Cohomology of automorphism groups of free groups with twisted coefficients

Galatius [12] has proved the remarkable theorem that the natural homomorphisms

both induce homology isomorphisms in degrees $2*\leq n-3$ with integral coefficients. His approach is to model $B\mathrm{Out}(F_{n})$ as the space $\mathcal{G}_{n}$ of graphs of the homotopy type of $\vee^{n}S^{1}$, and $B\mathrm{Aut}(F_{n})$ as the space $\mathcal{G}^{1}_{n}$ of pointed graphs of the same homotopy type. He then produces a natural map from such spaces of graphs to the infinite loop space $Q_{0}(S^{0})$, which he shows has a certain homological connectivity. One consequence of this is that $H^{i}(\mathrm{Aut}(F_{n});\mathbb{Q})=0$ for $0<i\leq\tfrac{n-3}{2}$.

At the same time, Satoh [26, 27] has studied the low dimensional (co)homology of $\mathrm{Aut}(F_{n})$ and $\mathrm{Out}(F_{n})$ with coefficients in the module $H\mathrel{\mathop{\mathchar 12346\relax}}=H_{1}(F_{n};\mathbb{Z})$ given by the abelianisation of $F_{n}$, and in the dual module $H^{*}\mathrel{\mathop{\mathchar 12346\relax}}=H^{1}(F_{n};\mathbb{Z})$. His methods are those of combinatorial group theory, and proceed by calculation with a presentation of these groups.

Our goal is to show that the stable cohomology of $\mathrm{Aut}(F_{n})$ and $\mathrm{Out}(F_{n})$ with twisted coefficients may also be approached with the geometric techniques used by Galatius, along with a little representation theory. The method we shall introduce is quite general and may be applied whenever a suitable Madsen–Weiss-type theorem has been proved; in Appendix B we will show how to use it to recover a theorem of Looijenga [20] on the stable cohomology of mapping class groups with twisted coefficients.

For $\mathrm{Aut}(F_{n})$ and $\mathrm{Out}(F_{n})$ we will consider cohomology with coefficients in the modules $H^{*}_{\mathbb{Q}}\mathrel{\mathop{\mathchar 12346\relax}}=H^{1}(F_{n};\mathbb{Q})$ and $H_{\mathbb{Q}}\mathrel{\mathop{\mathchar 12346\relax}}=H_{1}(F_{n};\mathbb{Q})$, and more generally with coefficients in $S_{\lambda}(H^{*}_{\mathbb{Q}})$ and $S_{\lambda}(H_{\mathbb{Q}})$, where $S_{\lambda}(-)$ is the Schur functor associated to a partition $\lambda\vdash q$, which we think of as being given by a Young diagram. To define this, recall that to such a partition there is an associated irreducible $\mathbb{Q}[\Sigma_{q}]$-module $S^{\lambda}$, the Specht module. For a $\mathbb{Q}$-vector space $V$ we may consider $V^{\otimes q}$ as a $\mathbb{Q}[\Sigma_{q}]$-module by permuting the factors, and we may hence form $S_{\lambda}(V)\mathrel{\mathop{\mathchar 12346\relax}}=\mathrm{Hom}_{\mathbb{Q}[\Sigma_{q}]}(S^{\lambda},V^{\otimes q})$. This construction defines the Schur functor $S_{\lambda}$. It is a basic result that $S_{\lambda}(V)$ is an irreducible representation of $GL(V)$.

In fact, our result is best expressed as calculating $H^{*}(\mathrm{Aut}(F_{n});H_{\mathbb{Q}}^{\otimes q})$ as a $\mathbb{Q}[\Sigma_{q}]$-module. The result for $S_{\lambda}(H_{\mathbb{Q}})$ may then be extracted as

We write $\mathbb{Q}^{-}$ for the sign representation of $\Sigma_{q}$.

Theorem A (i) may also be deduced from work of Djament–Vespa [9]. We believe that Theorem A (ii) may also be obtained by combining work of Djament [8] and Vespa [28].

However, more important than these particular results is our technique, which is of very general applicability. For example, it can easily be modified to obtain results for $\mathrm{Out}(F_{n})$.

We give tables listing the dimensions of the groups $H^{|\lambda|}(\mathrm{Aut}(F_{n});S_{\lambda}(H_{\mathbb{Q}}))$ and $H^{|\lambda|}(\mathrm{Out}(F_{n});S_{\lambda}(H_{\mathbb{Q}}))$ for $|\lambda|\leq 6$ in Appendix C.

