Embedded Cobordism Categories and Spaces of Manifolds
Algebraic Topology
2011-09-23 v1 Geometric Topology
Abstract
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.
Keywords
Cite
@article{arxiv.0912.2505,
title = {Embedded Cobordism Categories and Spaces of Manifolds},
author = {Oscar Randal-Williams},
journal= {arXiv preprint arXiv:0912.2505},
year = {2011}
}