\partial(D^{2n})\rar& B\mathrm{\widetilde{Diff}}_\partial(W_{g-1,1})\rar& B\mathrm{\widetilde{Diff}}_\partial(W_{g,1}\natural D^n\times S^n)\rar&B\mathrm{\widetilde{Diff}}_\partial(W_{g,1}). \end{tikzcd}\tag{47}\] This is advantageous since the manifolds featuring in the right columns of \(\square_{V_{g-1,g}}\), \(\square_{W_{g-1, g-1/2}}\), and \(\square_{W_{g-1/2,g}}\) are obtained from those in the respective left columns by attaching a single handle. As we shall see in the next subsections, the total homotopy fibres of squares of this form can be expressed in terms of (block) embedding spaces of discs.

4.2 Splitting the handles↩︎

This part of the argument is not particular to the manifolds \(V_g\) and \(W_{g,1}\), so we shall phrase it more generally.

Let \(N\) be a compact \(d\)-manifold, possibly with boundary, and \(M=N\cup \{(d-k)\text{-handle}\}\) a handle attachment of index \(d-k\ge 3\). Smoothing corners, \(N\) is diffeomorphic to the complement in \(M\) of a tubular neighborhood \(D^k\times D^{n-k}\subset M\) of the cocore \(D^k\subset M\), so the isotopy extension theorem (see [38] for the block-case) provides a map of fibre sequences \[\begin{tikzcd} \mathrm{Emb}_{\partial D^k\times D^{d-k}}(D^{k}\times D^{d-k},M)_N\dar\rar&B\mathrm{Diff}_{\partial}(N)\rar\dar&B\mathrm{Diff}_\partial(M)\dar\\ \mathrm{\widetilde{Emb}}_{\partial D^k\times D^{d-k}}(D^{k}\times D^{d-k},M)_N\rar&B\mathrm{\widetilde{Diff}}_{\partial}(N)\rar&B\mathrm{\widetilde{Diff}}_\partial(M), \end{tikzcd}\] where \((-)_N\) indicates that we pass to the appropriate collection of components, so the total homotopy fibre of right-hand square is equivalent to \[\mathrm{hofib}\big(\mathrm{Emb}_{\partial D^k\times D^{d-k}}(D^{k}\times D^{d-k},M)_N\rightarrow\mathrm{\widetilde{Emb}}_{\partial D^k\times D^{d-k}}(D^{k}\times D^{d-k},M)_N\big).\] Inclusion induces a map of this homotopy fibre to \(\mathrm{Emb}^{(\sim)}_{\partial D^k\times D^{n-k}}(D^{k}\times D^{d-k},M)\) which is an equivalence since the latter is connected (this uses the assumption \(d-k\ge3\), c.f.[39]). Restricting embeddings to the cocore \(D^k\) induces an equivalence \[\label{equ:no-normal-data} \mathrm{Emb}^{(\sim)}_{\partial D^k\times D^{d-k}}(D^{k}\times D^{d-k},M)\overset{\simeq}{\longrightarrow}\mathrm{Emb}^{(\sim)}_{\partial}(D^{k},M),\tag{48}\] since the homotopy fibres of the individual maps between the spaces of embeddings and between the spaces of block embeddings are, by taking differentials, both equivalent to the space of automorphisms of the normal bundle of the cocore \(D^k\subset M\) which cover the identity and are standard near the boundary.

Returning to the factorisation 46 , we specialise to \(N=V_{g-1}\) and \(M=V_{g-1}\cup\{n\text{-handle}\}\cong V_g\) to derive an equivalence \[\label{equ:tohofib-embeddings-Vg} \mathrm{tohofib}\big(\square_{V_{g-1,g}}\big)\simeq\mathrm{Emb}^{(\sim)}_\partial(D^{n+1},V_g),\tag{49}\] where \(D^{n+1}\times\{*\}\subset V_{g-1}\natural (D^{n+1}\times S^n)=V_g\) is the cocore of the “last” handle. Similarly, we obtain \[\begin{align} \label{equ:tohofib-embeddings-Wg} \begin{aligned} \mathrm{tohofib}\big(\square_{W_{g-1,g-1/2}}\big) &\simeq\mathrm{Emb}^{(\sim)}_\partial(D^{n},W_{g-1,1}\natural (D^n\times S^n))\\ \mathrm{tohofib}\big(\square_{W_{g-1/2,g-1}}\big) &\simeq\mathrm{Emb}^{(\sim)}_\partial(D^{n},W_{g,1}) \end{aligned} \end{align}\tag{50}\] of the total homotopy fibre of the middle and right square of 47 , using the cocores \[D^n\times\{*\}\subset W_{g-1,1}\natural (D^n\times S^n)\quad \text{and}\quad \{*\}\times D^n\subset W_{g-1,1}\natural (D^n\times S^n\cup_{D^n\times D^n}S^n\times D^n).\]

4.3 The delooping trick and disjunction↩︎

To analyse the right-hand sides of 49 and 50 , we use a convenient trick to relate embedding spaces of discs of different dimension, sometimes called the delooping trick. It applies to several variants of embedding spaces and has various incarnations (see e.g. [40] or [38]). Here is a version suitable for our needs, which we later apply to 49 and 50 :

Let \(M\) be a compact \(d\)-manifold with an embedded neat disc \(D^k\subset M\), i.e. \(D^k\) is transverse to \(\partial M\) and \(D^k\cap\partial M=\partial D^k\). We assume \(d-k\ge3\). Let \(D^{k-1}\subset D^k\) be an equatorial disc separating \(D^k\) into half-discs \(D^k=D^k_+\cup D^k_-\). We write \(S^{k-1}_{\pm}=D^k_{\pm}\cap \partial D^{k}\) for the boundary hemispheres. We have a map of fibre sequences \[\begin{tikzcd} \mathrm{Emb}_{S^{k-1}_+\cup D^{k-1}}(D^k_+,M)\dar\rar&\mathrm{Emb}_{S^{k-1}_+}(D^k_+,M)\rar\dar& \mathrm{Emb}_{\partial}(D^{k-1},M)\dar\\ \mathrm{\widetilde{Emb}}_{S^{k-1}_+\cup D^{k-1}}(D^k_+,M)\rar&\mathrm{\widetilde{Emb}}_{S^{k-1}_+}(D^k_+,M)\rar& \mathrm{\widetilde{Emb}}_{\partial}(D^{k-1},M) \end{tikzcd}\] by restricting embeddings of \(D^k_+\) to the equatorial disc. Shrinking \(D^k_+\) to a small neighborhood of \(S^{k-1}_+\) shows that the spaces forming the middle column are contractible, so taking vertical homotopy fibres yields a preferred (up to contractible choices) equivalence \[\mathrm{Emb}_{S^{k-1}_+\cup D^{k-1}}^{(\sim)}(D^k_+,M)\simeq \Omega\mathrm{Emb}_\partial^{(\sim)}(D^{k-1},M).\] Now let \(T\subset M\) be a closed tubular neighborhood of \(D^{k-1}\). Restricting embeddings induces a map of fibre sequences \[\begin{tikzcd}[column sep=0.25cm] \mathrm{Emb}_{\partial}\big(D^k_+\backslash (\mathrm{int}(T)\cap D^k_+),M\backslash \mathrm{int}(T) \big)\dar\rar&\mathrm{Emb}_{S^{k-1}_+\cup D^{k-1}}(D^k_+,M)\rar\dar& \mathrm{Emb}_{D^{k-1}}(T\cap D^k_+,M)\arrow[d]\\ \mathrm{\widetilde{Emb}}_{\partial} \big(D^k_+ \backslash(\mathrm{int}(T)\cap D^k_+),M\backslash \mathrm{int}(T) \big)\rar&\mathrm{\widetilde{Emb}}_{S^{k-1}_+\cup D^{k-1}}(D^k_+,M)\rar& \mathrm{\widetilde{Emb}}_{D^{k-1}}(T\cap D^k_+,M) \end{tikzcd}\] whose right column is an equivalence, since by taking derivatives both spaces are equivalent to the space of normal vector fields of \(D^{k-1}\). Up to smoothing corners, \(D^k_+\backslash (\mathrm{int}(T)\cap D^k_+)\) is a \(k\)-disc and \(M\backslash \mathrm{int}(T)\) is up to diffeomorphism obtained from \(M\) by attaching an \((d-k)\)-handle \(H\) (see [40] for an enlightening picture in the situation of concordance embeddings), so we arrive at equivalences \[\label{equ:delooping-trick} \mathrm{Emb}^{(\sim)}_{\partial}(D^k,M\cup H)\simeq \mathrm{Emb}_{S^{k-1}_+\cup D^{k-1}}^{(\sim)}(D^k_+,M)\simeq \Omega\mathrm{Emb}_\partial^{(\sim)}(D^{k-1},M).\tag{51}\] Viewing \(M\cong (M\cup H)\backslash \nu(D^k)\) as being obtained from \(M\cup H\) by removing an open tubular neighborhood \(\nu(D^k)\subset H\) of the cocore, we have an inclusion \(\mathrm{Emb}^{(\sim)}_\partial (D^k,M)\subset \mathrm{Emb}^{(\sim)}_{\partial}(D^k,M\cup H)\), which gives, via the equivalence above, a map of the form \[\label{equ:delooping} \mathrm{Emb}^{(\sim)}_\partial(D^k,M)\longrightarrow\Omega\mathrm{Emb}_\partial^{(\sim)}(D^{k-1},M).\tag{52}\] By a form of Morlet’s lemma of disjunction [38] (and an improvement by one degree in some cases due to Goodwillie [40]), the inclusion \[\mathrm{Emb}^{(\sim)}_\partial (D^k, (M\cup H)\backslash \nu(D^k))\subset \mathrm{Emb}^{(\sim)}_{\partial}(D^k,M\cup H)\] is \((2d-2k-4)\)-connected, so in this case the map 52 is \((2d-2k-4)\)-connected as well. Iterating this argument, we arrive at the following (c.f. [38]).

From work of Haefliger, one can deduce the following (see [38]).

In view of 40, this has the following corollary (see [38]).

Applied to the cases 49 and 50 of our interest, we conclude that the total homotopy fibres of the squares \(\square_{V_{g-1,g}}\), \(\square_{W_{g-1,g-1/2}}\), and \(\square_{W_{g-1/2,g-1}}\) are \((2n-5)\)-connected, so an induction on \(g\) using the factorisations 46 and 47 shows that the same holds for the squares \(\square_{V_{g}}\) and \(\square_{W_{g}}\). Together with 39, this proves 37 in the weaker range \(*<2n-5\).

4.4 Secondary disjunction↩︎

To go beyond this range, we use a relative version of 40 which can be seen as a special case of a multi-relative disjunction lemma for \(\mathrm{Emb}^{(\sim)}_\partial(-,-)\). The latter can be derived from Goodwillie’s multi-relative disjunction lemma for concordance embeddings [40] by a relatively straight-forward induction (see [39]).

In the situation of 4.3, suppose we are given an additional neat submanifold \(N^m\subset M^d\) disjoint from the previously fixed disc \(D^k\). Applying the above procedure to the pair \((M,M\backslash \nu(N))\) where \(\nu(N)\) is an open tubular neighbourhood of \(N\subset M\) yields a square \[\begin{tikzcd} \mathrm{Emb}^{{(\sim)}}_\partial(D^k,M\backslash \nu(N))\dar\rar&\Omega\mathrm{Emb}^{{(\sim)}}_\partial(D^{k-1},M\backslash \nu(N))\dar\\ \mathrm{Emb}^{{(\sim)}}_\partial(D^k,M)\rar&\Omega\mathrm{Emb}^{{(\sim)}}_\partial(D^{k-1},M) \end{tikzcd}\] which commutes up to preferred homotopy and agrees up to equivalence with the square induced by inclusion \[\begin{tikzcd}[column sep=0.3cm] \mathrm{Emb}^{(\sim)}_\partial\big(D^k, (M\cup H)\backslash (\nu(D^k)\cup \nu(N))\big)\rar\dar& \mathrm{Emb}^{(\sim)}_\partial \big(D^k, (M\cup H)\backslash \nu(N)\big)\dar\\ \mathrm{Emb}^{(\sim)}_\partial\big(D^k, (M\cup H)\backslash \nu(D^k)\big)\rar&\mathrm{Emb}^{(\sim)}_{\partial}\big(D^k,M\cup H\big) \end{tikzcd}\] which is \((3d-2k-m-6)\)-cartesian if \(d-k\ge3\) and \(d-m\ge3\) by an application of [39]. Iterating this argument yields the following relative from of 40.

In 4.6, we will use 43 to compare the embedding spaces 49 and 50 to spaces of embeddings of discs. In the next section, we explain how to compute the homotopy groups of the relevant embedding spaces of discs.

4.5 The case of discs↩︎

As a next step, we compute the rational homotopy groups of the spaces \(\mathrm{Emb}_\partial^{(\sim)}(D^k,D^d)\) based at the standard inclusion in a range by combining 40 with computations for the spaces \(\mathrm{Emb}_{\partial}(D^k,D^d)\) of long embeddings due to Arone–Turchin and Fresse–Turchin–Willwacher [41], [42]. Note that \(\mathrm{Emb}_\partial^{(\sim)}(D^k,D^d)\) carries an \(H\)-space structure induced by stacking, so its fundamental group is abelian and may be rationalised.

During the proof of this proposition, we write \[\mathrm{G}(m)\coloneq \mathrm{hAut}(S^{m-1}) \quad\text{and}\quad G \coloneq {\mathop{\mathrm{hocolim}}}_m\mathrm{G}(m)\] where the colimit is taking over the maps induced by unreduced suspension. The standard action of \(\mathrm{O}(m)\) on \(S^{m-1}\) defines a map \(\mathrm{O}(m)\rightarrow\mathrm{G}(m)\) which is compatible with the stabilisation maps given by inclusion and unreduced suspension. Moreover, this map fits into a commutative diagram of horizontal fibre sequences \[\begin{tikzcd} \mathrm{O}(m-1)\rar{\subset}\dar &\mathrm{O}(m)\dar\rar&S^{m-1}\arrow[d,equal]\\ \Omega^{m-1}_{\pm1}S^{m-1}=\mathrm{hAut}_*(S^{m-1})\rar{\subset}&G(m)\rar{\mathrm{ev}}&S^{m-1}. \end{tikzcd}\] Using the induced map of long exact sequences of homotopy groups, one computes that \[\label{equ:homotopy-g40n41} \pi_*(G(m))_\mathbf{Q}\cong \begin{cases} \mathbf{Q}[m-1]&m\text{ even}\\ \mathbf{Q}[2m-3]&m \text{ odd} \end{cases}\quad\text{for }*>0,\tag{53}\] a fact we shall make use of in the proof of the proposition.

We now turn towards the proof of the main result of the section, 37.

4.6 Proof of 37↩︎

The claim regarding the finiteness of the fundamental groups has been dealt with as part of 39, so we are left with showing that the higher rational homotopy groups are as claimed, which we do for \(V_g\) first. By 39, these homotopy groups agree up to a shift by \(1\) with the rational homotopy groups of the total homotopy fibre of the square \(\square_{V_{g}}\) in 44 , so our task is to compute \[\pi_*(\mathrm{tohofib}(\square_{V_{g}}))_\mathbf{Q}\cong \begin{cases} \mathbf{Q}^{\frac{g(g-1)}{2}}[2n-3]&n\text{ even}\\ \mathbf{Q}^{\frac{g(g+1)}{2}}[2n-3]&n\text{ odd} \end{cases}\quad\text{for }0<*<3n-6.\] Using the factorisation in 46 and the equivalence 49 , these homotopy groups fits into a long exact sequence of the form \[\ldots\rightarrow\pi_*(\mathrm{tohofib}(\square_{V_{g-1}}))\rightarrow\pi_*(\mathrm{tohofib}(\square_{V_{g}}))\rightarrow\pi_{*}(\mathrm{Emb}^{(\sim)}_\partial(D^{n+1},V_g))\rightarrow\ldots,\] where \(D^{n+1}\subset V_g\) is the cocore of the last handle. Given that \(\mathrm{tohofib}(\square_{V_{0}})\) is evidently contractible, the claim for \(V_g\) will follow by induction on \(g\) once we show \[\pi_*\mathrm{Emb}^{(\sim)}_\partial(D^{n+1},V_{g})_\mathbf{Q}\cong \begin{cases}\mathbf{Q}^{g-1}[2n-3]&n\text{ even}\\\mathbf{Q}^g[2n-3]&n\text{ odd}\end{cases}\quad\text{for }0<* < 3n-6.\] To this end, we apply 43 with \(M=V_g\) and \(N=\coprod_{i=1}^gD^{n+1}\) a choice of cocores of the \(g\) handles, disjoint from the previously fixed cocore \(D^{n+1}\subset V_g\) of the last handle. In this case, \(M\backslash \nu(N)\cong D^{2n+1}\) and this diffeomorphism can be chosen such that \(D^{n+1}\subset M\backslash \nu(N)\cong D^{2n+1}\) is the standard inclusion. We thus have a \((3n-6)\)-cartesian square \[\label{equ:disunction-square-vg} \begin{tikzcd} \mathrm{Emb}^{{(\sim)}}_\partial(D^{n+1},D^{2n+1})\arrow[d,hook]\rar&\Omega^{n+1}\mathrm{Emb}^{{(\sim)}}(*,D^{2n+1})\arrow[d,hook]\\ \mathrm{Emb}^{{(\sim)}}_\partial(D^{n+1},V_g)\rar& \Omega^{n+1}\mathrm{Emb}^{{(\sim)}}(*,V_g). \end{tikzcd}\tag{58}\] By 41, \(\Omega^{n+1}\mathrm{Emb}^{{(\sim)}}(*,D^{2n+1})\) is \((3n-4)\)-connected. From 44, we obtain \[\pi_*(\mathrm{Emb}^{(\sim)}(D^{n+1},D^{2n+1}))_\mathbf{Q}\cong \begin{cases} \mathbf{Q}[2n-4]&n\text{ even}\\ 0&n\text{ odd}\\ \end{cases}\quad\text{for }0<*<3n-6\] and in the next section we will compute \(\pi_*(\mathrm{Emb}^{(\sim)}(*,V_g))_\mathbf{Q}\cong \mathbf{Q}^g[3n-2]\) for \(0<*< 4n-5\) as part of 45. Putting these computations together, we see that in the range \(0<*<3n-6\), the square 58 induces an exact sequence of the form \[\ldots\overset{\partial}{\longrightarrow} \begin{cases} \mathbf{Q}[2n-4]&n\text{ even}\\ 0&n\text{ odd}\\ \end{cases}\longrightarrow\pi_*(\mathrm{Emb}^{{(\sim)}}_\partial(D^{n+1},V_g))_\mathbf{Q}\longrightarrow\mathbf{Q}^g[2n-3] \longrightarrow\ldots,\] which leaves us with showing that the boundary map \(\partial\) is surjective in degree \(2n-4\) if \(n\) is even. As the square 58 is natural (up to homotopy) with respect to the inclusion \(V_g\subset V_{g+1}\), it suffices to check surjectivity for \(g=1\), where we have \[\mathrm{Emb}_\partial^{(\sim)}(D^{n+1},V_1)\simeq \Omega\mathrm{Emb}_\partial^{(\sim)}(D^{n},D^{2n+1})\] by applying the delooping trick 51 for \(M=D^{2n+1}\). From 44, we see that this space has no rational homotopy in degree in \(2n-4\), so \(\partial\) must be surjective and the proof in the case of \(V_g\) is finished.

