On the Homotopy Type of Certain Cobordism Categories of Surfaces

Abstract

Let be the cobordism category of Riemann surfaces whose connected components are diffeomorphic to either S(1)xI 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 category and the associated cobordism category of its connected components is the cobordism category of complex annuli, which was considered by Costello, and is homotopy equivalent to the positive-boundary one-dimensional embedded cobordism category of Galatius-Madsen-Tillmann-Weiss. We identify their homotopy types with the infinite loop spaces associated with certain Thom spectra.