Algebraic and Geometric Topology 2 (2002),
paper no. 21, pages 433-447.
Every orientable 3-manifold is a B\Gamma
Danny Calegari
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 .
Keywords.
Foliation, classifying space, groupoid, germs of homeomorphisms
AMS subject classification.
Primary: 57R32.
Secondary: 58H05.
DOI: 10.2140/agt.2002.2.433
E-print: arXiv:math.GT/0206066
Submitted: 25 March 2002.
Accepted: 28 May 2002.
Published: 29 May 2002.
Notes on file formats
Danny Calegari
Department of Mathematics, Harvard University
Cambridge MA, 02138, USA
Email: dannyc@math.harvard.edu
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