The argument for \(W_{g,1}\) proceeds analogously: using 39, the factorisation 47 and the equivalences 50 , it suffices to show that the spaces \[\mathrm{Emb}^{(\sim)}_\partial(D^{n},W_{g-1,1}\natural (D^n\times S^n))\quad\text{and}\quad\mathrm{Emb}^{(\sim)}_\partial(D^{n},W_{g,1})\] are rationally \((3n-7)\)-connected. We then apply 43 in both cases to the cocores of the manifolds to reduce the claim to showing that the spaces \[\mathrm{Emb}^{(\sim)}_\partial(D^{n},D^{2n}),\quad\Omega^{n}\mathrm{Emb}^{(\sim)}_\partial(*,W_{g-1,1}\natural (D^n\times S^n)),\quad\text{and}\quad\Omega^{n}\mathrm{Emb}^{(\sim)}_\partial(*,W_{g,1})\] are rationally \((3n-7)\)-connected, which follows from 44 and 45 below.

5 Block embeddings of a point↩︎

This section serves to explain a method to compute rational homotopy groups of the spaces \(\mathrm{Emb}^{(\sim)}(*,M)\) (see 4.2 for the notation) for \(1\)-connected compact manifolds \(M\) and to use it to provide the following missing ingredient in the proof of 37. Note that spaces involved are \(1\)-connected by 41, so rationalising their homotopy groups is no issue.

We first translate the problem into unstable homotopy theory. To this end, we fix more generally a compact \(d\)-manifold \(M\) together with a fixed embedded disc \(D^d\subset \mathrm{int}(M)\) and denote by \(\mathrm{hAut}_{\partial\sqcup *}(M)/\mathrm{hAut}_\partial(M^\circ)\) the homotopy fibre of the map \(B\mathrm{hAut}_{\partial M^\circ}(M^\circ)\rightarrowB\mathrm{hAut}_{\partial M\sqcup *}(M)\) induced by extending homotopy equivalences by the identity, where \(*\in \mathrm{int}(M)\) is the centre of the disc and \(M^\circ\coloneq M\backslash\mathrm{int}(D^d)\). Note that \(\partial M^\circ=\partial M\sqcup \partial D^d\).

5.1 Pointed Poincaré embeddings of discs↩︎

In view of 47, we are interested in the rational homotopy groups of the space \[\Omega \mathrm{hAut}_{\partial\sqcup *}(M)/\mathrm{hAut}_\partial(M^\circ)\simeq \mathrm{hofib}_{\mathrm{id}}\big(\mathrm{Map}_\partial(M^\circ,M^\circ)\longrightarrow\mathrm{Map}_{\partial \sqcup*}(M,M)\big).\] for a \(1\)-connected \(d\)-manifold \(M\) with nonempty boundary and \(d\ge6\). We assume the boundary to be connected for simplicity. To explain our method, we fix a disc \(D^d\subset \mathrm{int}(M)\) with centre \(*\in D^d\) and recall that \(M^\circ=M\backslash \mathrm{int}(D^d)\), so \(\partial(M^\circ)=\partial M\sqcup S^{d-1}\). We denote by \(\iota^{\circ}\colon S^{d-1}\hookrightarrow M^\circ\) the inclusion of the additional boundary sphere, and by \(\iota_\partial \colon \partial M \hookrightarrow M^\circ\) the inclusion of the original boundary. Extension by the identity induces a map of horizontal fibre sequences \[\label{boundaryrestriction} \begin{tikzcd}[column sep=1.5cm] \mathrm{Map}_\partial(M^{\circ},M^{\circ})\rar\dar&\mathrm{Map}_{S^{2n-1}}(M^{\circ},M^{\circ})\rar{(-)\circ \iota_\partial}\dar\arrow[d, phantom, shift left=1.9cm, "\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}"]&\mathrm{Map}(\partial M,M^{\circ})\arrow[d,"\mathrm{inc}\circ (-)"]\\ \mathrm{Map}_{\partial \sqcup*}(M,M)\rar&\mathrm{Map}_{*}(M,M)\rar{(-)\circ\iota_\partial}&\mathrm{Map}(\partial M,M), \end{tikzcd}\tag{63}\] which we consider as a diagram of based spaces using the standard inclusions. Our goal is to compute the rational homotopy groups of the homotopy fibre of the left vertical map, i.e. those of the total homotopy fibre \[\label{equ:hAut-as-tohofib} \mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}})\simeq \Omega \mathrm{hAut}_{\partial\sqcup *}(M)/\mathrm{hAut}_\partial(M^\circ).\tag{64}\] Fixing inverse homotopy equivalences \[\begin{tikzcd} M\vee S^{d-1}\arrow[rr,"{j}","\simeq"']&&M^{\circ}& M^{\circ}\arrow[rr,"p","\simeq"']&&M\vee S^{d-1}\\ &S^{d-1}\arrow[ul,"\mathrm{inc}_2"]\arrow[ur,"\iota^{\circ}",swap]&&&S^{d-1}\arrow[ul,"\iota^{\circ}"]\arrow[ur,"\mathrm{inc}_2",swap]& \end{tikzcd}\] over \(S^{d-1}\)supported in a disc in \(M\), there is a diagram \[\begin{tikzcd}[column sep=1.5cm] \mathrm{Map}_{S^{d-1}}(M^{\circ},M^{\circ})\dar{p\circ(- )\circ (j\circ \mathrm{inc}_{1})}[swap]{\simeq}\arrow[rr,"(-)\circ\iota_\partial"]&&\mathrm{Map}(\partial M,M^{\circ})\dar{p\circ(-)}[swap]{\simeq}\\ \mathrm{Map}_{*}(M,M\vee S^{d-1})\arrow[r,"(-)\circ \iota_\partial+ \mathrm{inc}_2"]&\mathrm{Map}_{*}(\partial M,M\vee S^{d-1})\arrow[r,"\subset"]&\mathrm{Map}(\partial M,M\vee S^{d-1}) \end{tikzcd}\] which commutes up to preferred homotopy. Here "+" denotes the \(\mathrm{Map}_*(S^{d-1},M\vee S^{d-1})\)-module structure on \(\mathrm{Map}_*(\partial M,M\vee S^{d-1})\) induced by a pinch map \(\partial M\rightarrow\partial M\vee S^{d-1}\) and all spaces inherit base points by the image of \(\mathrm{id}_{M^{\circ}}\); for instance the lower left corner is based at the inclusion \(M\subset M\vee S^{d-1}\). Consequently, the square \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}\) factors up to equivalence as the composition of two squares commuting up to preferred homotopy \[\begin{tikzcd}[column sep=1.5cm] \mathrm{Map}_{*}(M,M\vee S^{d-1})\arrow[r,"(-)\circ \iota_\partial+ \mathrm{inc}_2"]\arrow[d,"\mathrm{pr}_1\circ(-)"]\arrow[d, phantom, shift left=2.2cm, "\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}"]&\mathrm{Map}_{*}(\partial M,M\vee S^{d-1})\arrow[r,"\subset"]\arrow[d,"\mathrm{pr}_1\circ(-)"]\arrow[d, phantom, shift left=2.5cm, "\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}"]&\mathrm{Map}(\partial M,M\vee S^{d-1})\arrow[d,"\mathrm{pr}_1\circ(-)"]\\ \mathrm{Map}_{*}(M,M)\arrow[r,"(-)\circ \iota_\partial"]&\mathrm{Map}_{*}(\partial M,M)\arrow[r,"\subset"]&\mathrm{Map}(\partial M,M), \end{tikzcd}\] so we obtain a fibre sequence \[\label{equ:lestohofib} \mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}})\longrightarrow\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}})\longrightarrow\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}).\tag{65}\] It will be sometimes convenient to replace \(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}})\) by the total homotopy fibre of the square \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'\) obtained from \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}\) by replacing the upper horizontal map \((-)\circ \iota_\partial+ \mathrm{inc}_2\) by \((-)\circ\iota_\partial\). These total homotopy fibres are equivalent via the map \[\label{equ:modified-square} \mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}})\overset{\simeq}{\longrightarrow}\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}')\tag{66}\] induced by acting with \(-\mathrm{inc}_2\).

The homotopy groups of \(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}})\) can easily be computed: the responsible square is obtained by taking horizontal fibres from the square \[\begin{tikzcd} \mathrm{Map}(\partial M,M\vee S^{d-1})\arrow[d,"\mathrm{pr}_1\circ(-)",swap]\rar{\mathrm{ev}}&M\vee S^{d-1}\dar{\mathrm{pr}_1}\\ \mathrm{Map}(\partial M,M)\rar{\mathrm{ev}}&M, \end{tikzcd}\] so we have an equivalence \(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}})\simeq \Omega\mathrm{hofib}(M\vee S^{d-1}\xrightarrow{\mathrm{pr}_1} M)\). Since the projection \(M\vee S^{d-1}\rightarrow M\) has an evident section, we conclude \[\label{equ:homotopy-tohofib3} \pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}))= \ker\big(\pi_{*+1}(M\vee S^{d-1})\xrightarrow{(\mathrm{pr}_1)_*} \pi_{*+1}(M)\big).\tag{67}\] In all our examples, \(M\) comes with an equivalence to a wedge of spheres, so employing the notation of Sections 3.1.1 and 3.3 we choose a splitting \(s_M\colon H_M\rightarrow\pi_{*+1}(M)_\mathbf{Q}\) of the Hurewicz map which induces an analogous splitting for \(M\vee S^{d-1}\) and thus provides isomorphisms \[\mathbf{L}(H_M)\cong \pi_{*+1}(M)_\mathbf{Q}\quad\text{and}\quad \mathbf{L}(H_M\oplus x)\cong \pi_{*+1}(M\vee S^{d-1})_\mathbf{Q},\] where \(\mathbf{L}(-)\) is the free graded Lie algebra and \(x\) is a class in degree \(d-2\). Then 67 becomes \[\label{equ:homotopy-tohofib3-rational} \pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}))_\mathbf{Q}\cong \widetilde{\mathbf{L}}(H_M\oplus x)\coloneq \ker\big(\mathbf{L}(H_M\oplus x)\xrightarrow{\mathrm{pr}_{H_M}}\mathbf{L}(H_M)\big).\tag{68}\]

We now apply this setup to the three manifolds featuring in 45.

5.2 Case I: \(M=W_{g,1}\)↩︎

Applying 26 we see that the square \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'\) has on rational homotopy groups in positive degrees the form (this uses that the projection \(W_{g,1}\vee S^{2n-1}\rightarrow W_{g,1}\) is compatible with the chosen splittings of the rational Hurewicz map) \[\begin{tikzcd} s^{-(2n-2)}\mathbf{L}(H_W\oplus x)\otimes H_W\rar{[-,-]}\dar{\mathrm{id}\otimes \mathrm{pr}_{H_W}}& s^{-(2n-2)}\mathbf{L}(H_W\oplus x)\dar{\mathrm{pr}_{H_W}}\\ s^{-(2n-2)}\mathbf{L}(H_W)\otimes H_W\rar{[-,-]}& s^{-(2n-2)}\mathbf{L}(H_W). \end{tikzcd}\] Using the notation introduced in 68 , the long exact sequence obtained by taking vertical homotopy fibres in \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'\) is thus in positive degrees of the form \[\ldots\rightarrow\pi_*\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}')_\mathbf{Q}\rightarrow s^{-(2n-2)} \widetilde{\mathbf{L}}(H_W\oplus x) \otimes H_W\xrightarrow{[-,-]}s^{-(2n-2)}\widetilde{\mathbf{L}}(H_W\oplus x) \rightarrow\ldots,\] Suppressing degree shifts from the notation, we see from 25 that in degrees \(0<*< 4n-4\), the bracket has the form \[\label{equ:decomp-HW} \begin{tikzcd}[column sep=-0.2cm] {\big(x\otimes H_W\big)}&\oplus& {\big([x,H_W]\otimes H_W\big)}&\oplus\dar{[-,-]}& {\big([H_W,[x,H_W]]\otimes H_W\big)}&&\\ {[x,H_W]}&\oplus&{[[x,H_W],H_W]}&\oplus&{ [[H_W,[x,H_W]],H_W]}&\oplus&{ [x,[x,H_W]]}. \end{tikzcd}\tag{69}\] As the first three direct summands of the target are evidently surjected upon by the corresponding summands of the source and also have the same dimension as a result of the dimension formulas in 25, we conclude \[\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'))_\mathbf{Q}\cong s^{-(2n-3)}[x,[x,H_W]]\quad\text{for }*< 4n-5.\] Moreover, we have \(\widetilde{\mathbf{L}}(H_W\oplus x)=x\oplus [x,H_W]\) in degrees \(*< 4n-4\), so using 66 and 68 , the long exact sequence induced by 65 has in degrees \(*< 4n-5\) the form \[\label{equ:les-tohofib1} \ldots \longrightarrow s(x\oplus [x,H_W])\overset{\partial}{\longrightarrow} s^{-(2n-3)}[x,[x,H_W]]\longrightarrow\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}))_\mathbf{Q}\longrightarrow\ldots.\tag{70}\] The class \(\partial(x)\) vanishes for the degree reasons, and moreover we claim that \(\partial\) maps \([x,H_W]\) surjectively onto \([x,[x,H_W]]\) (and hence isomorphically, as they are equidimensional by 25). Chasing through the computation, we see that this map is given by the composition \[s[x,H_W]\subset s\widetilde{\mathbf{L}}(H_W\oplus x)\longrightarrow s^{-(2n-3)}\widetilde{\mathbf{L}}(H_W\oplus x) \longrightarrow s^{-(2n-3)}[x,[x,H_W]],\] where the first arrow is the restriction of the boundary map of the evaluation fibration \[s\mathbf{L}(H_W\oplus x)\cong\pi_*(W_{g,1}\vee S^{2n-1})_\mathbf{Q}\rightarrow\pi_{*-1}(\mathrm{Map}_*(\partial W_{g,1},W_{g,1}\vee S^{2n-1});\iota_\partial+\mathrm{inc}_2)_\mathbf{Q}\cong s^{-(2n-3)}\mathbf{L}(H_W\oplus x).\] and the second is the projection onto \([x,[x,H_W]]\) with respect to the above decomposition of \(\widetilde{\mathbf{L}}(H_W\oplus x)\) in a range. The boundary map of the evaluation fibration is up to signs given by taking bracket with the class \((\iota_\partial+\mathrm{inc}_2)\in \pi_{*+1}(W_{g,1}\vee S^{2n-1})_\mathbf{Q}\cong \mathbf{L}(H_W\oplus x)\) (see [44]). The inclusion \(\mathrm{inc}_2\) corresponds to the class \(x\) and \(\iota_\partial\) corresponds to an element \(\omega\in[H_W,H_W]\), so the image of \(\partial\) in 70 is given by the projection of \([x+\omega,[x,H_W]]\) onto \([x,[x,H_W]]\) with respect to the decomposition of \(\widetilde{\mathbf{L}}(H_W\oplus x)\) in 69 . As the projection of \([\omega,[x,H_W]]\subset [[H_W,H_W],[x,H_W]]\subset [[H_W,[x,H_W]],H_W]\) is trivial (see 25), this is indeed surjective. Going back to the exact sequence 70 , we conclude \(\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}))_\mathbf{Q}\cong \mathbf{Q}[2n-2]\) for \(0<*< 4n-5\), so using the equivalence 64 , the long exact sequence resulting from 47 has in degrees \(*< 4n-5\) the form \[\ldots\longrightarrow\pi_{*+2}(\mathrm{SO}/\mathrm{SO}(2n))_\mathbf{Q}\overset{\partial}{\longrightarrow} \mathbf{Q}[2n-2]\longrightarrow\pi_*(\mathrm{Emb}^{(\sim)}(*,W_{g,1})_\mathbf{Q}\longrightarrow\ldots .\] In this range, we have \(\pi_{*+2}(\mathrm{SO}/\mathrm{SO}(2n))_\mathbf{Q}\cong\mathbf{Q}[2n-2]\), as already noted in 57 , so given that \(\mathrm{Emb}^{(\sim)}(*,W_{g,1})\) is \((3n-4)\)-connected by 41, the boundary map \(\partial\) has to be an isomorphism and we conclude that \(\mathrm{Emb}^{(\sim)}(*,W_{g,1})\) is rationally \((4n-6)\)-connected, which finishes the proof of 45 for \(W_{g,1}\).

5.3 Case II: \(M=V_{g}\)↩︎

To study the right vertical arrow of \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}\) (tentatively denoted as \(\psi\)) for \(M=V_g\) and \(\partial M=\partial V_g=W_g\), we use the map of fibre sequences \[\begin{tikzcd} \mathrm{Map}_*(W_g,V_g\vee S^{2n})\arrow[d,"(\mathrm{pr}_1\circ (-))\eqcolon\psi"]\arrow[r]&\mathrm{Map}_*(W_{g,1},V_g\vee S^{2n})\dar{(\mathrm{pr}_1\circ (-))}\rar&\mathrm{Map}_*(S^{2n-1},V_g\vee S^{2n})\dar{(\mathrm{pr}_1\circ (-))}\\ \mathrm{Map}_*(W_g,V_g)\rar&\mathrm{Map}_*(W_{g,1},V_g)\rar&\mathrm{Map}_*(S^{2n-1},V_g). \end{tikzcd}\] induced by the cofibre sequence \(S^{2n-1}\rightarrow W_{g,1}\rightarrow W_g\). Using 26 for \(M=W_{g,1}\) and the splittings of the vertical arrows induced by the inclusion \(V_g\subset V_{g}\vee S^{2n-1}\), we see that the short exact sequence of the fibration obtained by taking vertical homotopy fibres at the inclusion \(\iota_\partial\colon W_g\hookrightarrow V_g\) is of the form \[\ldots\longrightarrow\pi_*(\mathrm{hofib}(\psi);\iota_\partial)_\mathbf{Q}\longrightarrow s^{-(2n-2)}\widetilde{\mathbf{L}}(H_V\oplus x)\otimes H_W\overset{[-,\iota(-)]}{\longrightarrow}s^{-(2n-2)}\widetilde{\mathbf{L}}(H_V\oplus x)\longrightarrow\ldots,\] where \(\iota\colon H_W\rightarrow H_V\) is the canonical projection and \(x\) is of degree \(2n-1\). As \(V_g\simeq \vee^gS^n\), the homotopy groups of the homotopy fibre of the left vertical arrow of \(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'\) are given by \[\widetilde{\mathbf{L}}(H_V\oplus x)\otimes H_{V}^\vee\cong s^{-(2n-2)}\widetilde{\mathbf{L}}(H_V\oplus x)\otimes K_W.\] Mapping in these groups into the previous long exact sequence and using \(H_{W}/K_W\cong H_V\), we arrive at a long exact sequence \[\ldots\longrightarrow\pi_{*-1}(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'))_\mathbf{Q}\longrightarrow s^{-(2n-2)}\widetilde{\mathbf{L}}(H_V\oplus x)\otimes H_V\overset{[-,-]}{\longrightarrow}s^{-(2n-2)}\widetilde{\mathbf{L}}(H_V\oplus x)\longrightarrow\ldots.\] Omitting degree shifts, we see from 25 that the bracket is of the form \[\begin{tikzcd}[column sep=-0.2cm] {\big(x\otimes H_V\big)}&\oplus&{\big([x,H_V]\otimes H_V\big)}&\oplus&{ \big([H_V,[x,H_V]]\otimes H_V\big)}\dar{[-,-]}& \oplus &{\big([x,x]\otimes H_V\big) }&&\\ {[x,H_V]}&\oplus&{[[x,H_V],H_V]}&\oplus &{ [[H_V,[x,H_V]],H_V]}&\oplus&{ [[x,x],H_V]}&\oplus&{[x,x]} \end{tikzcd}\] in degrees \(1<*< 4n-3\). In addition to 25, this uses \([H_V,[x,x]]=[x,[x,H_V]]\) which follows from an application of the Jacobi identity. As for \(W_{g,1}\) we combine this with the dimension formulas in 25 to conclude \(\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'))_\mathbf{Q}\cong s^{-2n}[x,x]\) for \(0<*< 4n-5\). From 68 , we see \(\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}}))_\mathbf{Q}\cong \widetilde{\mathbf{L}}(H_V\oplus x)=x\oplus [x,H_V]\) for \(*< 4n-4\), so using 66 the long exact sequence of 65 has in the range \(0<*< 4n-5\) the form \[\ldots \longrightarrow s^{-1}(x\oplus [x,H_V]) \longrightarrow s^{-2n}[x,x] \longrightarrow\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}))_\mathbf{Q}\longrightarrow\ldots.\] By a similar argument as for \(W_{g,1}\), one shows that the first map in this sequence maps \(s^{-1}x\) isomorphically to \(s^{-2n}[x,x]\), so we deduce \(\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {1}}}))_\mathbf{Q}\cong [x,H_V]\) for \(0<*< 4n-5\). As \(\mathrm{SO}/\mathrm{SO}(2n+1)\) is rationally \((4n+2)\)-connected, we can combine this computation with 64 and 47 to conclude that \(\pi_*(\mathrm{Emb}^{(\sim)}(*,V_g))_\mathbf{Q}\cong [x,H_V]\cong\mathbf{Q}^g[3n-2]\) for \(*<4n-5\), which proves 45 for \(V_g\).

