[Paper Review] Sur l'homologie des groupes d'automorphismes des groupes libres \\`a coefficients polynomiaux

Aurélien Djament, Christine Vespa|arXiv (Cornell University)|Oct 15, 2012

Homotopy and Cohomology in Algebraic Topology20 references3 citations

TL;DR

This paper establishes that the stable homology of automorphism groups of free groups vanishes when coefficients are given by a reduced polynomial covariant functor. For contravariant polynomial functors factoring through abelianization, it computes the stable first homology group as isomorphic to the functor's tensor product with the identity functor, revealing non-trivial stable homology in this case.

ABSTRACT

We study in this article stable homology of automorphism groups of free groups with coefficients twisted by a poynomial functor. We show that this homology is zero for a reduced covariant polynomial functor. For a reduced contravariant functor, we compute the first homology group, which is in general non zero. Our methods relie on the use of functor categories. ---On \\'etudie dans cet article l'homologie stable des groupes d'automorphismes des groupes libres \\`a coefficients tordus par un foncteur polynomial. On montre que cette homologie est nulle pour un foncteur polynomial covariant r\\'eduit. Dans le cas d'un foncteur polynomial r\\'eduit contravariant, on calcule le premier groupe d'homologie, qui n'est g\\'en\\'eralement pas nul. Nos m\\'ethodes reposent sur l'utilisation de cat\\'egories de foncteurs.

Motivation & Objective

• To determine the stable homology of automorphism groups of free groups with twisted coefficients given by polynomial functors.
• To extend known results on stable homology—previously obtained via topological methods—to a broader algebraic framework.
• To compute the stable first homology group for contravariant polynomial functors that factor through abelianization.
• To clarify the distinction between covariant and contravariant polynomial functors in the context of stable homology of Aut(F_n).
• To provide an algebraic proof of a result previously established via topological and geometric techniques.

Proposed method

• Use of polynomial functors on the category of finitely generated free groups, with a focus on covariant and contravariant functors.
• Application of colimit techniques to analyze stable homology, particularly in the limit as n → ∞.
• Employment of algebraic methods to prove vanishing of stable homology for reduced polynomial covariant functors.
• Utilization of the natural action of Aut(F_n) on F_n and its quotient GL_n(Z) to define twisted coefficients.
• Establishment of an isomorphism between the stable first homology group and the tensor product F ⊗_ab Id via functorial homological algebra.
• Leveraging known results on functor homology and cohomology, including connections to Betley’s work on linear groups and Galatius’s theorem on symmetric groups.

Experimental results

Research questions

• RQ1 Does the stable homology of Aut(F_n) vanish when coefficients are given by a reduced polynomial covariant functor?
• RQ2 How does the stable first homology of Aut(F_n) behave when coefficients arise from a contravariant polynomial functor factoring through abelianization?
• RQ3 Can the result of Hatcher and Wahl on abelianization coefficients be recovered via purely algebraic methods?
• RQ4 What is the precise structure of the stable first homology group for contravariant polynomial coefficients?
• RQ5 How do the behaviors of covariant and contravariant polynomial functors differ in the stable homology of Aut(F_n)?

Key findings

• The stable homology of Aut(F_n) with coefficients in a reduced polynomial covariant functor F is trivial: colim_n H_*(Aut(F_n); F(F_n)) = 0.
• For the abelianization functor (degree 1 polynomial), the result recovers a theorem of Hatcher and Wahl, now proven via algebraic methods.
• For contravariant polynomial functors F factoring through abelianization, the stable first homology group is isomorphic to F ⊗_ab Id.
• This stable first homology is generally non-zero, contrasting sharply with the vanishing result for covariant functors.
• The distinction between covariant and contravariant functors in this context is significant, as the automorphism group lacks the duality symmetry present in linear groups.
• The result generalizes Betley’s theorem on stable homology of linear groups with polynomial coefficients to the setting of free group automorphisms.