1 Introduction↩︎

Let \(\mathcal{K}\) be a simplicial complex on the vertex set \([m]=\{1,\dots,m\}\). The corresponding Davis–Januszkiewicz space \[\mathop{\mathit{DJ}}(\mathcal{K}):=(\mathbb{C}P^\infty,\ast)^\mathcal{K}=\bigcup_{I\in\mathcal{K}}(\mathbb{C}P^\infty)^{\times I}\] is a CW-subcomplex in \((\mathbb{C}P^\infty)^m\) and a particular case of the polyhedral product construction. These spaces appear in toric topology [1] as the Borel constructions for torus actions on quasitoric manifolds and smooth toric varieties. The cohomology ring of \(\mathop{\mathit{DJ}}(\mathcal{K})\) is identified with the Stanley–Reisner ring of \(\mathcal{K}\), providing a link to combinatorial commutative algebra.

The second homotopy group \(\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\cong\mathbb{Z}^m\) has \(m\) canonical generators represented by the maps \[t_i\colon S^2 \longrightarrow \mathbb{C}P^\infty \longrightarrow (\mathbb{C}P^\infty)^{\vee m}=\mathop{\mathit{DJ}}(\mathcal{L}) \longrightarrow \mathop{\mathit{DJ}}(\mathcal{K}),\quad 1\le i\le m,\] where \(\mathcal{L}\) is the simplicial complex consisting of the \(m\) disjoint vertices, the left map is the inclusion of the bottom cell, the middle map is the inclusion of the \(i\)th wedge summand, and the right map is the map of polyhedral products induced by the simplicial inclusion \(\mathcal{L}\to\mathcal{K}\).

The homotopy groups \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) form a quasi-Lie algebra [2] with respect to the Whitehead product, and form a \(\Pi\)-algebra [3] with respect to the Whitehead product and compositions with elements of the homotopy groups of spheres.

Rationally, the only relations satisfied by the canonical elements \(t_1,\ldots,t_m\in\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\otimes\mathbb{Q}\) are \([t_i,t_i]=0\) and \([t_i,t_j]=0\) for \(\{i,j\}\in\mathcal{K}\) (see [4] and also [5]). Integrally, the part of the homotopy groups \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) generated by the canonical elements is much more subtle, and can be studied in two formats.

These questions are interesting already in the simplest case, where \(\mathcal{K}=\mathcal{L}\) is the disjoint union of \(m\) points and \(\mathop{\mathit{DJ}}(\mathcal{L})=(\mathbb{C}P^\infty)^{\vee m}\) is the \(m\)-fold wedge.

It is helpful to consider the homotopy fibration \[\label{eq:zk-dj32fibration} \mathcal{Z}_\mathcal{K}\overset{i}\longrightarrow\mathop{\mathit{DJ}}(\mathcal{K})\overset{p} \longrightarrow(\mathbb{C}P^\infty)^m,\tag{1}\] where \(p\) is the standard inclusion and \(\mathcal{Z}_\mathcal{K}=(D^2,S^1)^\mathcal{K}\) is the moment-angle complex corresponding to \(\mathcal{K}\). Since the looped map \(\Omega p\) has a right homotopy inverse, we obtain an isomorphism of graded abelian groups \[\label{eq:pi32dj326132pi32bt324332pi32zk} \pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\cong\pi_*(\mathbb{C}P^\infty)^{\oplus m}\oplus\pi_*(\mathcal{Z}_\mathcal{K}).\tag{2}\] However, this isomorphism does not preserve the Whitehead product, and therefore does not give a splitting of \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) as a quasi-Lie algebra. In view of Problems 1 and 2, it is therefore important to describe the Whitehead products of the canonical elements \(t_1,\ldots,t_m\in\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\) with elements from \(\pi_*(\mathcal{Z}_\mathcal{K})\).

Although in general the homotopy type of \(\mathcal{Z}_\mathcal{K}\) might be complicated, in many cases \(\mathcal{Z}_\mathcal{K}\) is homotopy equivalent to a wedge of simply connected spheres, and the fibre inclusion \(i\colon\mathcal{Z}_\mathcal{K}\to\mathop{\mathit{DJ}}(\mathcal{K})\) can be described as a wedge of iterated (higher) Whitehead products of the canonical elements \(t_1,\dots,t_m\in\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\). This holds, for example, for the following classes of simplicial complexes:

1. \(\mathcal{K}\) is a flag complex with chordal \(1\)-skeleton [6];

2. \(\mathcal{K}\) is totally homology fillable [7];

3. \(\mathcal{K}\) is a substitution complex corresponding to an iterated higher Whitehead product [8], [9].

If \(\mathcal{Z}_\mathcal{K}\simeq\bigvee_{x\in X}S^{n_x}\) is a wedge of spheres, the Hilton–Milnor theorem [10] provides an additive description of the homotopy groups of \(\mathcal{Z}_\mathcal{K}\). In this case, the Whitehead product of any two elements of \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) can be calculated once we know the elements \([t_i,x]\in\pi_{n_x+1}(\mathcal{Z}_\mathcal{K})\), as described in §2.12.

We also note that in some cases \(\Omega\mathcal{Z}_\mathcal{K}\) is homotopy equivalent to a finite-type product of loops on spheres (by [11]–[14], see [15]). In particular, \(\pi_*(\mathcal{Z}_\mathcal{K})\) can be described in terms of the homotopy groups of spheres for any flag complex \(\mathcal{K}\) by [15]. After localisation away from a finite number of primes, this holds for any complex \(\mathcal{K}\), see [16].

The case of flag simplicial complexes is of particular importance (\(\mathcal{K}\) is flag if any set of pairwise connected vertices spans a simplex). This is precisely the case when no higher Whitehead products appear in \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\). In the flag case, \(\mathcal{Z}_\mathcal{K}\) is homotopy equivalent to a wedge of spheres precisely when the 1-skeleton of \(\mathcal{K}\) is a chordal graph [6]. In this case, we have \[\mathcal{Z}_\mathcal{K}\simeq\bigvee_{x\in GPTW}S^{|x|}\] and the inclusion \(\mathcal{Z}_\mathcal{K}\to\mathop{\mathit{DJ}}(\mathcal{K})\) is identified with the wedge sum of maps \[\{S^{|x|}\to\mathop{\mathit{DJ}}(\mathcal{K}),\, x\in GPTW\}\] representing iterated Whitehead products of \(t_1,\dots,t_m\) indexed by the Grbić–Panov–Theriault–Wu elements, which are certain iterated commutators without repeating indices (see §2.3 for details).

In the chordal flag case, by the Hilton–Milnor Theorem applied to the wedge decomposition of \(\mathcal{Z}_\mathcal{K}\) and 2 , \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) is a direct sum of homotopy groups of spheres, namely, \[\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\cong \mathbb{Z}^m\oplus\pi_*\Bigl(\bigvee_{x\in GPTW}S^{|x|}\Bigr) \cong\mathbb{Z}^m\oplus\bigoplus_{b\in\mathop{\mathit{BC}}(GPTW)}\pi_*(S^{|b|}).\] Here \(\mathop{\mathit{BC}}(GPTW)\) are the basic commutators forming the standard basis of the free Lie algebra on the set \(GPTW\). In more detail, \[\label{eq:homotopy32groups32of32dj32for32wedges} \pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\cong\mathbb{Z}\langle t_1,\dots,t_m\rangle \oplus\bigoplus_{b\in\mathop{\mathit{BC}}(GPTW)} w_b\circ \pi_*(S^{|b|})\tag{3}\] where, if \(b\in\mathop{\mathit{BC}}(GPTW)\), then \(w_b\in \pi_{|b|}(\mathop{\mathit{DJ}}(\mathcal{K}))\) is the corresponding iterated Whitehead product of the composite maps \(S^{|x|}\hookrightarrow\mathcal{Z}_\mathcal{K}\to\mathop{\mathit{DJ}}(\mathcal{K})\), \(x\in GPTW\).

In particular, for flag complexes \(\mathcal{K}\) with chordal \(1\)-skeleton the GPTW elements in \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) do not satisfy any algebraic relations, except for the universal relations that hold in the homotopy groups of any topological space. For general \(\mathcal{K}\), the GPTW elements are subject to Lie relations corresponding to chordless cycles, i. e., to elements of the first homology groups \(H_1(\mathcal{K}^f_J;\mathbf{k})\) of the flagifications of full subcomplexes \(\mathcal{K}_J\). These relations are described in Section 4.

To approach Problems 1 and 2 for general \(\mathcal{K}\), we develop the commutator calculus of iterated Whitehead products. In Section 3 we express any iterated Whitehead product \(x\) of the canonical elements \(t_1,\ldots,t_m\in\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\) through GPTW elements. If no \(t_i\) appears in \(x\) twice (there are no repeating indices), then \(x\) can be written as a Lie polynomial in GPTW elements according to  [6]. When there are repeating indices, one needs to compose GPTW elements with the Hopf elements \(\eta^k\in\pi_{n+k}(S^n)\). In the most basic case, this is expressed by the key identity \[[t_i,[t_i,t_j]]=[t_i,t_j]\circ\eta\in\pi_4(\mathop{\mathit{DJ}}(\mathcal{K}))\] (Proposition 27). The general case is summarised in the following result.