5.4 Case III: \(M=W_{g,1}\natural T_h\)↩︎

To ease the notation, we abbreviate \(M\coloneq W_{g,1}\natural T_h\) and write \[\overline{\mathrm{Map}}_A(X,M\vee S^{2n-1})\coloneq \mathrm{hofib}\big(\mathrm{Map}_A(X,M\vee S^{2n-1})\xrightarrow{\mathrm{pr}\circ-} \mathrm{Map}_A(X,M)\big)\] for a space \(X\) together with a subspace \(A\subset X\); the fixed map \(A\rightarrow M\vee S^{2n-1}\) and the basepoints will be clear from the context. Restriction along the system of inclusions \[\begin{tikzcd}[row sep=-0.2cm] &U_{h,1}\arrow[dr,hook]&\\ S^{2n-2}\arrow[ur,hook]\arrow[dr,hook]&&W_{g,1}\natural T_h=M\\ &D^{2n-1}\arrow[ur,hook]& \end{tikzcd},\] where the left arrows are boundary inclusions, \(D^{2n-1}=\partial W_{g,1}\cap \partial(W_{g,1}\natural T_h)\), and \(U_{h,1}=\partial T_h\cap \partial (W_{g,1}\natural T_h)\) induces a commutative diagram of horizontal and vertical fibre sequences \[\begin{tikzcd}[column sep=0.4cm,row sep=0.4cm] \mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}')\dar\rar&\overline{\mathrm{Map}}_{D^{2n-1}}(M,M\vee S^{2n-1})\arrow[d]\rar&\overline{\mathrm{Map}}_*(U_h,M\vee S^{2n-1})\dar\\ \overline{\mathrm{Map}}_{U_{h,1}}(M,M\vee S^{2n-1})\dar\rar&\overline{\mathrm{Map}}_*(M,M\vee S^{2n-1})\dar\rar&\overline{\mathrm{Map}}_*(U_{h,1},M\vee S^{2n-1})\dar\\ \overline{\mathrm{Map}}_{S^{2n-2}}(D^{2n-1},M\vee S^{2n-1})\rar&\overline{\mathrm{Map}}_*(D^{2n-1},M\vee S^{2n-1})\rar&\overline{\mathrm{Map}}_*(S^{2n-2},M\vee S^{2n-1}). \end{tikzcd}\] The long exact sequence of the middle row has the form \[\ldots \rightarrow\pi_*(\overline{\mathrm{Map}}_{U_{h,1}}(M,M\vee S^{2n-1}))_{\mathbf{Q}}\rightarrow\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_{WT}^\vee\rightarrow\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_U^\vee\rightarrow\ldots,\] where the right map is induced by the map \(H_U\rightarrow H_{WT}\) induced by inclusion. The image of the latter is \(H_T\subset H_{WT}\), its kernel is \(K_U\), and its cokernel is \(H_W\), so the long exact sequence yields a short exact sequence of the form \[\label{equ:intermediate-ses-Map-Uh} 0\rightarrow s^{-1} \widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes K_U^\vee \rightarrow\pi_*(\overline{\mathrm{Map}}_{U_{h,1}}(M,M\vee S^{2n-1}))_{\mathbf{Q}}\rightarrow\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_{W}^\vee\rightarrow 0.\tag{71}\] Extension by the identity along the inclusion \(W_{g,1}\subset W_{g,1}\natural T_h=M\) induces a map \[\overline{\mathrm{Map}}_{D^{2n-1}}(W_{g,1},M\vee S^{2n-1})\longrightarrow\overline{\mathrm{Map}}_{U_{h,1}}(M,M\vee S^{2n-1})\] (here \(D^{2n-1}\subset \partial W_{g,1}\) is the disc at which the boundary connected sum \(W_{g,1}\natural T_h\) is taken) which on homotopy groups induces a splitting of 71 which combined with the isomorphisms \(K_U^\vee\cong s^{-(2n-3)}H_T\) and \(H_W^\vee\cong s^{-(2n-2)}H_W\) provides an isomorphism \[\pi_*(\overline{\mathrm{Map}}_{U_{h,1}}(M,M\vee S^{2n-1}))_{\mathbf{Q}}\cong s^{-(2n-2)}\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_{WT}.\] The long exact sequence of the leftmost column therefore has the form \[\label{equ:les-tohofib2-WgTh} \ldots\rightarrow\pi_*(\mathrm{tohofib}(\raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {2}}}'))_\mathbf{Q}\rightarrow s^{-(2n-2)}\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_{WT}\to s^{-(2n-2)}\widetilde{\mathbf{L}}(H_{WT}\oplus x)\rightarrow\ldots,\tag{72}\] and we claim that the right map is given by bracketing. On the subspace \(s^{-(2n-2)}\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_T\), this map is a quotient of the induced map on \(\pi_{*+1}(-)_\mathbf{Q}\) of the lower right vertical arrow in the above diagram, which is given by \([-,\iota(-)]\colon s^{-(2n-3)}\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_U\rightarrow\widetilde{\mathbf{L}}(H_{WT}\oplus x)\) by 26. Secondly, on the subspace \(s^{-(2n-2)}\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_W\), this map is the induced map on homotopy groups by the composition \[\overline{\mathrm{Map}}_{D^{2n-1}}(W_{g,1},M\vee S^{2n-1})\longrightarrow\overline{\mathrm{Map}}_{U_{h,1}}(M,M\vee S^{2n-1})\longrightarrow\overline{\mathrm{Map}}_{D^{2n-1}}(S^{2n-1},M\vee S^{2n-1}),\] which is induced by restricting along the boundary inclusion \(S^{2n-1}\subset W_{g,1}\), so is given by \([-,\iota(-)]\colon s^{-(2n-2)}\widetilde{\mathbf{L}}(H_{WT}\oplus x)\otimes H_W\rightarrow\widetilde{\mathbf{L}}(H_{WT}\oplus x)\) by another application of 26.

Using the exact sequence 72 , the rest of the argument proceeds as for \(W_{g,1}\) in 5.2 and eventually shows that \(\mathrm{Emb}^{(\sim)}(*,W_{g,1}\natural T_h)\) is rationally \((4n-6)\)-connected.

6 Claspers↩︎

By Theorem 37, the dimensions of the vector spaces (see 3.1.1 for the notation) \[\pi_{2n-2}(\mathrm{hAut}^{\cong,\ell}_\partial(V_g)/\mathrm{Emb}^{\mathrm{sfr},\cong}_{\frac{1}{2}\partial}(V_g)_\ell)_\mathbf{Q}\;\;\text{and}\;\; S_{2}(H_{V_g})_{2(n-1)}=\begin{cases} \Lambda^2(\bar{H}_{V_g})&n\text{ even}\\ \mathrm{Sym}^2(\bar{H}_{V_g})&n\text{ odd} \end{cases}\] agree for \(n>3\). Both sides are acted upon by the semi-direct product of the group \[\check{\Lambda}_{V} \coloneq \pi_1(B\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2} D^{2n}}(V_g,W_{g,1})_\ell)\cong \pi_1(B\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2} D^{2n}}(V_g,W_{g,1})_\ell)\] acted upon by the reflection involution \(\langle \rho\rangle\cong \mathbf{Z}/2\) (see Sections 2.7 and 2.8); the action on \(S_{2}(H_{V_g})\) is trivial since \(\rho\) fixes \(H_{V_g}\). This section serves to improve this identification of the underlying vector spaces to an equivariant statement.

The arguments in Section 4 break the symmetry of the manifold \(V_g\) in an essential way, so we will not try to trace the equivariance through that section. Rather, we shall use a variant of Watanabe’s parametrised form [45] of the Goussarov–Habiro theory of claspers to produce an equivariant map from the right to the left term in 48, use Watanabe’s results to deduce that this map is non-zero, and then argue that it has to be an isomorphism.

We begin with a brief recollection on framed embedding spaces.

6.1 Framed embeddings↩︎

Given \(d\)-manifolds \(M\) and \(N\) with fixed framings \(\ell_M\) and \(\ell_N\), the space of framed embeddings \(\mathrm{Emb}^{\mathrm{fr}}((M,\ell_M),(N,\ell_N))\) is the homotopy fibre \[\mathrm{Emb}^{\mathrm{fr}}((M,\ell_M),(N,\ell_N))\coloneq \mathrm{hofib}_{\ell_M}\big(\mathrm{Emb}(M,N)\xrightarrow{(-)^*\ell_N}\mathrm{Bun}(\tau_M,\varepsilon^{d} \big)\big).\] of the map given by taking the derivative followed by postcomposition with the framing \(\ell_N\). The usual composition of embeddings extends to a composition map \[\mathrm{Emb}^{\mathrm{fr}}((M,\ell_M),(N,\ell_N))\times \mathrm{Emb}^{\mathrm{fr}}((N,\ell_N),(K,\ell_K)) \longrightarrow\mathrm{Emb}^{\mathrm{fr}}((M,\ell_M),(K,\ell_K))\] which is unital and associative if one uses Moore-paths in the definition of the homotopy fibres, so it gives rise to a topologically enriched category of framed \(d\)-manifolds and framed embeddings. In particular \(\mathrm{Emb}^{\mathrm{fr}}(M,\ell_M)\coloneq \mathrm{Emb}^{\mathrm{fr}}((M,\ell_M),(M,\ell_M))\) is a topological monoid. Replacing embeddings by diffeomorphisms in this definition yields a group-like monoid \(\mathrm{Diff}^{\mathrm{fr}}(M,\ell_M)\) which is related to \(B\mathrm{Diff}^\mathrm{fr}(M)\) as defined in 2.1.1 by an equivalence \(\mathrm{Diff}^{\mathrm{fr}}(M,\ell_M)\simeq \Omega B\mathrm{Diff}^\mathrm{fr}(M)\) where the loop space is based at the fixed framing \(\ell_M\).

Basing \(B\mathrm{Diff}^\mathrm{fr}_\partial(M)\) and \(B\mathrm{Diff}^\mathrm{fr}_\partial(N)\) at \(\ell_M\) and \(\ell_N\), extension by the identity induces a map \[\pi_0(\mathrm{Emb}^\mathrm{fr}(M,N))\longrightarrow[B\mathrm{Diff}^\mathrm{fr}_\partial(M),B\mathrm{Diff}^\mathrm{fr}_\partial(N)]_*\] where \([-,-]_*\) denotes based homotopy classes². This is functorial, so in particular \(\pi_0(\mathrm{Emb}^\mathrm{fr}(M,M))\) acts on \(B\mathrm{Diff}^\mathrm{fr}_\partial(M)\) in the based homotopy category. Restricting this action along the map \[\pi_1(B\mathrm{Diff}^\mathrm{fr}(M),\ell_M)\cong \pi_0(\mathrm{Diff}^\mathrm{fr}(M,\ell_M))\longrightarrow\pi_0(\mathrm{Emb}^\mathrm{fr}(M,\ell_M))\] agrees with the action of \(\pi_1(B\mathrm{Diff}^\mathrm{fr}(M),\ell_M)\) on \(B\mathrm{Diff}^\mathrm{fr}_\partial(M)\) induced by conjugation.

6.1.1 Framed embeddings between \(V_g\)↩︎

We will occasionally need to produce framed embeddings of the type \(V_a \sqcup V_b \hookrightarrow V_g\), and it will be convenient to do so by indicating where (thickenings of) the cores of the handles go and ignoring the rest. Let \(U_g = \sqcup^g V_1\), so that \(V_g\) is obtained from \(U_g\) by attaching \((g-1)\) 1-handles: more precisely, using the notation of 2.5 we write \(U_g \coloneq \psi(\sqcup^g V_1) \subset V_g\), and denote \(\mathrm{inc}_i : V_1 \hookrightarrow V_3\) for the inclusion of the \(i\)th thickened core.

6.2 Splitting the Weiss fibre sequence↩︎

The proof of 48 makes use of the framed Weiss fibre sequence \(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n+1}) \rightarrowB\mathrm{Diff}^{\mathrm{fr}}_{\partial}(V_g) \rightarrowB\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(V_g)\) from 2.3, which has the following convenient property that distinguishes it from the Weiss fibre sequence for \(W_{g,1}\): the framed standard embedding \(V_g \subset D^{2n+1}\) splits the inclusion of the fibre up to equivalence, so it induces an equivalence \[\label{eq:FrWeissSplit} B\mathrm{Diff}^{\mathrm{fr}}_{\partial}(V_g) \overset{\simeq}\longrightarrowB\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n+1}) \times B\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(V_g).\tag{73}\] This is natural up to homotopy with respect to framed embeddings \(V_g\hookrightarrow V_h\) that commute with the standard framed embedding into \(D^{2n+1}\) up to isotopy of framed embeddings.

We shall be particularly interested in the induced splitting on \(\pi_{n-1}(-)_\mathbf{Q}\), \[\label{equ:splitting-homotopy-group} \pi_{n-1}(B\mathrm{Diff}^{\mathrm{fr}}_{\partial}(V_g))_\mathbf{Q}\cong \pi_{n-1}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n+1}))_\mathbf{Q}\oplus \pi_{n-1}(B\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(V_g))_{\mathbf{Q}},\tag{74}\] which one can further simplify by using the isomorphisms \[\pi_{n-1}(B\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(V_g))_\mathbf{Q}\overset{\cong}{\longrightarrow} \pi_{n-1}(B\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2}\partial}(V_g))_\mathbf{Q}\overset{\cong}{\longrightarrow} \pi_{n-1}(B\mathrm{hAut}_\partial(V_g))_\mathbf{Q}\] resulting from 18 and 37. Furthermore, 33 gives an isomorphism \[\label{equ:iso-first-haut-group} \kappa_V\colon \pi_{n-1}(B\mathrm{hAut}_\partial(V_g))_\mathbf{Q}\overset{\cong}{\longrightarrow} S_{1^3}(H_{V_g})_{3(n-1)}=\begin{cases} \Lambda^3(\bar{H}_{V_g})&n\text{ odd}\\ \mathrm{Sym}^3(\bar{H}_{V_g})&n\text{ even}\end{cases}\tag{75}\] which is \(\pi_0\mathrm{hAut}_{D^{2n}}(V_g,W_{g,1})\)-equivariant and natural with respect to orientation-preserving embeddings \(V_g\hookrightarrow V_h\) by 35. The splitting 74 thus becomes \[\label{equ:splitting-homotopy-group2} \pi_{n-1}(B\mathrm{Diff}^{\mathrm{fr}}_{\partial}(V_g))_\mathbf{Q}\cong \pi_{n-1}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n+1}))_\mathbf{Q}\oplus S_{1^3}(H_{V_g})_{3(n-1)}\tag{76}\] and is natural with respect to framed embeddings \(V_g\hookrightarrow V_h\) that commute with the standard framed embedding into \(D^{2n+1}\) up to isotopy of framed embeddings.

6.3 Watanabe’s Borromean construction↩︎

In [45] Watanabe has constructed classes in the homotopy groups \(\pi_{k(n-1)}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n+1}))\) for \(k\ge2\) and \(n\ge3\) indexed by trivalent graphs with \(k\) vertices (and some extra data). The basic building block of his construction is a class \[\label{equ:alpha-wat}\alpha_{\text{Wat}} \in \pi_{n-1}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_3))\tag{77}\] which he produces (see Section 4.1 loc.cit.) by first constructing an explicit \(V_3\)-bundle over \(S^{n-1}\) using a high-dimensional version of the Borromean rings, giving rise to \(\alpha_{\text{Bor}} \in \pi_{n-1}(B\mathrm{Diff}_\partial(V_3))\) (denoted \(\delta(\alpha')\) on p. loc.cit.), and then showing by obstruction theory that, as long as \(n\) is odd, this class may be lifted along \(B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_3)\rightarrowB\mathrm{Diff}_\partial(V_3)\) after perhaps scaling by some non-zero integer (see Proposition 4.2 loc.cit.).

6.3.1 The Borromean property↩︎

The crucial feature of the class \(\alpha_{\text{Bor}}\), which can be seen directly from Watanabe’s construction (see in particular the proof of Lemma 4.3 loc.cit.), is the following “Borromean property”: attaching a handle to any of the three summands of \(V_3 = \natural^3 S^n \times D^{n+1}\) along \(S^n \times D^n_{-} \subset S^n \times S^n\) where \(D^n_-\subset D^n\) is a hemisphere gives “handle-filling” embeddings \[\label{equ:handle-filing}n_i \colon V_3 \lhook\joinrel\longrightarrow V_2\quad\text{for } i=1,2,3,\tag{78}\] and under any of the induced maps \((n_i)_* \colon \pi_{n-1}(B\mathrm{Diff}_\partial(V_3)) \to \pi_{n-1}(B\mathrm{Diff}_\partial(V_2))\) the class \(\alpha_{\text{Bor}}\) is sent to zero. In the corrigendum [46] Watanabe shows that there is a choice of lift 77 to the classifying space for framed smooth bundles which still satisfies this Borromean property, and he also removes the assumption that \(n\) be odd (see Lemma A and Remark 6 loc.cit.).

For our purposes, it is convenient to work with the following axiomatic description of Watanabe’s building block 77 . This alternative description is slightly surprising, as it is by definition symmetric in the three handles of \(V_3\), whereas Watanabe’s construction favours one.

6.4 The construction↩︎

We begin the construction of the isomorphism in 48 by choosing once and for all a framed embedding \[b \colon V_3 \sqcup V_3 \hookrightarrow V_2\] as depicted in 2. More precisely, this is constructed as follows: inside the standard model \[V_2 \cong \big(S^n \times D^{n+1} \big) \cup ([0,1] \times D^{2n}) \cup \big(S^n \times D^{n+1}\big)\] of 2.5 there are disjoint framed embeddings \(V_1 \hookrightarrow V_2\) given by the two copies of \(S^n \times \tfrac{1}{2}D^{n+1}\), and inside two disjoint discs away from these we can form Hopf linked framed embeddings \(V_1 \sqcup V_1 \hookrightarrow D^{2n+1}\), where each individual embedding \(V_1 \hookrightarrow D^{2n+1}\) is framed isotopic to the tautological framed embedding. Appealing to 50, these are then joined together as shown in 2.

