Stable moduli spaces of odd dimensional manifolds

Understanding characteristic classes of manifolds bundles is one of the key topics in the
study of manifolds and their automorphisms. The advent of cobordism categories through
the work of Madsen, Tillmann and Weiss introduced a new method for studying such groups.
Its application in high dimensions was spearheaded by Galatius and Randal-Williams in the
case of highly connected even-dimensional manifolds by identifying groups of stable
characteristic classes with the cohomology of certain computationally accessible infinite
loopspaces. Results of Ebert, however, sharply limited the efficacy of usual cobordism
categories in odd dimensions.
I will report on joint work with Nathan Perlmutter on an enhancement of odd dimensional
cobordism categories, that avoid these difficulties. In particular our results imply that stable
characteristic classes also form the cohomology of a certain infinite loopspace in this case.
In contrast to the even dimensional situation, however, the homotopy type of this space
remains largely unidentified and its exploration is work in progress.