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The Farrell-Jones isomorphism conjecture for 3-manifold groups. (English) Zbl 1160.57018

Let FIC mean the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor (see F. T. Farrell and L. E. Jones [J. Am. Math. Soc. 6, No. 2, 249–297 (1993; Zbl 0798.57018)]). In this fine paper, the author proves that the FIC is true for fundamental groups of a large class of 3-manifolds. He also proves 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 proved by the author which states 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 the FIC for the fundamental group of a graph of groups, where all the vertex and edge groups satisfy the FIC. A consequence of the first result gives a partial solution to a problem in the problem list of R. Kirby [Problems in low-dimensional topology, Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2(pt.2), 35–473 (1997; Zbl 0888.57014)]. The author also deduces that the FIC is true for a class of virtually \(PD_3\)-groups.
Another main aspect of this paper is to prove the FIC for all Haken 3-manifold groups assuming that the FIC is true for \(B\)-groups. Recall that a group \(G\) is called a \(B\)-group if it contains the fundamental group of a compact orientable irreducible 3-manifold with all the boundary components of higher genus (that is, when the genus is greater than or equal 2) and is incompressible (nonempty) as a subgroup of finite index. Finally, the author proves the FIC for a large class of \(B\)-groups, and moreover, using a recent result of L. E. Jones [A paper for F. T. Farrell on his 60’th birthday, High-dimensional manifold topology, River Edge, NJ: World Scientific. 200–260 (2003; Zbl 1044.57011)], he shows that the surjective part of the FIC is true for any \(B\)-group.

MSC:

57N37 Isotopy and pseudo-isotopy
19J10 Whitehead (and related) torsion
19D35 Negative \(K\)-theory, NK and Nil
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