This is proved in Section 3 through a sequence of lemmas, allowing for an algorithmic expression of \(x\) in terms of GPTW and Hopf elements.

To address Problem 2, in Section 5 we identify \(S(\mathcal{K})\) with the image of the homomorphism induced by the inclusion of the canonical elements and the map \(g_\mathcal{K}\colon\bigvee_{x\in GPTW}S^{|x|}\to \mathop{\mathit{DJ}}(\mathcal{K})\) given by the iterated Whitehead products corresponding to GPTW elements. The part of \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) generated by iterated Whitehead products is identified in the following result.

Restating this in terms of \(\mathcal{Z}_\mathcal{K}\) through decomposition 2 , we obtain that \(\pi_*(\mathcal{Z}_\mathcal{K})\) is generated as a \(\Pi\)-algebra by GPTW elements if and only if \(\mathcal{K}\) is flag and there are no relations between the GPTW elements if and only if \(\mathcal{K}^1\) is chordal. Elaborating on Theorem 44, we show that \(\mathcal{K}\) is flag if and only if \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) is generated by \(t_1,\dots,t_m\) as a \(\Pi\)-algebra.

For \(\mathcal{K}=\mathcal{L}\), Problem 2 can be thought of as determining the image of the standard inclusion \((S^2)^{\vee m}\to (\mathbb{C}P^\infty)^{\vee m}\) on the level of homotopy groups. More generally, we consider the map of polyhedral products \(t_\mathcal{K}\colon(S^2,\ast)^\mathcal{K}\to(\mathbb{C}P^\infty,\ast)^\mathcal{K}=\mathop{\mathit{DJ}}(\mathcal{K})\) induced by the inclusions \(S^2\to\mathbb{C}P^\infty\). Then the homotopy fibration 1 can be included in a diagram of homotopy fibrations: \[\label{eq:main32diagram32for32dj} \xymatrix{ (C\Omega S^2,\Omega S^2)^\mathcal{K} \ar[d]^-{r_{\mathcal{K}}} \ar[r]^-{} & (S^2,\ast)^\mathcal{K} \ar[r] \ar[d]^-{t_\mathcal{K}} & (S^2)^{\times m} \ar[d]\\ (CS^1,S^1)^\mathcal{K} \ar@<1ex>@/^/[u]^-{q_{\mathcal{K}}} \ar[r]^-{} & (\mathbb{C}P^\infty,\ast)^\mathcal{K} \ar[r] & (\mathbb{C}P^\infty)^{\times m}, }\tag{4}\] where the map \(r_{\mathcal{K}}\) has a right homotopy inverse \(q_{\mathcal{K}}\) with both induced by the natural retraction \(S^1\to\Omega S^2\to S^1\).

When \(\mathcal{K}=\mathcal{L}\), both \(\mathcal{Z}_\mathcal{K}\) and \((C\Omega S^2,\Omega S^2)^\mathcal{K}\) are finite type wedges of spheres, and 4 takes the following form: \[\xymatrix{ \bigvee\limits_{\alpha\in\mathbb{Z}_{\geq 0}^m}(S^{|\alpha|+1})^{\vee (|\mathop{\mathrm{supp}}\alpha|-1)} \ar[d]^-{r_{\mathcal{L}}} \ar[rr]^-{f_\mathcal{L}} && (S^2,\ast)^{\vee m} \ar[r] \ar[d] & (S^2)^{\times m} \ar[d]\\ \bigvee\limits_{J\subset[m]}(S^{|J|+1})^{\vee (|J|-1)} \ar@<1ex>@/^/[u]^-{q_{\mathcal{L}}} \ar[rr]^-{g_\mathcal{L}} && (\mathbb{C}P^\infty,\ast)^{\vee m} \ar[r] & (\mathbb{C}P^\infty)^{\times m}. }\] Here, \(g_\mathcal{L}\) is a wedge of iterated Whitehead products corresponding to GPTW elements for \(\mathcal{Z}_\mathcal{L}=(CS^{1},S^{1})^{\mathcal{L}}\) (Proposition 42). The other maps in this diagram are described in Section 5.

Problem 1 is further addressed in Section 6. For \(n\geq 2\), let \(A_n\subset\pi_*(S^n)\) be the quasi-Lie subalgebra generated by the identity map \(\iota\colon S^n\to S^n\), compositions with suspended Hopf elements, and Whitehead products. The additive structure of \(A_n\) is known, see Appendix 8. In particular, \(A_n\) is a finite direct sum of copies of \(\mathbb{Z}\), \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\). Let \(Q(\mathcal{K})\) be the subgroup \[Q(\mathcal{K})=\mathbb{Z}\langle t_1,\ldots,t_m\rangle \oplus\bigoplus_{b\in\mathop{\mathit{BC}}(GPTW)}A_{|b|}\] of the group \[\mathbb{Z}\langle t_1,\ldots,t_m\rangle \oplus\bigoplus_{b\in\mathop{\mathit{BC}}(GPTW)}\pi_*(S^{|b|})\cong \mathbb{Z}^m\oplus\pi_*\Big(\bigvee_{x\in GPTW}S^{|x|}\Big),\] and let \[\phi\colon Q(\mathcal{K})\to\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\] be the restriction of the map \(\varPhi\colon \mathbb{Z}^m\oplus\pi_*(\bigvee_{x\in GPTW} S^{|x|})\to\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\).

The quasi-Lie algebra \(QL(\mathcal{K})\) is described as follows.

We also describe an algorithm that allows us to express any element of \(QL(\mathcal{K})\) as an element in \(\mathop{\mathrm{Im}}\phi\) and to compute the bracket in \(QL(\mathcal{K})\) in these terms.

Problems 1 and 2 can be generalised. The homotopy fibration 1 is a special case of the homotopy fibration of polyhedral products \[(C\Omega{\underline{X}},\Omega{\underline{X}})^\mathcal{K}\to({\underline{X}},\ast)^\mathcal{K}\to\prod X_i.\] The canonical generators \(t_{i}\) generalise to the composites \[t_{{\underline{X}},i}\colon \Sigma\Omega X_i\xrightarrow{\mathop{\mathrm{ev}}\nolimits_{X_i}} X_i\hookrightarrow ({\underline{X}},\ast)^\mathcal{K}\] where \(\mathop{\mathrm{ev}}\nolimits_X\colon \Sigma\Omega X\to X\) is the standard evaluation map and the right map is induced by including the vertex \(i\) into \(\mathcal{K}\). In Section 7 we study the iterated generalised Whitehead products of the maps \(t_{{\underline{X}},i}\).

Let \(E_Y\colon Y\to\Omega\Sigma Y\) be the suspension map, defined as the adjoint of the identity map on \(\Sigma Y\). Diagram 4 generalises to the commutative diagram of fibrations \[\xymatrix{ (C\Omega\Sigma\Omega{\underline{X}},\Omega\Sigma\Omega{\underline{X}})^\mathcal{K} \ar[d]^-{r_{\mathcal{K}}=(C\Omega\mathop{\mathrm{ev}}\nolimits_{\underline{X}},\Omega\mathop{\mathrm{ev}}\nolimits_{\underline{X}})^\mathcal{K}} \ar[rr] && (\Sigma\Omega{\underline{X}},\ast)^\mathcal{K} \ar[d]^-{(\mathop{\mathrm{ev}}\nolimits_{\underline{X}},\ast)^\mathcal{K}} \ar[r] & \prod_{i=1}^m\Sigma\Omega X_i \ar[d]^-{\prod \mathop{\mathrm{ev}}\nolimits_{\underline{X}}} \\ (C\Omega{\underline{X}},\Omega{\underline{X}})^\mathcal{K} \ar@<1ex>@/^/[u]^-{q_{\mathcal{K}}=(CE,E)^\mathcal{K}} \ar[rr] && ({\underline{X}},\ast)^\mathcal{K} \ar[r] & \prod_{i=1}^m X_i. }\]

When \(\mathcal{K}=\mathcal{L}\), the fibre inclusions in the diagram above are identified with wedges of generalised Whitehead products: \[\xymatrix { \bigvee_{\alpha\in\mathbb{Z}_{\geq 0}^m} \bigl(\Sigma(\Omega{\underline{X}})^{\wedge \alpha}\bigr)^{\vee (|\mathop{\mathrm{supp}}\alpha|-1)} \ar[r] \ar[d]^{r'_\mathcal{L}} & \bigvee_{i=1}^m \Sigma\Omega X_i \ar[r] \ar[d] & \prod_{i=1}^m\Sigma\Omega X_i\ar[d]\\ \bigvee_{J\subset[m]} \bigl(\Sigma(\Omega{\underline{X}})^{\wedge J}\bigr)^{\vee (|J|-1)} \ar[r] \ar@<1ex>@/^/[u]^-{q'_{\mathcal{L}}} & \bigvee_{i=1}^m X_i\ar[r] & \prod_{i=1}^m X_i, }\] as described in Theorem 62 and Proposition 63.

