Embedded Cobordism Categories and Spaces of Manifolds

Details Embedded Cobordism Categories and Spaces of Manifolds Embedded Cobordism Categories and Spaces of Manifolds

2009-12-13

arxiv.org/abs/0912.2505v1

Int. Math. Res. Not. IMRN 2011, no. 3, 572-608

Mathematics

Algebraic Topology

Scientific paper

10.1093/imrn/rnq072

Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the classifying space of the cobordism category with objects (d-1)-dimensional manifolds embedded in R^\infty. In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius, to identify the homotopy type of the cobordism category with objects (d-1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincare duality, relating a space of submanifolds of M to a space of functions on M.

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