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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05474-7

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 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.

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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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05474-7

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.