Generalising the identity \([[t_i,t_j],t_j]=[t_i,t_j]\circ\eta\), we obtain \[[[[t_{{\underline{X}},i},t_{{\underline{X}},j}],t_{{\underline{X}},j}]\simeq [t_{{\underline{X}},i},t_{{\underline{X}},j}]\circ({\mathrm{id}}_{ \Omega X_i}\wedge H\mu_{X_j}) \colon \Sigma(\Omega X_i\wedge\Omega X_j\wedge\Omega X_j)\to ({\underline{X}},\ast)^\mathcal{K},\] i.e. the role of the Hopf element \(\eta\in\pi_3(S^2)\) is played by the Hopf construction \[H\mu_X\colon \Sigma(\Omega X\wedge\Omega X)\to\Sigma\Omega X\] of the multiplication \(\mu_X\colon \Omega X\times\Omega X\to\Omega X\). This formula was mentioned by Baues [17]; we give a complete proof in Appendix 9. However, the general case is much less approachable: Davis–Januszkiewicz spaces are simpler because \(\Omega X\) is a co-H-space when \(X=\mathbb{C}P^\infty\). On the other hand, our results easily generalise from the \(\mathbb{C}P^\infty\) case to the \(\mathbb{H}P^\infty\) case.

2 Preliminaries↩︎

This section collects together many of the tools that will be used subsequently and describes their relevant properties.

2.1 Polyhedral products↩︎

Let \(\mathcal{K}\) be a simplicial complex on the set \([m]=\{1,2,\ldots,m\}\). Let \((\underline X,\underline A)=\{(X_{i},A_{i})\}_{i=1}^{m}\) be the sequence of pairs of pointed \(CW\)-complexes, \(A_{i}\subset X_{i}\). For each simplex \(I=(i_1,\ldots,i_k)\in\mathcal{K}\), let \((\underline X,\underline A)^I\) be the subspace of \(\prod_{i=1}^{m} X_{i}\) defined by \[(\underline X,\underline A)^I=\prod_{i=1}^{m} Y_{i}\qquad where\qquad Y_{i}=\left\{\begin{array}{ll} X_{i} & if i\in I \\ A_{i} & if i\notin I. \end{array}\right.\] The polyhedral product determined by \((\underline X,\underline A)\) and \(\mathcal{K}\) is \[(\underline X,\underline A)^\mathcal{K}=\bigcup_{I\in\mathcal{K}} (\underline X,\underline A)^I \subset\prod_{i=1}^{m} X_{i}.\] For example, suppose each \(A_{i}\) is a point. If \(\mathcal{K}\) is a disjoint union of \(m\) points then \((\underline{X},\ast)^{\mathcal{K}}\) is the wedge \(X_{1}\vee\cdots\vee X_{m}\), and if \(\mathcal{K}\) is the standard \((m-1)\)-simplex then \((\underline{X},\ast)^{\mathcal{K}}\) is the product \(X_{1}\times\cdots\times X_{m}\).

The moment-angle complex is the polyhedral product \(\mathcal{Z}_\mathcal{K}=(D^2,S^1)^\mathcal{K}\) and the Davis–Januszkiewicz space is \(\mathop{\mathit{DJ}}(\mathcal{K})=(\mathbb{C}P^\infty,\ast)^K\).

As an example, taking each \(X_{i}=\mathbb{C}P^{\infty}\) and noting that there are homotopy equivalences of pairs \((C\Omega\mathbb{C}P^{\infty},\mathbb{C}P^{\infty})\simeq (CS^{1},S^{1})\simeq (D^{2},S^{1})\), we obtain the homotopy fibration 1 .

2.2 Whitehead products↩︎

Given two maps \(f\colon S^{k+1}\to Z\) and \(g\colon S^{l+1}\to Z\), their Whitehead product is the composite \[[f,g]\colon S^{k+l+1}\longrightarrow S^{k+1}\vee S^{l+1}\xrightarrow{f\vee g} Z,\] where the first map is the attaching map of the top cell in the product \(S^{k+1}\times S^{l+1}\).

The adjoint \([f,g]'\colon S^{k+l}\to\Omega Z\) is homotopic up to sign to the composite \[S^{k+l}\cong S^k\wedge S^l\xrightarrow{f'\wedge g'} \Omega Z\wedge\Omega Z\xrightarrow{c}\Omega Z,\] where \(f'\colon S^k\to\Omega Z\) and \(g'\colon S^l\to\Omega Z\) are the adjoints of \(f\) and \(g\) respectively, and \(c\colon \Omega Z\wedge\Omega Z\to\Omega Z\) is the commutator, \((x,y)\mapsto x^{-1}y^{-1}xy\). The sign depends on the identification of \(S^{k+l}\) with \(S^k\wedge S^l\); we use the sign convention from [20].

The Whitehead product of maps defines an operation on homotopy groups, \[\alpha\in\pi_{k+1}(X),~\beta\in\pi_{\ell+1}(X)~\leadsto~[\alpha,\beta]\in\pi_{k+\ell+1}(X),\] which is also referred to as the Whitehead product. We denote by \(|\alpha|\) the topological degree: \(|\alpha|=i\) if \(\alpha\in\pi_i(X)\).

The (generalised) Whitehead product of based maps \(f\colon\Sigma X\to Z\) and \(g\colon\Sigma Y\to Z\) is the based map \([f,g]\colon \Sigma(X\wedge Y)\to Z\) defined as the adjoint of the composite \[X\wedge Y\xrightarrow{f'\wedge g'} \Omega Z\wedge\Omega Z\xrightarrow{c}\Omega Z.\] It coincides with the classical Whitehead product when \(X=S^k\) and \(Y=S^l\), with the appropriate sign convention.

2.3 GPTW elements↩︎

Consider the canonical elements \(t_i\colon S^2\to\mathop{\mathit{DJ}}(\mathcal{K})=(\mathbb{C}P^\infty,\ast)^\mathcal{K}\) for \(i=1,\ldots,m\).

A GPTW element is an iterated Whitehead product of the form \[[t_{i_1},[t_{i_2},\dots[t_{i_k},t_j]\dots]]\colon S^{k+2}\to\mathop{\mathit{DJ}}(\mathcal{K}),\] where \(k>0\), \(i_1<\cdots<i_k>j\), \(j\notin \{i_1,\ldots,i_k\}\) and \(j\) is the smallest vertex in a connected component not containing \(i_k\) of the subcomplex \(\mathcal{K}_{\{i_1,\ldots,i_k,j\}}\). For a given \(\mathcal{K}\), denote the set of all GPTW elements by \(GPTW\). Then \[|GPTW|=\sum_{J\subset[m]}\widetilde{b}_0(\mathcal{K}_J),\] where \(\widetilde{b}_0(\mathcal{K}_J)\) is one less the number of connected components of \(\mathcal{K}_J\) and \(\widetilde{b}_0(\varnothing)=0\).

A GPTW element lifts uniquely to a map \(S^{k+2}\to\mathcal{Z}_\mathcal{K}\) through the homotopy fibration 1 . The GPTW elements are viewed as elements in both \(\pi_{k+2}(\mathop{\mathit{DJ}}(\mathcal{K}))\) and \(\pi_{k+2}(\mathcal{Z}_\mathcal{K})\) with \(k>0\).

A proof of Theorem 10 (3) is included as a consequence of the stronger Proposition 42, which also describes the map \(\mathcal{Z}_\mathcal{K}\longrightarrow\mathop{\mathit{DJ}}(\mathcal{K})\).

2.4 Basic commutators↩︎

Let \(X=\{x_1>\dots>x_m\}\) be an ordered set. Choose any linear ordering on the set of all iterated commutators on \(X\) so that the following condition holds:

• If \(c_1\) is longer than \(c_2,\) then \(c_1>c_2.\)

Now define the set \(\mathop{\mathit{BC}}(X)\) of basic commutators [2], [10] on \(X\) as follows:

• \(x_1,\dots,x_m\in\mathop{\mathit{BC}}(X);\)

• \(c=[c',c'']\in\mathop{\mathit{BC}}(X)\) if and only if the following conditions hold:

  1. \(c',c''\in\mathop{\mathit{BC}}(X);\)

  2. \(c'<c'';\)

  3. if \(c''=[c_1,c_2],\) then \(c'\geq c_1.\)

For example, \([[x_i,x_j],x_k]\) is never a basic commutator, since \([x_i,x_j]>x_k\). On the other hand, for nested commutators we have \[[x_{i_1},[\dots[x_{i_{k-2}},[x_{i_{k-1}},x_{i_k}]]\dots]]\in\mathop{\mathit{BC}}(X)\] if and only if \(i_1\leq\dots\leq i_{k-1}>i_k\). It follows that \(GPTW\subset\mathop{\mathit{BC}}(t_1,\dots,t_m)\).

It is well known (see [10], for example) that basic commutators form a basis of the free Lie ring \(\mathop{\mathit{FL}}\nolimits(x_1,\dots,x_m)\) generated by \(X=\{x_1,\dots,x_m\}\), so that \(\mathop{\mathit{FL}}\nolimits(X)\cong\mathbb{Z}\langle\mathop{\mathit{BC}}(X)\rangle\) as free abelian groups.

