Acta Mathematica

The homotopy type of the cobordism category

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].

Note

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.

Article information

Source
Acta Math., Volume 202, Number 2 (2009), 195-239.

Dates
Received: 30 April 2007
Revised: 30 September 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892397

Digital Object Identifier
doi:10.1007/s11511-009-0036-9

Mathematical Reviews number (MathSciNet)
MR2506750

Zentralblatt MATH identifier
1221.57039

Rights
2009 © Institut Mittag-Leffler

Citation

Galatius, Søren; Madsen, Ib; Tillmann, Ulrike; Weiss, Michael. The homotopy type of the cobordism category. Acta Math. 202 (2009), no. 2, 195--239. doi:10.1007/s11511-009-0036-9. https://projecteuclid.org/euclid.acta/1485892397

References

• B auer, T., An infinite loop space structure on the nerve of spin bordism categories. Q. J. Math., 55 (2004), 117–133.
• B röcker, T. & J änich, K., Introduction to Differential Topology. Cambridge University Press, Cambridge, 1982.
• C ohen, R. & M adsen, I., Surfaces in a background space and the homology of mapping class groups. Preprint, 2006.
• G alatius, S., Mod 2 homology of the stable spin mapping class group. Math. Ann., 334 (2006), 439–455.
• H arer, 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.
• I vanov, 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.
• K riegl, A. & M ichor, P.W., The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, 53. Amer. Math. Soc., Providence, RI, 1997.
• M adsen, I. & T illmann, U., The stable mapping class group and $Q\left( {\mathbb{C}P^{\infty } } \right)$. Invent. Math., 145 (2001), 509–544.
• M adsen, I. & W eiss, M., The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math., 165 (2007), 843–941.
• M cD uff, D. & S egal, G., Homology fibrations and the “group-completion” theorem. Invent. Math., 31 (1975/76), 279–284.
• M ilnor, J.W., The geometric realization of a semi-simplicial complex. Ann. of Math., 65 (1957), 357–362.
• M ilnor, J.W. & S tasheff, J. D., Characteristic Classes. Annals of Mathematics Studies, 76. Princeton University Press, Princeton, NJ, 1974.
• P hillips, A., Submersions of open manifolds. Topology, 6 (1967), 171–206.
• S egal, 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.
• S teenrod, N., The Topology of Fibre Bundles. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1951.
• T illmann, U., On the homotopy of the stable mapping class group. Invent. Math., 130 (1997), 257–275.
• W ahl, N., Homological stability for the mapping class groups of non-orientable surfaces. Invent. Math., 171 (2008), 389–424.