Assembling homology classes in automorphism groups of free groups

James Conant, Allen Hatcher, Martin Kassabov, Karen Vogtmann

Published 2015-01-10Version 1

We introduce a new approach to understanding the rational homology of $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$ by constructing homology classes as products of classes from subgroups $\Gamma_{k,s}\subset Aut(F_n)$ of self-homotopy equivalences of graphs of rank $k<n$ with $s$ leaves. We calculate $H_*(\Gamma_{k,s};\mathbb Q)$ completely for $k\leq 2$ and use this information to recover all of the classes which are known to be nontrivial. The new viewpoint brings the power of representation theory into the picture since the symmetric group $\mathfrak S_s$ acts on $\Gamma_{k,s}$ by permuting the leaves. It also makes elementary geometric arguments possible in some situations. These tools lead to simpler proofs of known theorems, and they also allow us to show that some of the previous constructions of homology classes for $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$ produce trivial classes, thus narrowing the search for nonzero classes. We also give some implications of our calculations for the homology of the Lie algebra of symplectic derivations.