Cobordism categories and moduli spaces of odd dimensional manifolds

TL;DR

This paper establishes a homology equivalence between the stable moduli space of certain high-dimensional manifolds and an infinite loopspace, using a novel cobordism category with surgery data.

Contribution

It introduces a new cobordism category incorporating surgery data, proving homology equivalence for a class of high-dimensional manifolds.

Findings

Homology equivalence to an infinite loopspace for specific manifolds

Development of a cobordism category with surgery data

Extension of results to manifolds with certain connectivity and parallelizability

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 $n \geq 4, n \neq 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.