As further piece of terminology, we say that a framed embedding \(\varphi\colon V_3\hookrightarrow V_g\) has a null handle if it factors through \(V_2\) via one of the handle-filling embeddings \(n_i\) in 78 , up to isotopy of framed embeddings. Note that the restrictions of \(b\) to both copies of \(V_3\) are of this form. By the Borromean property of 51 [it:alphaExists2], it follows that the class \(\varphi_*(\alpha)\in \pi_{n-1}(B\mathrm{Diff}^\mathrm{fr}_\partial(V_g))\) obtained from \(\alpha\) by extension along a framed embedding \(\varphi\) with a null-handle vanishes.

None

The embedding \(b\) induces by extension with the identity a map \[b \colon B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_3) \times B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_3) \longrightarrowB\mathrm{Diff}^{\mathrm{fr}}_\partial(V_2)\] which we combine with the class from 51 to define a class \[\alpha^{(2)} \coloneq b_*(\alpha \otimes \alpha) \in \mathrm{H}_{2(n-1)}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_2)_\ell;\mathbf{Q}).\]

In the following, we abbreviate the image of the rational Hurewicz map by \[h_i(-)_\mathbf{Q}\coloneq \mathrm{im}\big(\pi_i(-)_\mathbf{Q}\to \mathrm{H}_i(-;\mathbf{Q}) \big)\quad\text{for }i>1.\] Given a framed embedding \(\varphi \colon V_2 \hookrightarrow V_g\) we say that \((\varphi \circ b) \colon V_3 \sqcup V_3 \hookrightarrow V_g\) is the pattern associated to \(e\). Using the previous lemma, this induces a class \[\varphi_*(\alpha^{(2)}) \in h_{2(n-1)}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_g))_\mathbf{Q},\] and this gives rise to a function \[c_{\mathrm{Diff}}\colon \pi_0(\mathrm{Emb}^{\mathrm{fr}}(V_2, V_g)) \longrightarrow h_{2(n-1)}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_g)_\ell)_\mathbf{Q}\] which is equivariant with respect to the action of \(\pi_1(B\mathrm{Diff}^\mathrm{fr}(V_g))\cong \pi_0\mathrm{Diff}^\mathrm{fr}(V_g,\ell_{V_g})\) by postcomposition in the source and conjugation in the target (see 6.1). From the discussion at the beginning of this section, it follows that \(c_{\mathrm{Diff}}\) vanishes on framed embeddings with null handles. Restricting the action along the morphism induced by relaxing the boundary condition \[\check{\Lambda}_V \rtimes \langle \rho\rangle \cong \pi_1(B\mathrm{Diff}^\mathrm{fr}_{D^{2n}}(V_g)) \rtimes \langle \rho\rangle \longrightarrow\pi_1(B\mathrm{Diff}^\mathrm{fr}(V_g))\] (see the proof of 19 for the first isomorphism) and forgetting from diffeomorphisms to embeddings, the function \(c_{\mathrm{Diff}}\) yields a \((\check{\Lambda}_V\rtimes\langle \rho\rangle)\)-equivariant function \[\label{equ:c-function}c\colon \pi_0(\mathrm{Emb}^{\mathrm{fr}}(V_2, V_g)) \longrightarrow h_{2(n-1)}(B\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}\partial}(V_g)_\ell)_\mathbf{Q}.\tag{80}\] This function is the subject of the following proposition central to the proof of 48.

Before proving 54, we explain how it can be used to accomplish the goal of this section, proving 48. In addition to 54 and 37, this relies on Watanabe’s main result of [45].

6.5 Proof of Theorem 48↩︎

Assume \(n>3\). Proposition 54 gives \(\mathbf{Q}[\check{\Lambda}_{V}\rtimes \langle \rho\rangle]\)-module maps \[\Phi: S_2(H_{V_g})_{2(n-1)} \longrightarrow h_{2n-2}(B\mathrm{Emb}_{\frac{1}{2}\partial}^{\mathrm{fr}, \cong}(V_g)_\ell)_\mathbf{Q}\longrightarrow h_{2n-2}(B\mathrm{Emb}_{\frac{1}{2}\partial}^{\mathrm{sfr}, \cong}(V_g)_\ell)_\mathbf{Q}.\] By 37 the homotopy fibre of the universal cover of \(F\coloneq \mathrm{hAut}^{\cong,\ell}_\partial(V_g)/\mathrm{Emb}^{\mathrm{sfr}, \cong}_{\frac{1}{2}\partial}(V_g)_\ell\) has trivial rational homology in degrees \(* < 2n-2\), so \(\mathrm{H}_{2n-2}(F;\mathbf{Q})\cong [\pi_{2n-2}(F)_\mathbf{Q}]_{\pi_1(F)}\) and the Serre spectral sequence provides an exact sequence of \(\mathbf{Q}[\check{\Lambda}_{V}\rtimes \langle \rho\rangle]\)-modules \[\cdots \overset{d^{2n-1}}\longrightarrow[\pi_{2n-2}(F)_\mathbf{Q}]_{\pi_1(F)} \longrightarrow\mathrm{H}_{2n-2}(B\mathrm{Emb}_{\frac{1}{2}\partial}^{\mathrm{sfr}, \cong}(V_g)_\ell;\mathbf{Q}) \longrightarrow\mathrm{H}_{2n-2}(B\mathrm{hAut}_{\partial}^{\cong,\ell}(V_g);\mathbf{Q}).\] The reflection \(\rho\) acts trivially on \(H_{V_g}\) and so also on the source of the composition \[S_2(H_{V_g})_{2(n-1)}\overset{\Phi}\longrightarrow h_{2n-2}(B\mathrm{Emb}_{\frac{1}{2}\partial}^{\mathrm{sfr}, \cong}(V_g))_\mathbf{Q}\longrightarrow h_{2n-2}(B\mathrm{hAut}_{\partial}^{\cong,\ell}(V_g))_\mathbf{Q},\] but it follows from 28 that \(\rho\) acts by \(-1\) on the target, so this composition must be trivial and we obtain a factorisation \[\Phi: S_2(H_{V_g})_{2(n-1)} \longrightarrow[\pi_{2n-2}(F)_\mathbf{Q}]_{\pi_1(F)}/\mathrm{im}(d^{2n-1}) \lhook\joinrel\longrightarrow h_{2n-2}(B\mathrm{Emb}_{\frac{1}{2}\partial}^{\mathrm{sfr}, \cong}(V_g))_\mathbf{Q}.\] By 37, \(\pi_{2n-2}(F)_\mathbf{Q}\) has the same dimension as \(S_2(H_{V_g})_{2(n-1)}\), so the dimension of the middle group in this composition is at most that of \(S_2(H_{V_g})_{2(n-1)}\). As the latter is zero or irreducible as a \(\check{\Lambda}_V\)-module (see 7.1.1 below) we conclude that either

(i) the composition \(\Phi\) is zero, or

(ii) the maps in the zig-zag of \(\mathbf{Q}[\check{\Lambda}_{V}\rtimes \langle \rho\rangle]\)-modules \[S_2(H_{V_g})_{2(n-1)} \longrightarrow[\pi_{2n-2}(F)_\mathbf{Q}]_{\pi_1(F)}/\mathrm{im}(d^{2n-1})\longleftarrow\pi_{2n-2}(F)_\mathbf{Q}\] are isomorphisms and 48 follows.

We wish to show that the latter case holds. Suppose for a contradiction that the former case holds and consider the framed inclusion \(\varphi \colon V_2 \hookrightarrow V_g\) as the first two handles, giving a class \[\xi\coloneq c_{\mathrm{Diff}}(\varphi)\in h_{2n-2}(B\mathrm{Diff}_\partial^{\mathrm{fr}}(V_g)_\ell)_\mathbf{Q}.\] Under the splitting \[h_{2(n-1)}(B\mathrm{Diff}_\partial^{\mathrm{fr}}(V_g)_\ell)_\mathbf{Q}\cong h_{2(n-1)}(B\mathrm{Diff}_\partial^{\mathrm{fr}}(D^{2n+1})_\ell)_\mathbf{Q}\oplus h_{2(n-1)}(B\mathrm{Emb}_{\frac{1}{2}\partial}^{\mathrm{fr},\cong}(V_g)_\ell)_\mathbf{Q}\] induced by 73 , the class \(\xi\) maps to \((c_{\mathrm{Diff}}(\mathrm{std}),c(\varphi))\) where \(\mathrm{std}\colon V_2\hookrightarrow D^{2n+1}\) is the standard embedding. As both of the components vanish—\(c_{\mathrm{Diff}}(\mathrm{std})\) because \(\mathrm{std}\) has a null-handle and \(c(\varphi)\) by our assumption that [it:dichotomy1] holds—the class \(\xi\) is trivial. On the other hand, we will now argue that Watanabe’s work [45] implies that \(\xi\) is nontrivial, which yields a contradiction and thus finishes the proof. To this end, we choose a nonstandard framed embedding \(\psi\colon V_g \hookrightarrow D^{2n+1}\) for which the cores of the first two handles link. Sending forwards the class \(\xi\) using this embedding gives a class in \(h_{2n-2}(B\mathrm{Diff}_\partial^\mathrm{fr}(D^{2n+1}))\) which agrees by definition with \(\theta_*(\alpha\otimes \alpha)\) where \(\theta\) is the composition of embeddings \((\psi\circ \varphi\circ b)\colon V_3 \sqcup V_3\hookrightarrow D^{2n+1}\) visualised in 3. As a result of the final part of 51, the class \(\theta_*(\alpha\otimes \alpha)\) agrees up to a nontrivial constant with the class \(\theta_*(\alpha_{\text{Wat}}\otimes \alpha_{\text{Wat}})\) where \(\alpha_{\text{Wat}}\in \pi_{n-1}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_3))_\mathbf{Q}\) is the class from [46]. Using the notation from [45], the class \(\theta_*(\alpha_{\text{Wat}}\otimes \alpha_{\text{Wat}})\) is thus by definition exactly the image of the theta-graph \(\Theta \in\mathcal{G}_{2,3}\) under the map \(\psi_2\) from Definition 1 on p. loc.cit.. But this image is nontrivial by Theorem 3.1 (i) loc.cit., since the image of \(\Theta \in\mathcal{G}_{2,3}\) in \(\mathcal{A}_{2,3}\cong\mathbf{Q}\) is a generator (see Remark 2 on p. loc.cit.and Figure 2 on p.). Note that the statement of Theorem 3.1 assumes that \(n\ge3\) is odd, but it also holds for even \(n\ge3\) by [46].

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6.6 Some clasper calculus↩︎

Before giving the proof of 54, we describe a linearity property of the class \(\alpha\) from 51. To this end, we consider framed embeddings \[\label{equ:linearity-embeddings} \varphi_3, \varphi_4, \varphi_{34} \colon V_3 \lhook\joinrel\longrightarrow V_4\tag{81}\] as depicted in 4, which are such that each embedding \(V_3 \overset{\varphi}\hookrightarrow V_4 \subset D^{2n+1}\) is framed isotopic to the tautological embedding, and whose induced maps on homology fix the classes \(f_1\) and \(f_2\) and satisfy \[(\varphi_3)_*(f_3) = f_3,\quad (\varphi_4)_*(f_3) = f_4,\quad\text{and}\quad(\varphi_{34})_*(f_3) = f_3+f_4.\] These embeddings induce classes \[\alpha_3 \coloneq (\varphi_3)_*(\alpha),\quad \alpha_4 \coloneq (\varphi_4)_*(\alpha),\quad\text{ and }\alpha_{34} \coloneq (\varphi_{34})_*(\alpha)\] in the group \(\pi_{n-1}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(V_4))_\mathbf{Q}\) which are related as follows.

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7 Homology of self-embedding spaces↩︎

Based on Theorems 28, 37, and 48, we determine in a range the rational homology of the self-embedding spaces appearing in 17. To state the result, note that these spaces are related to certain stabilised arithmetic groups by compositions \[B\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_\ell\;\rightarrow B G_W\rightarrow B\mathrm{OSp}_\infty(\mathbf{Z})\;\;\;\text{and}\;\;\; B\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}D^{2n}}(V_g, W_{g,1})_\ell\;\rightarrow B\mathrm{GL}(\bar{H}_V^\mathbf{Z})\rightarrow B\mathrm{GL}_\infty(\mathbf{Z})\] where the first maps are given by the action on homology (see 2.7) and the second maps use the standard bases from 2.5 and the stabilisation maps induced by block-inclusion.

The plan to prove 56 is:

1. Compute (in a range) the homotopy Lie algebras \[\pi_*(\Omega_0B\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_\ell)_\mathbf{Q}\quad\text{and}\quad\pi_*(\Omega_0B\mathrm{Emb}^{\mathrm{fr}}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_\ell)_\mathbf{Q}.\]

2. Use [enum:homotopy-selfemb] to compute (in a range and up to some ambiguity) the rational homology of the universal covers \[\def\arraystretch{1.5} \begin{array}{r@{\hskip 0.2cm} c@{\hskip 0.2cm} l@{\hskip 0.2cm} } B\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_\ell^\sim&\longrightarrow&B\mathrm{Emb}^{\mathrm{fr},\cong}_{\frac{1}{2}\partial}(W_{g,1})_\ell\\ B\mathrm{Emb}^{\mathrm{fr}}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_\ell^\sim&\longrightarrow&B\mathrm{Emb}^{\mathrm{fr}}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_\ell. \end{array}\]

3. Run the universal cover spectral sequence to prove 56, based on [enum:homology-cover-selfemb]. In doing so, we show that the ambiguity in [enum:homology-cover-selfemb] does not affect the end result.

7.1 Representations of \(G_W\) and \(G_V\) and their cohomology↩︎

We begin this section with some representation-theoretic and group-cohomological preliminaries on the groups \(\check{\Lambda}_V\) and \(\check{\Lambda}_W\) mapping to \(G_V \cong M_V^\mathbf{Z}\rtimes \mathrm{GL}(\bar{H}^\mathbf{Z}_V)\) and \(G_W\) respectively. For this, it is worth recalling the explicit descriptions of \(G_W\) and \(G_V\) as matrix groups from 2.7 in terms of \(\mathrm{Sp}^q_{2g}(\mathbf{Z})\), \(\mathrm{O}_{g,g}(\mathbf{Z})\), and \(\mathrm{GL}_g(\mathbf{Z})\). Furthermore, recall from 2.8.2 that these groups are acted upon by the reflection involution \(\rho\) (though \(\rho\) acts trivially on the subgroup \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\le G_V\)) and that the intersection form induces isomorphisms (see 2.8.3) \(H_W\cong s^{2n-2}H_W^\vee \otimes \mathbf{Q}^-\) as a \((G_W\rtimes\langle \rho\rangle)\)-module and an isomorphism \(H_W\cong H_V\oplus s^{2n-2}H_V^\vee\otimes\mathbf{Q}^-\) as a module over the subgroup \(\mathrm{GL}(\bar{H}_V^\mathbf{Z})\times\langle \rho\rangle\le G_V\rtimes \langle \rho\rangle\).

7.1.1 Some representation theory↩︎

Recall from classical representation theory (see e.g. [47]) that the \(\mathbf{C}[\mathrm{GL}_g(\mathbf{C})]\)-module \(S_\mu(\mathbf{C}^g)=S_\mu(\mathbf{C}^g[0])\) for a partition \(\mu\) (see 3.1) is nonzero if and only if \(\mu\) has at most \(g\) rows (when thought of as a Young diagram), and in this case it is irreducible. Moreover, for partitions \(\mu\neq \lambda\), the \(\mathbf{C}[\mathrm{GL}_g(\mathbf{C})]\)-modules \(S_\mu(\mathbf{C}^g)\) and \(S_{\lambda}(\mathbf{C}^g)\) are distinct if they are nonzero. By dualising, the same discussion applies when replacing the \(S_\mu(\mathbf{C}^g)\)’s with their duals \(S_\mu(\mathbf{C}^g)^\vee\cong S_\mu((\mathbf{C}^g)^\vee)\). Similarly, the \(\mathbf{C}[\mathrm{OSp}_{2g}(\mathbf{C})]\)-module \(V_\mu(\mathbf{C}^g)\) is nonzero for \(n\) odd if and only if \(\mu\) has at most \(g\) rows, and for \(n\) even if and only if the first two columns of \(\mu\) have in sum at most \(g\) boxes, and in these cases \(V_\mu(\mathbf{C}^{2g})\) is an irreducible \(\mathbf{C}[\mathrm{OSp}_{2g}(\mathbf{C})]\)-module. As in the \(\mathrm{GL}_g(\mathbf{C})\)-case, for \(\mu\neq\lambda\) the \(\mathbf{C}[\mathrm{OSp}_{2g}(\mathbf{C})]\)-modules \(V_\mu(\mathbf{C}^{2g})\) and \(V_{\lambda}(\mathbf{C}^{2g})\) are distinct if they are nonzero.

Using that the representations discussed above are algebraic (i.e.extend to representations of the ambient algebraic groups \(\mathrm{GL}_g(-)\) or \(\mathrm{OSp}_{g}(-)\)) and the subgroups \(\mathrm{O}_{g,g}(\mathbf{Z})\subset \mathrm{O}_{g,g}(\mathbf{C})\) and \(\mathrm{Sp}_{2g}^q(\mathbf{Z})\subset\mathrm{Sp}_{2g}(\mathbf{Z})\subset \mathrm{Sp}_{2g}(\mathbf{C})\) are Zariski dense, the analogous properties hold for the \(\mathbf{Q}[G_W]\)-modules \(V_\mu(\bar{H}_{W})\) and thus also for their graded versions \(V_\mu(H_W)\); this remains true after pulling back along any epimorphism onto \(G_W\), such as \(\check{\Lambda}_W\twoheadrightarrow G_W\).

Unfortunately \(\mathrm{GL}_g(\mathbf{Z}) \subset \mathrm{GL}_g(\mathbf{C})\) is not Zariski dense (its Zariski closure consists of the complex matrices of determinant \(\pm 1\)) so more care is needed in this setting. It is still true that the \(\mathbf{Q}[\mathrm{GL}_g(\mathbf{Z})]\)-modules \(S_\mu(\mathbf{Q}^g)\) and \(S_\mu(\mathbf{Q}^g)^\vee\) are irreducible when they are non-zero, because \(S_\mu(\mathbf{C}^g)\) remains irreducible as a \(\mathrm{SL}_g(\mathbf{C})\)-module (as every element of \(\mathrm{GL}_g(\mathbf{C})\) is a scalar multiple of an element of \(\mathrm{SL}_g(\mathbf{C})\)) and \(\mathrm{SL}_g(\mathbf{Z}) \subset \mathrm{SL}_g(\mathbf{C})\) is Zariski dense. Thus the \(\mathbf{Q}[\mathrm{GL}(\bar{H}^\mathbf{Z}_V)]\)-modules \(S_\mu(\bar{H}_V)\) and \(S_\mu(\bar{H}_V)^\vee\) are still zero or irreducible, as are their graded versions \(S_\mu(H_V)\) and \(S_\mu(H_V)^\vee\); this remains true after pulling back these modules along any epimorphism onto \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\), such as \(\check{\Lambda}_V\twoheadrightarrow\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\), or \(G_V\twoheadrightarrow \mathrm{GL}(\bar{H}^\mathbf{Z}_V)\). What is no longer true is that these are distinct irreducible modules for distinct partitions: the square of the determinant representation of \(\mathrm{GL}_g(\mathbf{C})\) is trivial when restricted to \(\mathrm{GL}_g(\mathbf{Z})\), so e.g.\(S_{(2^g)}(\mathbf{Q}^g) \cong S_0(\mathbf{Q}^g)\) as \(\mathbf{Q}[\mathrm{GL}_g(\mathbf{Z})]\)-modules. But we may still decompose algebraic \(\mathrm{GL}_g(\mathbf{Z})\)-representations into irreducibles by decomposing some \(\mathrm{GL}_g(\mathbf{C})\)-representation which induces it into irreducibles.

