Abstract
The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].
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Bauer, T., An infinite loop space structure on the nerve of spin bordism categories. Q. J. Math., 55 (2004), 117–133.
Bröcker, T. & Jänich, K., Introduction to Differential Topology. Cambridge University Press, Cambridge, 1982.
Cohen, R. & Madsen, I., Surfaces in a background space and the homology of mapping class groups. Preprint, 2006. arXiv:math/0601750v4 [math.GT].
Galatius, S., Mod 2 homology of the stable spin mapping class group. Math. Ann., 334 (2006), 439–455.
Harer, J. L., Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math., 121 (1985), 215–249.
— Stability of the homology of the moduli spaces of Riemann surfaces with spin structure. Math. Ann., 287 (1990), 323–334.
Ivanov, N.V., Stabilization of the homology of Teichmüller modular groups. Algebra i Analiz, 1:3 (1989), 110–126 (Russian); English translation in Leningrad Math J., 1 (1990), 675–691.
Kriegl, A. & Michor, P.W., The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, 53. Amer. Math. Soc., Providence, RI, 1997.
Madsen, I. & Tillmann, U., The stable mapping class group and \( Q\left( {\mathbb{C}P^{\infty } } \right) \). Invent. Math., 145 (2001), 509–544.
Madsen, I. & Weiss, M., The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math., 165 (2007), 843–941.
McDuff, D. & Segal, G., Homology fibrations and the “group-completion” theorem. Invent. Math., 31 (1975/76), 279–284.
Milnor, J.W., The geometric realization of a semi-simplicial complex. Ann. of Math., 65 (1957), 357–362.
Milnor, J.W. & Stasheff, J. D., Characteristic Classes. Annals of Mathematics Studies, 76. Princeton University Press, Princeton, NJ, 1974.
Phillips, A., Submersions of open manifolds. Topology, 6 (1967), 171–206.
Segal, G., Classifying spaces and spectral sequences. Inst. Hautes Études Sci. Publ. Math., 34 (1968), 105–112.
— The definition of conformal field theory, in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., 308, pp. 421–577. Cambridge Univ. Press, Cambridge, 2004.
Steenrod, N., The Topology of Fibre Bundles. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1951.
Tillmann, U., On the homotopy of the stable mapping class group. Invent. Math., 130 (1997), 257–275.
Wahl, N., Homological stability for the mapping class groups of non-orientable surfaces. Invent. Math., 171 (2008), 389–424.
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S. Galatius was partially supported by NSF grant DMS-0505740 and the Clay Institute.
M. Weiss was partially supported by the Royal Society and the Mittag-Leffler Institute.
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Galatius, S., Madsen, I., Tillmann, U. et al. The homotopy type of the cobordism category. Acta Math 202, 195–239 (2009). https://doi.org/10.1007/s11511-009-0036-9
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DOI: https://doi.org/10.1007/s11511-009-0036-9