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The topology of the category of open and closed strings. (English) Zbl 1108.55008

Ádem, Alejandro (ed.) et al., Recent developments in algebraic topology. A conference to celebrate Sam Gitler’s 70th birthday, San Miguel de Allende, México, December 3–6, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3676-5/pbk). Contemporary Mathematics 407, 11-26 (2006).
Author’s summary: We study the topology of the cobordism category \({\mathcal S}^{\text{oc}}\) of open and closed strings. This is a 2-category in which the objects are compact one-manifolds whose boundary components are labeled by an indexing set (the set of “\(D\)-branes”), the one-morphisms are cobordisms of manifolds with boundary, and the two-morphisms are diffeomorphisms of the surface cobordisms. Our methods and techniques are direct generalizations of those used by U. Tillmann in her study of the category of closed strings. We input the striking theorem of Madsen and Weiss regarding the topology of the stable mapping class group to identify the homotopy type of the geometric realization \(|{\mathcal S}^{\text{oc}}|\) as an infinite loop space.
For the entire collection see [Zbl 1094.55500].

MSC:

55P47 Infinite loop spaces
57M99 General low-dimensional topology
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