7.1.2 Manipulating Schur functors↩︎

At various points in this section, we have to calculate tensor products, plethysm, or restrictions of (symplectic or orthogonal) Schur functors, such as \(s^{-(2n-2)}S_{1^3}(H_W)\cong s^{-(2n-2)}V_{1^3}(H_W)\oplus H_W\) as \(G_W\)-modules or \(S_2(H_V)\otimes H_V=S_{2,1}(H_V)\oplus S_3(H_V)\) as \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\)-modules (and thus also as \(G_V\) or \(\check{\Lambda}_V\)-modules). In all cases we performed these calculations using the SymmetricFunctions package provided by SageMath [48].

7.1.3 Stable cohomology of \(G_W\) and \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\)↩︎

Recall from [5] that \(\mathrm{H}^*(B\mathrm{GL}_{\infty}(\mathbf{Z});\mathbf{Q})\) is an exterior algebra with generators in degrees \(4i+1\) for \(i\ge1\), and that \(\mathrm{H}^*(B\mathrm{OSp}_\infty(\mathbf{Z});\mathbf{Q})\) is a polynomial algebra with generators in degrees \(4i\) for \(i\ge1\) if \(n\) is even and in degrees \(4i-2\) for \(i\ge1\) if \(n\) is odd. For \(\mathrm{H}^*(B\mathrm{OSp}_\infty(\mathbf{Z});\mathbf{Q})\), it will be convenient for us to use explicit generators \[\mathrm{H}^*(B\mathrm{OSp}_\infty(\mathbf{Z});\mathbf{Q})\cong \begin{cases} \mathbf{Q}[\sigma_{4},\sigma_8,\ldots]&n\text{ even}\\ \mathbf{Q}[\sigma_{2},\sigma_6,\ldots]&n\text{ odd}\end{cases}\] defined in the terms of the signature as explained e.g.in [13]. The definition in terms of the signature makes apparent that the reflection \(\rho\) of 2.8.2 acts on the \(\sigma_i\)’s as \(-1\) whereas it acts trivially on \(\mathrm{H}^*(\mathrm{GL}({\bar{H}_{V_g}}^\mathbf{Z});\mathbf{Q})\) for all \(g\) and thus also on \(\mathrm{H}^*(B\mathrm{GL}_\infty(\mathbf{Z});\mathbf{Q})\).

Combined with further work of Borel [49], the calculation of \(\mathrm{H}^*(B\mathrm{GL}_{\infty}(\mathbf{Z});\mathbf{Q})\) and \(\mathrm{H}^*(B\mathrm{OSp}_\infty(\mathbf{Z});\mathbf{Q})\) allows for a computation of the \(G_W\)- and \(\mathrm{GL}(\bar{H}^\mathbf{Z}_V)\)-homology with coefficients in large class of \(\mathbf{Q}[G_{W}]\)-modules \(M_W\) and \(\mathbf{Q}[\mathrm{GL}(\bar{H}^\mathbf{Z}_V)]\)-modules \(M_V\) in a range of degrees increasing with \(g\) in terms of the invariants. Concretely, the maps \[\label{equ:borel-vanishing-maps}\begin{gather}\mathrm{H}^*( \mathrm{OSp}_\infty(\mathbf{Z});\mathbf{Q})\otimes M_W^{G_{W}}\overset{\mathrm{res}\otimes\mathrm{id}}{\longrightarrow}\mathrm{H}^*(G_W;\mathbf{Q})\otimes M_W^{G_{W}}\overset{\mathrm{cup}}{\longrightarrow} \mathrm{H}^*(G_{W};M_W)\quad\text{and}\\ \mathrm{H}^*( \mathrm{GL}_\infty(\mathbf{Z});\mathbf{Q})\otimes M_V^{\mathrm{GL}(\bar{H}_{V})}\overset{\mathrm{res}\otimes\mathrm{id}}{\longrightarrow}\mathrm{H}^*(\mathrm{GL}(\bar{H}_{V});\mathbf{Q})\otimes M_V^{\mathrm{GL}(\bar{H}_{V})}\overset{\mathrm{cup}}{\longrightarrow} \mathrm{H}^*(\mathrm{GL}(\bar{H}_{V});M_V).\end{gather}\tag{82}\] are often isomorphisms in a range of degrees. The following instance will be sufficient for us:

7.1.4 Stable homology of \(G_V\)↩︎

In 98, as a replacement for 58 for the group \(G_V\), we establish an isomorphism of bigraded groups (the first grading is the homological degree and the second the degree of the coefficients) \[\label{equ:stable-gv-homology}\mathrm{H}_*\big(G_V; S_\lambda({H}_{V}) \otimes S_\mu({H}_{V}^\vee)\big)\cong\mathrm{H}_*(\mathrm{GL}_\infty(\mathbf{Z});\mathbf{Q}[0]) \otimes \chi(\lambda, \mu)\tag{84}\] in a range of homological degrees increasing with \(g\) for partitions \(\lambda\) and \(\mu\). Here \(\chi(\lambda, \mu)\) vanishes if \(|\lambda|-|\mu|\) is odd or negative, and if \(|\lambda|-|\mu|=2t \geq 0\) then \[\chi(\lambda, \mu) \coloneq \left[ \mathrm{Ind}_{\Sigma_{|\mu|} \times \Sigma_{2t}}^{\Sigma_{|\lambda|}} \left( (\mu) \boxtimes \mathrm{Ind}^{\Sigma_{2t}}_{\Sigma_t \wr \Sigma_2} (1^t) \wr (2) \right) \otimes (\lambda) \right]^{\Sigma_{|\lambda|}} [t, (|\lambda|-|\mu|)(n-1)].\] We use this to prove the following lemma, which collects all computations of \(G_V\)-homology groups necessary for this section. In the statement (and the subsequent sections), in addition to Schur functors applied to \(H_V\) and \(H_V^\vee\) we will make use of the graded \(\mathbf{Q}[G_V\rtimes \langle \rho\rangle ]\)-modules \[K_\mu \coloneq \ker\big(S_{\mu}(H_{W}) \twoheadrightarrow S_{\mu}(H_{V})\big)\] associated to the canonical projection \(H_W\rightarrow H_V\).

We now begin executing the outlined plan to prove 56.

7.2 Step [enum:homotopy-selfemb]↩︎

We first settle the \(W_{g,1}\)-case.

The \(V_g\)-case of Step [enum:homotopy-selfemb] is harder. We first establish two preparatory results.

The second ingredient in the proof of the \(V_g\)-case of Step [enum:homotopy-selfemb] is the following:

Equipped with 63 and Lemma 65, we establish the \(V_g\)-part of Step [enum:homotopy-selfemb].

7.3 Step [enum:homology-cover-selfemb]↩︎

Propositions 62 and 66 allow us to carry out Step [enum:homology-cover-selfemb].

7.4 Step [enum:homology-selfemb]↩︎

For Step [enum:homology-selfemb], we first compute the homology of \(G_W\) and \(G_V\) with coefficients in the modules featuring in Corollaries 67 and 68. As before, \(W_{g,1}\) is easier and comes first.

Putting together the pieces, we finish the proof of this section’s main result, 56.

8 Rational homotopy groups of diffeomorphism groups of discs↩︎

Based on 56, we compute the rational homotopy groups of \(B\mathrm{Diff}_\partial(D^d)\) and \(B\mathrm{Diff}_{D^d}(D^{d+1})\) in degrees \(*\lesssim \tfrac{3}{2} d\) and deduce a computation of the rational homotopy groups of various related spaces in a similar range. In particular, we prove 1 and 2.

8.1 Transgression and Pontryagin classes↩︎

Given a sequence of based spaces \(F\rightarrow E\rightarrow B\) whose composition is constant and an abelian group \(\mathbf{k}\), consider the maps \[\mathrm{H}^k(B;\mathbf{k})\cong \mathrm{H}^k(B,*;\mathbf{k})\longrightarrow\mathrm{H}^k(E,F;\mathbf{k})\longleftarrow \mathrm{H}^{k-1}(F;\mathbf{k})\quad\text{for }k>0.\] By the long exact sequence of a pair, the image in \(\mathrm{H}^k(E,F;\mathbf{k})\) of any class \(c\in \mathrm{H}^k(B;\mathbf{k})\) that vanishes in \(\mathrm{H}^k(E;\mathbf{k})\) lifts to a (usually non-unique) class \(\tilde{c}\in \mathrm{H}^{k-1}(F;\mathbf{k})\). This has an obvious naturality property: if the sequence \(F\rightarrow E\rightarrow B\) receives compatible maps from another sequence \(F'\rightarrow E'\rightarrow B'\) of the same type, then the pullback of a lift \(\tilde{c}\) to \(\mathrm{H}^{k-1}(F;\mathbf{k})\) is a lift of the pullback of \(c\) to \(\mathrm{H}^k(B;\mathbf{k})\). Moreover, note that if \(\mathrm{H}^{k-1}(E;\mathbf{k})=0\), then the rightmost map in the above zig-zag is injective, so the lift \(\tilde{c}\) is unique; in this case we write \(c^\tau\coloneq \tilde{c}\in \mathrm{H}^{k-1}(F;\mathbf{k})\). If \(F\rightarrow E\rightarrow B\) is a fibration sequence, then \(c\) is the transgression of \(c^\tau\) in the usual sense, so in this case it enjoys the following the well-known properties:

1. The pullback of \(c^\tau\) along the induced map \(\Omega B\rightarrow F\) agrees with \(\Omega c\in \mathrm{H}^{k-1}(\Omega B;\mathbf{k})\).

2. For \(\mathbf{k}=\mathbf{Z}\) and \(x \in \pi_k(B)\) we have \(\langle c, x \rangle = \langle c^\tau, \partial(x)\rangle\) for the boundary map \(\partial\colon \pi_k(B) \to \pi_{k-1}(F)\), where \(\langle -, - \rangle\colon \pi_k(X)\otimes \mathrm{H}^k(X;\mathbf{Z})\rightarrow\mathbf{Z}\) is the canonical pairing.

8.1.1 Transgressed Pontryagin and Euler classes↩︎

We will use this to define classes \[\begin{align} \label{equ:transgressed-classes} \begin{aligned} p_i^\tau&\in \mathrm{H}^{4i-1}(\mathrm{Top}(d)/\mathrm{O}(d);\mathbf{Q})\text{ for }i>\lfloor \tfrac{d}{2}\rfloor,\\ (p_n-e^2)^\tau&\in \mathrm{H}^{4n-1}(\mathrm{Top}(2n)/\mathrm{O}(2n);\mathbf{Q}),\text{ and}\\ (p_n-E)^\tau&\in \mathrm{H}^{4n-1}(\mathrm{Top}(2n+1)/\mathrm{O}(2n+1);\mathbf{Q}) \end{aligned} \end{align}\tag{92}\] such that the \(p_i^\tau\) are preserved by pullback along the stabilisation map \(\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow\mathrm{Top}(d+1)/\mathrm{O}(d+1)\), and \((p_n-E)^\tau\) pulls back to \((p_n-e^2)^\tau\) for \(d=2n\). First, recall that the map \(B\mathrm{O}\rightarrow B\mathrm{Top}\) is a rational homotopy equivalence [1], so there are well-defined Pontryagin classes \(p_i\in\mathrm{H}^{4i}(B\mathrm{Top};\mathbf{Q})\) which we may pull back to classes \(p_i\in\mathrm{H}^{4i}(B\mathrm{Top}(d);\mathbf{Q})\). The first classes in 92 result from these by applying the above principle to \[\label{equ:o-to-top} \mathrm{Top}(d)/\mathrm{O}(d)\longrightarrow B\mathrm{O}(d)\longrightarrow B\mathrm{Top}(d),\tag{93}\] using the fact that \(p_i\in \mathrm{H}^{4i}(B\mathrm{O}(d);\mathbf{Q})\) vanishes in \(\mathrm{H}^{4i}(B\mathrm{O}(d);\mathbf{Q})\) for \(i>\lfloor \tfrac{d}{2}\rfloor\). To construct the other two classes, we write \(\mathrm{G}(d) = \mathrm{hAut}(S^{d-1})\) whose delooping receives a map from \(B\mathrm{Top}(d)\) classifying the underlying spherical fibration of an \(\mathbf{R}^d\)-bundle. Passing to the orientation preserving submonoid \(\mathrm{SG}(d)\subset\mathrm{G}(d)\) we see from the computation of the rational homotopy groups in 53 that there are rational homotopy equivalences \[\label{equ:homotopy-type-g40n41} B\mathrm{SG}(2n) \simeq_\mathbf{Q}K(\mathbf{Q}^-, 2n) \quad\quad\quad B\mathrm{SG}(2n+1) \simeq_\mathbf{Q}K(\mathbf{Q}^+, 4n).\tag{94}\] The first of these is induced by the Euler class \(e \in\mathrm{H}^{2n}(B\mathrm{SG}(2n);\mathbf{Q})\) and the second is induced by the unique rational cohomology class \[E \in\mathrm{H}^{4n}(B\mathrm{SG}(2n+1);\mathbf{Q})\] that pulls back to \(e^2 \in \mathrm{H}^{4n}(B\mathrm{SG}(2n);\mathbf{Q})\) under the stabilisation map \(B\mathrm{SG}(2n)\rightarrow B\mathrm{SG}(2n+1)\). The \(\pm\)-superscripts indicate the effect of the involution on \(B\mathrm{SG}(d)\) induced by conjugation with a reflection in \(\mathbf{R}^d\). Using the map \(B\mathrm{STop}(2n+1) \to B\mathrm{SG}(2n+1)\) we form the class \(p_n - E \in\mathrm{H}^{4n}(B\mathrm{STop}(2n+1);\mathbf{Q})\), which vanishes in \(B\mathrm{SO}(2n)\) as \(E\) becomes \(e^2=p_n\). As the map \(B\mathrm{SO}(2n)\rightarrow B\mathrm{SO}(2n+1)\) is injective on rational cohomology, it also vanishes on \(B\mathrm{SO}(2n+1)\), so the orientation preserving analogue of the fibration 93 induces the third class in 92 . Its pullback to \(\mathrm{Top}(2n)/\mathrm{O}(2n)\) agrees with the second class in 92 , which is obtained by performing the same construction to \(p_n-e^2\in\mathrm{H}^{4n}(B\mathrm{STop}(2n);\mathbf{Q})\).

8.2 Smoothing theory, Morlet style↩︎

By smoothing theory and the Alexander trick there is a homotopy commutative diagram \[\label{equ:smoothing-theory} \begin{tikzcd} \Omega^{d}\mathrm{O}(d)\arrow[d,equal]\rar&B\mathrm{Diff}^\mathrm{fr}_\partial(D^d)\rar\dar&B\mathrm{Diff}_\partial(D^d)\dar\\ \Omega^{d}\mathrm{O}(d)\rar&\Omega^{d}\mathrm{Top}(d)\rar&\Omega^d\mathrm{Top}(d)/\mathrm{O}(d) \end{tikzcd}\tag{95}\] whose vertical maps are \(0\)-coconnected as long as \(d\neq 4\) by a result due to Morlet [1]. The induced isomorphisms \[\label{equ:smoothing-theory-component} \pi_*(B\mathrm{Diff}^\mathrm{fr}_\partial(D^d)_\ell)\cong \pi_{*+d}(\mathrm{Top}(d))\quad\text{and}\quad \pi_*(B\mathrm{Diff}_\partial(D^d))\cong \pi_{*+d}(\mathrm{Top}(d)/\mathrm{O}(d))\tag{96}\] in positive degrees are anti-equivariant with respect to the reflection involution on the source and the involution on the target that is induced by conjugation with a reflection in \(\mathbf{R}^d\). For the first isomorphism (the second one is similar), this can be seen by chasing through the smoothing theory equivalence and observing that: (i) the induced isomorphism \(\pi_*(B\mathrm{Diff}^\mathrm{fr}_\partial(D^d)_\ell)\cong \pi_{*}(\Omega^d_0\mathrm{Top}(d))\) is equivariant with respect to the reflection involution on the source and the involution on the target induced by reflection in \(\mathrm{Top}(d)\) and a reflection in the loop coordinate, and (ii) reflection in the loop coordinate induces multiplication by \(-1\) on homotopy groups. More precisely, the effect of the involution on a class represented by a map \((D^d,\partial D^d)\rightarrow(\mathrm{Top}(d),*)\) is by precomposition with the reflection \(\rho\colon (D^d,S^{d-1})\rightarrow(D^d,S^{d-1})\) and postcomposition with the selfmap of \(\mathrm{Top}(d)\) induced by conjugation with a reflection of \(\mathbf{R}^d\) in one of the coordinates.

Similarly, there is an equivalence \[\label{equ:smoothing-theory-concordances}B\mathrm{Diff}_{D^d}(D^{d+1})\overset{\simeq}{\longrightarrow} \Omega^d_0\mathrm{hofib}\big(\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow\mathrm{Top}(d+1)/\mathrm{O}(d+1)\big)\tag{97}\] for \(d\neq 4,5\) involving the stabilisation map \(\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow\mathrm{Top}(d+1)/\mathrm{O}(d+1)\) induced by \((-)\times\mathbf{R}\), and this equivalence is compatible up to homotopy with the second equivalence in 96 for \(d\) and \(d+1\) via the fibre sequence \[B\mathrm{Diff}_{\partial}(D^{d+1})\longrightarrowB\mathrm{Diff}_{D^d}(D^{d+1})\longrightarrowB\mathrm{Diff}^\mathrm{id}_{\partial}(D^{d})\] induced by restricting diffeomorphisms of \(D^{d+1}\) to the moving part of the boundary. The induced isomorphism in positive degrees \[\label{equ:smoothing-theory-on-homotopy-conc}\pi_*(B\mathrm{Diff}_{D^d}(D^{d+1}))\cong \pi_{*+d}(\mathrm{hofib}\big(\mathrm{Top}(d)/\mathrm{O}(d)\rightarrow\mathrm{Top}(d+1)/\mathrm{O}(d+1)\big)\tag{98}\] is anti-equivariant with respect to the reflection involution on the source and the involution on the target induced by conjugation with reflections of \(\mathbf{R}^d\) and \(\mathbf{R}^{d+1}\).

