Monoids of moduli spaces of manifolds

Joint work with Soren Galatius. We study categories C of d-dimensional cobordisms, from the perspective of Galatius, Madsen, Tillmann and Weiss. Their main result is the determination of the homotopy type of the classifying-space of such cobordism categories, as the infinite loop space of a certain Thom spectrum. One can investigate subcategories D of C having the property that the classifying-space BD is equivalent to BC, the smaller such D one can find the better.

We prove that in may cases of interest, D can be taken to be a homotopy commutative monoid. As a consequence, the stable cohomology of many moduli spaces of surfaces can be identified with that of the infinite loop space of certain Thom spectra.