Monoids of moduli spaces of manifolds

By Søren Galatius, O. Randal-Williams. 2009

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4d38cf556317024c8a3098a9363a21b554a0c0bd

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Homology of the moduli spaces and mapping class groups of framed, r‐Spin and Pin surfaces (2010, background): “The left-hand side admits the structure of a topological monoid, under the ‘pair of pants’ product, and Galatius and the author [11] have shown that this map is a group-completion.”
Resolutions of moduli spaces and homological stability (2009, background): “As is now well known [16, 8, 5, 9], in both cases the stable homology coincides with that of the infinite loop space of a certain Thom spectrum.”
Semisimplicial spaces (2017, methodology): “Semi-simplicial spaces play an important role in the theory of moduli spaces of manifolds, beginning with [6] and [26], and continuing in [7], [9], [10].”
Homological stability for mapping class groups of surfaces (2010, background): “Let A(S,∆) be the simplicial complex whose vertices are isotopy classes of non-trivial arcs in S with boundary in ∆.”