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

Abstract

This article has two purposes. In \cite{R3} (math.KT/0405211) we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by \cal 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 \cal 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 in {\cal C} there is a closed 3-manifold P such that either P is Seifert fibered or is a nonpositively curved 3-manifold and π_1(M) is a subgroup of π_1(P). As a consequence this proves that the FIC is true for any B-group (see definition 3.2 in \cite{R3}). Therefore, the FIC is true for any Haken 3-manifold group and hence for any 3-manifold group (using the reduction theorem of \cite{R3}) provided we assume the Geometrization conjecture. The above result also proves the FIC for a class of 4-manifold groups (see \cite{R2}(math.GT/0209119)). The second aspect of this article is to relax a condition in the definition of strongly poly-surface group (\cite{R1} (math.GT/0209118)) and define a new class of groups (we call them {\it 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 \cite{R1}.