Each partition of $\{1,2,\ldots,q\}$ may be expressed as a partition of a smaller set with no parts of size 1 along with the set of parts of size 1, which translates to the expression

as long as $n\geq 4q+3$. Applying $\mathrm{Hom}_{\Sigma_{q}}(S^{\lambda},-)$ and using the Pieri rule gives the pleasant formula

as long as $n\geq 4|\lambda|+3$, where $\rho(\lambda)$ denotes the set of Young diagrams which may be obtained from $\lambda$ by removing at most one box from each row.

The observations of this section are no doubt known to some experts. The groups $\mathrm{Aut}(F_{n})$ and $\mathrm{Out}(F_{n})$ fit into a more general family of groups denoted $\Gamma_{n,s}$ by Hatcher–Vogtmann [16], where $\mathrm{Aut}(F_{n})=\Gamma_{n,1}$ and $\mathrm{Out}(F_{n})=\Gamma_{n,0}$. Classifying spaces for these may be taken to be the spaces $\mathcal{G}_{n}^{s}$ of graphs of the homotopy type of $\vee^{n}S^{1}$ equipped with $s$ distinct ordered marked points.

Hatcher and Vogtmann [16] prove that the map $\mathcal{G}_{n}^{s}\to\mathcal{G}_{n+1}^{s}$ (defined for $s>0$) that adds a loop at a particular marked point induces an integral homology isomorphism in degrees $2*\leq n-2$ (and induces a rational homology isomorphism in degrees $5*\leq 4n-10$). Furthermore, the map $\mathcal{G}_{n}^{s}\to\mathcal{G}_{n}^{s-1}$ that forgets a marked point is an integral homology isomorphism in degrees $2*\leq n-3$ (or $2*\leq n-4$ if it is the last marked point).

We can make an immediate observation regarding cohomology with coefficients in $H^{*}$ from this homology stability result. There is an extension $F_{n}\to\mathrm{Aut}(F_{n})\to\mathrm{Out}(F_{n})$, and the corresponding Leray–Hochschild–Serre spectral sequence has two rows. However, as the projection is a homology equivalence in a range of degrees we deduce

Similarly, there is a fibration with section $\vee^{n}S^{1}\to\mathcal{G}_{n}^{2}\to\mathcal{G}_{n}^{1}$ and the projection map is a homology equivalence in a range of degrees, so

More generally, the map $\mathcal{G}_{n}^{k+1}\to\mathcal{G}_{n}^{1}$ has fibre $(\vee^{n}S^{1})^{k}$ and $\mathrm{Aut}(F_{n})$ acts on its homology diagonally. Thus for $k=2$ the Serre spectral sequence has three rows, with

and

Using the fact that the projection map is a homology equivalence in degrees $2*\leq n-2$, we deduce that $H^{*}(\mathrm{Aut}(F_{n});H^{*}\otimes H^{*})=0$ for $2*\leq n-8$. Continuing in this way for higher $k$, we establish the following proposition.

By Galatius’ theorem, the groups $\mathrm{Aut}(F_{n})$ are closely related to the symmetric groups, but also share many properties with mapping class groups of surfaces. These three families of groups are known to exhibit homological stability for integral homology, but symmetric groups and mapping class groups also exhibit homological stability for certain systems of coefficients, those of “finite degree”, a notion that is originally due to Dwyer [10] in his study of homological stability for general linear groups with coefficient systems.

This notion of degree may be formalised, in the context of free groups, as follows.

In a 2010 preprint [23], we conjectured (based on the analogy with general linear groups [10] and mapping class groups [17]) that the groups $H_{*}(\mathrm{Aut}(F_{n});V(F_{n}))$ should exhibit homological stability in degrees $2*\leq n-k-2$ when $V$ is a polynomial coefficient system of degree $\leq k$. Since then, in joint work with Wahl [25] we have established a quite general homological stability theorem with polynomial coefficients, and using the highly-connected simplicial complexes of [15] it applies in this case. The result obtained is as follows.

One advantage of the category $\mathfrak{Gr}$ over the more naïve category $\mathfrak{gr}$ of finitely generated free groups and injective homomorphisms is that there are more functors out of it.

There are some easy consequences of the definition of polynomiality which allow us to compute or estimate degrees of coefficient systems. Firstly, a summand of a polynomial functor of degree $\leq k$ is again polynomial of degree $\leq k$, and an extension of two polynomial functors of degree $\leq k$ is again polynomial of degree $\leq k$. Secondly, if $V$ and $W$ are coefficient systems which are polynomial of degree $k$ and $\ell$ respectively, and if they both take values in flat $\mathbb{Z}$-modules, then $V\otimes_{\mathbb{Z}}W$ is polynomial of degree $\leq(k+\ell)$. More generally, if $\mathbb{F}$ is a field and $V,W\mathrel{\mathop{\mathchar 12346\relax}}\mathfrak{Gr}\to\mathbb{F}\text{-mod}\to\mathbf{Ab}$ are coefficient systems which are polynomial of degree $k$ and $\ell$ respectively and factor through the category of $\mathbb{F}$-modules, then $V\otimes_{\mathbb{F}}W$ is polynomial of degree $\leq(k+\ell)$.

