Monoids of moduli spaces of manifolds

Abstract

We study categories of d -dimensional cobordisms from the perspective of Tillmann [14] and Galatius, Madsen, Tillman and Weiss [6].There is a category C  of closed smooth .d1/-manifolds and smooth d -dimensional cobordisms, equipped with generalised orientations specified by a map ÂW X !BO.d /.The main result of [6] is a determination of the homotopy type of the classifying space BC  .The goal of the present paper is a systematic investigation of subcategories D  C  with the property that BD ' BC  , 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  -structure is the cohomology of the infinite loop space of a certain Thom spectrum MT .This was known for certain special  , using homological stability results; our work is independent of such results and covers many more cases.

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