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Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds. (English) Zbl 0885.57008

Summary: The main theorem shows that if \(M\) is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space \(B \text{Diff}(M \text{ rel } \partial M)\) of the space of diffeomorphisms of \(M\) which restrict to the identity map on boundary \(\partial M\) 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 \(\mathcal H(M \text{ rel } \partial M)\) 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.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
57M99 General low-dimensional topology
58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
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References:

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