Embedded Cobordism Categories and Spaces of Manifolds

TL;DR

This paper extends the understanding of cobordism categories by identifying their homotopy types for submanifolds within a fixed background manifold, using advanced space-of-manifolds techniques and Poincare duality concepts.

Contribution

The paper applies space-of-manifolds techniques to determine the homotopy type of cobordism categories with submanifolds in a fixed manifold, generalizing previous results.

Findings

Homotopy type of cobordism category with submanifolds identified

Description in terms of sections of a bundle over M

Relation to Poincare duality between submanifolds and functions

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.