Volume 14, issue 3 (2010)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 22
Issue 4, 1893–2510
Issue 3, 1267–1891
Issue 2, 645–1266
Issue 1, 1–644

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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