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

Hatcher, Allen¹ ; McCullough, Darryl²

¹ Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
² Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA

Geometry & topology, Tome 1 (1997) no. 1, pp. 91-109.

Résumé

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 $\mathcal{\mathscr{H}}\left(Mrel\partial M\right)$ 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.

Détail

DOI : 10.2140/gt.1997.1.91

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

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