1 Introduction↩︎

1.1 Gauge groups and their homotopy types↩︎

Let \(G\) be a compact, connected Lie group, let \(X\) be a pointed CW complex, and let \(P\to X\) be a principal \(G\)–bundle. The gauge group \(\mathcal{G}(P)\) is the topological group of \(G\)–equivariant bundle automorphisms of \(P\) that cover the identity on \(X\). When \(X\) is finite, Crabb–Sutherland showed that there are only finitely many homotopy types among the gauge groups \(\mathcal{G}(P)\) as \(P\) ranges over all principal \(G\)–bundles over \(X\) [1]. In the fundamental case \(X=S^4\) and \(G\) simply connected and simple, isomorphism classes of \(G\)–bundles are classified by \(\pi_4(BG)\cong \mathbb{Z}\), and we write \(\mathcal{G}_k(G)\) for the gauge group of the bundle classified by \(k\in \mathbb{Z}\).

The problem of determining the homotopy types of these gauge groups for specific Lie groups \(G\) has received sustained attention; see for example [2]–[4]. In many cases the classification is governed by the order of a fundamental Samelson product in \(G\).

The case \(G=G_2\) is particularly interesting because of the exceptional nature of \(G_2\). Kishimoto, Theriault and Tsutaya  [2] proved that the classification for \(G_2\)–gauge groups reduces to determining the order of the Samelson product \[\langle i_3,1\rangle\in [\Sigma^3G_2,G_2],\] where \(i_3\colon S^3\to G_2\) is the inclusion of the bottom cell and \(1\) denotes the identity map. Moreover, their work determines the number of homotopy types up to one factor of \(2\) [2]. Kameko subsequently resolved the remaining \(2\)–primary ambiguity by proving that the order of \(\langle i_3,1\rangle\) is exactly \(84\) [5]. Consequently, the \(p\)–local homotopy type of \(\mathcal{G}_k(G_2)\) depends only on the greatest common divisor \((k,84)\). Thus the homotopy types of the gauge groups are completely understood up to \(p\)–localization. It is natural to ask how far this information is reflected in the (co)homology of their classifying spaces.

1.2 Cohomology of \(B\mathcal{G}_k\) and the Steenrod algebra↩︎

Gottlieb identified the classifying space of \(\mathcal{G}_k\) with a component of a mapping space [6]: specifically, \(B\mathcal{G}_k\) is homotopy equivalent to \(\mathrm{Map}_k(S^4,BG)\), the path component of \(\mathrm{Map}(S^4,BG)\) containing a classifying map representing \(k\in \pi_4(BG)\cong \mathbb{Z}\). This identification yields a fundamental evaluation fibration \[\Omega^3_0 G \;\longrightarrow\; B\mathcal{G}_k \xrightarrow{\;\mathrm{ev}\;} BG.\] For \(G=G_2\), this expresses \(B\mathcal{G}_k\) as a twisted mapping space over the base \(BG_2\).

The mod \(2\) cohomology of \(BG_2\) is classically known: Borel showed that \[H^*(BG_2;\mathbb{F}_2) \cong \mathbb{F}_2[x_4,x_6,x_7]\] is a polynomial algebra on generators in degrees \(4,6,7\) [7]. The action of the Steenrod algebra \(\mathcal{A}\) on \(H^*(BG_2)\) can be described explicitly via Wu formulas, and from the viewpoint of the hit problem one is interested in minimal generating sets for the unstable \(\mathcal{A}\)–module \(\mathbb{F}_2[x_4,x_6,x_7]\).

By contrast, there seems to be no published description of the full mod \(2\) cohomology ring \(H^*(B\mathcal{G}_k;\mathbb{F}_2)\) for nontrivial \(G_2\)–bundles over \(S^4\), nor of its structure as an unstable \(\mathcal{A}\)–module. There are, however, important partial precedents. Choi studied the mod \(p\) homology of classifying spaces of \(\mathrm{Sp}(n)\)–gauge groups over \(S^4\) [8], and he also computed the homology of the \(G_2\)–gauge groups themselves using Eilenberg–Moore and Serre spectral sequences [9]. Our aim here is to give a precise structural description of \(H^*(B\mathcal{G}_k;\mathbb{F}_2)\) in low degrees, in terms of the evaluation fibration and the first nontrivial cohomology of the triple loop space \(\Omega^3_0G_2\).

