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Monoids of moduli spaces
of manifolds
Søren Galatius and Oscar Randal-Williams |
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Geometry & Topology 14 (2010) 1243–1302
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Abstract |
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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. |
Keywords
cobordism category, surface bundles, topological monoids
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Mathematical Subject Classification 2000
Primary: 57R90, 57R15, 57R56, 55P47
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References |
Publication
Received: 29 July 2009
Accepted: 19 April 2010
Published: 23 May 2010
Proposed: Haynes Miller
Seconded: Jesper Grodal, Bill Dwyer
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Authors |
| Søren Galatius | |
| Department of Mathematics Stanford University Stanford CA, 94305 United States |
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| Oscar Randal-Williams | |
| Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB United Kingdom |
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