2.5 Notation for iterated Whitehead products↩︎

Let \(\{e_i\}_{i=1}^{m}\) be the standard basis of \(\mathbb{Z}^m\). We consider multi-indices \(\alpha=\sum_{i=1}^m\alpha_ie_i\in\mathbb{Z}_{\geq 0}^m\) and denote \[|\alpha|:=\sum\alpha_i,\quad \mathop{\mathrm{supp}}\alpha:=\{i\in [m]:\alpha_i\neq 0\}.\] A subset \(I\subset[m]\) is identified with the multi-index \(\sum_{i\in I}e_i\), so \(I=\mathop{\mathrm{supp}}I\). We also write \(\alpha-i:=\alpha-e_i\) for \(i\in\mathop{\mathrm{supp}}\alpha\).

For \({\underline{X}}=(X_1,\dots,X_m)\), we denote \({\underline{X}}^{\wedge\alpha}:=\bigwedge_{i=1}^m X_i^{\wedge\alpha_i}\), so \({\underline{X}}^{\wedge I}=\bigwedge_{i\in I}X_i\) and \({\underline{X}}^{\wedge\alpha}\wedge{\underline{X}}^{\wedge\beta}={\underline{X}}^{\wedge(\alpha+\beta)}\) for any \(\alpha,\beta\in\mathbb{Z}_{\geq 0}^m\), where we formally set \({\underline{X}}^{\wedge 0}:=S^0\).

Let \(f_i\colon\Sigma X_i\to Z\), \(i=1,\dots,m\), be based maps. For a multi-index \(\alpha\in\mathbb{Z}_{\geq 0}^m\) and an index \(j\in[m]\), we consider the iterated generalised Whitehead product \[c(\alpha,j;\underline{f}):=[\underbrace{f_1,[\dots,[f_1}_{\alpha_1\text{ times}},[f_2,\dots,[\underbrace{f_m,[\dots[f_m}_{\alpha_m\text{ times}},f_j]\dots]]\dots]]\dots]] \colon \Sigma{\underline{X}}^{\wedge\alpha}\wedge X_j\to Z.\] For example, \(c(3e_2+e_5,4;\underline{f})=[f_2,[f_2,[f_2,[f_5,f_4]]]].\)

For \(i\in[m]\) and \({\underline{X}}=(X_1,\dots,X_m)\), we consider the maps \[\mathop{\mathrm{incl}}\nolimits_{{\underline{X}},i}\colon X_i\hookrightarrow ({\underline{X}},\ast)^\mathcal{K},\quad t_{{\underline{X}},i}:=\mathop{\mathrm{incl}}\nolimits_{{\underline{X}},i}\circ \mathop{\mathrm{ev}}\nolimits_{X_i}\colon \Sigma\Omega X_i\to X_i\to ({\underline{X}},\ast)^\mathcal{K}\] and their iterated Whitehead products \[\label{cprod} \begin{gather} c(\alpha,j;t_{{\underline{X}}})\colon \Sigma(\Omega{\underline{X}})^{\wedge\alpha}\wedge\Omega X_j\to ({\underline{X}},\ast)^\mathcal{K},\\ c(\alpha,j;\mathop{\mathrm{incl}}\nolimits_{\Sigma{\underline{Y}}})\colon \Sigma{\underline{Y}}^{\wedge\alpha}\wedge Y_j\to (\Sigma{\underline{Y}},\ast)^\mathcal{K}. \end{gather}\tag{5}\] In the case \(X_i=\mathbb{C}P^\infty\) we omit the subscript \({\underline{X}}\) and obtain the maps \[t_i\colon S^2\cong\Sigma\Omega\mathbb{C}P^\infty\to\mathop{\mathit{DJ}}(\mathcal{K}),\quad c(\alpha,j;\underline t)\colon S^{|\alpha|+2}\cong\Sigma S^{|\alpha|}\wedge S^1\to\mathop{\mathit{DJ}}(\mathcal{K}).\] Similarly, for \(Y_i=S^1\) we denote \(s_i=\mathop{\mathrm{incl}}\nolimits_{\Sigma S^1,i}\) and obtain the maps \[\label{cscom} s_i\colon S^2\to (S^2,\ast)^\mathcal{K},\quad c(\alpha,j;\underline{s})\colon S^{|\alpha|+2}\cong\Sigma S^{|\alpha|}\wedge S^1\to (S^2,\ast)^\mathcal{K}.\tag{6}\]

For a simplicial complex \(\mathcal{K}\) on \([m]\) and a subset \(J\subset[m]\), denote by \(\Theta_\mathcal{K}(J)\) the \(\widetilde{b}_0(\mathcal{K}_J)\)-element subset of \(J\) which is used in the definition of GPTW elements: \(j\in \Theta_\mathcal{K}(J)\) if and only if \(j\) and \(\max(J)\) are in different connected components of \(\mathcal{K}_J\) and \(j\) is the smallest element in its connected component of \(\mathcal{K}_J\). A GPTW element in \(\mathop{\mathit{DJ}}(\mathcal{K})\) is therefore \(c(J\setminus j,j;\underline t)\colon S^{|J|+1}\to\mathop{\mathit{DJ}}(\mathcal{K})\) with \(J\subset[m]\) and \(j\in\Theta_\mathcal{K}(J)\). We define the set of GPTW elements in the polyhedral product \(({\underline{X}},\ast)^\mathcal{K}\) as \[\label{GPTWTheta} GPTW=\{c(J\setminus j,j;t_{\underline{X}})\colon J\subset[m], j\in\Theta_\mathcal{K}(J)\}.\tag{7}\] Clearly, \(\Theta_\mathcal{K}(\varnothing)=\varnothing\) and \(|\Theta_\mathcal{K}(J)|=\widetilde{b}_0(\mathcal{K}_J)\) for \(J\neq\varnothing\). For the discrete \(m\) point complex \(\mathcal{L}\) we have \(\Theta_\mathcal{L}(J)=J\setminus\{\max(J)\}\).

We also denote \(\Theta_\mathcal{K}(\alpha):=\Theta_\mathcal{K}(\mathop{\mathrm{supp}}\alpha)\) and \(\mathcal{K}_\alpha:=\mathcal{K}_{\mathop{\mathrm{supp}}\alpha}\) for \(\alpha\in\mathbb{Z}_{\geq 0}^m\).

2.6 Quasi-Lie rings↩︎

For a graded module \(A\) and an element \(a\in A\), write \(\deg(a)\) for the degree of \(a\) and \(A_{even}, A_{odd}\) for the submodules of \(A\) consisting of elements of even or odd degree respectively.

The next lemma follows from the identities in Definition 12.

The following result of Hilton describes the additive structure of free quasi-Lie rings.

2.7 The Hilton–Milnor Theorem↩︎

For based maps \(f\colon\Sigma X\to Z\), \(g\colon\Sigma Y\to Z\), let \([f,g]\colon\Sigma(X\wedge Y)\to Z\) be their generalised Whitehead product.

Given maps \(f_i\colon\Sigma X_i\to Y\), \(i=1,\dots,m\), we can associate a map with each iterated commutator of elements \(X=\{x_1,\dots,x_m\}\). For example, the commutator \(b=[x_2,[x_5,x_1]]\) corresponds to the map \[b(\underline{f}):=[f_2,[f_5,f_1]]\colon\Sigma X_2\wedge X_5\wedge X_1\to Y.\] In particular, for any \(b\in\mathop{\mathit{BC}}(X)\) we obtain a map \(b(\underline{f})\colon\Sigma\underline{X}^{\wedge \mathrm{mdeg}(b)}\to Y,\) where \(\underline{X}^{\wedge\alpha}:=\bigwedge_{i=1}^mX_i^{\wedge \alpha_i}\) for \(\alpha\in\mathbb{Z}_{\geq 0}^m\), and \(\mathrm{mdeg}(b)\in\mathbb{Z}_{\geq 0}^m\) counts the number of occurrences of \(x_i\) in \(b\).

Less precisely, the factors correspond to a basis in a free Lie algebra with generators corresponding to the wedge summands.

In the case \(X_i=S^{n_i}\), we obtain a homotopy equivalence \[\Omega(S^{n_1+1}\vee\dots\vee S^{n_m+1})\simeq\prod_{b\in\mathop{\mathit{BC}}(X)}\Omega S^{1+\sum_i \mathop{\mathrm{mdeg}}(b)_i\cdot n_i},\] implying there is an isomorphism of homotopy groups \[\label{piwesp} \pi_*(S^{n_1+1}\vee\dots\vee S^{n_m+1})\cong\bigoplus_{b\in\mathop{\mathit{BC}}(X)}\pi_*\bigl(S^{1+\sum_i \mathop{\mathrm{mdeg}}(b)_i\cdot n_i}\bigr)\tag{8}\] where the right side is generated by elements of the form \(w_b\circ\alpha\), where \(w_b\colon S^{|w_b|}\to S^{n_1+1}\vee\dots\vee S^{n_m+1}\) is the Whitehead product corresponding to a basic commutator \(b\in\mathop{\mathit{BC}}(X)\) and \(\alpha\in\pi_*(S^{|w_b|})\).

2.8 Composition products in homotopy groups↩︎

We always assume that the following formulas only involve \(\pi_n\) for \(n\geq 2\). Our main references are the books by G. W. Whitehead [20] and Baues [21].

