Monoids of moduli spaces of manifolds

Soren Galatius, Oscar Randal-Williams

Published 2009-05-18, updated 2009-07-27Version 2

We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_\theta$ of closed smooth (d-1)-manifolds and smooth d-dimensional cobordisms, equipped with generalised orientations specified by a fibration $\theta : X \to BO(d)$. The main result of GMTW is a determination of the homotopy type of the classifying space $BC_\theta$. The goal of the present paper is a systematic investigation of subcategories $D$ of $C_\theta$ having classifying space homotopy equivalent to that of $C_\theta$, the smaller such $D$ the better. We prove that in most cases of interest, $D$ can be chosen to be a homotopy commutative monoid. As a consequence we prove that the stable cohomology of many moduli spaces of surfaces with $\theta$-structure is the cohomology of the infinite loop space of a certain Thom spectrum. This was known for certain special $\theta$, using homological stability results; our work is independent of such results and covers many more cases.