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# The Fibered Isomorphism Conjecture for Complex Manifolds

Acta Mathematica Sinica (Impact Factor: 0.48). 09/2002; 23(4). DOI: 10.1007/s10114-005-0759-2

Source: arXiv

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**ABSTRACT:**In this paper we generalize the notion of strongly poly-free group to a larger class of groups, we call them strongly poly-surface groups and prove that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for any virtually strongly poly-surface group. A consequence is that the Whitehead group of a torsion free subgroup of any virtually strongly poly-surface group vanishes. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this note, we point out an error in the above paper. We also refer to some papers where this error is corrected partially and describe a positive approach to correct it completely. AMS Classification 57N37, 19J10; 19D35 - [Show abstract] [Hide abstract]

**ABSTRACT:**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.