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The Farrell-Jones isomorphism conjecture for 3-manifold groups

Published online by Cambridge University Press:  30 November 2007

S.K. Roushon
S.K. Roushon
Affiliation:
[email protected] of Mathematics, Tata Institute, Homi Bhabha Road, Mumbai 400005, India
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Abstract

We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for 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, 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 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. We also deduce that the FIC is true for a class of virtually P D3-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 ≥ 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.

Keywords

3-manifoldvirtual fiberingfibered isomorphism conjectureWhitehead groupstable topological pseudo-isotopy functor
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 1 , February 2008 , pp. 49 - 82
Copyright
Copyright © ISOPP 2008

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References

1.Anderson, R. and Hsiang, W.-C., The functors K −i and pseudoisotopies of polyhedra, Ann. of Math. (2) 105 (1977), 201–223CrossRefGoogle Scholar
2.Berkove, E., Farrell, F. T., Juan-Pineda, D. and Pearson, K., The Farrell-Jones isomorphism conjecture for finite co-volume hyperbolic actions and the algebraic K-theory of Bianchi groups, Trans. Amer. Math. Soc., 352 (2000), 5689–5702CrossRefGoogle Scholar
3.do Carmo, M.P., Riemannian Geometry, English edition, Birkhäuser, Boston, Basel, Berlin, 1991Google Scholar
4.Dicks, W. and Dunwoody, M.J., Groups acting on graphs, Cambridge University Press, Cambridge, 1989Google Scholar
5.Eckmann, B. and Linnell, P., Poincaré duality groups of dimension two. II, Comment. Math. Helv., 58 (1983), 111–114CrossRefGoogle Scholar
6.Eckmann, B. and Müller, H., Poincaré duality groups of dimension two, Comment. Math. Helv., 55 (1980), 510–520CrossRefGoogle Scholar
7.Epstein, D.B.A., ‘Ends’ in Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., (1962), 110–117Google Scholar
8.Farrell, F.T. and Jones, L.E., Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc., 6 (1993), 249–297Google Scholar
9.Farrell, F.T. and Jones, L.E., K-theory and dynamics. I, Ann. of Math. (2), 124 (1986), no. 3, 531–569CrossRefGoogle Scholar
10.Farrell, F.T. and Jones, L.E., K-theory and dynamics. II, Ann. of Math. (2) (2), 126 (1987), no. 3, 451–493Google Scholar
11.Farrell, F.T. and Jones, L.E., Stable pseudoisotopy spaces of compact non-positively curved manifolds, J. Differential Geom., 34 (1991), no. 3, 769–834CrossRefGoogle Scholar
12.Farrell, F.T. and Linnell, P.A., K-Theory of solvable groups, Proc. London Math. Soc. (3), 87 (2003), 309–336CrossRefGoogle Scholar
13.Farrell, F.T. and Roushon, Sayed K., The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices, no. 10 (2000), 515–526Google Scholar
14.Fathi, A., Laudenbach, F., Poenaru, V., Travaux de Thurston sur les surfaces, Astérisque, 66–67, 1979Google Scholar
15.Hatcher, A., Concordance spaces, higher simple homotopy theory, and applications, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, (1978), 2–21Google Scholar
16.Hempel, J., 3-manifolds, Annals of Mathematics Studies, Princeton University Press, 1976Google Scholar
17.Hicks, N.J., Notes on Differential Geometry, D. Van Nostrand Company, Inc., Princeton, New Jersey, Toronto, New York, London, 1965Google Scholar
18.Hillman, J.A., Four-manifolds, geometries and knots, Geometry and Topology Monographs, Geometry and Topology Publications, University of Warwick, UK, 2002Google Scholar
19.Jaco, W. and Shalen, P., Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc., vol. 220 (1979)Google Scholar
20.Jones, L.E., A paper for F.T. Farrell on his 60'th birthday, High-dimensional manifold topology, 200–260, World Sci. Publishing, River Edge, NJ, 2003Google Scholar
21.Kapovich, M. and Leeb, B., Actions of discrete groups on nonpositively curved spaces, Math. Ann., 306 (1996), 341–352Google Scholar
22.Kirby, R., Problems in low-dimensional topology, Geometric Topology (Athens, GA, 1993), Part 2, Amer. Math. Soc., Providence, RI, (1997), 35–473Google Scholar
23.Leeb, B., 3-manifolds with(out) metrices of nonpositive curvature, Invent. Math., 122 (1995), 277–289Google Scholar
24.Luecke, J., Finite covers of 3-manifolds containing essential tori, Trans. of Amer. Math. Soc., 310 (1988), 381–391Google Scholar
25.Luecke, J. and Wu, Y-Q, Relative Euler number and finite covers of graph manifolds, Geometric topology, AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, (1993), 80–103Google Scholar
26.Maillot, S., Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds, Trans. of Amer. Math. Soc., 355 (2003), 4595–4638CrossRefGoogle Scholar
27.Meeks, W. III, Simon, L. and Yau, S-T, Embedded minimal surfaces, exotic sphere, and manifolds with positive Ricci curvature, Ann. of Math., 116 (1982), 621–659Google Scholar
28.Otal, Jean-Pierre, Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3. (French. French summary) [The hyperbolization theorem for fibered manifolds of dimension 3], Astérisque, no. 235, 1996Google Scholar
29.Plotnick, S., Vanishing of Whitehead groups for Seifert manifolds with infinite fundamental group, Comment. Math. Helv. 55 (1980), 654–667CrossRefGoogle Scholar
30.Poénaru, V., Classification des difféomorphismes des surfaces, Astérisque no. 66–67 (1979), 159–180Google Scholar
31.Quinn, F., Ends of maps. II, Invent. Math., 68 (1982), 353–424Google Scholar
32.Quinn, F., Controlled K-theory I: basic theory, math.KT/0402396Google Scholar
33.Roushon, S.K., The Fibered isomorphism conjecture for complex manifolds, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 4, 639–658CrossRefGoogle Scholar
34.Roushon, S.K., The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, J. K-Theory, to appearGoogle Scholar
35.Thurston, W.P., Hyperbolic structures on 3-manifolds II. Surface groups and manifolds which fiber over the circle, Princeton Univ. preprint, 1986Google Scholar
36.Waldhausen, F., Algebraic K-theory of generalized free products, Parts 1 and 2, Ann. of Math., 108 (1978), 135–256CrossRefGoogle Scholar