The rational stable homology of mapping class groups of universal nil-manifolds
[L’homologie rationnelle stable des groupes modulaires des variétés nil universelles]
Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 783-803.

Nous calculons l’homologie rationnelle stable des groupes d’automorphismes de groupes nilpotents libres. Ces groupes s’intercalent entre les groupes généraux linéaires sur l’anneau des entiers et les groupes d’automorphismes de groupes libres, et nous employons l’homologie de foncteurs pour nous réduire au cas abélien. A titre d’application, nous calculons également l’homologie rationnelle stable des groupes d’automorphismes extérieurs et des groupes modulaires des variétés asphériques associées dans les catégories TOP, PL, et DIFF.

We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.

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DOI : https://doi.org/10.5802/aif.3258
Classification : 20J05,  20E36,  19M05,  18A25,  18G40
Mots clés : homologie stable, groupes d’automorphismes, groupes nilpotents, catégories de foncteurs, homologie de Hochschild, K-théorie stable, suites spectrales
@article{AIF_2019__69_2_783_0,
     author = {Szymik, Markus},
     title = {The rational stable homology of mapping class groups of universal nil-manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {783--803},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {2},
     year = {2019},
     doi = {10.5802/aif.3258},
     zbl = {07067419},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3258/}
}
Szymik, Markus. The rational stable homology of mapping class groups of universal nil-manifolds. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 783-803. doi : 10.5802/aif.3258. http://archive.numdam.org/articles/10.5802/aif.3258/

[1] Andreadakis, S. On the automorphisms of free groups and free nilpotent groups, Proc. Lond. Math. Soc., Volume 15 (1965), pp. 239-268 | Article | MR 188307 | Zbl 035.04502

[2] Balcerak, W.; Hajduk, Bogusław Homotopy type of automorphism groups of manifolds, Colloq. Math., Volume 45 (1981) no. 1, pp. 1-33 | Article | MR 652596 | Zbl 0493.57015

[3] Bestvina, Mladen Homological stability of Aut(F n ) revisited, Hyperbolic geometry and geometric group theory (Advanced Studies in Pure Mathematics), Volume 73, Mathematical Society of Japan, 2017, pp. 1-11

[4] Betley, Stanisław Homology of Gl(R) with coefficients in a functor of finite degree, J. Algebra, Volume 150 (1992) no. 1, pp. 73-86 | MR 1174889 | Zbl 0808.20042

[5] Betley, Stanisław; Pirashvili, Teimuraz Stable K-theory as a derived functor, J. Pure Appl. Algebra, Volume 96 (1994) no. 3, pp. 245-258 | Article | MR 1303284 | Zbl 0812.19002

[6] Borel, Armand Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Supér., Volume 7 (1974), pp. 235-272 | Article | MR 387496 | Zbl 0316.57026

[7] Burghelea, Dan Automorphisms of manifolds, Algebraic and geometric topology (Stanford, 1976), Part 1 (Proceedings of Symposia in Pure Mathematics), Volume 32, American Mathematical Society, 1978, pp. 347-371 | Article | Zbl 0402.57033

[8] Charney, Ruth M. Homology stability for GL n of a Dedekind domain, Invent. Math., Volume 56 (1980) no. 1, pp. 1-17 | MR 557579 | Zbl 0427.18013

[9] Djament, Aurélien Sur l’homologie des groupes unitaires à coefficients polynomiaux, J. K-Theory, Volume 10 (2012) no. 1, pp. 87-139 | MR 2990563 | Zbl 1281.19004

[10] Fiedorowicz, Zbigniew; Pirashvili, Teimuraz; Schwänzl, Roland; Vogt, Rainer; Waldhausen, Friedhelm Mac Lane homology and topological Hochschild homology, Math. Ann., Volume 303 (1995) no. 1, pp. 149-164 | MR 1348360 | Zbl 0846.55007

[11] Franjou, Vincent; Friedlander, Eric M.; Scorichenko, Alexander; Suslin, Andreĭ A. General linear and functor cohomology over finite fields, Ann. Math., Volume 150 (1999) no. 2, pp. 663-728 | MR 1726705 | Zbl 0952.20035

[12] Franjou, Vincent; Pirashvili, Teimuraz On the Mac Lane cohomology for the ring of integers, Topology, Volume 37 (1998) no. 1, pp. 109-114 | Article | MR 1480880 | Zbl 0889.16002

[13] Franjou, Vincent; Pirashvili, Teimuraz Stable K-theory is bifunctor homology (after A. Scorichenko), Rational representations, the Steenrod algebra and functor homology (Panoramas et Synthèses), Volume 16, Société Mathématique de France, 2003, pp. 107-126 | Zbl 1063.19002

[14] Galatius, Søren Stable homology of automorphism groups of free groups, Ann. Math., Volume 173 (2011) no. 2, pp. 705-768 | MR 2784914 | Zbl 1268.20057

