Monoids of moduli spaces of manifolds

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

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 C¿ of closed smooth
(d - 1)–manifolds and smooth d–dimensional cobordisms, equipped with generalised orientations specified by a map
¿: X ¿ BO(d). The main result of [Acta Math. 202 (2009) 195–239] 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.
OriginalsprogEngelsk
TidsskriftGeometry & Topology
Vol/bind14
Sider (fra-til)1243-1302
ISSN1465-3060
DOI
StatusUdgivet - 2010
Eksternt udgivetJa

Bibliografisk note

Paper id:: 10.2140/gt.2010.14.1243

ID: 22502949