We consider three homotopy operations:

1. the suspension homomorphism \[E^k:\pi_*(X)\to\pi_{*+k}(\Sigma^k X),~k\geq 0;\]

2. the Whitehead product \[\alpha\in\pi_{k+1}(X),~\beta\in\pi_{\ell+1}(X)~\leadsto~[\alpha,\beta]\in\pi_{k+\ell+1}(X);\]

3. the composition product \[\alpha\in\pi_k(X),~\beta\in\pi_\ell(S^k) ~\leadsto~\alpha\circ\beta\in\pi_\ell(X).\]

Comparing the relations in Proposition 9 and Definition 12, we obtain

We use the same Greek letter to denote iterated suspensions of the same element in the homotopy groups of spheres; the subscript indicates the codomain. For example, \(\eta_2\in\pi_3(S^2)\) is the Hopf element; we have \(\pi_3(S^2)=\mathbb{Z}\langle\eta_2\rangle\) and \(\pi_{k+1}(S^k)=\mathbb{Z}_2\langle\eta_k\rangle\) for \(k\geq 3\), where \(\eta_k:=E^{k-2}\eta_2.\) We omit the subscript when the codomain is clear from the context. The identity map \({\mathrm{id}}_{S^n}\colon S^n\to S^n\) corresponds to the element \(\iota_n\in\pi_n(S^n)\).

For \(\alpha\in\pi_{k+i}(S^k)\) we write \(\alpha^n:=\underbrace{\alpha\circ\dots\circ\alpha}_{n\text{ times}}\in\pi_{k+in}(S^k).\) For example, \(\eta_2^3=\eta_2\circ\eta_3\circ\eta_4\in\pi_5(S^2).\)

2.9 James–Hopf invariants↩︎

2.10 Distributivity laws↩︎

The following important result is originally due to Barcus and Barratt [23], who stated it in terms of Hilton–Hopf invariants instead of James–Hopf invariants. We use a restatement from the book of Baues [21]. Since his sign convention may be different from Whitehead [20], we do not specify the signs.

Note that the sum is finite since \(\gamma_n(\zeta)=0\) for \(n> |\zeta|\). A special case of this formula was obtained by Barratt and Hilton as an application of Blakers–Massey relative Whitehead products [24].

2.11 Composition with iterated Hopf maps↩︎

In what follows, we denote by \(\eta\) the suspended Hopf map \(S^{n+1}\to S^n\), \(n\ge 2\), as well as the corresponding element in \(\pi_{n+1}(S^n)\). We also consider the \(k\)-fold iteration \(\eta^k\colon S^{n+k}\to S^n\), \(k\ge0\). The properties of the composition operations \((-)\circ\eta^k\colon\pi_n(X)\to \pi_{n+k}(X)\) are summarised next.

2.12 The case when \(\mathcal{Z}_\mathcal{K}\) is a wedge of spheres↩︎

Let \(\mathcal{K}\) be a simplicial complex on \([m]\) such that \(\mathcal{Z}_\mathcal{K}\simeq\bigvee_{x\in X}S^{n_x}\) is homotopy equivalent to a wedge of simply connected spheres, \(x\in\pi_{n_x}(\mathcal{Z}_\mathcal{K})\). Then there is an additive isomorphism \[\pi_*(\mathcal{Z}_\mathcal{K})\cong\bigoplus_{b\in\mathop{\mathit{BC}}(X)}w_b\circ\pi_*(S^{|b|})\] by the Hilton–Milnor theorem.

The homotopy fibration 1 gives an extension of quasi-Lie algebras \[0\to \pi_*(\mathcal{Z}_\mathcal{K})\to\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\to\mathbb{Z}\langle t_1,\dots,t_m\rangle\to 0\] and an isomorphism of abelian groups \[\label{piDJdec} \pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\cong\mathbb{Z}\langle t_1,\dots,t_m\rangle\oplus\bigoplus_{b\in\mathop{\mathit{BC}}(X)}w_b\circ\pi_*(S^{|b|}).\tag{9}\] The nontriviality of the extension above is expressed in the Whitehead products \([t_i,x]\) for \(x\in X\). (Note that \(n_x\geq 2\), so \([t_i,x]\in\pi_{n_x+1}(\mathop{\mathit{DJ}}(\mathcal{K}))\cong\pi_{n_x+1}(\mathcal{Z}_\mathcal{K})\).) Identifying the elements \([t_i,x]\) is the most important part of the description of the quasi-Lie algebra structure of \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\).

If \(\mathcal{K}\) is a flag complex then \(\mathcal{Z}_\mathcal{K}\) is a wedge of spheres if and only if \(\mathcal{K}^1\) is a chordal graph. In this case, the wedge summands correspond to GPTW elements, i. e., \(X=GPTW\), and the description of the elements \([t_i,x]\), \(x\in GPTW\) is given in Section 3 as part of the proof of Theorem 26.

Once the elements \([t_i,x]\in\pi_{n_x+1}(\mathcal{Z}_\mathcal{K})\) are known, there is the following algorithmic description of the Whitehead product of any two elements in \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\). There are Whitehead products of five types in 9 :

• \([t_i,t_j]\), \(i,j\in [m]\);

• \([w_b,w_{b'}]\), \(b,b'\in\mathop{\mathit{BC}}(X)\);

• \([w_b\circ\alpha,w_{b'}\circ\alpha']\), \(\alpha\in\pi_*(S^{|b|})\), \(\alpha'\in\pi_*(S^{|b'|});\)

• \([t_i,w_b]\), \(i\in[m]\), \(b\in\mathop{\mathit{BC}}(X)\);

• \([t_i,w_b\circ\alpha]\), \(\alpha\in\pi_*(S^{|b|})\).

Type (1): \([t_i,t_j]\) is zero if \(\{i,j\}\in\mathcal{K}\) and a GPTW element if \(\{i>j\}\in\mathcal{K}\).

Types (2) and \((2')\) are computed in the homotopy groups of the wedge of spheres \(\mathcal{Z}_\mathcal{K}\simeq\bigvee_{x\in X}S^{n_x}\), which is a classical computation. More explicitly, by Corollary 16, an element of type (2) is a linear combination of elements of the form \(w_{b''}\), \([w_{b''},w_{b''}]=w_{b''}\circ [{\mathrm{id}},{\mathrm{id}}]\) and \([w_{b''},[w_{b''},w_{b''}]]=w_{b''}\circ[{\mathrm{id}},[{\mathrm{id}},{\mathrm{id}}]]\) for \(b''\in\mathop{\mathit{BC}}(X)\). Type \((2')\) is reduced to type (2) by repeated use of Propositions 23 and 9.

Type (3): induction on \(|b|\). For the base case, \(b=x\in X\), and \([t_i,x]\) is known by assumption. For the inductive step, we have \(b=[b',b'']\), \(b',b''\in\mathop{\mathit{BC}}(X)\). Then \([t_i,w_b]=[t_i,[w_{b'},w_{b''}]]=\pm [[t_i,w_{b'}],w_{b''}]\pm [[t_i,w_{b''}],w_{b'}]\). A decomposition \([t_i,w_{b'}]=\sum_sw_{b_s}\circ \alpha_{s}\) is known by inductive assumption, and then \([w_{b_s}\circ\alpha_{s},w_{b''}]\) is an expression of type \((2')\). The computation of \([[t_i,w_{b''}],w_{b'}]\) is similar.

Type \((3')\): by Proposition 23, \[[t_i,w_b\circ\alpha]=\sum_{k\geq 1}[\dots[[t_i,\underbrace{w_b],w_b]\dots,w_b}_{k\text{ times}}]\circ E\gamma_k(\alpha).\] The summand with \(k=1\) is of type (3), while the other summands are computed by induction on \(k\), where the inductive step is a computation of type \((2')\).

3 Composing iterated Whitehead products with Hopf elements↩︎

In this section we develop the commutator calculus needed to approach Problem 1. Namely, we prove that any bracket formed from \(t_1,\ldots,t_m\in\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\) can be expressed via GPTW and Hopf elements.

The proof will provide an algorithmic expression of commutators. It will depend on a series of lemmas that give explicit expressions for iterated Whitehead products of the canonical elements \(t_1,\ldots,t_m\in\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\) with repeating indices. The most basic of commutator expressions is of special interest.

Let \(\mathcal{K}=\{\varnothing,\{1\},\{2\}\}\) be a two point complex. Then the homotopy fibration 1 becomes \[S^3\to \mathbb{C}P^\infty\vee\mathbb{C}P^\infty\to\mathbb{C}P^\infty\times\mathbb{C}P^\infty\] and the inclusion \(\mathcal{Z}_\mathcal{K}\to\mathop{\mathit{DJ}}(\mathcal{K})\) identifies with the Whitehead product \([t_1,t_2]\colon S^3\to\mathbb{C}P^\infty\vee\mathbb{C}P^\infty\). We therefore have \(\pi_*(\mathbb{C}P^\infty\vee\mathbb{C}P^\infty)\cong \mathbb{Z}\langle t_1, t_2\rangle \oplus[t_1,t_2]\circ\pi_*(S^3)\).