1.3 Main result: a low-degree description↩︎

We work \(2\)–locally throughout and write \(H^*(-)\) for \(H^*(-;\mathbb{F}_2)\). Consider the Serre spectral sequence of the evaluation fibration \[\label{eq:Serre-SS} E_2^{s,t}(k)\;\cong\; H^s(BG_2;H^t(\Omega^3_0G_2)) \;\Longrightarrow\; H^{s+t}(B\mathcal{G}_k).\tag{1}\] The \(E_2\)–term is a module over \(H^*(BG_2)\) and an unstable module over the Steenrod algebra. The twisting of this spectral sequence depends on \(k\) through the connecting map \[\partial_k\colon G_2\longrightarrow \Omega^3_0 G_2,\] which, by Lang’s description of the boundary map of the evaluation fibration and its formulation in the gauge-group setting [2], [10], is the triple adjoint of the Samelson product \(\langle k\cdot i_3,1\rangle\). Kameko’s calculation of the \(2\)–primary order of \(\langle i_3,1\rangle\) implies that, after localization at \(2\), the map \(\partial_k\) depends only on \(k\bmod 4\).

The cohomology of the fibre \(\Omega^3_0 G\) can in principle be computed by iterated Eilenberg–Moore spectral sequences applied to the path-loop fibrations of \(BG\) and \(G\). Choi and Yoon conjectured that, for any simply connected finite \(H\)–space \(X\), the Eilenberg–Moore spectral sequences converging to the mod \(p\) homology of \(\Omega^2 X\) and \(\Omega^3 X\) collapse at \(E_2\) [11]. This conjecture was subsequently proved by Lin [12]; in particular, for any compact simple Lie group \(G\) and any prime \(p\), the Eilenberg–Moore spectral sequences computing \(H_*(\Omega^2 G;\mathbb{F}_p)\) and \(H_*(\Omega^3 G;\mathbb{F}_p)\) collapse at \(E_2\). Combined with the classical description of \(H^*(BG_2)\), this implies that \(H^*(\Omega^3_0G_2)\) is of finite type in each degree and that its generators occur in bounded degrees. For our purposes, however, we only need to know the very first nontrivial cohomology group of \(\Omega^3_0G_2\).

Let \[M \;=\; H^*(\Omega^3_0G_2;\mathbb{F}_2)\] and write \(M^{\le N}=\bigoplus_{j\le N}M^j\) for its truncation in degrees \(\le N\). Our main theorem describes the Serre spectral sequence 1 in total degree \(\le 10\) and singles out a distinguished scalar \(\varepsilon(k)\in\mathbb{F}_2\) which already captures the effect of \(k\) on the first nontrivial fibre class.

The paper is organised as follows. In Section 2, we review the necessary background on gauge groups, Samelson products, the mod \(2\) cohomology of \(BG_2\) and the Steenrod algebra. In Section 3, we describe the evaluation fibration and its Serre spectral sequence, and we recall the relationship between the connecting map and the Samelson product. Finally, Section 4 contains the proof of Theorem 1, including a detailed analysis of the bidegrees for \(j\le 10\) and the vanishing of \(H^j(\Omega^3_0G_2)\) for \(1\le j\le 4\).

2 Preliminaries↩︎

2.1 Gauge groups and mapping spaces↩︎

Let \(G\) be a compact, connected Lie group and \(P\to S^4\) a principal \(G\)–bundle. The associated gauge group \(\mathcal{G}(P)\) is the group of \(G\)–equivariant automorphisms of \(P\) covering the identity on \(S^4\), with the compact-open topology. When \(G\) is simply connected and simple, its classifying space \(BG\) is \(3\)–connected and \(\pi_4(BG)\cong \mathbb{Z}\); the integer \(k\in \mathbb{Z}\) corresponding to \([P]\) plays the role of the second Chern class.

