The homotopy type of the cobordism category

Soren Galatius, Ib Madsen, Ulrike Tillmann, Michael Weiss

Research output: Contribution to journalArticle

58 Citations (Scopus)

Abstract

We define cobordism categories of manifolds, and show that their classifying spaces are weakly homotopy equivalent to infinite loop spaces arising from Thom spectra. We generalise this to manifolds with tangential structures, for example oriented manifolds or manifolds with maps to a fixed space X. As an application, a new and shorter proof is given of the generalised Mumford conjecture concerning mapping class groups of surfaces.
Original languageEnglish
Pages (from-to)195-239
Number of pages45
JournalActa Mathematica
Volume202
Issue number2
DOIs
Publication statusPublished - Jun 2009

    Fingerprint

Cite this

Galatius, S., Madsen, I., Tillmann, U., & Weiss, M. (2009). The homotopy type of the cobordism category. Acta Mathematica, 202(2), 195-239. https://doi.org/10.1007/s11511-009-0036-9