For the two point complex \(\mathcal{K}\), the only GPTW element is \([t_2,t_1]\), so Proposition 27 together with the algorithm in §2.12 can be used to describe the Whitehead product in \(\mathop{\mathit{DJ}}(\mathcal{K})=\mathbb{C}P^\infty\vee\mathbb{C}P^\infty\) completely.

Now we give a sequence of lemmas leading to the proof of Theorem 26.

Given an ordered subset \(J=\{j_1<\dots<j_k\}\subset[m]\) and an element \(y\in\pi_N(\mathop{\mathit{DJ}}(\mathcal{K}))\), we write \[c(J,y):=[t_{j_1},[t_{j_2},\dots[t_{j_k},y]\dots]]\in\pi_{N+k}(\mathop{\mathit{DJ}}(\mathcal{K})).\] Note that \(c(J,t_j)=c(J,j;\underline t)\) in the notation of §2.5.

For \(A,B\subset[m]\), consider the Koszul sign \[\theta(A,B):=\#\{(i,j)\in A\times B:i>j\}.\]

::: {#lmm:c(A,[x,y]) .lmm} Lemma 31. Given \(P\subset[m]\) and \(x,y\in\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\), we have \[(-1)^{|P|}c(P,[x,y])=\sum_{P=A\sqcup B}(-1)^{\theta(A,B)+|x|\cdot|B|}\bigl[c(A,x),c(B,y)\bigr].\] In particular, \((-1)^{|P|}c(P,[t_i,y])=\sum_{P=A\sqcup B}(-1)^{\theta(A,B)}\bigl[c(A,t_i),c(B,y)\bigr].\) :::

::: {#lmm:[j,c(Q,j)] .lmm} Lemma 32. Let \(j\in [m],\) \(Q\subset[m]\), \(Q\neq\varnothing\). Then \[\bigl[t_j,c(Q,t_j)\bigr]=c(Q,t_j)\circ\eta-\sum_{\begin{smallmatrix} Q=A\sqcup B,\\ A\neq\varnothing,\\ \max(Q)\in B \end{smallmatrix}} (-1)^{\theta(A,B)}\bigl[c(A,t_j),c(B,t_j)\bigr].\] :::

We use the notation \[I_{<i}:=\{k\in I:k<i\},\quad I_{\geq i}=\{k\in I:k\geq i\}.\]

The following expressions for iterated Whitehead products with repeating indices will be used in Section 5.

For a multi-index \(\alpha\in\mathbb{Z}_{\geq 0}^m\) and an index \(j\in[m]\) we use the notation \(c(\alpha,t_j)=c(\alpha,j;\underline t)\colon S^{|\alpha|+2}\to\mathop{\mathit{DJ}}(\mathcal{K})\), see 5 .

4 Relations between iterated Whitehead products↩︎

Iterated Whitehead products and, in particular, GPTW elements in \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) are subject to a set of Lie relations, which are described in this section.

When \(\mathcal{K}\) is a flag complex, the loop homology algebra \(H_*(\Omega\mathcal{Z}_\mathcal{K};\mathbf{k})\) is minimally generated by the adjoints of GPTW elements [6]. A minimal set of relations between the adjoints of GPTW elements in \(H_*(\Omega\mathcal{Z}_\mathcal{K};\mathbf{k})\) was described in [15]. These relations correspond to generators \(\kappa\in H_1(\mathcal{K}_J;\mathbf{k})\) for \(J\subset[m]\) and have the form \(r_\kappa=0\), where \(r_\kappa\) is a quadratic Lie polynomial in iterated Whitehead products \(c(A,t_i)\). Here we prove the stronger result that the relations \(r_\kappa=0\) hold already in \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\).

For ordered subsets \(A=\{a_1<\cdots<a_k\}\) an \(B=\{b_1<\cdots<b_l\}\) of \([m]\) and \(x\in\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\), we recall the notation from the previous section: \[c(A,x)=[t_{a_1},[\dots[t_{a_k},x]\dots]],\qquad \theta(A,B)=\#\{(a,b)\in A\times B\colon a>b\}.\]

We shall need the following version of identities from Lemma 31.

5 A criterion for when \(\pi_{\ast}(\mathop{\mathit{DJ}}(\mathcal{K}))\) is generated by iterated Whitehead products↩︎

We consider the map given by the iterated Whitehead products corresponding to GPTW elements and its lift to \(\mathcal{Z}_\mathcal{K}\): \[g_\mathcal{K}\colon\bigvee_{x\in GPTW}S^{|x|}\to \mathop{\mathit{DJ}}(\mathcal{K}),\qquad \widehat g_\mathcal{K}\colon \bigvee_{x\in GPTW}S^{|x|}\to\mathcal{Z}_\mathcal{K}.\] Define the homomorphism \[\label{varPhi} \varPhi\colon \mathbb{Z}\langle t_1,\ldots,t_m\rangle \oplus \pi_*\Bigl(\bigvee_{x\in GPTW}S^{|x|}\Bigr)\to \pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\tag{10}\] by mapping each \(t_i\) to the corresponding canonical element of \(\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\) and using the map induced by \(g_\mathcal{K}\) on the wedge of spheres.

We start with the following equivalent description of the \(\Pi\)-subalgebra \(S(\mathcal{K})\) of \(\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\) featuring in Problem 2.

A simplicial complex \(K\) is flag if any set of vertices of \(\mathcal{K}\) which are pairwise connected by edges spans a simplex. Any flag complex \(\mathcal{K}\) is the clique complex of its 1-skeleton (graph) \(\mathcal{K}^1\). The flagification of \(\mathcal{K}\) is the minimal flag complex \(\mathcal{K}^{f}\) on the same vertex set \([m]\) that contains \(\mathcal{K}\).

A graph is called chordal if each of its cycles with \(\ge 4\) vertices has a chord (an edge joining two vertices that are not adjacent in the cycle). Equivalently, a graph is chordal if it has no induced cycles of length more than three.

Proposition 42 implies that the map \(\varPhi\) in 10 is an isomorphism if \(\mathcal{K}\) is a flag complex with chordal \(1\)-skeleton. The main result of this section is the following.

When \(\mathcal{K}\) is flag, Theorem 44 together with Proposition 41 imply the following.

Theorem 46 solves Problem 2 in the case of flag complexes. We now consider an approach to the case of a general simplicial complex by comparing it to its flagification.

If the answer to the question above is positive, then \(S(\mathcal{K})\to S(\mathcal{K}^{f})\) is an isomorphism, and \(\mathop{\mathrm{Im}}\bigl(\pi_*((S^2)^{\vee m})\to\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\bigr) \cong \pi_*(\mathop{\mathit{DJ}}(\mathcal{K}^{f}))\).

