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

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

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 [email protected]
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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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References

Références

Barratt, M. G. and Eccles, P. J., 𝛤+ -structures. III. The stable structure of 𝛺 ∞ 𝛴 ∞ A , Topology 13 (1974), 199–207.Google Scholar
Bousfield, A. K. and Friedlander, E. M., Homotopy theory of 𝛤-spaces, spectra, and bisimplicial sets , in Geometric applications of homotopy theory (Proc. Conf., Evanston, IL, 1977), II, Lecture Notes in Mathematics, vol. 658 (Springer, Berlin, 1978), 80–130.Google Scholar
Cohen, F. R., May, J. P. and Taylor, L. R., Splitting of certain spaces CX , Math. Proc. Cambridge Philos. Soc. 84 (1978), 465–496.Google Scholar
Djament, A., Sur l’homologie des groupes unitaires à coefficients polynomiaux , J. K-Theory 10 (2012), 87–139.Google Scholar
Djament, A., Pirashvili, T. and Vespa, C., Cohomologie des foncteurs polynomiaux sur les groupes libres , Doc. Math. J. DMV 21 (2016), 205–222.Google Scholar
Djament, A. and Vespa, C., Sur l’homologie des groupes orthogonaux et symplectiques à coefficients tordus , Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 395–459.Google Scholar
Djament, A. and Vespa, C., Sur l’homologie des groupes d’automorphismes des groupes libres à coefficients polynomiaux , Comment. Math. Helv. 90 (2015), 33–58.Google Scholar
Dold, A., Zur Homotopietheorie der Kettenkomplexe , Math. Ann. 140 (1960), 278–298.Google Scholar
Eilenberg, S. and Mac Lane, S., On the groups H (𝛱, n). II. Methods of computation , Ann. of Math. (2) 60 (1954), 49–139.Google Scholar
Franjou, V. and Pirashvili, T., Stable K-theory is bifunctor homology (after A. Scorichenko) , in Rational representations, the Steenrod algebra and functor homology, Panoramas et Synthèses, vol. 16 (Société Mathématique de France, Paris, 2003), 107–126.Google Scholar
Galatius, S., Stable homology of automorphism groups of free groups , Ann. of Math. (2) 173 (2011), 705–768.Google Scholar
Hartl, M., Pirashvili, T. and Vespa, C., Polynomial functors from algebras over a set-operad and nonlinear Mackey functors , Int. Math. Res. Not. IMRN 2015 (2015), 1461–1554.Google Scholar
Hatcher, A. and Vogtmann, K., Homology stability for outer automorphism groups of free groups , Algebr. Geom. Topol. 4 (2004), 1253–1272.Google Scholar
Hatcher, A., Vogtmann, K. and Wahl, N., Erratum to: ‘Homology stability for outer automorphism groups of free groups’ [Algebr. Geom. Topol. 4 (2004), 1253–1272 (electronic)] by Hatcher and Vogtmann , Algebr. Geom. Topol. 6 (2006), 573–579 (electronic).Google Scholar
Hatcher, A. and Wahl, N., Stabilization for the automorphisms of free groups with boundaries , Geom. Topol. 9 (2005), 1295–1336 (electronic).Google Scholar
Hatcher, A. and Wahl, N., Erratum to: ‘Stabilization for the automorphisms of free groups with boundaries’ [Geom. Topol. 9 (2005), 1295–1336; 2174267] , Geom. Topol. 12 (2008), 639–641.Google Scholar
Hoare, A. H. M., On length functions and Nielsen methods in free groups , J. Lond. Math. Soc. (2) 14 (1976), 188–192.Google Scholar
Jibladze, M. and Pirashvili, T., Cohomology of algebraic theories , J. Algebra 137 (1991), 253–296.Google Scholar
Kahn, D. S., On the stable decomposition of 𝛺 ∞ S ∞ A , in Geometric applications of homotopy theory (Proc. Conf., Evanston, IL, 1977), II, Lecture Notes in Mathematics, vol. 658 (Springer, Berlin, 1978), 206–214.Google Scholar
Kawazumi, N., Cohomological aspects of Magnus expansions, Preprint (2006), arXiv:math.GT/0505497.Google Scholar
Kawazumi, N., Twisted Morita–Mumford classes on braid groups , in Groups, homotopy and configuration spaces, Geometry & Topology Monographs, vol. 13 (Geometry & Topology Publications, Coventry, 2008), 293–306.Google Scholar
Loday, J.-L., Cyclic homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, second edition (Springer, Berlin, 1998). Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili.Google Scholar
Lyndon, R. C., Length functions in groups , Math. Scand. 12 (1963), 209–234.Google Scholar
Lyndon, R. C. and Schupp, P. E., Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89 (Springer, Berlin, 1977).Google Scholar
Morita, S., Cohomological structure of the mapping class group and beyond , in Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74 (American Mathematical Society, Providence, RI, 2006), 329–354.10.1090/pspum/074/2264550Google Scholar
Nakaoka, M., Decomposition theorem for homology groups of symmetric groups , Ann. of Math. (2) 71 (1960), 16–42.Google Scholar
Nakaoka, M., Homology of the infinite symmetric group , Ann. of Math. (2) 73 (1961), 229–257.10.2307/1970333Google Scholar
Nielsen, J., Über die Isomorphismen unendlicher Gruppen ohne Relation , Math. Ann. 79 (1918), 269–272.Google Scholar
Petresco, J., Sur les groupes libres , Bull. Sci. Math. (2) 80 (1956), 6–32.Google Scholar
Pirashvili, T., Hodge decomposition for higher order Hochschild homology , Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 151–179.Google Scholar
Quillen, D., Higher algebraic K-theory. I , in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics, vol. 341 (Springer, Berlin, 1973), 85–147.Google Scholar
Randal-Williams, O., The stable cohomology of automorphisms of free groups with coefficients in the homology representation, Preprint (2010), arXiv:1012.1433.Google Scholar
Randal-Williams, O., Cohomology of automorphism groups of free groups with twisted coefficients , Selecta Math. (N.S.) 24 (2018), 1453–1478.Google Scholar
Randal-Williams, O. and Wahl, N., Homological stability for automorphism groups , Adv. Math. 318 (2017), 534–626.Google Scholar
Satoh, T., Twisted first homology groups of the automorphism group of a free group , J. Pure Appl. Algebra 204 (2006), 334–348.Google Scholar
Satoh, T., Twisted second homology groups of the automorphism group of a free group , J. Pure Appl. Algebra 211 (2007), 547–565.10.1016/j.jpaa.2007.02.003Google Scholar
Scorichenko, A., Stable $K$ -theory and functor homology over a ring, PhD thesis, Northwestern University, Evanston, IL (2000).Google Scholar
Segal, G., Categories and cohomology theories , Topology 13 (1974), 293–312.Google Scholar
Snaith, V. P., A stable decomposition of 𝛺 n S n X , J. Lond. Math. Soc. (2) 7 (1974), 577–583.Google Scholar
Thomason, R. W., Homotopy colimits in the category of small categories , Math. Proc. Cambridge Philos. Soc. 85 (1979), 91–109.Google Scholar
Vespa, C., Extensions between functors from free groups , Bull. Lond. Math. Soc. 50 (2018), 401–419.Google Scholar
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).Google Scholar