8.2.1 Pontryagin classes on diffeomorphism of discs↩︎

Combining 95 with the canonical equivalence \(\Omega^{d}\mathrm{Top}(d)\simeq \Omega^{d+1}B\mathrm{STop}(d)\), we may pull back the appropriate loopings of the classes considered in 8.1.1 along the vertical maps to obtain classes \[\Omega^{d+1}p_i\in \mathrm{H}^{4i-d-1}(B\mathrm{Diff}^\mathrm{fr}_\partial(D^d);\mathbf{Q})\;\;\text{ and }\;\;\Omega^{d}p_i^\tau\in \mathrm{H}^{4i-d-1}(B\mathrm{Diff}_\partial(D^d);\mathbf{Q})\text{ for }i>\lfloor d/2\rfloor\] and classes \[\Omega^{2n}(p_n-e^2)^\tau\in \mathrm{H}^{2n-1}(B\mathrm{Diff}_\partial(D^{2n});\mathbf{Q})\;\;\text{ and }\;\;\Omega^{2n+1}(p_n-E)^\tau\in \mathrm{H}^{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1});\mathbf{Q}).\]

8.3 Diffeomorphisms of discs↩︎

Recall that for a path-connected homotopy commutative \(H\)-space \(X\), there are canonical cohomology classes in \(\mathrm{H}^i(X;\pi_i(X)_\mathbf{Q})\) for \(i\ge1\): they are given by the composition \(\mathrm{H}_i(X;\mathbf{Q})\rightarrow Q(\mathrm{H}_*(X;\mathbf{Q}))_i\cong P(\mathrm{H}_*(X;\mathbf{Q}))_i\cong \pi_i(X)_\mathbf{Q}\) where the second isomorphism is induced by the Hurewicz map and the first is the usual isomorphism between the primitives and indecomposables of the Hopf algebra \(\mathrm{H}_*(X;\mathbf{Q})\) (see [51]). Applying this to \(B\mathrm{GL}_{\infty}(\mathbf{Z})^+\) and \(B\mathrm{OSp}_{\infty}(\mathbf{Z})^+\) and writing \[K_i(\mathbf{Z})\coloneq \pi_i(B\mathrm{GL}_{\infty}(\mathbf{Z})^+)\quad\text{and}\quad K\mathrm{OSp}_i(\mathbf{Z})\coloneq \pi_i(B\mathrm{OSp}_{\infty}(\mathbf{Z})^+)\quad\text{ for }i>0,\] we obtain classes \[\xi_i\in \mathrm{H}^i(B\mathrm{GL}_{\infty}(\mathbf{Z});K_i(\mathbf{Z})_\mathbf{Q})\quad\text{and}\quad \xi^{\mathrm{OSp}}_i\in \mathrm{H}^i(B\mathrm{OSp}_{\infty}(\mathbf{Z});K\mathrm{OSp}_i(\mathbf{Z})_\mathbf{Q})\] for \(i>0\). We now consider the framed Weiss fibre sequences 26 \[\label{equ:framed-weiss-fs-ready-for-trangression} \begin{gather} B\mathrm{Diff}_{D^{2n}}(D^{2n+1})\longrightarrowB\mathrm{Diff}^{\mathrm{fr}}_{D^{2n}}(V_{g})_{\ell}\longrightarrowB\mathrm{Emb}^{\mathrm{fr}}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_{\ell}\\ B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n})_B\longrightarrowB\mathrm{Diff}^{\mathrm{fr}}_\partial(W_{g,1})_{\ell}\longrightarrowB\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(W_{g,1})_{\ell} \end{gather}\tag{99}\] whose total spaces have trivial rational homology in a range of degrees with \(g\) by 16 (assuming \(n\ge4\) for the first fibration and \(n\ge3\) for the second fibration). Pulling back the classes \(\xi_i\) and \(\xi^{\mathrm{OSp}}\) along the compositions \[B\mathrm{Emb}^{\mathrm{fr}}_{\frac{1}{2}D^{2n}}(V_g,W_{g,1})_{\ell}\rightarrow B G_{V}\rightarrow B\mathrm{GL}_\infty(\mathbf{Z})\quad\text{and}\quad B\mathrm{Emb}^{\mathrm{fr}, \cong}_{\frac{1}{2}\partial}(W_{g,1})_{\ell}\rightarrow B G_{W}\rightarrow B\mathrm{OSp}_\infty(\mathbf{Z})\] induced by the homology actions and choosing \(g\gg i\) large enough, we thus obtain classes \[\label{equ:xis} \xi_i^\tau\in \mathrm{H}^{i-1}(B\mathrm{Diff}_{D^{2n}}(D^{2n+1});K_i(\mathbf{Z})_\mathbf{Q})\quad\text{and}\quad (\xi_i^{\mathrm{OSp}})^\tau\in \mathrm{H}^{i-1}(B\mathrm{Diff}^{\mathrm{fr}}_\partial(D^{2n})_\ell;K\mathrm{OSp}_i(\mathbf{Z})_\mathbf{Q}).\tag{100}\] Before stating the first result of this section, let us recall that all spaces and groups above are compatibly acted upon by the reflection involution \(\rho\). As pointed out in 2.8.2 this action is trivial on the group \(\mathrm{GL}_\infty(\mathbf{Z})\) and hence also on \(K_*(\mathbf{Z})_\mathbf{Q}\). On \(K\mathrm{OSp}_*(\mathbf{Z})_\mathbf{Q}\) however \(\rho\) acts by \(-1\), because it does so on the indecomposables of \(\mathrm{H}^*(B\mathrm{OSp}_\infty(\mathbf{Z});\mathbf{Q})\), as explained in 7.1.3.

For even discs 71 can be reinterpreted in terms of Pontryagin classes in the sense of 8.2.1. To do so, recall from 7.1.3 that for \(i>0\) the group \(K\mathrm{OSp}_i(\mathbf{Z})_\mathbf{Q}\) is trivial unless \(i=4j-2n\) for some \(j\) and in this case we have the canonical isomorphism \(\sigma_{4j-2n}\colon K\mathrm{OSp}_{4j-2n}(\mathbf{Z})_\mathbf{Q}\rightarrow\mathbf{Q}^-\) respecting the action of the reflection involution, given by evaluating the classes \(\sigma_{4j-2n}\in\mathrm{H}^{4j-2n}(B\mathrm{OSp}_\infty(\mathbf{Z});\mathbf{Q})\) described in 7.1.3.

To compute \(\pi_*(-)_\mathbf{Q}\) of \(B\mathrm{Diff}_\partial(D^{2n+1})\) and \(B\mathrm{Diff}^\mathrm{fr}_\partial(D^{2n+1})_\ell\), we consider the fibre sequence \[\label{equ:fibration-diff-vs-concordances} B\mathrm{Diff}_\partial(D^{d+1})\longrightarrowB\mathrm{Diff}_{D^{d}}(D^{d+1})\longrightarrowB\mathrm{Diff}^{\mathrm{id}}_\partial(D^{d}),\tag{103}\] for \(d=2n\), acted upon by the reflection involution. The answer involves the classes \(\Omega^{2n+1}p_i^\tau\in\mathrm{H}^{4i-2n-2}(B\mathrm{Diff}_\partial(D^{d+1});\mathbf{Q})\) for \(i>n\) and \(\Omega^{2n+1}(p_n-E)^\tau\in\mathrm{H}^{2n-2}(B\mathrm{Diff}_\partial(D^{d+1});\mathbf{Q})\) from 8.2, and the pullbacks of the \((2n+2)\)th loopings of \(E\) and \(p_i\) on \(B\mathrm{Top}(2n+1)\) along the map \(B\mathrm{Diff}^\mathrm{fr}_\partial(D^{2n+1})\rightarrow\Omega^{2n+2}B\mathrm{STop}(2n+2)\). Together with 73 this proves 1.

Finally, denoting the pullback of the cohomology classes appearing in the previous corollary along \(B\mathrm{Diff}_{D^{2n+1}}(D^{2n+2})\rightarrowB\mathrm{Diff}_\partial(D^{2n+1})\) by the same name, we conclude the following by combining 73 and 74 with the fibration 103 for \(d=2n+1\).

There is a different description of the class \(\Omega^{2n+1}(p_n-E)^\tau\in\mathrm{H}^{2n-2}(B\mathrm{Diff}_{D^{2n+2}}(D^{2n+1});\mathbf{Q})\) which is sometimes more convenient. Recall that smoothing theory provides a fibre sequence \[B\mathrm{Diff}_{D^{2n+1}}(D^{2n+2})\longrightarrow\Omega^{2n+1}\mathrm{O}(2n+2)/\mathrm{O}(2n+1)\longrightarrow\Omega^{2n+1}\mathrm{Top}(2n+2)/\mathrm{Top}(2n+1).\] Pulling back \(E\in \mathrm{H}^{4n}(B\mathrm{Top}(2n+1);\mathbf{Q})\) to \(\mathrm{Top}(2n+2)/\mathrm{Top}(2n+1)\) gives a class which vanishes in \(\mathrm{O}(2n+2)/\mathrm{O}(2n+1)\) (as \(E=p_n\) in \(B\mathrm{O}(2n+1)\)), so we obtain a class \(E^{\tau}\) of degree \(4n-1\) in the cohomology of the fibre of \(\mathrm{O}(2n+2)/\mathrm{O}(2n+1)\rightarrow\mathrm{Top}(2n+2)/\mathrm{Top}(2n+1)\). This gives a class \(\Omega^{2n+1}E^\tau\in\mathrm{H}^{2n-2}(B\mathrm{Diff}_{D^{2n+1}}(D^{2n+2});\mathbf{Q})\), which we now show agrees with the negative of the class \(\Omega^{2n+1}(p_n-E)^\tau\) featuring in 75:

8.4 Concordances of discs↩︎

Smoothing corners induces an identification \[\label{equ:smoothing-corners}\mathrm{Diff}_{D^d}(D^{d+1})\cong \mathrm{C}(D^d)\tag{104}\] of \(\mathrm{Diff}_{D^d}(D^{d+1})\) with the group of concordances \(\mathrm{C}(M)\) for \(M=D^d\), where \[\mathrm{C}(M)\coloneq \mathrm{Diff}_{M\times \{0\}\cup \partial M\times [0,1]}(M\times [0,1])\] for a compact \(d\)-manifold \(M\). Restricting a concordance to \(M\times \{1\}\) induces a fibre sequence \[\label{equ:concordance-fibration} \mathrm{Diff}_\partial(M\times [0,1])\longrightarrow\mathrm{C}(M)\longrightarrow\mathrm{Diff}_\partial(M)\tag{105}\] which for \(M=D^d\) agrees with the looping of 103 under the appropriate identifications.

8.4.1 Two involutions↩︎

For any manifold \(M\), there is a concordance involution on \(\mathrm{C}(M)\) given by sending a concordance \(\phi\) to \((\mathrm{id}_M\times r)\circ (\phi|_{M\times\{1\}}^{-1}\times\mathrm{id}_{[0,1]})\circ \phi\circ (\mathrm{id}_M\times r)\) where \(r(t)=(1-t)\), which makes 105 an equivariant fibre sequence when equipping the base space with the involution given by inversion in the group \(\mathrm{Diff}_\partial(M)\) and the fibre with the involution induced by conjugation with \((\mathrm{id}_M\times r)\). If \(M=M'\times [0,1]\), then there is another involution on \(\mathrm{C}(M'\times [0,1])\) given by conjugation with \((\mathrm{id}_{M'} \times r\times \mathrm{id}_{[0,1]})\), to which we refer to as the reflection involution. On the subspace \(\mathrm{Diff}_\partial(M'\times[0,1]\times [0,1])\subset \mathrm{C}(M'\times [0,1])\) the two involutions agree up to homotopy, since \(\pi_0(\mathrm{Diff}([0,1]^2))=\mathbf{Z}/2\). We call the second involution the reflection involution, because it agrees for \(M=D^d\cong D^{d-1}\times [0,1]\) via the identification 104 with the reflection involution on \(\mathrm{Diff}_{D^d}(D^{d+1})\) we considered so far, given by conjugation with a diffeomorphism \(\rho\colon D^{d+1}\rightarrow D^{d+1}\) given by reflection in one coordinate that also induces a reflection on the chosen half-disc \(D^d\subset \partial D^{d+1}\) (note that there are several choices for \(\rho\), but they are all homotopic as \(\pi_0\mathrm{Diff}(D^{d+1},D^d)\cong \mathbf{Z}/2\) for \(d\ge5\)).

71 and 75 determine \(\pi_*(\mathrm{C}(D^d))_\mathbf{Q}\) in a range with respect to the reflection involution. This can be upgraded to incorporate the concordance involution:

8.4.2 Stabilisation of concordances↩︎

Concordance spaces can be stabilised: taking products with \([0,1]\) and “bending” the result suitably to make it satisfy the boundary condition induces a map \[\label{equ:conc-stabilisation} \mathrm{C}(M)\longrightarrow\mathrm{C}(M\times [0,1]),\tag{106}\] often called the Hatcher suspension map (see e.g. [52], where this map is called the upper suspension map). With respect to the equivalence 104 , this map agrees for \(M=D^d\) with the map induced by \((-)\times \mathrm{id}_{[-1,1]}\) and identifying \(D^{d+1}\times[-1,1]\cong D^{d+2}\). This is equivariant with respect to the reflection involution and anti-equivariant with respect to the concordance involution (see e.g. [52]). Choosing the reflection involution in the target of 106 to be conjugation with the reflection in the stabilised coordinate, we see that the image of this map on homotopy groups lands in the \((+1)\)-eigenspace of \(\mathrm{C}(D^{d+1})\) with respect to the reflection involution, so 106 is trivial on \((-1)\)-eigenspaces with respect to the reflection involution. Its effect on \((+1)\)-eigenspaces is the content of the following proposition.

Combining 78 with the calculations in Corollaries 71, 73, and 75 with the long exact sequence for the rational homotopy groups of a pair, one deduces the following corollary. It involves the composition \(\pi_{2n-2}(\mathrm{C}(D^\infty),\mathrm{C}(D^{2n+1}))\rightarrow\mathbf{Q}^-\) of the boundary map \(\partial\) to \(\pi_{2n-3}(\mathrm{C}(D^{2n+1}))\) with the evaluation against the class \(\Omega^{2n+2}E^\tau\in\mathrm{H}^{2n-2}(\mathrm{C}(D^{2n+1});\mathbf{Q})^{(-)}\) resulting from 104 and the discussion below 75. We write \(\mathrm{C}(D^\infty)\coloneq \mathop{\mathrm{hocolim}}_k\mathrm{C}(D^k)\) for the space of stable concordances whose homotopy groups are isomorphic to \(K_{*+2}(\mathbf{Z})_\mathbf{Q}\) as a result of 77 and 78.

Using the long exact sequence of the triple \((\mathrm{C}(D^\infty),\mathrm{C}(D^{d+1}),\mathrm{C}(D^d))\), this yields 2.

9 The rational homotopy type of \(B\mathrm{STop}(d)\) and orthogonal calculus↩︎

We conclude this work by determining the rational homotopy type of \(B\mathrm{STop}(d)\) in a range and by spelling out the consequences of our results to orthogonal calculus.

9.1 The rational homotopy type of \(B\mathrm{STop}(d)\)↩︎

Our formula for the rational homotopy type of \(B\mathrm{STop}(d)\) in a range involves the forgetful map to \(B\mathrm{SG}(d)\) and the stabilisation map to \(B\mathrm{STop}\) \[g\colon B\mathrm{STop}(d)\longrightarrow B\mathrm{SG}(d)\quad\text{and}\quad s\colon B\mathrm{STop}(d)\longrightarrow B\mathrm{STop}.\] In odd dimensions it also involves a certain spectrum \(\Theta\mathsf{Bt}^{(1)}\simeq_\mathbf{Q}K(\mathbf{Z})\).

9.1.1 The rational homotopy type of \(B\mathrm{Top}(2n)\)↩︎

Even dimensions are simpler, so they come first.

9.1.2 \(\Theta\mathsf{Bt}^{(1)}\) and the rational homotopy type of\(\mathrm{Top}(d+1)/\mathrm{Top}(d)\)↩︎

Before turning to \(B\mathrm{Top}(2n+1)\), we study the homotopy fibres \(\mathrm{Top}(d+1)/\mathrm{Top}(d)\) of the stabilisation map \(B\mathrm{Top}(d)\rightarrow B\mathrm{Top}(d+1)\) induced by \((-)\times \mathbf{R}\) by comparison to similarly defined fibres \(\mathrm{G}(d+1a)/\mathrm{G}(d)\). So far we considered \(\mathrm{G}(d)\) as the space of homotopy automorphisms of \(S^{d-1}\) and the stabilisation map \(\mathrm{G}(d)\rightarrow\mathrm{G}(d+1)\) to be induced by unreduced suspension, but for the remainder of this section it is convenient to work with a different model with better functoriality, namely as the space of proper homotopy equivalences of \(\mathbf{R}^d\), related to \(\mathrm{hAut}(S^{d-1})\) by an equivalence \[\mathrm{hAut}(S^{d-1}) \longrightarrow\mathrm{hAut}(\mathbf{R}^d)^{\mathrm{prop}}\] given by taking open cones. With this model for \(\mathrm{G}(d)\) the comparison map \(\mathrm{Top}(d)\rightarrow\mathrm{G}(d)\) is simply given by inclusion, the stabilisation map \(\mathrm{G}(d)\rightarrow\mathrm{G}(d+1)\) by taking product with \(\mathbf{R}\), and we have compatible analogues of the loop-suspension map \[\label{equ:loop-suspension}S^d=\mathrm{O}(d+1)/\mathrm{O}(d)\longrightarrow\Omega\mathrm{O}(d+2)/\mathrm{O}(d+1)=S^{d+1},\tag{108}\] for \(\mathrm{Top}(d)\) and \(\mathrm{G}(d)\) in the form of stabilisation maps \[\label{equ:loop-suspension-gtop} \def\arraystretch{1.5} \begin{array}{c@{\hskip 0.1cm} l@{\hskip 0.1cm} c@{\hskip 0.1cm} } \mathrm{Top}(d+1)/\mathrm{Top}(d)&\longrightarrow&\Omega(\mathrm{Top}(d+2)/\mathrm{Top}(d+1))\\ \mathrm{G}(d+1)/\mathrm{G}(d)&\longrightarrow&\Omega(\mathrm{G}(d+2)/\mathrm{G}(d+1)) \end{array}\tag{109}\] induced by conjugation with rotations in a \(2\)-plane (see [22]). These maps define sequential spectra with \(d\)th spaces \(\mathrm{Top}(d+1)/\mathrm{Top}(d)\) and \(\mathrm{G}(d+1)/\mathrm{G}(d)\), and for reasons that will be become clear later when we discuss orthogonal calculus we denote these spectra by \(\Theta\mathsf{Bt}^{(1)}\) and \(\Theta\mathsf{Bg}^{(1)}\), respectively. To have a uniform notation, we denote the sphere spectrum thought of as a sequential spectrum using 108 by \(\Theta\mathsf{Bo}^{(1)}\). By construction, the inclusions \(\mathrm{O}(d)\subset\mathrm{Top}(d)\subset\mathrm{G}(d)\) induce maps of spectra \[\Theta\mathsf{Bo}^{(1)} \longrightarrow\Theta\mathsf{Bt}^{(1)}\longrightarrow\Theta\mathsf{Bg}^{(1)}\] whose composition turns out to be an equivalence; this is a consequence of the fact that the map \(\mathrm{O}(d+1)/\mathrm{O}(d)\rightarrow\mathrm{G}(d+1)/\mathrm{G}(d)\) is \((2d-3)\)-connected for \(d>2\) by [54]. We thus have preferred equivalences \[\label{equ:first-derivatives} \mathbf{S}= \Theta\mathsf{Bo}^{(1)}\simeq \Theta\mathsf{Bg}^{(1)}\quad\text{and}\quad \Theta\mathsf{Bt}^{(1)}\simeq \mathbf{S}\vee (\Theta\mathsf{Bt}^{(1)}/\mathbf{S}).\tag{110}\] The homotopy type of \(\Theta\mathsf{Bt}^{(1)}\) was determined by Waldhausen as his \(A(*)\) [20], but instead of using his result we reproduce it rationally from our point view.

