Skip to main content Accessibility help
×
Home
Hostname: page-component-768ffcd9cc-727vs Total loading time: 0.327 Render date: 2022-12-03T07:31:59.878Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Homology stability for symmetric diffeomorphism and mapping class groups

Published online by Cambridge University Press:  02 December 2015

ULRIKE TILLMANN*
Affiliation:
Mathematical Institute, Oxford University, Andrew Wiles Building, Oxford, OX2 6GG. e-mail: Ulrike.Tillmann@maths.ox.ac.uk

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
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. 174187.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), 5173.CrossRefGoogle Scholar
[ES]Earle, C. F. and Schatz, A.Teichmüller theory for surfaces with boundary. J. Differential Geom. 4 (1970) 169185.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, 195239.CrossRefGoogle Scholar
[GRW1]Galatius, S. and Randal-Williams, O.Stable moduli spaces of high dimensional manifolds. Acta Math. 212 (2014), no. 2, 257377.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, 553607.CrossRefGoogle Scholar
[HW]Hatcher, A. and Wahl, N.Stabilization for mapping class groups of 3-manifolds. Duke Math. J. 155 (2010), no. 2, 205269.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), 163164.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, 843941.CrossRefGoogle Scholar
[MT]Manthorpe, R. and Tillmann, U.Tubular configurations: equivariant scanning and splitting. Jour. London Math. Soc. 90 (2014), 940962.CrossRefGoogle Scholar
[McD]McDuff, D.Configuration spaces of positive and negative particles. Topology 14 (1975), 91107.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), 211264.CrossRefGoogle Scholar
[Pal]Palais, R.Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34 (1960) 305312.CrossRefGoogle Scholar
[P]Palmer, M.Homological stability for oriented configuration spaces. Trans. Amer. Math. Soc. 365 (2013), no. 7, 36753711.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, 303326.CrossRefGoogle Scholar
[RW2]Randal-Williams, O.‘Group-completion’, local coefficient systems and perfection. Q. J. Math. 64 (2013), no. 3, 795803.CrossRefGoogle Scholar
[Se]Segal, G.The topology of spaces of rational functions. Acta Math. 143 (1979), no. 1–2, 3972.CrossRefGoogle Scholar
[Sm1]Smale, S.Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc. 10 (1959), 621626.Google Scholar
[Sm2]Smale, S.Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math. (2) 74 (1961) 391406.CrossRefGoogle Scholar
[W]Waldhausen, F.Algebraic K-theory of spaces. In Algebraic and geometric topology (New Brunswick, N.J., 1983), 318419, Lecture Notes in Math. no. 1126 (Springer, Berlin, 1985).CrossRefGoogle Scholar
3
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Homology stability for symmetric diffeomorphism and mapping class groups
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Homology stability for symmetric diffeomorphism and mapping class groups
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Homology stability for symmetric diffeomorphism and mapping class groups
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *