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

Details Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds

1997-12-20

arxiv.org/abs/math/9712260v2

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

Mathematics

Geometric Topology

19 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol1/paper7.abs.html

Scientific paper

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 dM) of the space of diffeomorphisms of M which restrict to the identity map on boundary(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 dM) 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.

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