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

The main theorem shows that if $M$ is an irreducible compact connected orientable 3–manifold with non-empty boundary, then the classifying space $BDiff (M rel \partial M)$ of the space of diffeomorphisms of $M$ which restrict to the identity map on $\partial M$ 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 $ℋ (M rel \partial M)$ 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.

Keywords

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

Mathematical Subject Classification

Primary: 57M99

Secondary: 55R35, 58D99

References

Forward citations

Publication

Received: 12 June 1997

Revised: 19 December 1997

Published: 21 December 1997

Proposed: Robion Kirby

Seconded: Joan Birman, David Gabai

Authors

Allen Hatcher
Department of Mathematics, Cornell University Ithaca New York 14853 USA

Darryl McCullough
Department of Mathematics, University of Oklahoma Norman Oklahoma 73019 USA

Volume 1, issue 1 (1997)

Allen Hatcher and Darryl McCullough