## Geometry & Topology

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

#### Abstract

The main theorem shows that if $M$ is an irreducible compact connected orientable 3–manifold with non-empty boundary, then the classifying space $BDiff(Mrel∂M)$ of the space of diffeomorphisms of $M$ which restrict to the identity map on $∂M$ 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 $ℋ(Mrel∂M)$ 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.

#### Article information

Source
Geom. Topol., Volume 1, Number 1 (1997), 91-109.

Dates
Received: 12 June 1997
Revised: 19 December 1997
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882847

Digital Object Identifier
doi:10.2140/gt.1997.1.91

Mathematical Reviews number (MathSciNet)
MR1486644

Zentralblatt MATH identifier
0885.57008

#### Citation

Hatcher, Allen; McCullough, Darryl. Finiteness of classifying spaces of relative diffeomorphism groups of 3–manifolds. Geom. Topol. 1 (1997), no. 1, 91--109. doi:10.2140/gt.1997.1.91. https://projecteuclid.org/euclid.gt/1513882847

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