On the homotopy type of certain cobordism categories of surfaces

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.