#### 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

Email:

hatcher@math.cornell.edu

dmccullough@math.ou.edu

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