Cobordism Categories and their Classifying Spaces

I will report on recent work with Ib Madsen, Soren Galatius, and Michael Weiss in which we show that the cobordism category of d - 1 dimensional closed, compact manifolds and their d dimensional cobordisms is the infinite loop space associated to the spectrum MO_d shifted by d - 1 dimensions. There is a similar theorem in the oriented case.

This work was motivated by the case d = 2 which plays an important role in topological conformal field theory and recently led to the proof of the Mumford conjecture by Madsen and Weiss.