We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones
corresponding to the stable topological pseudoisotopy functor is true for the
fundamental groups of a large class of 3-manifolds. We also prove that if the
FIC is true for irreducible 3-manifold groups then it is true for all
3-manifold groups. In fact, this follows from a more general result we prove
here, namely we show that if the FIC is true for each vertex group of a graph
of groups with trivial edge groups then the FIC is true for the fundamental
group of the graph of groups. This result is part of a program to prove FIC for
the fundamental group of a graph of groups where all the vertex and edge groups
satisfy FIC. A consequence of the first result gives a partial solution to a
problem in the problem list of R. Kirby. We also deduce that the FIC is true
for a class of virtually PD_3-groups.
Another main aspect of this article is to prove the FIC for all Haken
3-manifold groups assuming that the FIC is true for B-groups. By definition a
B-group contains a finite index subgroup isomorphic to the fundamental group of
a compact irreducible 3-manifold with incompressible nonempty boundary so that
each boundary component is of genus \geq 2. We also prove the FIC for a large
class of B-groups and moreover, using a recent result of L.E. Jones we show
that the surjective part of the FIC is true for any B-group.Comment: 35 pages, 1 figure (.eps file), AMS Latex file, final version.
accepted for publication in K-theor