Monoids of moduli spaces of manifolds

TL;DR

This paper investigates the algebraic structures of moduli spaces of manifolds via cobordism categories, revealing that many such spaces have homotopy equivalent classifying spaces that form homotopy commutative monoids, leading to new cohomological insights.

Contribution

It systematically studies subcategories of cobordism categories with equivalent classifying spaces, showing they can often be modeled as homotopy commutative monoids, and extends cohomological results to many new cases.

Findings

Subcategories with classifying spaces homotopy equivalent to the original can be chosen as homotopy commutative monoids.

Stable cohomology of moduli spaces aligns with the cohomology of infinite loop spaces of Thom spectra.

Results extend cohomological understanding to many cases beyond those covered by homological stability.

Abstract

We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_\theta$ of closed smooth (d-1)-manifolds and smooth d-dimensional cobordisms, equipped with generalised orientations specified by a fibration $\theta : X \to BO(d)$. The main result of GMTW is a determination of the homotopy type of the classifying space $BC_\theta$. The goal of the present paper is a systematic investigation of subcategories $D$ of $C_\theta$ having classifying space homotopy equivalent to that of $C_\theta$, the smaller such $D$ the better. We prove that in most cases of interest, $D$ can be chosen to be a homotopy commutative monoid. As a consequence we prove that the stable cohomology of many moduli spaces of surfaces with $\theta$-structure is the cohomology of the infinite loop space of a certain Thom spectrum. This was known for certain special $\theta$, using homological stability results; our work is independent of such results and covers many more cases.