Following the notation of [2], [5], we write \(\mathcal{G}_k=\mathcal{G}_k(G_2)\) for the gauge group of the principal \(G_2\)–bundle classified by \(k\in \pi_4(BG_2)\cong \mathbb{Z}\). By a classical result of Gottlieb [6], the classifying space \(B\mathcal{G}_k\) is homotopy equivalent to \(\mathrm{Map}_k(S^4, BG_2)\), the connected component of the mapping space \(\mathrm{Map}(S^4, BG_2)\) corresponding to the class \(k\in \pi_4(BG_2)\cong \mathbb{Z}\).

Under this identification, the evaluation map \(\mathrm{ev}\colon \mathrm{Map}(S^4, BG_2) \to BG_2\) restricts to a fibration \[\mathrm{Map}_*(S^4, BG_2)_k \longrightarrow B\mathcal{G}_k \xrightarrow{\;\mathrm{ev}\;} BG_2,\] where \(\mathrm{Map}_*(S^4, BG_2)_k\) denotes the path component of the based mapping space \(\mathrm{Map}_*(S^4, BG_2)\) corresponding to \(k\in \pi_4(BG_2)\cong \mathbb{Z}\). For simply connected \(G\), this fibre is homotopy equivalent to \(\Omega_0^3 G\).

When \(G\) is simply connected, the based mapping space \(\mathrm{Map}_*(S^4,BG)_k\) is homotopy equivalent to the triple loop space \(\Omega^3_0G\) for every \(k\), and via Gottlieb’s equivalence we obtain the fibration \[\Omega^3_0 G \;\longrightarrow\; B\mathcal{G}_k(G) \xrightarrow{\;\mathrm{ev}\;} BG.\]

We henceforth specialise to \(G=G_2\).

2.2 Samelson products and the connecting map↩︎

Let \(G\) be an \(H\)–space. The Samelson product \[\langle -, -\rangle\colon \pi_m(G)\otimes \pi_n(G) \longrightarrow \pi_{m+n}(G)\] is defined by the composite \[S^m\wedge S^n \xrightarrow{f\wedge g} G\wedge G \xrightarrow{[-,-]} G,\] where \(f,g\) represent the chosen homotopy classes and \([-,-]\) is the commutator in \(G\). For a compact simple Lie group \(G\) the order of certain Samelson products controls the classification of gauge groups; see for example [2]–[4] for concrete instances of this phenomenon.

In the present context the relevant Samelson product is \[\langle i_3,1\rangle\in [\Sigma^3G,G],\] where \(i_3\colon S^3\to G\) is the inclusion of the bottom cell and \(1\) is the identity. Lang [10] showed that the boundary map of the evaluation fibration is identified with the adjoint of a Whitehead product. In the present gauge-group setting, this implies that the connecting map \[\partial_k\colon G\longrightarrow \Omega^3_0G\] in the homotopy fibration \[\mathcal{G}_k(G)\;\longrightarrow\; G \xrightarrow{\;\partial_k\;} \Omega^3_0G \longrightarrow B\mathcal{G}_k(G) \xrightarrow{\;\mathrm{ev}\;} BG\] is the triple adjoint of the Samelson product \(\langle k\cdot i_3,1\rangle\); see [2] for the formulation used here. In particular \(\partial_k\simeq k\cdot \partial_1\), and the order of \(\partial_1\) is equal to the order of \(\langle i_3,1\rangle\).

For \(G=G_2\), the work of Kishimoto–Theriault–Tsutaya [2] and Kameko [5] implies that the \(2\)–primary component of \(\langle i_3,1\rangle\) has order \(4\). Thus, after localizing at \(2\), the homotopy class of \(\partial_k\) depends only on \(k\bmod 4\). This \(2\)–primary information is the only input from the classification of \(G_2\)–gauge groups that will enter our cohomological analysis.

3 The evaluation fibration and its spectral sequence↩︎

3.1 The Serre spectral sequence↩︎

Specialising the evaluation fibration to \(G=G_2\), we obtain a fibration of simply connected spaces \[\Omega^3_0G_2\;\longrightarrow\; B\mathcal{G}_k\xrightarrow{\;\mathrm{ev}\;} BG_2\] and hence a mod \(2\) cohomology Serre spectral sequence \[E_2^{s,t}(k) = H^s(BG_2;H^t(\Omega^3_0G_2;\mathbb{F}_2)) \;\Longrightarrow\; H^{s+t}(B\mathcal{G}_k;\mathbb{F}_2).\] We abbreviate \(M = H^*(\Omega^3_0G_2;\mathbb{F}_2)\); it is a graded commutative algebra and an unstable \(\mathcal{A}\)–module.

