Volume 1, issue 1 (1997)

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

Allen Hatcher and Darryl McCullough

Geometry & Topology 1 (1997) 91–109

arXiv: math.GT/9712260


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.

3–manifold, diffeomorphism, classifying space, mapping class group, homeotopy group, geometrically finite, torsion
Mathematical Subject Classification
Primary: 57M99
Secondary: 55R35, 58D99
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Received: 12 June 1997
Revised: 19 December 1997
Published: 21 December 1997
Proposed: Robion Kirby
Seconded: Joan Birman, David Gabai
Allen Hatcher
Department of Mathematics, Cornell University
New York 14853
Darryl McCullough
Department of Mathematics, University of Oklahoma
Oklahoma 73019