Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T02:19:55.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Homology stability for symmetric diffeomorphism and mapping class groups

Published online by Cambridge University Press:  02 December 2015

ULRIKE TILLMANN
ULRIKE TILLMANN*
Affiliation:
Mathematical Institute, Oxford University, Andrew Wiles Building, Oxford, OX2 6GG. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 121 - 139
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[B]Bödigheimer, C.-F.Stable splittings of mapping spaces. Algebraic Topology (Seattle, Wash., 1985) Lecture Notes in Math. vol. 1286 (Springer, Berlin, 1987), pp. 174–187.CrossRefGoogle Scholar
[BT]Bödigheimer, C.-F. and Tillmann, U.Stripping and splitting decorated mapping class groups. Cohomological Methods in Homotopy Theory (Bellaterra, 1998), 47–57. Progr. Math. 196, (Birkhuser, Basel, 2001).Google Scholar
[CP]Cantero, F. and Palmer, M. On homological stability for configuration spaces on closed background manifolds, arXiv:1406.4916Google Scholar
[C1]Cerf, J.Sur les difféomorphismes de la sphere de dimension trois. Γ4 = 0. Lecture in Notes Math. 53 (Springer 1968).Google Scholar
[C2]Cerf, J.La stratification naturelle des espaces de fonctions différentiables réelles et le théorme de la pseudo-isotopie. (French) Inst. Hautes Études Sci. Publ. Math. 39 (1970), 5–173.CrossRefGoogle Scholar
[ES]Earle, C. F. and Schatz, A.Teichmüller theory for surfaces with boundary. J. Differential Geom. 4 (1970) 169–185.CrossRefGoogle Scholar
[GMTW]Galatius, S., Madsen, I., Tillmann, U. and Weiss, M.The homotopy type of the cobordism category. Acta Math. 202 (2009), no. 2, 195–239.CrossRefGoogle Scholar
[GRW1]Galatius, S. and Randal-Williams, O.Stable moduli spaces of high dimensional manifolds. Acta Math. 212 (2014), no. 2, 257–377.CrossRefGoogle Scholar
[GRW1]Galatius, S. and Randal-Williams, O. Homological stability for moduli spaces of high dimensional manifolds. arXiv:1203.6830Google Scholar
[H]Hatcher, A.A proof of the Smale conjecture. Diff(S 3) ≃ O(4). Ann. of Math. (2) 117 (1983), no. 3, 553–607.CrossRefGoogle Scholar
[HW]Hatcher, A. and Wahl, N.Stabilization for mapping class groups of 3-manifolds. Duke Math. J. 155 (2010), no. 2, 205–269.CrossRefGoogle Scholar
[I]Igusa, K.The stability theorem for smooth pseudoisotopies. K-Theory 2 (1988), no. 1–2, vi+355ppCrossRefGoogle Scholar
[KM]Kupers, A. and Miller, J.E n-cell attachments and a local-to-global principle for homological stability. arXiv:1405.7087Google Scholar
[L]Lima, E. L.On the local triviality of the restriction map for embeddings. Comment. Math. Helv. 38 (1964), 163–164.CrossRefGoogle Scholar
[MW]Madsen, I. and Weiss, M.The stable moduli space of Riemann surfaces: Mumford's conjecture. Ann. of Math. (2) 165 (2007), no. 3, 843–941.CrossRefGoogle Scholar
[MT]Manthorpe, R. and Tillmann, U.Tubular configurations: equivariant scanning and splitting. Jour. London Math. Soc. 90 (2014), 940–962.CrossRefGoogle Scholar
[McD]McDuff, D.Configuration spaces of positive and negative particles. Topology 14 (1975), 91–107.CrossRefGoogle Scholar
[MP]Miller, J. and Palmer, M. A twisted homology fibration criterion and the twisted group-completion theorem. arXiv:1409.4389.Google Scholar
[MM]Milnor, J. W. and Moore, J. C.On the structure of Hopf algebras. Ann. of Math. (2) 81 (1965), 211–264.CrossRefGoogle Scholar
[Pal]Palais, R.Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34 (1960) 305–312.CrossRefGoogle Scholar
[P]Palmer, M.Homological stability for oriented configuration spaces. Trans. Amer. Math. Soc. 365 (2013), no. 7, 3675–3711.CrossRefGoogle Scholar
[Pe]Perlmutter, N. Homological stability for the moduli spaces of products of spheres. To be published in Trans. Amer. Math. Soc.Google Scholar
[RW1]Randal-Williams, O.Homological stability for unordered configuration spaces. Q. J. Math. 64 (2013), no. 1, 303–326.CrossRefGoogle Scholar
[RW2]Randal-Williams, O.‘Group-completion’, local coefficient systems and perfection. Q. J. Math. 64 (2013), no. 3, 795–803.CrossRefGoogle Scholar
[Se]Segal, G.The topology of spaces of rational functions. Acta Math. 143 (1979), no. 1–2, 39–72.CrossRefGoogle Scholar
[Sm1]Smale, S.Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc. 10 (1959), 621–626.CrossRefGoogle Scholar
[Sm2]Smale, S.Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math. (2) 74 (1961) 391–406.CrossRefGoogle Scholar
[W]Waldhausen, F.Algebraic K-theory of spaces. In Algebraic and geometric topology (New Brunswick, N.J., 1983), 318–419, Lecture Notes in Math. no. 1126 (Springer, Berlin, 1985).CrossRefGoogle Scholar