Singular cobordism categories

TL;DR

This paper extends the homotopy type identification of cobordism categories to include manifolds with prescribed singularities, linking singular cobordism categories to the Gromov h-principle and advancing understanding of their topological structure.

Contribution

It generalizes the main theorem on cobordism categories to singular manifolds and explores their relation to the Gromov h-principle.

Findings

Extended the homotopy type result to singular cobordism categories.

Connected singular cobordism categories to the Gromov h-principle.

Provided new insights into the topology of manifolds with singularities.

Abstract

Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded d-dimensional manifolds [7] for each positive integer d. Their result lead to a new proof of the generalized standard Mumford conjecture. We extend the main theorem of [7] to the case of cobordism categories of embedded d-dimensional manifolds with prescribed singularities, and explain the relation of singular cobordism categories to the bordism version of the Gromov h-principle.