Geometric interpretation↩︎

The Whitehead products generating the subgroup \(S(\mathcal{K})\) in the homotopy groups of \(\mathop{\mathit{DJ}}(\mathcal{K})\) are interpreted geometrically as a map \(g_\mathcal{K}\) from a wedge of spheres corresponding to GPTW elements. Other related maps are shown in the following diagram: \[\label{s2djfib} \xymatrix{ \bigvee\limits_{\alpha\in\mathbb{Z}_{\geq 0}^m}(S^{|\alpha|+1})^{\vee \widetilde{b}_0(\mathcal{K}_\alpha)} \ar[rrd]^-{f_\mathcal{K}} \ar[rd]^-{\widehat f_\mathcal{K}} \ar[d]^-{r'_{\mathcal{K}}}\\ \bigvee\limits_{J\subset[m]}(S^{|J|+1})^{\vee \widetilde{b}_0(\mathcal{K}_J)} \ar[rd]^-{\widehat g_\mathcal{K}} \ar@<1ex>@/^/[u]^-{q'_{\mathcal{K}}} & (C\Omega S^2,\Omega S^2)^\mathcal{K} \ar[d]^-{r_{\mathcal{K}}} \ar[r] & (S^2,\ast)^\mathcal{K} \ar[r] \ar[d]^-{t_\mathcal{K}} & (S^2)^{\times m} \ar[d]\\ & (CS^1,S^1)^\mathcal{K} \ar@<1ex>@/^/[u]^-{q_{\mathcal{K}}} \ar[r]^-{} & (\mathbb{C}P^\infty,\ast)^\mathcal{K} \ar[r] & (\mathbb{C}P^\infty)^{\times m} }\tag{13}\] Here, the horizontal lines are homotopy fibrations of polyhedral products (see Proposition 8). The map \(t_\mathcal{K}\) is induced by the inclusion \(S^2\to\mathbb{C}P^\infty\). The map \(r_{\mathcal{K}}\) has a right homotopy inverse \(q_{\mathcal{K}}\) and both are induced by the natural retraction \(S^1\to\Omega S^2\to S^1\). The map \(\widehat g_\mathcal{K}\) is the lift to \(\mathcal{Z}_\mathcal{K}=(CS^1,S^1)^\mathcal{K}\) of the map \(g_\mathcal{K}\) given by the wedge of GPTW elements. In the notation 7 the latter is \[g_\mathcal{K}=\bigvee_{J\subset[m]} \bigvee_{j\in\Theta_\mathcal{K}(J)}c(J\setminus j,j;\underline t)\colon \bigvee_{J\subset[m]}\bigvee_{j\in\Theta_\mathcal{K}(J)}S^{|J|+1}\to (\mathbb{C}P^\infty,\ast)^\mathcal{K}\] where each GPTW element \(c(J\setminus j,j;\underline t)\) is an iterated Whitehead product of the maps \(t_i\) with no repeating indices. The map \(f_\mathcal{K}\) is defined as a similar wedge of iterated Whitehead products \(c(\alpha-j,j;\underline{s})\colon S^{|\alpha|+1}\to (S^2,\ast)^\mathcal{K}\), this time with repeating indices: \[f_\mathcal{K}=\bigvee_{\alpha\in\mathbb{Z}_{\geq 0}^m} \bigvee_{j\in\Theta_\mathcal{K}(\alpha)}c(\alpha-j,j;\underline{s})\colon \bigvee_{\alpha\in\mathbb{Z}_{\geq 0}^m} \bigvee_{j\in\Theta_\mathcal{K}(\alpha)} S^{|\alpha|+1}\to (S^2,\ast)^\mathcal{K},\] and the map \(\widehat f_\mathcal{K}\) is the lift of \(f_\mathcal{K}\) to \((C\Omega S^2,\Omega S^2)^\mathcal{K}\). The map \(q'_\mathcal{K}\) is the inclusion of wedge summands corresponding to squarefree multi-indices. Finally, \(r'_\mathcal{K}\) is the map expressing each iterated Whitehead product \(c(\alpha-j,j;\underline{s})\) via GPTW elements. Namely, we express the iterated Whitehead product \[\label{cts} c(\alpha-j,j;\underline{t})=t_\mathcal{K}\circ c(\alpha-j,j;\underline{s})\colon S^{|\alpha|+1}\to(\mathbb{C}P^\infty,\ast)^\mathcal{K}\tag{14}\] as a sum of polynomials in GPTW elements \(c(I\setminus i,i;\underline t)\) composed with \(\eta,\eta^2,\eta^3\), using Theorem 26 (an explicit expression can be obtained using Lemma 35 and [15]). This expression for \(c(\alpha-j,j;\underline{t})\) defines a map \(S^{|\alpha|+1}\to \bigvee\limits_{J\subset[m]}(S^{|J|+1})^{\vee \widetilde{b}_0(\mathcal{K}_J)}\), and \(r'_\mathcal{K}\) is the wedge of these maps over \(\alpha\in\mathbb{Z}^m_{\ge0}\) and \(j\in\Theta_\mathcal{K}(\alpha)\). It is clear from this definition that \(r'_\mathcal{K}\circ q'_\mathcal{K}={\mathrm{id}}\), and \(r'_\mathcal{K}\) is nontrivial only on a finite number of wedge summands. Furthermore, 14 together with the functoriality of Whitehead products implies that \(g_\mathcal{K}\circ r'_\mathcal{K}=t_\mathcal{K}\circ f_\mathcal{K}\). The lifts \(\widehat f_\mathcal{K}\) and \(\widehat g_\mathcal{K}\) are unique up to homotopy, which implies that \(\widehat g_\mathcal{K}\circ r'_\mathcal{K}\simeq r_\mathcal{K}\circ \widehat f_\mathcal{K}\). By [28] or Proposition 61 below, \(\widehat{f}_\mathcal{K}\circ q'_\mathcal{K}\simeq q_\mathcal{K}\circ \widehat{g}_\mathcal{K}\). Thus diagram 13 is homotopy commutative.

The map \(\widehat g_\mathcal{K}\) in 13 is surjective in homotopy groups when \(\mathcal{K}\) is flag, and it is a homotopy equivalence when \(\mathcal{K}\) is flag with chordal \(\mathcal{K}^1\) (Proposition 42). Furthermore, for the disjoint \(m\) point complex \(\mathcal{L}\), the map \(\widehat f_\mathcal{L}\) is also a homotopy equivalence by [32]. (Conjecturally, the map \(\widehat f_\mathcal{K}\) is a homotopy equivalence when \(\mathcal{K}\) is flag with chordal \(\mathcal{K}^1\).) Here is the explicit description of the maps in diagram 13 for the case \(\mathcal{K}=\mathcal{L}\) that uses the identity from Lemma 35.

6 Quasi-Lie algebras and Problem 1↩︎

In this section we address Problem 1 by studying the quasi-Lie subring \[QL(\mathcal{K})\subset \pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\] generated by the standard elements \(t_1,\dots,t_m\in\pi_2(\mathop{\mathit{DJ}}(\mathcal{K}))\).

Consider the graded abelian group \[A_n=\mathbb{Z}\bigl\langle\iota,\eta^k,[\iota,\iota]\circ\eta^k,[\iota,[\iota,\iota]]\circ\eta^k,\;0\leq k\leq 3\bigr\rangle\subset\pi_*(S^n)\] described in Appendix 8. Let \[Q(\mathcal{K})=\mathbb{Z}\langle t_1,\ldots,t_m\rangle \oplus\bigoplus_{b\in\mathop{\mathit{BC}}(GPTW)}A_{|b|}\] be the subgroup of the group \[\mathbb{Z}\langle t_1,\ldots,t_m\rangle \oplus\bigoplus_{b\in\mathop{\mathit{BC}}(GPTW)}\pi_*(S^{|b|})\cong \mathbb{Z}^m\oplus\pi_*\Big(\bigvee_{x\in GPTW}S^{|x|}\Big),\] and let \[\phi\colon Q(\mathcal{K})\to\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\] be the restriction of the map \(\varPhi\colon \mathbb{Z}^m\oplus\pi_*(\bigvee_{x\in GPTW} S^{|x|})\to\pi_*(\mathop{\mathit{DJ}}(\mathcal{K}))\).

If \(\mathcal{K}^1\) is a chordal graph, Theorem 52 provides an additive description of \(QL(\mathcal{K})=QL(\mathcal{K}^f)\), and the proof gives an algorithm that computes the bracket of any two elements. However, we do not have an explicit description of the multiplicative structure of \(QL(\mathcal{K})\), e.g., by generators and relations, except for the simplest case considered in the next example.

We now collect some general phenomena seen in this case.

7 Generalisation to polyhedral products↩︎

This section describes how the results on iterated Whitehead products in the homotopy groups of \(\mathop{\mathit{DJ}}(\mathcal{K})=(\mathbb{C} P^\infty,\ast)^\mathcal{K}\) can be generalised to polyhedral products of the form \(({\underline{X}},\ast)^{\mathcal{K}}\), and in a refined way to polyhedral products of the form \((\Sigma{\underline{Y}},\ast)^{\mathcal{K}}\).

We use notation 5 for iterated Whitehead products in a polyhedral product.

We now define the map \[g_\mathcal{K}=\bigvee c(J\setminus j,j;t_{\underline{X}})\colon \bigvee_{J\subset[m]}\bigvee_{j\in\Theta_\mathcal{K}(J)} \Sigma(\Omega{\underline{X}})^{\wedge J} \to ({\underline{X}},\ast)^\mathcal{K}\] given by the wedge of GPTW elements (see §2.5), and its lift \[\widehat g_\mathcal{K}=\bigvee \widehat{c}(J\setminus j,j;t_{\underline{X}})\colon \bigvee_{J\subset[m]}\bigvee_{j\in\Theta_\mathcal{K}(J)} \Sigma(\Omega{\underline{X}})^{\wedge J} \to (C\Omega{\underline{X}},\Omega{\underline{X}})^\mathcal{K}.\] These maps appear in a natural generalisation of diagram 13 . We have the unit and counit maps \(\mathop{\mathrm{ev}}\nolimits_X\colon\Sigma\Omega X\to X\) and \(E_Y\colon Y\to\Omega\Sigma Y\) for the pair of adjoint functors \(\Sigma\) and \(\Omega\). The maps \(r_\mathcal{K}:=(C\Omega \mathop{\mathrm{ev}}\nolimits_{\underline{X}},\Omega\mathop{\mathrm{ev}}\nolimits_{\underline{X}})^\mathcal{K}\) and \(t_\mathcal{K}:=(\mathop{\mathrm{ev}}\nolimits_{{\underline{X}}},\ast)^{\mathcal{K}}\) define a map between the homotopy fibrations in the rows of the following diagram: \[\label{phpfib} \xymatrix@R=0.8pc{ & (C\Omega\Sigma\Omega{\underline{X}},\Omega\Sigma\Omega{\underline{X}})^\mathcal{K} \ar[dd]^-{r_{\mathcal{K}}} \ar[r]^-{\iota} & (\Sigma\Omega{\underline{X}},\ast)^\mathcal{K} \ar[dd]^-{t_\mathcal{K}} \ar[r] & \prod_{i=1}^m\Sigma\Omega X_i \ar[dd]^-{\prod\mathop{\mathrm{ev}}\nolimits_{X_i}} \\ \Sigma(\Omega{\underline{X}})^{\wedge\alpha} \ar@/^0.8pc/[ur]^-{\widehat{c}(\alpha-j,j;\mathop{\mathrm{incl}}\nolimits_{\Sigma\Omega{\underline{X}}}) \;\;\;} \ar@/_0.8pc/[dr]_-{\widehat{c}(\alpha-j,j;t_{\underline{X}})\;}\\ &(C\Omega{\underline{X}},\Omega{\underline{X}})^\mathcal{K} \ar@<1ex>@/^/[uu]^-{q_{\mathcal{K}}} \ar[r]^-{} & ({\underline{X}},\ast)^\mathcal{K} \ar[r] & \prod_{i=1}^m X_i. }\tag{16}\] The map \(r_\mathcal{K}\) has a right homotopy inverse \(q_\mathcal{K}:=(C E_{\Omega{\underline{X}}},E_{\Omega{\underline{X}}})^\mathcal{K}\), since the composite \(\Omega\mathop{\mathrm{ev}}\nolimits_X\circ E_{\Omega X}\colon\Omega X\to\Omega\Sigma\Omega X\to\Omega X\) is the identity map.

