Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T05:04:40.733Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE COHOMOLOGY OF TORELLI GROUPS

Part of: Fiber spaces and bundles Categories with structure Discontinuous groups and automorphic forms Linear algebraic groups and related topics

Published online by Cambridge University Press:  13 April 2020

ALEXANDER KUPERS  and
OSCAR RANDAL-WILLIAMS
ALEXANDER KUPERS
Affiliation:
Department of Mathematics, One Oxford Street, Cambridge, MA02138, USA; kupers@math.harvard.edu
OSCAR RANDAL-WILLIAMS
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, UK; o.randal-williams@dpmms.cam.ac.uk

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $\#^{g}S^{n}\times S^{n}$ relative to a disc in a stable range, for $2n\geqslant 6$. Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.

MSC classification

Primary: 55R40: Homology of classifying spaces, characteristic classes 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Secondary: 11F75: Cohomology of arithmetic groups 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 20G05: Representation theory
Type
Topology
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e7
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Akazawa, H., ‘Symplectic invariants arising from a Grassmann quotient and trivalent graphs’, Math. J. Okayama Univ. 47 (2005), 99–117.Google Scholar
Akita, T., ‘Homological infiniteness of Torelli groups’, Topology 40(2) (2001), 213–221.CrossRefGoogle Scholar
Atiyah, M. F., ‘The signature of fibre-bundles’, inGlobal Analysis (Papers in Honor of K. Kodaira) (University Tokyo Press, Tokyo, 1969), 73–84.Google Scholar
Berglund, A. and Bergström, J., ‘Hirzebruch L-polynomials and multiple zeta values’, Math. Ann. 372(1–2) (2018), 125–137.CrossRefGoogle Scholar
Berglund, A. and Madsen, I., ‘Rational homotopy theory of automorphisms of manifolds’, Acta Math. 224(1) (2020), 67–185.CrossRefGoogle Scholar
Boldsen, S. K., ‘Improved homological stability for the mapping class group with integral or twisted coefficients’, Math. Z. 270(1–2) (2012), 297–329.CrossRefGoogle Scholar
Boldsen, S. K. and Hauge Dollerup, M., ‘Towards representation stability for the second homology of the Torelli group’, Geom. Topol. 16(3) (2012), 1725–1765.CrossRefGoogle Scholar
Borel, A., ‘Density properties for certain subgroups of semi-simple groups without compact components’, Ann. of Math. (2) 72 (1960), 179–188.CrossRefGoogle Scholar
Borel, A., ‘Stable real cohomology of arithmetic groups’, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 235–272.CrossRefGoogle Scholar
Borel, A., ‘Stable real cohomology of arithmetic groups. II’, inManifolds and Lie Groups (Notre Dame, Ind., 1980), Progress in Mathematics, 14 (Birkhäuser, Boston, MA, 1981), 21–55.CrossRefGoogle Scholar
Borel, A. and Harish-Chandra, ‘Arithmetic subgroups of algebraic groups’, Ann. of Math. (2) 75 (1962), 485–535.CrossRefGoogle Scholar
Church, T. and Farb, B., ‘Parameterized Abel–Jacobi maps and abelian cycles in the Torelli group’, J. Topol. 5(1) (2012), 15–38.CrossRefGoogle Scholar
Ebert, J. and Randal-Williams, O., ‘Generalised Miller–Morita–Mumford classes for block bundles and topological bundles’, Algebr. Geom. Topol. 14(2) (2014), 1181–1204.CrossRefGoogle Scholar
Ebert, J. and Randal-Williams, O., ‘Torelli spaces of high-dimensional manifolds’, J. Topol. 8(1) (2015), 38–64.CrossRefGoogle Scholar
Galatius, S. and Randal-Williams, O., ‘Stable moduli spaces of high-dimensional manifolds’, Acta Math. 212(2) (2014), 257–377.CrossRefGoogle Scholar
Galatius, S. and Randal-Williams, O., ‘Homological stability for moduli spaces of high dimensional manifolds. II’, Ann. of Math. (2) 186(1) (2017), 127–204.CrossRefGoogle Scholar
Galatius, S. and Randal-Williams, O., ‘Homological stability for moduli spaces of high dimensional manifolds. I’, J. Amer. Math. Soc. 31(1) (2018), 215–264.CrossRefGoogle Scholar
Garoufalidis, S. and Getzler, E., ‘Graph complexes and the symplectic character of the Torelli group’, Preprint, 2017, arXiv:1712.03606.Google Scholar
Garoufalidis, S. and Nakamura, H., ‘Some IHX-type relations on trivalent graphs and symplectic representation theory’, Math. Res. Lett. 5(3) (1998), 391–402.CrossRefGoogle Scholar
Habegger, N. and Sorger, C., ‘An infinitesimal presentation of the Torelli group of a surface with boundary’, http://www.math.sciences.univ-nantes.fr/∼habegger/PS/inf180300.ps, 2000.Google Scholar
Hain, R., ‘Infinitesimal presentations of the Torelli groups’, J. Amer. Math. Soc. 10(3) (1997), 597–651.CrossRefGoogle Scholar
Hain, R., ‘Finiteness and Torelli spaces’, inProblems on Mapping Class Groups and Related Topics, Proceedings of Symposia in Pure Mathematics, 74 (American Mathematical Society, Providence, RI, 2006), 57–70.CrossRefGoogle Scholar
Harer, J. L., ‘Stability of the homology of the mapping class groups of orientable surfaces’, Ann. of Math. (2) 121(2) (1985), 215–249.CrossRefGoogle Scholar
