Volume 14, issue 3 (2010)

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Monoids of moduli spaces of manifolds

Søren Galatius and Oscar Randal-Williams

Geometry & Topology 14 (2010) 1243–1302
Abstract

We study categories of $d$–dimensional cobordisms from the perspective of Tillmann [Invent. Math. 130 (1997) 257–275] and Galatius, Madsen, Tillman and Weiss [Acta Math. 202 (2009) 195–239]. There is a category ${\mathsc{C}}_{\theta }$ of closed smooth $\left(d-1\right)$–manifolds and smooth $d$–dimensional cobordisms, equipped with generalised orientations specified by a map $\theta :X\to BO\left(d\right)$. The main result of [Acta Math. 202 (2009) 195–239] is a determination of the homotopy type of the classifying space $B{\mathsc{C}}_{\theta }$. The goal of the present paper is a systematic investigation of subcategories $\mathsc{D}\subseteq {\mathsc{C}}_{\theta }$ with the property that $B\mathsc{D}\simeq B{\mathsc{C}}_{\theta }$, the smaller such $\mathsc{D}$ the better.

We prove that in most cases of interest, $\mathsc{D}$ 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 $MT\theta$. This was known for certain special $\theta$, using homological stability results; our work is independent of such results and covers many more cases.

Keywords
cobordism category, surface bundles, topological monoids
Mathematical Subject Classification 2000
Primary: 57R90, 57R15, 57R56, 55P47