Algebraic & Geometric Topology

Every orientable 3–manifold is a $\mathrm{B}\Gamma$

Danny Calegari

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We show that every orientable 3–manifold is a classifying space BΓ where Γ is a groupoid of germs of homeomorphisms of . This follows by showing that every orientable 3–manifold M admits a codimension one foliation such that the holonomy cover of every leaf is contractible. The we construct can be taken to be C1 but not C2. The existence of such an 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= BΓ for some C groupoid Γ.

Article information

Algebr. Geom. Topol., Volume 2, Number 1 (2002), 433-447.

Received: 25 March 2002
Accepted: 28 May 2002
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10]
Secondary: 58H05: Pseudogroups and differentiable groupoids [See also 22A22, 22E65]

foliation classifying space groupoid germs of homeomorphisms


Calegari, Danny. Every orientable 3–manifold is a $\mathrm{B}\Gamma$. Algebr. Geom. Topol. 2 (2002), no. 1, 433--447. doi:10.2140/agt.2002.2.433.

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