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.

Notes on file formats

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

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


GT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to