Geometry & Topology

Finiteness of classifying spaces of relative diffeomorphism groups of 3–manifolds

Allen Hatcher and Darryl McCullough

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The main theorem shows that if M is an irreducible compact connected orientable 3–manifold with non-empty boundary, then the classifying space BDiff(MrelM) of the space of diffeomorphisms of M which restrict to the identity map on M has the homotopy type of a finite aspherical CW–complex. This answers, for this class of manifolds, a question posed by M Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group (MrelM) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.

Article information

Geom. Topol., Volume 1, Number 1 (1997), 91-109.

Received: 12 June 1997
Revised: 19 December 1997
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M99: None of the above, but in this section
Secondary: 55R35: Classifying spaces of groups and $H$-spaces 58D99: None of the above, but in this section

3–manifold diffeomorphism classifying space mapping class group homeotopy group geometrically finite torsion


Hatcher, Allen; McCullough, Darryl. Finiteness of classifying spaces of relative diffeomorphism groups of 3–manifolds. Geom. Topol. 1 (1997), no. 1, 91--109. doi:10.2140/gt.1997.1.91.

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