Hodge decomposition for stable homology of automorphism groups of free groups

TL;DR

This paper introduces a Hodge decomposition for the stable homology of automorphism groups of free groups with polynomial coefficients, enabling explicit computations and extending previous results through advanced categorical and homological methods.

Contribution

It provides a new decomposition framework for stable homology of automorphism groups of free groups using functor homology and categorical techniques.

Findings

Explicit computations of stable homology using the decomposition

Extension of previous results by Randal-Williams

Application of Gamma-spaces and Snaith splitting methods

Abstract

We establish a decomposition of stable homology of automorphism groups of free groups with polynomial contravariant coefficients in term of functor homology. This allows several explicit computations, intersecting results obtained by independent methods by O. Randal-Williams and extending some of them.Our methods rely on the investigation of Kan extensions associated to several categories of free groups, the extension of a cancellation criterium for homology with polynomial coefficients due to Scorichenko, Galatius Theorem identifying the stable homology of automorphism groups of free groups to the one of symmetric groups, the machinery of Gamma-spaces and the Snaith splitting.