Having determined \(\Theta\mathsf{Bt}^{(1)}\), we express the rational homotopy type of \(\mathrm{Top}(d+1)/\mathrm{Top}(d)\) in a range in terms of \(\Theta\mathsf{Bt}^{(1)}\) and \(\mathrm{G}(d+1)/\mathrm{G}(d)\). Being the \(d\)th space in the spectrum \(\Theta\mathsf{Bt}^{(1)}\), the space \(\mathrm{Top}(d+1)/\mathrm{Top}(d)\) has a canonical map to \(\Omega^\infty( S^d\wedge \Theta\mathsf{Bt}^{(1)})\), and similarly for \(\mathrm{G}(d+1)/\mathrm{G}(d)\).

9.1.3 The rational homotopy type of \(B\mathrm{Top}(2n+1)\)↩︎

In view of 110 85 equivalently says that the map \[\label{equ:description-btop-odd-single-map}\mathrm{Top}(d+1)/\mathrm{Top}(d)\longrightarrow\mathrm{G}(d+1)/\mathrm{G}(d)\times \Omega^\infty (S^d\wedge \Theta\mathsf{Bt}^{(1)}/\mathbf{S})\tag{113}\] is rationally highly connected. In order to use this to provide a description of \(B\mathrm{STop}(2n+1)\), we rely on the following lemma. We postpone its proof to 9.2.3, as it will involve some orthogonal calculus which we have not yet discussed (but see 87).

Fixing any factorisation \(\tilde{k}\) promised by 86 the odd-dimensional analogue of 83 reads as follows. Together with 83 it proves 3, using 84 and rational the equivalences 94 .

9.2 Orthogonal calculus and the second derivative of \(B\mathrm{Top}(-)\)↩︎

The assignment \(V \mapsto B\mathrm{Top}(V)\) defines an orthogonal functor \(\mathsf{Bt}\), i.e.a continuous functor from the topologically enriched category of finite-dimensional inner product spaces to the category of pointed spaces. Similarly \(B\mathrm{O}(-)\) and \(B\mathrm{G}(-)\) define orthogonal functors \(\mathsf{Bo}\) and \(\mathsf{Bt}\) (using the model \(\mathrm{G}(V)=\mathrm{hAut}(V)^{\mathrm{prop}}\) discussed in 9.1.2), and there are maps of orthogonal functors \[\label{equ:the-orthogonal-stars}\mathsf{Bo}\longrightarrow\mathsf{Bt}\longrightarrow\mathsf{Bg}\tag{114}\] induced by the inclusions \(\mathrm{O}(-)\subset\mathrm{Top}(-)\subset\mathrm{G}(-)\). By taking homotopy fibres we can also form the orthogonal functor \(\mathsf{t}\mskip-2mu/\mskip-2mu\mathsf{o}\mskip 1mu\) having \(\mathsf{t}\mskip-2mu/\mskip-2mu\mathsf{o}\mskip 1mu(V) = \mathrm{Top}(V)/\mathrm{O}(V)\). Such functors are the input for Weiss’ theory of orthogonal calculus [17] as recalled in the introduction. Before studying the derivatives of 114 , we add a little more to this recollection. We refer to [17] for details.

9.2.1 Orthogonal derivatives↩︎

Given an orthogonal functor \(\mathsf{E}\), one can use the inclusion \(V\subset \mathbf{R}\oplus V\) to define a new functor \(\mathsf{E}^{(1)}\) by \(\mathsf{E}^{(1)}(V)\coloneq \mathrm{hofib}(\mathsf{E}(V)\rightarrow\mathsf{E}(\mathbf{R}\oplus V))\). This new functor comes with additional structure in the form of natural maps of the form \[\label{equ:structure-maps-first-derivative} \sigma\colon \mathsf{E}^{(1)}(V)\longrightarrow\Omega \mathsf{E}^{(1)}(\mathbf{R}\oplus V),\tag{115}\] so gives rise to a spectrum \(\Theta \mathsf{E}^{(1)}\)—the first derivative of \(\mathsf{E}\)—whose \(d\)th space is \(\mathsf{E}^{(1)}(\mathbf{R}^d)\). Moreover, the reflection along \(\{0\}\) in \(\mathbf{R}\) induces a \(\mathrm{O}(1)\)-action on \(\mathsf{E}^{(1)}(V)\) with respect to which the maps 115 are equivariant if \(\Omega(-)\) is formed with respect to the standard \(\mathrm{O}(1)\)-representation. By a standard untwisting argument (see [17]), this makes \(\Theta \mathsf{E}^{(1)}\) into a (naïve) \(\mathrm{O}(1)\)-spectrum, with \(\Omega^\infty \Theta \mathsf{E}^{(1)} \simeq \mathop{\mathrm{hocolim}}_{d} \Omega^d \mathsf{E}^{(1)}(\mathbf{R}^d)\) on which \(\mathrm{O}(1)\) acts as described above on \(\mathsf{E}^{(1)}(\mathbf{R}^d)\) and by reflection in each of the \(d\) coordinates of \(\Omega^d\). Proceeding as above, one sets \(\mathsf{E}^{(2)}(V)\coloneq \mathrm{hofib}(\sigma\colon \mathsf{E}^{(1)}(V)\rightarrow\Omega \mathsf{E}^{(1)})\), which again admits natural maps, this time of the form \[\sigma\colon \mathsf{E}^{(2)}(V)\longrightarrow\Omega^2 \mathsf{E}^{(2)}(\mathbf{R}\oplus V),\] and these give rise to the second derivative \(\Theta \mathsf{E}^{(2)}\); the \(2d\)th space is \(\mathsf{E}^{(2)}(\mathbf{R}^{2d})\). There is an alternative description of \(\mathsf{E}^{(2)}(V)\) which endows it with an \(\mathrm{O}(2)\)-action, which by the untwisting trick mentioned above leads to an \(\mathrm{O}(2)\)-spectrum structure on \(\Theta \mathsf{E}^{(2)}\) (see [17]). This procedure can be continued and yields spectra \(\Theta \mathsf{E}^{(k)}\) with \(\mathrm{O}(k)\)-action, one for each \(k\ge1\).

9.2.2 The first two rational derivatives of \(B\mathrm{O}(-)\), \(B\mathrm{Top}(-)\), and \(B\mathrm{G}(-)\)↩︎

In 9.1.2 we preemptively introduced the notation \(\Theta\mathsf{Bo}^{(1)}\), \(\Theta\mathsf{Bt}^{(1)}\), and \(\Theta\mathsf{Bg}^{(1)}\) for what we can now call the first derivatives of these functors, and in 110 and 84 we described the rational homotopy types of these spectra. Determining their rational homotopy types as \(\mathrm{O}(1)\)-spectra amounts to understanding the \(\mathrm{O}(1)\)-action on their rational homotopy groups, which is as follows. We write \(\mathbf{Q}^\pm\) for \(\mathbf{Q}\) acted upon by \(\mathrm{O}(1)\) via multiplication by \(\pm1\).

As for second derivatives, for \(\mathsf{Bo}\) we already mentioned in the introduction that the metastable EHP-description of the loop-suspension map \(S^d= \mathsf{Bo}^{(1)}(\mathbf{R}^d)\to \Omega\mathsf{Bo}^{(1)}(\mathbf{R}^{d+1})=\Omega S^{d+1}\) shows that \(\Theta\mathsf{Bo}^{(2)} \simeq \mathbf{S}^{-1}\), with trivial \(\mathrm{O}(2)\)-action (see [17] for details). Based on 94 there are several ways to determine the second derivative of \(\mathsf{Bg}\); Reis–Weiss [25] did it by describing an equivalence \[\label{eq:ThetaBg2} \Theta\mathsf{Bg}^{(2)} \simeq_\mathbf{Q}\mathrm{Map}(S^1_+, \mathbf{S}^{-1})\tag{116}\] under which the map \(\Theta\mathsf{Bo}^{(2)} \to \Theta\mathsf{Bg}^{(2)}\) is the inclusion of the constant maps and \(\mathrm{O}(2)\) acts on \(\mathrm{Map}(S^1_+, \mathbf{S}^{-1})\) through the canonical action on \(S^1\). We complete this picture with the following.

Combining 90 with Borel’s calculation \(K_{*>0}(\mathbf{Z})_\mathbf{Q}\cong \oplus_{k \geq 1} \mathbf{Q}[4k+1]\) (see 7.1.3), and 91 with 116 , we can summarise our discussion of the first two derivatives of \(\mathsf{Bt}\) as follows. This in particular includes 4 from the introduction.

We close this subsection by making good for the postponed proof of 86.

9.2.3 Proof of 86↩︎

During the proof we use the Taylor tower of an orthogonal functor as summarised in the introduction (see [17] for details). We furthermore consider the orientation preserving analogue \(\mathsf{Bst}(V) = B\mathrm{STop}(V)\) of \(\mathsf{Bt}\), the functor \(\mathsf{F}(V) = \mathrm{Top}(V \oplus \mathbf{R})/\mathrm{Top}(V)\) and the canonical morphism \(\mathsf{F}\to \mathsf{Bst}\). Note that \(\mathsf{F}= \mathsf{Bst}^{(1)}\), and that \(\mathsf{Bst}\) and \(\mathsf{Bt}\) differ by the constant functor on \(B\mathbf{Z}^\times\) and thus have the same derivatives, which we continue to denote by \(\Theta \mathsf{Bt}^{(k)}\). Now \(T_0 \mathsf{F}\) has contractible values and by general properties of orthogonal calculus we have a preferred equivalence \(\Theta \mathsf{F}^{(1)} \simeq \mathrm{Ind}_e^{\mathrm{O}(1)} \mathrm{Res}_e^{\mathrm{O}(1)}\Theta \mathsf{Bt}^{(1)}\) with respect to which the map of \(\mathrm{O}(1)\)-spectra \(\Theta \mathsf{F}^{(1)} \to \Theta \mathsf{Bt}^{(1)}\) is adjoint to the identity map. We consider the diagram \[\begin{tikzcd}[column sep=0.4cm, ar symbol/.style = {draw=none,"\textstyle#1" description,sloped}, equivalent/.style = {ar symbol={\simeq}}] \mathrm{Top}(2n+2)/\mathrm{Top}(2n+1) \dar \rar&[-0.25cm] T_1 \mathsf{F}(\mathbf{R}^{2n+1}) \dar &[-0.25cm] \Omega^\infty(S^{2n+1} \wedge_{\mathrm{O}(1)} {\Theta \mathsf{F}^{(1)}}) \arrow[l,equivalent ]&[-0.3cm]\Omega^\infty(S^{2n+1} \wedge {\Theta \mathsf{Bt}^{(1)}}) \dar\arrow[l,equivalent] \\ B\mathrm{STop}(2n+1) \rar& T_1 \mathsf{Bst}(\mathbf{R}^{2n+1}) \arrow[rr, dashed]& & \Omega^\infty(S^{2n+1} \wedge {\Theta \mathsf{Bt}^{(1)}}/\mathbf{S})_\mathbf{Q} \end{tikzcd}\] where upper left equivalence is because \(T_0 \mathsf{F}\) has contractible values and the right because \({\Theta \mathsf{F}^{(1)}}\) is induced from \({\Theta \mathsf{Bt}^{(1)}}\). As the top row of this diagram is compatible with the canonical map from the statement, it reduces the problem to producing the dashed map—a problem about first Taylor approximations. To construct the dashed map we consider the diagram \[\begin{tikzcd}[column sep=0.5cm] \Omega^\infty(S^{2n+1} \wedge_{\mathrm{O}(1)} {\Theta \mathsf{Bt}^{(1)}}) \rar \arrow[rd, dotted]&T_1 \mathsf{Bst}(\mathbf{R}^{2n+1}) \rar \arrow[d, dashed] &[-0.5cm] T_0 \mathsf{Bst}(\mathbf{R}^{2n+1}) \simeq B\mathrm{STop}\\ \Omega^\infty(S^{2n+1} \wedge {\Theta \mathsf{Bt}^{(1)}}) \uar \rar & \Omega^\infty(S^{2n+1} \wedge {\Theta \mathsf{Bt}^{(1)}}/\mathbf{S})_\mathbf{Q} \end{tikzcd}\] and first produce a dotted map making the lower triangle commute. From 92 we get \[\begin{align} \pi_*(S^{2n+1} \wedge {\Theta \mathsf{Bt}^{(1)}})_\mathbf{Q}&\cong \textstyle{\bigoplus_{k \geq 1} \mathbf{Q}[2n+1+4k+1]\oplus \mathbf{Q}[2n+1]}\\ \pi_*(S^{2n+1} \wedge {\Theta \mathsf{Bt}^{(1)}}/\mathbf{S})_\mathbf{Q}&\cong \textstyle{\bigoplus_{k \geq 1} \mathbf{Q}[2n+1+4k+1]}\\ \pi_*(S^{2n+1} \wedge_{\mathrm{O}(1)} {\Theta \mathsf{Bt}^{(1)}})_\mathbf{Q}&\cong \textstyle{\bigoplus_{k \geq 1} \mathbf{Q}[2n+1+4k+1]}, \end{align}\] so there is indeed a unique homotopy class of dotted infinite loop map making the lower triangle commute. To extend this to a map as indicated by the dashed arrow, we observe that \(\Omega^\infty(S^{2n+1} \wedge {\Theta \mathsf{Bt}^{(1)}}/\mathbf{S})_\mathbf{Q}\simeq \prod_{k \geq 1} K(\mathbf{Q}, 2n+4k+2)\), so the problem is to show that certain cohomology classes in \(\mathrm{H}^{2n+4k+2}(\Omega^\infty(S^{2n+1} \wedge_{\mathrm{O}(1)} {\Theta \mathsf{Bt}^{(1)}});\mathbf{Q})\) are permanent cycles in the Serre spectral sequence for the top row. But this spectral sequence collapses, as \(B\mathrm{STop}\) is simply-connected and has rational cohomology supported in even degrees. 0◻

9.3 The algebraic \(K\)-theory Euler class↩︎

The purpose of this final subsection is twofold: firstly, we explain how a more sophisticated use of orthogonal calculus yields an improvement on 86 in the form of a preferred map \[\tilde{k} : B\mathrm{STop}(2n+1) \longrightarrow\Omega^\infty (S^{2n+1} \wedge \Theta\mathsf{Bt}^{(1)}/\mathbf{S})_\mathbf{Q}=\Omega^\infty (S^{2n+1} \wedge (\Theta\mathsf{Bt}^{(1)}/\Theta\mathsf{Bo}^{(1)}))_\mathbf{Q},\] together with a nullhomotopy of its precomposition with \(B\mathrm{SO}(2n+1) \to B\mathrm{STop}(2n+1)\). Secondly, we show how this provides a reinterpretation of our results on \(B\mathrm{STop}(d)\) in terms of obstructions to finding vector bundle structures on topological \(\mathbf{R}^d\)-bundles. Throughout this section, we freely use the theory of orthogonal calculus and refer to [17] for details.

9.3.1 The intrinsic Euler class↩︎

There is a homotopy cofibre sequence of based \(\mathrm{O}(1)\)-spaces \[S^V \longrightarrow S^{\mathbf{R}\oplus V} \longrightarrow\mathrm{Ind}_e^{\mathrm{O}(1)} S^1 \wedge S^V,\] where the \(\mathrm{O}(1)\)-action on the left-hand and middle spaces is by negation in \(V\) and \(\mathbf{R}\oplus V\) respectively. If \(\mathsf{E}\) is an orthogonal functor with first derivative the \(\mathrm{O}(1)\)-spectrum \(\Theta \mathsf{E}^{(1)}\), then the above gives a fibration sequence \[\Omega^\infty(S^V \wedge_{\mathrm{O}(1)} \Theta \mathsf{E}^{(1)}) \longrightarrow\Omega^\infty(S^{\mathbf{R}\oplus V} \wedge_{\mathrm{O}(1)} \Theta \mathsf{E}^{(1)}) \longrightarrow\Omega^\infty(S^1 \wedge S^V \wedge \Theta \mathsf{E}^{(1)} ),\] so as \(T_0\mathsf{E}(V)\simeq T_0\mathsf{E}(\mathbf{R}\oplus V)\) on first Taylor approximations there is a homotopy fibre sequence \[\Omega^{\infty}(S^V \wedge \Theta \mathsf{E}^{(1)}) \longrightarrow T_1 \mathsf{E}(V) \longrightarrow T_1 \mathsf{E}(\mathbf{R}\oplus V).\] This can be expressed in the following more useful way. The \(\mathrm{O}(1)\)-spectrum \(\Theta \mathsf{E}^{(1)}\) arises as the fibre of a parametrised \(\mathrm{O}(1)\)-spectrum \(\underline{\Theta \mathsf{E}}^{(1)}\) over \(\mathsf{E}(\mathbf{R}^\infty) \simeq T_0 \mathsf{E}(\mathbf{R}\oplus V)\) (i.e. a parametrised spectrum with a fibrewise \(\mathrm{O}(1)\)-action), and there is a homotopy cartesian square \[\label{eq:The32UnivEulerClass} \begin{tikzcd} T_1 \mathsf{E}(V) \rar\dar & T_0 \mathsf{E}(\mathbf{R}\oplus V) \dar{z}\\ T_1 \mathsf{E}(\mathbf{R}\oplus V) \rar{e_\mathsf{E}} & \underline{\Omega}^\infty(S^{\mathbf{R}\oplus V} \wedge \underline{\Theta \mathsf{E}}^{(1)}), \end{tikzcd}\tag{117}\] where \(\underline{\Omega}^\infty(-)\) denotes the fibrewise infinite loop space of a parametrised spectrum, \(z\) is the basepoint section, and \(e_\mathsf{E}\) is a map covering the standard map \(T_1 \mathsf{E}(\mathbf{R}\oplus V) \to T_0 \mathsf{E}(\mathbf{R}\oplus V)\). This square is given by [17].

Generally speaking, if \(\underline{\mathrm{T}}\) is a parameterised spectrum over a space \(X\), then homotopy classes of sections of \(\underline{\Omega}^\infty(\underline{\mathrm{T}}) \to X\) may be seen as twisted cohomology classes on \(X\) with coefficients in \(\underline{\mathrm{T}}\), and the set of them is denoted \(H^0(X ; \underline{\mathrm{T}})\). If the parameterised spectrum \(\underline{\mathrm{T}}\) is pulled back from a (parametrised) spectrum \(\mathrm{T}\) over a point, i.e. a regular spectrum, then this notion recovers \(\mathrm{T}\)-cohomology in the usual sense. The basepoint section \(z : X \to \underline{\Omega}^\infty(\underline{\mathrm{T}})\) represents the zero cohomology class. If \((X,Y)\) is a space pair then a section \(s\) of \(\underline{\Omega}^\infty(\underline{\mathrm{T}}) \to X\) together with a homotopy \(s\vert_Y \simeq z\vert_Y\) fibrewise over \(X\) may be seen as a relative twisted cohomology class, and the set of homotopy classes of such is denoted \(H^0(X, Y ; \underline{\mathrm{T}})\).