The next proposition, generalising [28], shows that the left triangles in 16 are also commutative.

Here is an analogue of Proposition 49 for more general polyhedral products:

The description of the map \(r'_\mathcal{L}\) (or map \(r_\mathcal{L}\) in 16 ) in terms of the wedge summands reduces to the following question.

By Proposition 63, the map in 17 is the inclusion of a wedge summand if \(\alpha=\mathop{\mathrm{supp}}\alpha\), i. e., if the iterated commutator \(c(\alpha-j,j;t_{\underline{X}})\) has no repeating indices. For the simplest iterated commutator with repeating indices, we have the following description that uses a formula of Baues (see Appendix 9).

Proposition 66 is stated for the left nested commutator \([[t_{{\underline{X}},i},t_{{\underline{X}},j}],t_{{\underline{X}},j}]\), whereas the right nested commutators \([t_i,[t_i,t_j]]\) are used in Proposition 27 and throughout the paper. A version of the Baues formula for the right nested commutators would involve a variant of the Hopf construction that is conjugate to the standard one.

8 The groups \(A_n\)↩︎

Here we describe the graded abelian groups \(A_n\subset\pi_*(S^n)\) generated by the identity \(\iota\colon S^n\to S^n\), its compositions with the (suspended) Hopf elements \(\eta\), and Whitehead products. Note that iterated Whitehead products of \(\iota\) with itself of length \(\geq 4\) vanish (see [20]). Together with the identities from Proposition 25, this implies that \(A_n\) is spanned by the finite set of elements \[\left\{\iota,\eta^k,[\iota,\iota]\circ\eta^k,[\iota,[\iota,\iota]]\circ\eta^k,\;0\leq k\leq 3\right\}\subset\pi_*(S^n).\]

::: {#thm:[i,[i,i]]_in_spheres .thm} Theorem 67. Let \(I_n\subset\pi_*(S^n)\) be the quasi-Lie subring generated by \(\iota_n\in\pi_n(S^n).\) Then \[I_n= \begin{cases} \mathbb{Z}\langle\iota_n\rangle&\text{for}\quad n=1,3,7;\\ \mathbb{Z}\langle\iota_n\rangle\oplus\mathbb{Z}_2\langle[\iota_n,\iota_n]\rangle &\text{for}\quad n=2k+1,~n\neq 1,3,7;\\ \mathbb{Z}\langle\iota_n\rangle\oplus\mathbb{Z}\langle[\iota_n,\iota_n]\rangle &\text{for}\quad n=2;\\ \mathbb{Z}\langle\iota_n\rangle\oplus\mathbb{Z}\langle[\iota_n,\iota_n]\rangle \oplus\mathbb{Z}_3\langle[\iota_n,[\iota_n,\iota_n]]\rangle &\text{for}\quad n=2k,~n\neq 2. \end{cases}\] :::

The following theorem follows from results of Hilton–Whitehead, Kristensen–Madsen, Mahowald, Oshima, Thomeier and Toda. We rely on the exposition by Golasinski and Mukai [35].

::: {#thm:[i,eta^k] .thm} Theorem 70. The following hold for any \(k\geq 0\):

1. \([\iota_n,\eta_n]=0\) if and only if \(n\in\{2,6,4k+3\}\);

2. \([\iota_n,\eta_n^2]=0\) if and only if \(n\in\{5,4k+2,4k+3\}\);

3. \([\iota_n,\eta_n^3]=0\) if and only if \(n\in\{4,12,4k+1,4k+2,4k+3\}\). :::

Combining Theorems 67 and 70, we obtain an additive description of the groups \(A_n\).

9 A formula of Baues↩︎

For based CW-complexes \(X\) and \(Y\), the suspended projection \(\Sigma q\colon\Sigma(X\times Y)\to \Sigma X\wedge Y\) has a right homotopy inverse \(\sigma\colon\Sigma X\wedge Y\to\Sigma(X\times Y)\), which comes from the homotopy decomposition \[\Sigma(X\times Y)\simeq \Sigma X\vee \Sigma Y\vee (\Sigma X\wedge Y).\] Equivalently, the identity map \({\mathrm{id}}\) on \(\Sigma(X\times Y)\) decomposes as \[\label{idsusp} {\mathrm{id}}\simeq (\Sigma i_X\circ\Sigma \mathop{\mathrm{pr}}\nolimits_X) +(\Sigma i_Y\circ\Sigma \mathop{\mathrm{pr}}\nolimits_Y)+(\sigma\circ\Sigma q)\tag{18}\] where \(i_X\colon X\to X\times Y\) is the inclusion \(x\mapsto(x,{\mathrm{pt}})\) and similarly for \(i_Y\colon Y\to X\times Y\).

The Whitehead product of based maps \(f\colon\Sigma X\to Z\) and \(g\colon\Sigma Y\to Z\) is the adjoint \([f,g]\colon \Sigma X\wedge Y\to Z\) of the composite \(X\wedge Y\xrightarrow{f'\wedge g'} \Omega Z\wedge\Omega Z\xrightarrow{c}\Omega Z\). Since \(\Sigma q\colon\Sigma(X\times Y)\to \Sigma X\wedge Y\) has a right homotopy inverse, the Whitehead product \([\alpha,\beta]\in[\Sigma (X\wedge Y),Z]\) of homotopy classes \(\alpha\in[\Sigma X,Z]\) and \(\beta\in[\Sigma X,Z]\) is uniquely determined by the identity \[\label{Whcom} [\alpha,\beta]\circ\Sigma q= (\![\alpha\circ\Sigma\mathop{\mathrm{pr}}\nolimits_X,\beta\circ\Sigma\mathop{\mathrm{pr}}\nolimits_Y]\!) \in [\Sigma(X\times Y), Z].\tag{19}\] Here \(\mathop{\mathrm{pr}}\nolimits_X\colon X\times Y\to X\) and \(\mathop{\mathrm{pr}}\nolimits_Y\colon X\times Y\to Y\) are the projections, and \[(\![\varphi,\psi]\!) :=-\varphi-\psi+\varphi+\psi\] is the commutator of elements \(\varphi\), \(\psi\) in the group \([\Sigma (X\times Y),Z]\). It is more convenient to use additive notation for the group operation, although the group \([\Sigma (X\times Y),Z]\) is noncommutative.

The Hopf invariant \(Hf\in [\Sigma X\wedge Y,\Sigma Z]\) of a based map \(f\colon X\times Y\to Z\) is defined as follows. Let \(f_X\colon X\to Z\) and \(f_Y\colon Y\to Z\) be the restrictions of \(f\). Then \(Hf\) is uniquely determined by the equation \[\label{Hopf} Hf\circ\Sigma q=-\Sigma f_X\circ\Sigma\mathop{\mathrm{pr}}\nolimits_X -\Sigma f_Y\circ\Sigma\mathop{\mathrm{pr}}\nolimits_Y+\Sigma f\in [\Sigma(X\times Y), \Sigma Z].\tag{20}\]

As a special case, let \(\mu\colon \Omega L\times\Omega L\to\Omega L\) be the loop multiplication, and \(\mathop{\mathrm{pr}}\nolimits_{\Omega L,1}\), \(\mathop{\mathrm{pr}}\nolimits_{\Omega L,2}\colon \Omega L\times\Omega L\to\Omega L\) be the projections. Then \(H\mu\in[\Sigma(\Omega L\wedge\Omega L),\Sigma\Omega L]\) is uniquely determined by the identity \[H\mu\circ\Sigma q=-\Sigma\mathop{\mathrm{pr}}\nolimits_{\Omega L,1}-\Sigma \mathop{\mathrm{pr}}\nolimits_{\Omega L,2} +\Sigma\mu\in[\Sigma(\Omega L\times\Omega L),\Sigma\Omega L].\] Composing with \(\sigma\) and noting that \(\Sigma\mathop{\mathrm{pr}}\nolimits_{\Omega L,1}\circ\sigma\simeq \Sigma\mathop{\mathrm{pr}}\nolimits_{\Omega L,2}\circ\sigma\simeq\ast\), we obtain \[H\mu=\Sigma\mu\circ\sigma\colon \Sigma\Omega L\wedge\Omega L\stackrel{\sigma}{\longrightarrow} \Sigma(\Omega L\times\Omega L)\xrightarrow{\Sigma\mu}{\Sigma\Omega L}.\]

The following formula is given in [17] with a note that its proof is similar to [37]. We provide a full proof below.

References↩︎