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Central extensions of smooth $2$–groups and a finite-dimensional string $2$–group

Christopher J Schommer-Pries

Geometry & Topology 15 (2011) 609–676

We provide a model of the String group as a central extension of finite-dimensional 2–groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naive 2–category of Lie groupoids, smooth functors, and smooth natural transformations. In particular this notion of smooth 2–group subsumes the notion of Lie 2–group introduced by Baez and Lauda [Theory Appl. Categ. 12 (2004) 423–491]. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by Segal [Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London (1970) 377–387], and our String 2–group is a special case of such extensions. There is a nerve construction which can be applied to these 2–groups to obtain a simplicial manifold, allowing comparison with the model of Henriques [arXiv:math/0603563]. The geometric realization is an A–space, and in the case of our model, has the correct homotopy type of String(n). Unlike all previous models, our construction takes place entirely within the framework of finite-dimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a canonical central extension of Spin(n).

$2$–group, stack, central extension, string group, gerbe
Mathematical Subject Classification 2000
Primary: 57T10, 22A22, 53C08
Secondary: 18D10
Received: 23 June 2010
Revised: 20 January 2011
Accepted: 18 February 2011
Published: 3 May 2011
Proposed: Haynes Miller
Seconded: Bill Dwyer, Ralph Cohen
Christopher J Schommer-Pries
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge MA 02139-4307