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The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups

Published online by Cambridge University Press:  30 November 2007

Sayed K. Roushon
Sayed K. Roushon
Affiliation:
roushon@math.tifr.res.inhttp://www.math.tifr.res.in/~roushon/paper.htmlSchool of Mathematics, Tata Institute, Homi Bhabha Road, Mumbai 400005, India
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Abstract

This article has two purposes. In [15] we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M ε C there is a closed 3-manifold P such that either P is Seifert fibered or is a nonpositively curved 3-manifold and π1(M) is a subgroup of π1(P). As a consequence it is obtained that the FIC is true for any B-group (see definition 4.2 in [15]). Therefore, the FIC is true for any Haken 3-manifold group and hence for any 3-manifold group (using the reduction theorem of [15]) provided we assume the Geometrization conjecture. The above result also proves the FIC for a class of 4-manifold groups (see [14]).

The second aspect of this article is to relax a condition in the definition of strongly poly-surface group ([13]) and define a new class of groups (we call them weak strongly poly-surface groups). Then using the above result we prove the FIC for any virtually weak strongly poly-surface group. We also give a corrected proof of the main lemma of [13].

Keywords

strongly poly-surface groupfibered isomorphism conjecture3-manifold groupselliptic surfaces4-manifold groupspseudoisotopy functor
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 1 , February 2008 , pp. 83 - 93
Copyright
Copyright © ISOPP 2008

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