#### 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
##### Abstract

The main theorem shows that if $M$ is an irreducible compact connected orientable 3–manifold with non-empty boundary, then the classifying space $BDiff\left(Mrel\partial M\right)$ 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 $\mathsc{ℋ}\left(Mrel\partial M\right)$ 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