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Cobordism categories of manifolds with corners

Author: Josh Genauer
Journal: Trans. Amer. Math. Soc. 364 (2012), 519-550
MSC (2010): Primary 57R90, 57R19, 55N22, 55P47
Published electronically: August 2, 2011
MathSciNet review: 2833590
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Abstract: In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if Cob$ _{d,\langle k\rangle}$ is the category whose objects are a fixed dimension $ d$, with corners of codimension $ \leq\;k$, then we identify the homotopy type of the classifying space $ B$Cob$ _{d,\langle k\rangle}$ 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 $ X$, as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez (2006) and Hanbury.

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Additional Information

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

Received by editor(s): March 9, 2010
Received by editor(s) in revised form: September 9, 2010
Published electronically: August 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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