Since \(BG_2\) is simply connected and the action of \(\pi_1(BG_2)\) on \(H^*(\Omega^3_0G_2)\) is trivial, the local coefficient system on the fibre is trivial, and we have \[E_2^{s,t}(k) \cong H^s(BG_2)\otimes M^t\] as bigraded vector spaces, with multiplicative structure induced from the tensor product.

The spectral sequence is natural with respect to maps of fibrations. In particular, the connecting map \(\partial_k\colon G_2\to \Omega^3_0G_2\) in the associated homotopy fibration controls the \(k\)–dependence of the differentials.

3.2 The triple loop space \(\Omega^3_0G_2\)↩︎

The homotopy groups of \(G_2\) in low degrees were computed by Mimura and Toda [13]. For our purposes we only need the following facts: \[\pi_3(G_2)\cong \mathbb{Z},\qquad \pi_4(G_2)=0,\qquad \pi_5(G_2)=0,\qquad \pi_6(G_2)\text{ is 3--torsion},\qquad \pi_7(G_2)=0,\] and \(\pi_8(G_2)\) contains \(2\)–torsion. The precise structure of \(\pi_6(G_2)\) and \(\pi_8(G_2)\) will not be needed.

We focus on the component \(\Omega^3_0G_2\) containing the constant map.

We can now determine the first few homology and cohomology groups of \(\Omega^3_0G_2\) using the Hurewicz theorem and universal coefficients.

For the remainder of the paper, we fix a generator \[u_5\in M^5=H^5(\Omega^3_0G_2;\mathbb{F}_2)\cong \mathbb{F}_2.\]

4 Proof of the low-degree theorem↩︎

In this section we analyse the Serre spectral sequence 1 in total degree \(j=s+t\le 10\) and prove Theorem 1. We work throughout with mod \(2\) coefficients and suppress them from the notation.

4.1 The \(E_2\)–page in low degrees↩︎

By the discussion in Section 3, the \(E_2\)–term has the form \[E_2^{s,t}(k)\cong H^s(BG_2)\otimes M^t,\] and \(H^*(BG_2)\cong \mathbb{F}_2[x_4,x_6,x_7]\) is generated by \(x_4,x_6,x_7\) in degrees \(4,6,7\). For \(0<j\le 10\) the only degrees \(s>0\) in which \(H^s(BG_2)\) can be nonzero are \[s\in\{4,6,7,8,10\},\] corresponding to monomials \(x_4\), \(x_6\), \(x_7\), \(x_4^2\) and \(x_4x_6\), respectively.

By Proposition 4, the fibre cohomology \(M^t\) vanishes for \(1\le t\le 4\). Consequently, in total degree \(j=s+t\le 10\) the only nonzero groups \(E_2^{s,t}\) with \(t>0\) occur when \(t\ge 5.\) In particular, the class \(u_5\in M^5\) gives rise to a class \[u = 1\otimes u_5\in E_2^{0,5}(k),\] and this is a generator of least positive degree in \(M\).

4.2 Bidegree constraints on differentials↩︎

The differentials in the Serre spectral sequence have bidegree \[d_r\colon E_r^{s,t}\longrightarrow E_r^{s+r,t-r+1}.\] We now examine which values of \(r\) can occur with source or target in total degree \(j\le 10\).

Let \((s,t)\) be a bidegree with \(s+t\le 10\) and \(E_r^{s,t}\neq 0\). If \(d_r\) is nonzero on \(E_r^{s,t}\), then its target lies in bidegree \((s+r,t-r+1)\) with total degree \[(s+r) + (t-r+1) = s+t+1\le 11.\] Moreover, for \(s>0\) we know that \(H^s(BG_2)\) can only be nonzero in degrees \[s\in\{4,6,7,8,10\},\] and by Proposition 4 the fibre cohomology satisfies \[M^0\cong\mathbb{F}_2,\qquad M^j=0\;\text{for}\;1\le j\le 4,\qquad M^5\neq 0.\]

In this subsection we restrict attention to differentials whose source is the first nontrivial fibre class \[u_5\in M^5 \cong E_2^{0,5}(k),\] and we determine on which page \(u_5\) can possibly support a nontrivial differential in total degree \(j\le 10\).

