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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.

##### 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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