On the homotopy type of certain cobordism categories of surfaces

TL;DR

This paper investigates the homotopy types of classifying spaces of cobordism categories of surfaces, revealing their equivalence to infinite loop spaces related to Thom spectra, thus advancing understanding of surface cobordisms in topology.

Contribution

It identifies the homotopy types of specific cobordism categories of surfaces with infinite loop spaces associated to Thom spectra, extending prior work on related categories.

Findings

Homotopy type of $ ext{B}\mathcal{A}_{0,2}$ matches that of an infinite loop space.

Homotopy equivalence of $ ext{B}\mathbb{A}_2$ with a positive boundary cobordism category.

Connection established between cobordism categories and Thom spectra.

Abstract

Let $\mathcal{A}_{g,d}$ be the (topological) cobordism category of orientable surfaces whose connected components are homeomorphic to either $S^1 \times I$ with one incoming and one outgoing boundary component or the surface $\Sigma_{g,d}$ of genus $g$ and $d$ boundary components that are all incoming. In this paper, we study the homotopy type of the classifying space of the cobordism categories $\mathcal{A}_{g,d}$ and the associated (ordinary) cobordism categories of their connected components $\mathbb{A}_d$. $\mathcal{A}_{0,2}$ is the cobordism category of complex annuli that was considered by Costello and $\mathbb{A}_2$ is homotopy equivalent with the positive boundary 1-dimensional embedded cobordism category of Galatius-Madsen-Tillmann-Weiss. We identify their homotopy type with the infinite loop spaces associated with certain Thom spectra.