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The space of immersed surfaces in a manifold

Published online by Cambridge University Press:  16 January 2013

OSCAR RANDAL–WILLIAMS
OSCAR RANDAL–WILLIAMS*
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB. e-mail: [email protected]
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Abstract

We study the cohomology of the space of immersed genus g surfaces in a simply-connected manifold. We compute the rational cohomology of this space in a stable range which goes to infinity with g. In fact, in this stable range we are also able to obtain information about torsion in the cohomology of this space, as long as we localise away from (g-1).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 3 , May 2013 , pp. 419 - 438
Copyright
Copyright © Cambridge Philosophical Society 2013

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References

REFERENCES

[Ada93]Adachi, M.Embeddings and Immersions. Transl. Math. Monogr, vol. 124 (American Mathematical Society, Providence, RI, 1993). Translated from the 1984 Japanese original by Kiki Hudson.Google Scholar
[BL83]Bass, H. and Lubotzky, A.Automorphisms of groups and of schemes of finite type Israel J. Math. 44 (1983), no. 1, 1–22.CrossRefGoogle Scholar
[Bol12]Boldsen, S.Improved homological stability for the mapping class group with integral or twisted coefficients. Math. Z. 270 (2012), 297–329.CrossRefGoogle Scholar
[CM09]Cohen, R. and Madsen, I.Surfaces in a background space and the homology of mapping class group. Proc. Symp. Pure Math. 80 (2009), no. 1, 43–76.CrossRefGoogle Scholar
[CM11]Cohen, R. and Madsen, I.Stability for closed surfaces in a background space. Homology Homotopy Appl. 13 (2011), no. 2, 301–313.CrossRefGoogle Scholar
[CMM91]Cervera, V., Mascaró, F. and Michor, P. W.The action of the diffeomorphism group on the space of immersions. Differential Geom. Appl. 1 (1991), no. 4, 391–401.CrossRefGoogle Scholar
[EE69]Earle, C. J. and Eells, J.A fibre bundle description of Teichmueller theory. J. Differential Geom. 3 (1969), 19–43.CrossRefGoogle Scholar
[ERW12]Ebert, J. and Randal-Williams, O.Stable cohomology of the universal Picard varieties and the extended mapping class group. Doc. Math. 17 (2012), 417–450.CrossRefGoogle Scholar
[GMTW09]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
[Joh80]Johnson, D.Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22 (1980), no. 2, 365–373.CrossRefGoogle Scholar
[Lai74]Lai, H. F.On the topology of the even-dimensional complex quadrics. Proc. Amer. Math. Soc. 46 (1974), 419–425.CrossRefGoogle Scholar
[Loo96]Looijenga, E.Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel–Jacobi map. J. Algebraic Geom. 5 (1996), no. 1, 135–150. 1358038 (97g:14026)Google Scholar
[MM05]Michor, P. W. and Mumford, D.Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10 (2005), 217–245 (electronic).CrossRefGoogle Scholar
[MT01]Madsen, I. and Tillmann, U.The stable mapping class group and Q(ℂ; ℙ∞+). Invent. Math. 145 (2001), no. 3, 509–544.CrossRefGoogle Scholar
[Mum83]Mumford, D.Towards an enumerative geometry of the moduli space of curves. Arithmetic and geometry. Vol. II, Progr. Math., vol. 36 (Birkhäuser Boston, Boston, MA, 1983), pp. 271–328.CrossRefGoogle Scholar
[MW07]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
[RW09]Randal–Williams, O. Resolutions of moduli spaces and homological stability. arXiv:0909.4278v3, 2009.Google Scholar
[RW10]Randal-Williams, O., Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces. arXiv:1001.5366, 2010.Google Scholar
[Sma59]Smale, S.The classification of immersions of spheres in Euclidean spaces. Ann. of Math. (2) 69 (1959), 327–344.CrossRefGoogle Scholar
[Swa60]Swan, R. G.The nontriviality of the restriction map in the cohomology of groups. Proc. Amer. Math. Soc. 11 (1960), 885–887.Google Scholar