Let M be a smooth manifold, let Diff(M) denote the group of diffeomorphisms of M, and let BDiff(M) denote the classifying space which in the literature is often called the moduli space of manifolds of type M. For any paracompact space X, there is a bijection between the set of homotopy classes of maps from X to BDiff(M) and the set of (smooth) fibre-bundles over X with fibre diffeomorphic to M. In order to understand fibre bundles it suffices to understand the homotopy/homology of the moduli space BDiff(M).

For arbitrary M, the space BDiff(M) is extraordinarily complicated and very little information is known about its homotopy/homology in general. With the work of Madsen and Weiss in their proof of the Mumford conjecture, and the recent work of Galatius and Randal-Williams on the moduli spaces of high dimensional manifolds, much information is now known about the cohomology H^{*}(BDiff(M)) for certain manifolds M of dimension 2n. In particular, the methods of the above mentioned authors establish a homological equivalence between the space BDiff((S^{n}x S^{n})^{#g}) and the infinite loopspace of a certain Thom spectrum, in the limit as g approaches infinity.

To this day, no theorem analogous to the one discussed above yet exists for the diffeomorphism groups of odd dimensional manifolds. Many of the techniques employed by the above mentioned authors break down in the odd dimensional case and much of the conceptual territory is left uncharted. In my talk I will discuss recent work of mine joint with F. Hebestreit, where we prove that BDiff(M) is homology equivalent to an infinite loopspace for a certain class of odd dimensional manifolds M, after a certain stabilization process. Our results can be viewed as the first step toward obtaining an analogue of the Madsen-Weiss theorem for odd dimensional manifolds. I plan to discuss how our constructions relate to the cobordism categories studied by Galatius, Madsen, Tillmann, and Weiss, and also how they relate to certain classical objects from surgery theory, such as L-theory and the surgery exact sequence.