Abstract
We compute the abelianisations of the mapping class groups of the manifolds \(W_g^{2n} = g(S^n \times S^n)\) for \(n \ge 3\) and \(g \ge 5\). The answer is a direct sum of two parts. The first part arises from the action of the mapping class group on the middle homology, and takes values in the abelianisation of the automorphism group of the middle homology. The second part arises from bordism classes of mapping tori and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable J-homomorphism. We relate its calculation to a purely homotopy theoretic problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Some authors denote the \((n-1)\)-connected cover of a space X by \(X\langle n \rangle \), and so write \(\mathbf {MO}\langle n \rangle \) for the Thom spectrum associated to the \((n-1)\)-connected cover of BO. We emphasise that our notation is different.
References
Bak, A.: On Modules with Quadratic Forms, Algebraic \(K\)-Theory and its Geometric Applications (Conf. Hull, 1969), pp. 55–66. Springer, Berlin (1969)
Bak, A.: \(K\)-Theory of Forms, Annals of Mathematics Studies, vol. 98. Princeton University Press, Princeton, N.J. (1981)
Brumfiel, G.: On the homotopy groups of BPL and PL/O. Ann. Math. 88(2), 291–311 (1968)
Brumfiel, G.: On the homotopy groups of BPL and PL/O II. Topology 8, 305–311 (1969)
Brumfiel, G.: On the homotopy groups of BPL and PL/O III. Michigan Math. J. 17, 217–224 (1970)
Charney, R.: A generalization of a theorem of Vogtmann. In: Proceedings of the Northwestern Conference on Cohomology of Groups (Evanston, Ill., 1985), vol. 44, pp. 107–125 (1987)
Chern, S.S., Hirzebruch, F., Serre, J.-P.: On the index of a fibered manifold. Proc. Am. Math. Soc. 8, 587–596 (1957)
Crowley, D.J.: On the mapping class groups of # \(_r(S^p\times S^p)\) for \(p=3,7\). Math. Z. 269(3–4), 1189–1199 (2011)
Gritsenko, V., Hulek, K., Sankaran, G.K.: Abelianisation of orthogonal groups and the fundamental group of modular varieties. J. Algebra 322(2), 463–478 (2009)
Giambalvo, V.: On \(\langle 8\rangle \)-cobordism. Illinois J. Math. 15, 533–541 (1971)
Galatius, S., Randal-Williams, O.: Homological stability for moduli spaces of high dimensional manifolds. I. arXiv:1403.2334 (2014)
Galatius, S., Randal-Williams, O.: Stable moduli spaces of high dimensional manifolds. Acta Math. 212(2), 257–377 (2014)
Haefliger, A.: Plongements différentiables de variétés dans variétés. Comment. Math. Helv. 36, 47–82 (1961)
Johnson, D., Millson, J.J.: Modular Lagrangians and the theta multiplier. Invent. Math. 100(1), 143–165 (1990)
Kervaire, M.A., Milnor, J.W.: Groups of homotopy spheres. I. Ann. Math. 77(2), 504–537 (1963)
Kreck, M.: Isotopy Classes of diffeomorphisms of \((k-1)\)-connected almost parallelizable \(2k\)-manifolds. In: Algebraic Topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, 1979, pp. 643–663. Springer, Berlin (1978)
Krylov, N.A.: On the Jacobi group and the mapping class group of \(S^3\times S^3\). Trans. Am. Math. Soc. 355(1), 99–117 (2003)
Meyer, W.: Die Signatur von Flächenbündeln. Math. Ann. 201, 239–264 (1973)
Milnor, J., Husemoller, D.: Symmetric Bilinear Forms. Springer, New York, 1973, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73
Milnor, J.: Spin structures on manifolds. Enseign. Math. 9(2), 198–203 (1963)
Milnor, J.W., Kervaire, M.A.: Bernoulli numbers, homotopy groups, and a theorem of Rohlin. In: Proc. Internat. Congress Math. 1958, pp. 454–458. Cambridge Univ. Press, New York (1960)
Nariman, S.: Homological stability and stable moduli of flat manifold bundles. arXiv:1406.6416 (2014)
Paechter, G.F.: The groups \(\pi _{r}(V_{n,\,m})\). I. Quart. J. Math. Oxford Ser. 7(2), 249–268 (1956)
Powell, J.: Two theorems on the mapping class group of a surface. Proc. Am. Math. Soc. 68(3), 347–350 (1978)
Putman, A.: The Picard group of the moduli space of curves with level structures. Duke Math. J. 161(4), 623–674 (2012)
Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Co., New York-Toronto, Ont.-London (1966)
Stolz, S.: Hochzusammenhängende Mannigfaltigkeiten und ihre Ränder. Lecture Notes in Mathematics, vol. 1116. Springer, Berlin (1985)
Thurston, W.: Foliations and groups of diffeomorphisms. Bull. Am. Math. Soc. 80, 304–307 (1974)
Wall, C.T.C.: Classification of \((n-1)\)-connected \(2n\)-manifolds. Ann. Math. 75(2), 163–189 (1962)
Wall, C.T.C.: Diffeomorphisms of 4-manifolds. J. London Math. Soc. 39, 131–140 (1964)
Acknowledgments
The authors are grateful to Diarmuid Crowley for several useful discussions, both concerning attempts to prove Conjecture A, and especially concerning the extension problems discussed in Sect. 7. S. Galatius was partially supported by NSF grants DMS-1105058 and DMS-1405001, O. Randal-Williams was supported by the Herchel Smith Fund, and both authors were supported by ERC Advanced Grant No. 228082 and the Danish National Research Foundation through the Centre for Symmetry and Deformation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Galatius, S., Randal-Williams, O. Abelian quotients of mapping class groups of highly connected manifolds. Math. Ann. 365, 857–879 (2016). https://doi.org/10.1007/s00208-015-1300-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-015-1300-2