The homotopy type of the topological cobordism category

TL;DR

This paper establishes the homotopy type of the topological cobordism category's classifying space, connecting it to Thom spectra and smoothing theory, extending previous smooth results to the topological setting.

Contribution

It proves the homotopy equivalence of the classifying space of the topological cobordism category to a Thom spectrum, generalizing smooth cobordism results to topological manifolds.

Findings

Classifying space of topological cobordism category is weakly equivalent to a Thom spectrum.

Extension of smooth cobordism results to topological manifolds.

Inclusion of tangential structures and boundary cases.

Abstract

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $\Omega^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum of the inverse of the canonical bundle over $BTop(d)$. We also give versions with tangential structures and boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius-Madsen-Tillman-Weiss.