Abstract: In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if Cob is the category whose objects are a fixed dimension , with corners of codimension , then we identify the homotopy type of the classifying space Cob as the zero space of a homotopy colimit of a certain diagram of the Thom spectra. We also identify the homotopy type of the corresponding cobordism category when an extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss (2009), and their proofs are an adaptation of the methods of their paper. As an application we describe the homotopy type of the category of open and closed strings with a background space , as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez (2006) and Hanbury.
[CM]Ralph
L. Cohen and Ib
Madsen, Surfaces in a background space and the homology of mapping
class groups, Algebraic geometry—Seattle 2005. Part 1, Proc.
Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009,
pp. 43–76. MR 2483932
(2010d:57002)
[Cohar]
Ralph Cohen. The floer homotopy type of the cotangent bundle. Pure and Applied Math. Quarterly, issue in honor of M. Atiyah and I. Singer (arxiv.org/abs/math/0702852) (to appear).
[Han]
Elizabeth Hanbury. Homological stability of mapping class groups and open-closed cobordism categories. University of Oxford, Ph.D. Thesis (to appear).
[Seg04]Graeme
Segal, The definition of conformal field theory, Topology,
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vol. 308, Cambridge Univ. Press, Cambridge, 2004,
pp. 421–577. MR 2079383
(2005h:81334)
N.A. Baas, Ralph Cohen, and A. Ramirez. The topology of the category of open and closed strings. Contemporary Math., AMS, 407:11-26, 2006. MR 2248970 (2007f:55006)
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Ralph Cohen and Ib Madsen. Surfaces in a background space and the homology of mapping class groups. Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009. MR 2483932 (2010d:57002)
Ralph Cohen. The floer homotopy type of the cotangent bundle. Pure and Applied Math. Quarterly, issue in honor of M. Atiyah and I. Singer (arxiv.org/abs/math/0702852) (to appear).
Kevin Costello. Topological conformal field theories and Calabi-Yau categories. Advances in Mathematics, 210 (1): 165-214, 2007. MR 2298823 (2008f:14071)
Søren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss. The homotopy type of the cobordism category. Acta Math. 202 (2): 195-239, 2009. MR 2506750
John M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Studies in Mathematics. Springer-Verlag New York, Inc., 2003. MR 1930091 (2003k:58001)
G. Moore. Some comments on branes, -flux, and -theory, in strings 2000. Proceedings of the International Superstrings Conference (Ann Arbor, MI), 16:936-944, 2001. MR 1827965 (2002c:81167)
Graeme Segal. The definition of conformal field theory, in topology, geometry and quantum field theory. London Math. Soc. Lecture Note Series, 308:421-577, 2004. First circulated in 1988. MR 2079383 (2005h:81334)
Josh Genauer Affiliation:
Department of Mathematics, CINVESTAV, Av. Instituto Politécnico Nacional No. 258, San Pedro Zacatenco, Mexico
Address at time of publication:
2023 7th Street, Apt. B, Berkeley, California 94710