Assembling homology classes in automorphism groups of free groups

TL;DR

This paper develops a method to assemble homology classes of automorphism groups of free groups from smaller building blocks, computes their symmetric group actions, and identifies known and potential new homology classes.

Contribution

It introduces a novel assembly process for homology classes in automorphism groups of free groups and computes their symmetric group module structures for low ranks.

Findings

Complete computation of symmetric group actions on homology for k ≤ 2.

Reproduction of all known nontrivial rational homology classes for automorphism groups.

Identification of new candidate classes and insights into their geometric and algebraic support.

Abstract

The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for groups $\Gamma_{k,s}$, where the $\Gamma_{k,s}$ generalize $Out(F_k)=\Gamma_{k,0}$ and $Aut(F_k)=\Gamma_{k,1}$. The symmetric group $\Sigma_s$ acts on $H_*(\Gamma_{k,s})$ by permuting leaves, and for trivial rational coefficients we compute the $\Sigma_s$-module structure on $H_*(\Gamma_{k,s})$ completely for $k \leq 2$. Assembling these classes then produces all the known nontrivial rational homology classes for $Aut(F_n)$ and $Out(F_n)$ with the possible exception of classes for $n=7$ recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of $Aut(F_n)$ and $Out(F_n)$ which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.