Assembling homology classes in automorphism groups of free groups

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 $H_{\ast }({\Gamma }_{k,s})$ by permuting leaves, and for trivial rational coefficients we compute the ${\Sigma }_s$-module structure on $H_{\ast }({\Gamma }_{k,s})$ completely for $k\le 2$. 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.

Final version for Commentarii Math. Helv

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Cited by in corpus (17)

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• Feynman amplitudes on moduli spaces of graphs
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• Moduli spaces of colored graphs
• The Euler characteristic of the moduli space of graphs
• Modular characteristic classes for representations over finite fields
• Higher operations in string topology of classifying spaces
• The stable cohomology of self-equivalences of connected sums of products of spheres
• Algebraic Interplay between Renormalization and Monodromy
• Multi-valued Feynman Graphs and Scattering Theory
• Higher structures in rational homotopy theory
• Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group
• The Lie Lie algebra
• Direct products of free groups in $\mathrm{Aut}(F_N)$
• The ${\mathbb{S}}_n$-equivariant Euler characteristic of the moduli space of graphs
• On the low dimensional cohomology groups of the IA-automorphism group of a free group of rank three
• Secondary characteristic classes for subgroups of automorphism groups of free groups