Cobordism categories and moduli spaces of odd dimensional manifolds

Abstract

We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n\ge 4,n\ne 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.

57 pages. Minor revision, to appear in Advances in Mathematics