Hebestreit, F., Land, M., Lück, W. and Randal-Williams, O., ‘A vanishing theorem for tautological classes of aspherical manifolds’, Geom. Topol., to appear, arXiv:1705.06232.Google Scholar
Ivanov, N. V., ‘On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients’, inMapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen, 1991/Seattle, WA, 1991), Contemporary Mathematics, 150 (American Mathematical Society, Providence, RI, 1993), 149–194.CrossRefGoogle Scholar
Johnson, D., ‘The structure of the Torelli group. III. The abelianization of 𝓣’, Topology 24(2) (1985), 127–144.CrossRefGoogle Scholar
Kawazumi, N., ‘A generalization of the Morita–Mumford classes to extended mapping class groups for surfaces’, Invent. Math. 131(1) (1998), 137–149.CrossRefGoogle Scholar
Kawazumi, N., ‘On the stable cohomology algebra of extended mapping class groups for surfaces’, inGroups of Diffeomorphisms, Advanced Studies in Pure Mathematics, 52 (Mathematical Society of Japan, Tokyo, 2008), 383–400.Google Scholar
Kawazumi, N. and Morita, S., ‘The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes’, Math. Res. Lett. 3(5) (1996), 629–641.CrossRefGoogle Scholar
Kawazumi, N. and Morita, S., ‘The primary approximation to the cohomology of the moduli space of curves and cocycles for the Mumford–Morita–Miller classes’, www.ms.u-tokyo.ac.jp/preprint/pdf/2001-13.pdf, 2001.Google Scholar
Krannich, M., ‘Homological stability of topological moduli spaces’, Geom. Topol. 23(5) (2019), 2397–2474.CrossRefGoogle Scholar
Kreck, M., ‘Isotopy classes of diffeomorphisms of (k - 1)-connected almost-parallelizable 2k-manifolds’, inAlgebraic Topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Mathematics, 763 (Springer, Berlin, 1979), 643–663.CrossRefGoogle Scholar
Kupers, A., ‘Some finiteness results for groups of automorphisms of manifolds’, Geom. Topol. 23(5) (2019), 2277–2333.CrossRefGoogle Scholar
Kupers, A. and Randal-Williams, O., ‘The cohomology of Torelli groups is algebraic’, Preprint, 2019, arXiv:1908.04724.Google Scholar
Madsen, I. and Weiss, M., ‘The stable moduli space of Riemann surfaces: Mumford’s conjecture’, Ann. of Math. (2) 165(3) (2007), 843–941.CrossRefGoogle Scholar
Margulis, G. A., Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17 (Springer, Berlin, 1991).CrossRefGoogle Scholar
Miller, J., Patzt, P. and Wilson, J. C. H., ‘Central stability for the homology of congruence subgroups and the second homology of Torelli groups’, Adv. Math. 354 (2019), 106740, 45.CrossRefGoogle Scholar
Milnor, J. W. and Moore, J. C., ‘On the structure of Hopf algebras’, Ann. of Math. (2) 81 (1965), 211–264.CrossRefGoogle Scholar
Milnor, J. W. and Stasheff, J. D., Characteristic Classes, Annals of Mathematics Studies, 76 (Princeton University Press, Princeton, NJ, 1974), University of Tokyo Press, Tokyo.CrossRefGoogle Scholar
Morita, S., ‘Families of Jacobian manifolds and characteristic classes of surface bundles. I’, Ann. Inst. Fourier (Grenoble) 39(3) (1989), 777–810.CrossRefGoogle Scholar
Morita, S., ‘The extension of Johnson’s homomorphism from the Torelli group to the mapping class group’, Invent. Math. 111(1) (1993), 197–224.CrossRefGoogle Scholar
Morita, S., ‘A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles’, inTopology and Teichmüller Spaces (Katinkulta, 1995) (World Sci. Publ., River Edge, NJ, 1996), 159–186.CrossRefGoogle Scholar
Procesi, C., Lie Groups, Universitext (Springer, New York, 2007).Google Scholar
Raghunathan, M. S., ‘Cohomology of arithmetic subgroups of algebraic groups. I, II’, Ann. of Math. (2) 86 (1967), 409–424. ibid. (2) 87 (1967), 279–304.CrossRefGoogle Scholar
Raghunathan, M. S., ‘A note on quotients of real algebraic groups by arithmetic subgroups’, Invent. Math. 4 (1967/1968), 318–335.CrossRefGoogle Scholar
Randal-Williams, O., ‘Resolutions of moduli spaces and homological stability’, J. Eur. Math. Soc. (JEMS) 18(1) (2016), 1–81.CrossRefGoogle Scholar
Randal-Williams, O., ‘Cohomology of automorphism groups of free groups with twisted coefficients’, Selecta Math. (N.S.) 24(2) (2018), 1453–1478.CrossRefGoogle Scholar
The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.7), 2019, https://www.sagemath.org.Google Scholar
Sakasai, T., ‘The Johnson homomorphism and the third rational cohomology group of the Torelli group’, Topology Appl. 148(1–3) (2005), 83–111.CrossRefGoogle Scholar
Sam, S. V. and Snowden, A., ‘Stability patterns in representation theory’, Forum Math. Sigma 3 (2015), e11, 108.Google Scholar
Serre, J.-P., ‘Arithmetic groups’, inHomological Group Theory (Proc. Sympos., Durham, 1977), London Mathematical Society Lecture Note Series, 36 (Cambridge University Press, Cambridge, New York, 1979), 105–136.CrossRefGoogle Scholar
Tshishiku, B., ‘Geometric cycles and characteristic classes of manifold bundles’, Preprint, 2017, arXiv:1711.03139, with an appendix by Manuel Krannich.Google Scholar
Tshishiku, B., ‘Borel’s stable range for the cohomology of arithmetic groups’, J. Lie Theory 29(4) (2019), 1093–1102.Google Scholar