Monoids of moduli spaces of manifolds

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 $B{C}_{\theta }$. 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.

52 pages, 5 figures; v2: extended discussion of applications

References in corpus (1)

• Stable homology of automorphism groups of free groups

Cited by in corpus (31)

• Homological stability for moduli spaces of high dimensional manifolds. I
• A note on the $(\infty ,n)$-category of cobordisms
• Semi-simplicial spaces
• The positive scalar curvature cobordism category
• Stable Moduli Spaces of High Dimensional Handlebodies
• The cohomology of Torelli groups is algebraic
• On the infinite loop space structure of the cobordism category
• A short exposition of the Madsen-Weiss theorem
• Stable homology of surface diffeomorphism groups made discrete
• The Picard group of the moduli space of r-Spin Riemann surfaces
• The stable cohomology of automorphisms of free groups with coefficients in the homology representation
• Homological stability for spaces of embedded surfaces
• Algebraic independence of generalized Morita-Miller-Mumford classes
• Cobordism Categories and Parametrized Morse Theory
• A geometric interpretation of the homotopy groups of the cobordism category
• The homotopy type of the PL cobordism category. I
• Lectures on invertible field theories
• Advanced topics in gauge theory: mathematics and physics of Higgs bundles
• The Equivariant Cobordism Category
• Stability phenomena in the topology of moduli spaces
• The bordism version of the h-principle
• Invariance properties of Miller-Morita-Mumford characteristic numbers of fibre bundles
• Parametrized Morse Theory and Positive Scalar Curvature
• The space of short ropes and the classifying space of the space of long knots
• Parametrized geometric cobordism and smooth Thom stacks
• The space of traces in symmetric monoidal infinity categories
• The classifying space of the one-dimensional bordism category and a cobordism model for TC of spaces
• The homotopy type of the PL cobordism category. II
• The weak b-principle: Mumford conjecture
• The Mumford conjecture (after Bianchi)
• Semisimple Field Theories Detect Stable Diffeomorphism