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Décomposition de Hodge pour l’homologie stable des groupes d’automorphismes des groupes libres

Part of: Homotopy theory General theory of categories and functors Connections with homological algebra and category theory Abelian categories Structure and classification of infinite or finite groups Homological algebra

Published online by Cambridge University Press:  07 August 2019

Aurélien Djament
Aurélien Djament*
Affiliation:
CNRS, laboratoire Paul Painlevé (UMR 8524), Cité scientifique, bât. M2, 59655 Villeneuve d’Ascq Cedex, France email djament@math.cnrs.fr
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Abstract

On établit une décomposition de l’homologie stable des groupes d’automorphismes des groupes libres à coefficients polynomiaux contravariants en termes d’homologie des foncteurs. Elle permet plusieurs calculs explicites, qui recoupent des résultats établis de manière indépendante par O. Randal-Williams et généralisent certains d’entre eux. Nos méthodes reposent sur l’examen d’extensions de Kan dérivées associées à plusieurs catégories de groupes libres, la généralisation d’un critère d’annulation homologique à coefficients polynomiaux dû à Scorichenko, le théorème de Galatius identifiant l’homologie stable des groupes d’automorphismes des groupes libres à celle des groupes symétriques, la machinerie des $\unicode[STIX]{x1D6E4}$-espaces et le scindement de Snaith.

We establish a decomposition of stable homology of automorphism groups of free groups with polynomial contravariant coefficients in terms 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 criterion for homology with polynomial coefficients due to Scorichenko, Galatius’s theorem identifying the stable homology of automorphism groups of free groups to that of symmetric groups, the machinery of $\unicode[STIX]{x1D6E4}$-spaces and the Snaith splitting.

Keywords

automorphism groups of free groupsgroup homologyfunctor homologypolynomial functors𝛤-spaces and spectraderived Kan extensions

MSC classification

Primary: 18A25: Functor categories, comma categories 18G15: Ext and Tor, generalizations, Künneth formula 20E36: Automorphisms of infinite groups 20J06: Cohomology of groups 55P42: Stable homotopy theory, spectra
Secondary: 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 18E30: Derived categories, triangulated categories 18G40: Spectral sequences, hypercohomology 55P47: Infinite loop spaces
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 9 , September 2019 , pp. 1794 - 1844
Copyright
© The Author 2019 

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