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Monoids of moduli spaces of manifolds

Søren Galatius and Oscar Randal-Williams

Geometry & Topology 14 (2010) 1243–1302

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 Cθ of closed smooth (d 1)–manifolds and smooth d–dimensional cobordisms, equipped with generalised orientations specified by a map θ: X BO(d). The main result of [Acta Math. 202 (2009) 195–239] is a determination of the homotopy type of the classifying space BCθ. The goal of the present paper is a systematic investigation of subcategories DCθ with the property that BD BCθ, 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 θ–structure is the cohomology of the infinite loop space of a certain Thom spectrum MTθ. This was known for certain special θ, using homological stability results; our work is independent of such results and covers many more cases.

cobordism category, surface bundles, topological monoids
Mathematical Subject Classification 2000
Primary: 57R90, 57R15, 57R56, 55P47
Received: 29 July 2009
Accepted: 19 April 2010
Published: 23 May 2010
Proposed: Haynes Miller
Seconded: Jesper Grodal, Bill Dwyer
Søren Galatius
Department of Mathematics
Stanford University
Stanford CA, 94305
United States
Oscar Randal-Williams
Mathematical Institute
24-29 St Giles’
United Kingdom