In the case at hand, the square 117 may thus be interpreted as a relative twisted cohomology class on the pair \((T_1 \mathsf{E}(\mathbf{R}\oplus V), T_1 \mathsf{E}(V))\) with coefficients in the pullback along \(T_1 \mathsf{E}(\mathbf{R}\oplus V) \to T_0 \mathsf{E}(\mathbf{R}\oplus V)\) of the parametrised spectrum \(\underline{\Theta \mathsf{E}}^{(1)}\) over \(T_0 \mathsf{E}(\mathbf{R}\oplus V)\). Neglecting that it is a relative class, and pulling back further along \(\mathsf{E}(\mathbf{R}\oplus V) \to T_1 \mathsf{E}(\mathbf{R}\oplus V)\), gives an absolute twisted cohomology class \[e_\mathsf{E}\in \mathrm{H}^{0}\big(\mathsf{E}(\mathbf{R}\oplus V) ; S^{\mathbf{R}\oplus V} \wedge \underline{\Theta \mathsf{E}}^{(1)}\big).\] If \(f : \mathsf{E}\to \mathsf{F}\) is a morphism of orthogonal functors, there is an induced map \(\Theta f^{(1)} : \underline{\Theta \mathsf{E}}^{(1)} \to f(\mathbf{R}^\infty)^*\underline{\Theta \mathsf{F}}^{(1)}\) of first derivatives (the target is pulled back along the map \(f(\mathbf{R}^\infty) : \mathsf{E}(\mathbf{R}^\infty) \to \mathsf{F}(\mathbf{R}^\infty)\)), and by naturality we have \[(T_1 f)^*(e_\mathsf{F}) = (\Theta f^{(1)})_*(e_\mathsf{E})\in \mathrm{H}^0\big(T_1 \mathsf{E}(\mathbf{R}\oplus V), T_1 \mathsf{E}(V) ; S^{\mathbf{R}\oplus V} \wedge f(\mathbf{R}^\infty)^*\underline{\Theta \mathsf{F}}^{(1)}\big).\] More precisely, there are preferred homotopies making the 3-cube given by the corresponding diagrams 117 commute, so again forgetting that \(e_\mathsf{E}\) and \(e_\mathsf{F}\) are relative classes and pulling back we obtain a twisted cohomology class \[(e_\mathsf{F}, e_\mathsf{E}) \in \mathrm{H}^0\big( \mathsf{F}(\mathbf{R}\oplus V), \mathsf{E}(\mathbf{R}\oplus V) ; S^{\mathbf{R}\oplus V} \wedge (\underline{\Theta \mathsf{F}}^{(1)}, \underline{\Theta \mathsf{E}}^{(1)})\big).\]

9.3.2 Application to \(\mathsf{Bo}\) and \(\mathsf{Bt}\)↩︎

Applying the above discussion to the map \(\mathsf{Bo}\to \mathsf{Bt}\) gives \[(e_\mathsf{Bt}, e_\mathsf{Bo}) \in \mathrm{H}^d(B\mathrm{Top}(d), B\mathrm{O}(d) ; (\underline{\Theta \mathsf{Bt}}^{(1)}, \underline{\Theta \mathsf{Bo}}^{(1)}).\]

The discussion in [25] shows that the parametrised spectrum \(\underline{\Theta \mathsf{Bo}}^{(1)}\) over \(B\mathrm{O}\), and the pullback of the parametrised spectrum \(\underline{\Theta \mathsf{Bt}}^{(1)}\) along \(B\mathrm{O}\to B\mathrm{Top}\), are both pulled back along the map \(B\mathrm{O}\to B\mathrm{G}\) from the parametrised spectrum given by the canonical action of \(\mathrm{G}= \mathrm{GL}_1(\mathbf{S})\) on \(\Theta \mathsf{Bo}^{(1)} = \Theta \mathsf{Bo}^{(1)} \wedge \mathbf{S}\) and \(\Theta \mathsf{Bt}^{(1)} = \Theta \mathsf{Bt}^{(1)} \wedge \mathbf{S}\). Upon rationalising this action factors over \(\mathrm{G}\to \mathbf{Z}^\times\), so \({\underline{\Theta \mathsf{Bo}}^{(1)}}_\mathbf{Q}\) and the pullback of \({\underline{\Theta \mathsf{Bt}}^{(1)}}_\mathbf{Q}\) to \(B\mathrm{O}\) are the pullbacks of \(\Theta \mathsf{Bo}^{(1)}_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}\) and \(\Theta \mathsf{Bt}^{(1)}_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}\) along the orientation map \(B\mathrm{O}\to B\mathbf{Z}^\times\), where \(\underline{\mathbf{S}}_\mathrm{sign}\) is the \(\mathbf{Z}^\times\)-spectrum given by \(\mathbf{S}\) with the sign involution. Using that \(\mathrm{O}\to \mathrm{Top}\) is a rational equivalence, it follows that also \(\underline{\Theta \mathsf{Bt}}^{(1)}_\mathbf{Q}\) itself is the pullback of \(\Theta \mathsf{Bt}^{(1)}_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}\) along the orientation map \(B\mathrm{Top}\to B\mathbf{Z}^\times\). Extending 117 there is a commutative diagram (similarly for \(\mathsf{Bo}\)) \[\begin{tikzcd}[ar symbol/.style = {draw=none,"\textstyle#1" description,sloped}, equivalent/.style = {ar symbol={\simeq}}] T_1 \mathsf{Bt}(V) \rar\dar & T_0 \mathsf{Bt}(\mathbf{R}\oplus V) \arrow[r,equivalent ] \dar{z} &[-1cm] B\mathrm{Top}\rar &[-1cm] B\mathbf{Z}^\times \dar{z}\\ T_1 \mathsf{Bt}(\mathbf{R}\oplus V) \rar{e_\mathsf{Bt}} & \underline{\Omega}^\infty(S^{\mathbf{R}\oplus V} \wedge \underline{\Theta \mathsf{Bt}}^{(1)}) \arrow[rr] & & \underline{\Omega}^\infty(S^{\mathbf{R}\oplus V} \wedge \Theta \mathsf{Bt}^{(1)}_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}), \end{tikzcd}\] where the left-hand square is homotopy cartesian, the right-hand square is rationally homotopy cartesian, and \(z\) is the basepoint section. In particular the lower map and its analogue for \(\mathsf{Bo}\) gives a diagram of the form \[\begin{tikzcd}[column sep=0.5cm] T_1 \mathsf{Bo}(\mathbf{R}^d) \dar \rar{e_\mathsf{Bo}} & \underline{\Omega}^\infty(S^d \wedge \underline{\Theta \mathsf{Bo}}^{(1)}) \rar \dar & \underline{\Omega}^\infty(S^{d} \wedge \Theta \mathsf{Bo}^{(1)}_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}) \dar \rar &[-0.2cm] B\mathbf{Z}^\times \dar{z}\\ T_1 \mathsf{Bt}(\mathbf{R}^d) \rar{e_\mathsf{Bt}} & \underline{\Omega}^\infty(S^{d} \wedge \underline{\Theta \mathsf{Bt}}^{(1)}) \rar & \underline{\Omega}^\infty(S^{d} \wedge \Theta \mathsf{Bt}^{(1)}_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}) \rar & \underline{\Omega}^\infty(S^{d} \wedge (\tfrac{\Theta \mathsf{Bt}^{(1)}}{\Theta \mathsf{Bo}^{(1)}})_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}), \end{tikzcd}\] where the right-hand square is developed by taking cofibres of (parametrised) spectra. When \(d\) is even the left-hand map in this diagram is a rational equivalence (over \(B\mathbf{Z}^\times\)), so the composition along the bottom is trivial. When \(d\) is odd, we consider the composition \[B\mathrm{Top}(d) \overset{e_{\mathsf{Bt}}}{\longrightarrow} \underline{\Omega}^\infty\big(S^{d} \wedge \underline{\Theta \mathsf{Bt}}^{(1)}\big) \longrightarrow\underline{\Omega}^\infty\big(S^{d} \wedge (\tfrac{\Theta \mathsf{Bt}^{(1)}}{\Theta \mathsf{Bo}^{(1)}}\big)_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}) \overset{\cdot 1/2}{\longrightarrow}\underline{\Omega}^\infty\big(S^{d} \wedge (\Theta \mathsf{Bt}_\mathbf{Q}^{(1)}/\Theta \mathsf{Bo}_\mathbf{Q}^{(1)})_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}\big)\] which qualifies as \(\tilde{k}\) in 86 when precomposed with \(B\mathrm{STop}(2n+1)\rightarrow B\mathrm{Top}(2n+1)\) as a result of the following lemma, using \(\mathbf{S}=\Theta\mathsf{Bo}^{(1)}\).

9.3.3 Obstruction-theoretic reinterpretation↩︎

It is now simply a reinterpretation of 83 and 88 that the square \[\begin{tikzcd} B\mathrm{O}(2n+1) \dar \rar & B\mathrm{Top}(2n+1) \dar{(p_n - E, \prod_{i>n} p_i, \tilde{k})}\\ B\mathbf{Z}^\times \rar& \prod_{i \geq n} K(\mathbf{Q}, 4i) \times \underline{\Omega}^\infty\big(S^{2n+1} \wedge (\Theta \mathsf{Bt}^{(1)}/\Theta \mathsf{Bo}^{(1)})_\mathbf{Q}\wedge \underline{\mathbf{S}}_\mathrm{sign}\big) \end{tikzcd}\] is rationally \((5n-6)\)-cartesian if \(d = 2n+1 \geq 5\), and similarly for \(d=2n \geq 6\) with the final factor omitted, \(B\mathbf{Z}^\times\) replaced by a point, and \(p_n-E\) replaced by \(p_n-e^2\). In particular, given \(d\ge6\) with \(d=2n\) or \(d=2n+1\) and a map of pairs \[\phi: \big(X, A\big) \longrightarrow\big(B\mathrm{Top}(d), B\mathrm{O}(d)\big)\] such that \((X,A)\) only has relative cells of dimension \(\leq 5n-6\) then a complete set of rational obstructions to compressing \(\phi\) (i.e. homotoping it into \((B\mathrm{O}(d),B\mathrm{O}(d))\) is:

1. \(\phi^*(p_i)\) \(\in\mathrm{H}^{4i}(X,A)\) for \(i > n\),

2. \(\phi^*(p_n-e^2)\) \(\in \mathrm{H}^{4n}(X,A)\) if \(d=2n\),

3. \(\phi^*(p_n-E)\) \(\in \mathrm{H}^{4n}(X,A)\) if \(d=2n+1\),

4. \(\phi^*(e_\mathsf{Bt}, e_\mathsf{Bo})\) \(\in \mathrm{H}^{d}(X, A ;(\underline{\Theta \mathsf{Bt}}^{(1)}, \underline{\Theta \mathsf{Bo}}^{(1)}))_\mathbf{Q}\) if \(d=2n+1\) (\((e_\mathsf{Bt}, e_\mathsf{Bo})\) vanishes for \(d=2n\)).

10 Some cohomology of \(G_{V}\)↩︎

As announced and used in 7.1.4 this appendix serves to compute the bigraded groups \[\mathrm{H}_*\big(G_V ; S_\lambda({H}_{V}) \otimes S_\mu({H}_V^\vee)\big)\] in a stable range of homological degrees. To do so we first pass to the ungraded setting and dualise: in 97 we calculate \(\mathrm{H}^*(G_V; (\bar{H}_{V}^\vee)^{\otimes p} \otimes \bar{H}_{V}^{\otimes q})\) in a stable range as a graded \(\mathbf{Q}[\Sigma_p \times \Sigma_q]\)-module, which we then use in 98 to answer the original question.

In fact it will be more convenient to calculate the groups \(\mathrm{H}^*(G_V; (\bar{H}_{V}^\vee)^{\otimes p} \otimes \bar{H}_{V}^{\otimes q})\) with arbitrary finite sets \(P\) and \(Q\) indexing the tensor powers. To state the answer (which also take the \(\mathrm{H}^*(\mathrm{GL}_\infty(\mathbf{Z});\mathbf{Q})\)-module structure induced by the surjection \(G_V \to \mathrm{GL}(\bar{H}_V^\mathbf{Z})\rightarrow\mathrm{GL}_\infty(\mathbf{Z})\) followed by the stabilisation map into account) we abbreviate \(\underline{k}\coloneq \{1,\ldots,k\}\) for \(k\ge0\), write \(\mathrm{Bij}(S,T)\) for the set of bijections from \(S\) to \(T\), and denote \((+)\) and \((-)\) for the trivial and sign representations of the symmetric group \(\Sigma_2\).

11 Relation to the work of Watanabe↩︎

In [8] Watanabe used a characteristic class associated to the “theta” graph defined by Kontsevich in terms of configuration space integrals \[\label{equ:Kontsevich-class}\zeta_{2}\in\mathrm{H}^{2n-2}(B\mathrm{Diff}_\partial^\mathrm{fr}(D^{2n+1});\mathbf{Q})\tag{121}\] to show that the group \(\pi_{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1}))_\mathbf{Q}\) is nontrivial for many values of \(n\ge2\) (see [8]). The purpose of this appendix is twofold:

1. In Section 11.1, we explain how one can replace Watanabe’s use of the Kontsevich class \(\zeta_2\) in [8] by an invariant with a significantly simpler description. This results a surprisingly elementary proof that \(\pi_{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1}))_\mathbf{Q}\) is nontrivial for many values of \(n\ge2\), which could have been discovered in the ‘70s. This also gives a simple argument for the odd-dimensional analogue \(p_n\neq E\in\mathrm{H}^{4n}(B\mathrm{Top}(2n+1);\mathbf{Q})\) of Weiss’ theorem \(p_n\neq e^2\in\mathrm{H}^{4n}(B\mathrm{Top}(2n+1);\mathbf{Q})\) described in the introduction.

2. In Section 11.2, we give an alternative description of \(\zeta_{2}\) which does not involve configuration space integrals and fits well with our computation of the rational homotopy type of the \((\approx 3n)\)-truncation of \(B\mathrm{Diff}_\partial^\mathrm{fr}(D^{2n+1})_{\ell}\) in 8 (but does not use it).

11.1 A “classical” proof that \(\pi_{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1}))_\mathbf{Q}\) is often nontrivial↩︎

On close inspection, the proof of the main theorem in [8] saying that \(\pi_{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1}))_\mathbf{Q}\) is nontrivial for “many” \(n\ge2\) hinges only on very few properties of the Kontsevich class \(\zeta_{2}\in\mathrm{H}^{2n-2}(B\mathrm{Diff}_\partial^\mathrm{fr}(D^{2n+1});\mathbf{Q})\). In fact, the only input needed for the final step in the argument (see Section 5 loc.cit.) to go through is the existence of a morphism \[\label{equ:alternative-invariant}\psi\colon \pi_{2n-2}(B\mathrm{Diff}^\mathrm{fr}_\partial(D^{2n+1})_\ell)\longrightarrow\mathbf{Q}\tag{122}\] with the following two properties:

1. \(\psi\) is integral, i.e.it has image in \(\mathbf{Z}\subset\mathbf{Q}\).

2. The composition \[\pi_{4n}(B\mathrm{O}(2n+1))\cong \pi_{2n-2}(\Omega^{2n+1}\mathrm{O}(2n+1))\overset{r}{\longrightarrow} \pi_{2n-2}(B\mathrm{Diff}^\mathrm{fr}_\partial(D^{2n+1})_\ell)\overset{\psi}{\longrightarrow}\mathbf{Q}\] agrees with \(\tfrac{1}{4}\cdot p_n\colon\pi_{4n}(B\mathrm{O}(2n+1))\rightarrow\mathbf{Q}\). Here \(r\) is induced by reframing.

The morphism induced by Kontsevich’s class \(\xi_2\in \mathrm{H}^{2n-2}(B\mathrm{Diff}_\partial^\mathrm{fr}(D^{2n+1});\mathbf{Q})\) as described in Section 2.5.2 loc.cit.satisfies [enum:psi-i] by construction and [enum:psi-ii] as a result of the proof of Theorem 2 loc.cit.. The class \(\xi_2\) is thus a valid choice for \(\psi\), but for the purpose of proving the main result in [8], any other \(\psi\) satisfying [enum:psi-i] and [enum:psi-ii] would do. Our favourite choice is the following.

11.1.1 An alternative \(\psi\)↩︎

Consider the composition \[\label{equ:framed-diff-to-G}B\mathrm{Diff}_\partial^\mathrm{fr}(D^{2n+1})_\ell\longrightarrow\Omega^{2n+1}\mathrm{Top}(2n+1)\longrightarrow\Omega^{2n+1}\mathrm{G}(2n+1)\simeq \Omega^{2n+2}B\mathrm{SG}(2n+1)\tag{123}\] where the first map is the “scanning map” appearing in smoothing theory and the second map is the canonical map \(\mathrm{Top}(2n+1)\rightarrow\mathrm{G}(2n+1)\) obtained by viewing \(\mathrm{Top}(2n+1)\) as homeomorphisms fixing the origin and \(\mathrm{G}(2n+1)\) as homotopy automorphisms of \(\mathbf{R}^{2n+1}\backslash \{0\}\). We construct \(\psi\) as the composition of the map induced by 123 on \(\pi_{2n-2}(-)\) with an invariant \[\label{equ:varepsilon-alternative}\varepsilon\colon \pi_{4n}(B\mathrm{SG}(2n+1))\longrightarrow\mathbf{Z}\tag{124}\] defined as follows: given an oriented fibration \(\pi\colon E\rightarrow S^{4n}\) together with an identification \(\pi^{-1}(*)\simeq S^{2n}\) of the preimage of the basepoint with \(S^{2n}\), we set \(\varepsilon(\pi)\coloneq \int_E \iota(v)^3\in \mathbf{Z}\) where \(v\in \mathrm{H}^{2n}(S^n;\mathbf{Z})\) is the standard generator. Here \(\iota\colon \mathrm{H}^{2n}(S^{2n};\mathbf{Z})\rightarrow\mathrm{H}^{2n}(E;\mathbf{Z})\) is the inverse of the map induced by restriction along \(S^{2n}\simeq \pi^{-1}(*)\subset E\), which is an isomorphism by the Serre spectral sequence. In the universal case, this defines 124 . The precomposition of \(\varepsilon\) with the map induced by 123 satisfies [enum:psi-i] by construction and [enum:psi-ii] by the following lemma.

Using this choice of \(\psi\) instead of \(\xi_2\), the argument in [8] shows that \(\pi_{2n-2}(B\mathrm{Diff}_\partial(D^{2n+1}))_\mathbf{Q}\) is nontrivial for many \(n\) without using configuration space integrals, in fact only relying on differential topology as developed in the ’60s and ’70s. To illustrate the simplicity of this alternative line of argument, we spell out a complete proof of:

11.2 An alternative description of the first Kontsevich class↩︎

Recall that there is a unique class \(E\in \mathrm{H}^{4n}(B\mathrm{SG}(2n+1);\mathbf{Q})\) that pulls back to the square of the Euler class \(e^2\in \mathrm{H}^{4n}(B\mathrm{SG}(2n);\mathbf{Q})\) along the stabilisation map \(B\mathrm{SG}(2n)\rightarrow B\mathrm{SG}(2n+1)\) (see 8.1.1). After looping this class \((2n+2)\) times, we may pull it back along the composition 123 to get a class \(\Omega^{2n+2}E\in \mathrm{H}^{2n-2}(B\mathrm{Diff}_\partial^\mathrm{fr}(D^{2n+1});\mathbf{Q})\). It turns out that this class agrees with the Kontsevich class \(\zeta_2\) up to the following explicit constant.

For the remainder of this section we assume familiarity with the construction of \(\zeta_2\in \mathrm{H}^{2n-2}(B\mathrm{Diff}^\mathrm{fr}_\partial(D^{2n+1});\mathbf{Z})\) as an integral class given in [8], and freely use the notation of that paper. For convenience we pass between \((D^d, \partial D^d)\)-bundles and \(\mathrm{int}(D^d)\)- or \(\mathbf{R}^d\)-bundles standard outside of a compact set without further mentioning. As a preparation to 102, we prove the following.

Glossary of notation↩︎

References↩︎

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1. There is a typo in [7]: The matrix should have \(AM\) instead of \(M\) as the upper right entry.↩︎

2. This map is not strictly base point preserving, but the image of the basepoint of the source is connected to the basepoint by a preferred path, and this is good enough.↩︎

3. See [56] for useful references on end spaces.↩︎