[15] Hatcher, Allen E. Concordance spaces, higher simple-homotopy theory, and applications, Algebraic and geometric topology (Stanford, 1976), Part 1 (Proceedings of Symposia in Pure Mathematics), American Mathematical Society, 1978, pp. 3-21 | Article | Zbl 0406.57031

[16] Hatcher, Allen E. Homological stability for automorphism groups of free groups, Comment. Math. Helv., Volume 70 (1995) no. 1, pp. 39-62 | Article | MR 1314940 | Zbl 0836.57003

[17] Hatcher, Allen E.; Vogtmann, Karen Cerf theory for graphs, J. Lond. Math. Soc., Volume 58 (1998) no. 3, pp. 633-655 | Article | MR 1678155 | Zbl 0922.57001

[18] Hatcher, Allen E.; Vogtmann, Karen Rational homology of Aut(F n ), Math. Res. Lett., Volume 5 (1998) no. 6, pp. 759-780 | Zbl 1040.20042

[19] Hsiang, Wu-Chung; Sharpe, Richard W. Parametrized surgery and isotopy, Pac. J. Math., Volume 67 (1976) no. 2, pp. 401-459 | Article | MR 494165 | Zbl 0348.57015

[20] Igusa, Kiyoshi What happens to Hatcher and Wagoner’s formulas for π 0 C(M) when the first Postnikov invariant of M is nontrivial?, Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982) (Lecture Notes in Mathematics), Volume 1046, Springer, 1984, pp. 104-172 | MR 750679 | Zbl 0546.57015

[21] Jibladze, Mamuka; Pirashvili, Teimuraz Cohomology of algebraic theories, J. Algebra, Volume 137 (1991) no. 2, pp. 253-296 | MR 1094244 | Zbl 0724.18005

[22] van der Kallen, Wilberd Homology stability for linear groups, Invent. Math., Volume 60 (1980) no. 3, pp. 269-295 | Article | MR 586429 | Zbl 0415.18012

[23] Kassel, Christian La K-théorie stable, Bull. Soc. Math. Fr., Volume 110 (1982) no. 4, pp. 381-416 | Article | Zbl 0507.18003

[24] Kassel, Christian Calcul algébrique de l’homologie de certains groupes de matrices, J. Algebra, Volume 80 (1983) no. 1, pp. 235-260 | Zbl 0511.18014

[25] Lazard, Michel Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Éc. Norm. Supér., Volume 71 (1954), pp. 101-190 | Article | MR 88496 | Zbl 0055.25103

[26] Loday, Jean-Louis Cyclic homology, Grundlehren der Mathematischen Wissenschaften, 301, Springer, 1998, xx+513 pages | MR 1600246 | Zbl 0885.18007

[27] Madsen, Ib; Milgram, R. James The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, 92, Princeton University Press; University of Tokyo Press, 1979, xii+279 pages | MR 548575 | Zbl 0446.57002

[28] Nomizu, Katsumi On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. Math., Volume 59 (1954), pp. 531-538 | Article | MR 64057 | Zbl 0058.02202

[29] Pickel, P. F. Rational cohomology of nilpotent groups and Lie algebras, Commun. Algebra, Volume 6 (1978) no. 4, pp. 409-419 | Article | MR 491902 | Zbl 0403.20031

[30] Pirashvili, Teimuraz; Waldhausen, Friedhelm Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra, Volume 82 (1992) no. 1, pp. 81-98 | Article | MR 1181095 | Zbl 0767.55010

[31] Schlesinger, James W. The semi-simplicial free Lie ring, Trans. Am. Math. Soc., Volume 122 (1966), pp. 436-442 | Article | MR 199861 | Zbl 0163.17901

[32] Suslin, Andreĭ A. Stability in algebraic K-theory, Algebraic K-theory, Part I (Oberwolfach, 1980) (Lecture Notes in Mathematics), Volume 966, Springer, 1982, pp. 304-333 | MR 689381 | Zbl 0498.18008

[33] Szymik, Markus Twisted homological stability for extensions and automorphism groups of free nilpotent groups, J. K-Theory, Volume 14 (2014) no. 1, pp. 185-201 | MR 3296808 | Zbl 1308.19001

[34] Waldhausen, Friedhelm Algebraic K-theory of generalized free products. III, IV, Ann. Math., Volume 108 (1978) no. 2, pp. 205-256 | MR 498808

[35] Waldhausen, Friedhelm Algebraic K-theory of topological spaces. I, Algebraic and geometric topology (Stanford, 1976), Part 1 (Proceedings of Symposia in Pure Mathematics), American Mathematical Society, 1978, pp. 35-60 | Article | MR 520492 | Zbl 0414.18010

[36] Wall, Charles T. C. Surgery on compact manifolds, 1, Academic Press Inc., 1970, x+280 pages (London Mathematical Society Monographs) | MR 431216 | Zbl 0219.57024

[37] Weiss, Michael; Williams, Bruce Automorphisms of manifolds, Surveys on surgery theory, Vol. 2 (Annals of Mathematics Studies), Volume 149, Princeton University Press, 2001, pp. 165-220 | MR 1818774 | Zbl 0971.57040