The homotopy type of the cobordism category. (English) Zbl 1221.57039
The authors study the homotopy type of embedded cobordism categories. These categories were introduced by G. Segal in order to study topological field theories. The main result identifies the homotopy type of the classifying space of the embedded \(d\)-dimensional cobordism category for all \(d\) together with a Thom spectrum. The proof is based upon a careful study of spaces of manifolds and a fantastic generalization of the Thom-Pontryagin collapse map. For \(d=2\), their results lead to a new proof of the generalized Mumford conjecture.
Reviewer: David Chataur (Villeneuve d’Ascq)
MSC:
| 57R90 | Other types of cobordism |
| 55P15 | Classification of homotopy type |
| 57R75 | \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism |
| 55P47 | Infinite loop spaces |
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