Since \(u_5\) lies in bidegree \((0,5)\), any differential with source \(u_5\) has the form \[d_r(u_5)\in E_r^{r,\,5-r+1}.\] We analyse the possible values of \(r\).

• If \(r\ge 7\), then \(5-r+1\le -1\), so the target bidegree has negative fibre degree and \(E_r^{r,5-r+1}=0\). Thus \(d_r(u_5)=0\) for all \(r\ge 7\) in total degree \(\le 10\).

• For \(2\le r\le 5\) we have the following possibilities:

  • \(r=2\): the target is \((2,4)\), but \(H^2(BG_2)=0\), so \(E_2^{2,4}=0\) and hence \(d_2(u_5)=0\).

  • \(r=3\): the target is \((3,3)\), and again \(H^3(BG_2)=0\), so \(d_3(u_5)=0\).

  • \(r=4\): the target is \((4,2)\); here \(H^4(BG_2)\cong\mathbb{F}_2\{x_4\}\) is nonzero, but \(M^2=0\) by Proposition 4, so \(E_2^{4,2}\cong H^4(BG_2)\otimes M^2=0\), and hence \(d_4(u_5)=0\).

  • \(r=5\): the target is \((5,1)\), and \(H^5(BG_2)=0\), so \(E_2^{5,1}=0\) and \(d_5(u_5)=0\).

  In particular, \(d_r(u_5)=0\) for all \(2\le r\le 5\).

• For \(r=6\) the target is bidegree \((6,0)\), and \[E_2^{6,0}\cong H^6(BG_2)\otimes M^0 \cong H^6(BG_2)\cong\mathbb{F}_2\{x_6\}\] is nonzero. There is therefore no a priori bidegree reason to force \(d_6(u_5)\) to vanish; this is the first page on which \(u_5\) can possibly support a nontrivial differential in total degree \(\le 10\).

We summarise the discussion in the following form, which will be sufficient for our later arguments.

Fix once and for all a generator \(x_6\in H^6(BG_2)\cong\mathbb{F}_2\). Since \(H^6(BG_2)\) is one-dimensional over \(\mathbb{F}_2\), there exists a uniquely determined scalar \(\varepsilon(k)\in\mathbb{F}_2\) such that \[d_6(u_5)=\varepsilon(k)\,x_6.\]

Extending this analysis slightly, we obtain a convenient description of the possible differentials on the \(H^*(BG_2)\)–submodule generated by \(u_5\) in low total degrees.

4.3 The \(E_\infty\)–page and completion of the proof↩︎

For each total degree \(j\le 10\), the Serre spectral sequence yields a finite filtration \[0=F^{j+1}H^j(B\mathcal{G}_k)\subseteq F^jH^j(B\mathcal{G}_k)\subseteq\cdots\subseteq F^0H^j(B\mathcal{G}_k)=H^j(B\mathcal{G}_k)\] whose associated graded object is \[\mathrm{gr}\,H^j(B\mathcal{G}_k)\cong \bigoplus_{s+t=j}E_\infty^{s,t}(k).\]

By Lemmas 2 and 3, the contribution of the \(H^*(BG_2)\)–submodule \[U=H^*(BG_2)\cdot u_5\subseteq E_2^{*,*}(k)\] to \(E_\infty^{*,*}(k)\) in total degree \(j\le 10\) is completely determined by the single differential \[d_6(u_5)=\varepsilon(k)x_6.\] More precisely, within \(U\) all differentials \(d_r\) for \(2\le r\le 5\) vanish, every \(d_6\) is obtained from \(d_6(u_5)\) by multiplication with classes from \(H^*(BG_2)\), and no \(d_r\) with \(r\ge 7\) contributes in total degree \(\le 10\).

We do not claim here that these are the only possible differentials in total degree \(\le 10\) on the whole spectral sequence, since additional fibre classes in degrees \(t\ge 6\) may in principle support further differentials. Thus Theorem 1 should be interpreted as a precise determination of the first \(k\)–dependent contribution coming from the fibre class \(u_5\).

References↩︎