Abstract
We study categories of –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 of closed smooth –manifolds and smooth –dimensional cobordisms, equipped with generalised orientations specified by a map . The main result of [Acta Math. 202 (2009) 195–239] is a determination of the homotopy type of the classifying space . The goal of the present paper is a systematic investigation of subcategories with the property that , the smaller such the better.
We prove that in most cases of interest, 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 . This was known for certain special , using homological stability results; our work is independent of such results and covers many more cases.
Citation
Søren Galatius. Oscar Randal-Williams. "Monoids of moduli spaces of manifolds." Geom. Topol. 14 (3) 1243 - 1302, 2010. https://doi.org/10.2140/gt.2010.14.1243
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