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Every orientable 3-manifold is a B\Gamma

Calegari, Danny (2002) Every orientable 3-manifold is a B\Gamma. Algebraic and Geometric Topology, 2 (21). pp. 433-447. ISSN 1472-2747. http://resolver.caltech.edu/CaltechAUTHORS:CALagt02

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Abstract

We show that every orientable 3-manifold is a classifying space B\Gamma where \Gamma is a groupoid of germs of homeomorphisms of R. This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. The F we construct can be taken to be C^1 but not C^2. The existence of such an F answers positively a question posed by Tsuboi [Classifying spaces for groupoid structures, notes from minicourse at PUC, Rio de Janeiro (2001)], but leaves open the question of whether M = B\Gamma for some C^\infty groupoid \Gamma.


Item Type:Article
Additional Information:Submitted: 25 March 2002. Accepted: 28 May 2002. Published: 29 May 2002. In August 2001, the P.U.C. in Rio de Janeiro held a conference on Foliations and Dynamics aimed at bringing together traditional foliators and 3-manifold topologists. At this conference, Takashi Tsuboi posed the question of the existence of typical foliations on 3-manifolds. I would very much like to thank Takashi for posing this question, for reading an early draft of this paper and catching numerous errors, and for introducting me to the beautiful subject of classical foliation theory. I'd also like to thank Curt McMullen for a useful conversation which helped clarify some analytic questions for me. E-print: arXiv:math.GT/0206066
Subject Keywords:Foliation, classifying space, groupoid, germs of homeomorphisms
Record Number:CaltechAUTHORS:CALagt02
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:CALagt02
Alternative URL:http://dx.doi.org/10.2140/agt.2002.2.433
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1249
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:06 Jan 2006
Last Modified:26 Dec 2012 08:43

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