Geometry & Topology, Vol. 1 (1997) Paper no. 7, pages 91-109.

Finiteness of Classifying Spaces of Relative Diffeomorphism Groups of 3-Manifolds

Allen Hatcher and Darryl McCullough

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(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.

Keywords. 3-manifold, diffeomorphism, classifying space, mapping class group, homeotopy group, geometrically finite, torsion

AMS subject classification. Primary: 57M99 Secondary: 55R35, 58D99

DOI: 10.2140/gt.1997.1.91

E-print: arXiv:math.GT/9712260

Submitted to GT on June 12, 1997. Revised 19 December, 1997. Accepted 20 December, 1997.

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Allen Hatcher
Department of Mathematics
Cornell University
Ithaca, NY 14853, USA

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

Email:
hatcher@math.cornell.edu
dmccullough@math.ou.edu

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