We will often be interested in cohomology rather than homology, for which we will use the following result of Universal Coefficient-type, which is surely standard but for which we could not find a reference.

Using the stability theorem and Lemma 3.5, we may improve Proposition 2.3 to the following.

This proves Theorem A (i). We remark that Theorem A (i) also follows from [9, Théorème 1] and Lemma 3.5, because the functor $H^{\otimes q}\mathrel{\mathop{\mathchar 12346\relax}}\mathfrak{Gr}\to\mathfrak{gr}\to\mathbf{Ab}$ is polynomial and vanishes on the trivial group (for $q\geq 1$), so by that theorem $H_{*}(\mathrm{Aut}(F_{n});H^{\otimes q})$ vanishes in the stable range.

Finally, we record the stability range for cohomology with coefficients in $S_{\lambda}(H_{\mathbb{Q}})$ with $\lambda\vdash q$. This follows from $S_{\lambda}(H_{\mathbb{Q}})\cong S_{\lambda}(H_{\mathbb{Q}}^{*})^{*}$, that $S_{\lambda}(H_{\mathbb{Q}}^{*})$ is polynomial of degree $\leq q$ as it is a summand of $(H_{\mathbb{Q}}^{*})^{\otimes q}$, the stability theorem, and Lemma 3.5.

Let $(X,x_{0})$ be a based space. Let us write $\mathcal{G}^{X}$ for the topological category whose objects are finite sets inside $\mathbb{R}^{\infty}$, and whose morphisms from $S\subset\mathbb{R}^{\infty}$ to $T\subset\mathbb{R}^{\infty}$ consist of a real number $t>0$ and a graph $\Gamma\subset[0,t]\times\mathbb{R}^{\infty}$ with leaves $\{0\}\times S$ and $\{t\}\times T$, equipped with a continuous map $f\mathrel{\mathop{\mathchar 12346\relax}}(\Gamma,S\cup T)\to(X,x_{0})$. This may be given a topology following [12]. Choosing once and for all an embedding $\{1,2,\ldots,s\}\subset\mathbb{R}^{\infty}$ the space $\mathcal{G}_{n}^{s}(X)$ may be defined as the subspace of $\mathcal{G}^{X}(\emptyset,\{1,2,\ldots,s\})$ consisting of those graphs which are connected and homotopy equivalent to $\vee^{n}S^{1}$.

Let $\mathcal{G}_{n}^{s}(X)_{*}\subset\mathcal{G}_{n}^{s}(X)\times\mathbb{R}^{\infty+1}$ be the subspace given by tuples $(t,\Gamma,f;p)$ where $p\in\Gamma$. Forgetting the point $p$ gives a map

which is a fibration with fibre over the point $(t,\Gamma,f)\in\mathcal{G}_{n}^{s}(X)$ given by the graph $\Gamma$. Furthermore, sending $(t,\Gamma,f,p)$ to $f(p)\in X$ defines a map

As the map $\pi$ is a fibration with homotopy-finite fibres, it admits a Becker–Gottlieb transfer map $\mathrm{trf}_{\pi}\mathrel{\mathop{\mathchar 12346\relax}}\Sigma^{\infty}_{+}\mathcal{G}_{n}^{s}(X)\to\Sigma^{\infty}_{+}\mathcal{G}_{n}^{s}(X)_{*}$. Composing this with $\Sigma^{\infty}_{+}e$ and taking the adjoint gives a map

to the free infinite loop space on $X$, landing in the path component indexed by $1-n$, the Euler characteristic of $\vee^{n}S^{1}$. This map is natural in $X$, and factors through the map $\tau^{0}_{n}\mathrel{\mathop{\mathchar 12346\relax}}\mathcal{G}_{n}(X)\to Q_{1-n}(X_{+})$ up to homotopy.

The point of this theorem is that the right-hand side may be computed: the rational cohomology of $Q_{1-n}(X_{+})$ may be described compactly as $S^{*}(\widetilde{H}^{*}(X;\mathbb{Q}))$, the free graded-commutative algebra on the reduced cohomology of $X$.

The main technical result we will require is the following homological stability theorem for the spaces $\mathcal{G}_{n}^{s}(X)$. We will deduce it from (two) arguments of Cohen–Madsen in the analogous situation of surfaces with maps to a background space.