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

Published online by Cambridge University Press:  30 November 2007

Sayed K. Roushon
Sayed K. Roushon
Affiliation:
[email protected]://www.math.tifr.res.in/~roushon/paper.htmlSchool of Mathematics, Tata Institute, Homi Bhabha Road, Mumbai 400005, India
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Abstract

This article has two purposes. In [15] we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by 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 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 ε C there is a closed 3-manifold P such that either P is Seifert fibered or is a nonpositively curved 3-manifold and π1(M) is a subgroup of π1(P). As a consequence it is obtained that the FIC is true for any B-group (see definition 4.2 in [15]). Therefore, the FIC is true for any Haken 3-manifold group and hence for any 3-manifold group (using the reduction theorem of [15]) provided we assume the Geometrization conjecture. The above result also proves the FIC for a class of 4-manifold groups (see [14]).

The second aspect of this article is to relax a condition in the definition of strongly poly-surface group ([13]) and define a new class of groups (we call them 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 [13].

Keywords

strongly poly-surface groupfibered isomorphism conjecture3-manifold groupselliptic surfaces4-manifold groupspseudoisotopy functor
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 1 , February 2008 , pp. 83 - 93
Copyright
Copyright © ISOPP 2008

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References

1.Aravinda, C.S., Farrell, F.T. and Roushon, S.K., Algebraic K-theory of pure braid groups, Asian J. Math. 4 (2000), 337–344CrossRefGoogle Scholar
2.Bartels, A. and Lück, W., Isomorphism Conjecture for homotopy K-theory and groups acting on trees, J. Pure Appl. Algebra 205, no. 3 (2006), 660–696CrossRefGoogle Scholar
3.Berkove, E., Juan-Pineda, D. and Lu, Q., Algebraic K-theory of mapping class groups, K-theory, 32 (2004), 83–100CrossRefGoogle Scholar
4.Casson, A.J. and Bleiler, S.A., Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, Cambridge, 1988Google Scholar
5.Farrell, F.T. and Jones, L.E., Isomorphism conjectures in algebraic K-theory, J. Amer.Math. Soc. 6 (1993), 249–297Google Scholar
6.Farrell, F.T. and Linnell, P.A., K-theory of solvable groups, Proc. London Math. Soc. 87 (3) (2003), 309–336CrossRefGoogle Scholar
7.Fathi, A., Laudenbach, F. and Poénaru, V., Travaux de Thurston sur les surfaces, Astérisque, 66–67 (1979)Google Scholar
8.Hempel, J. 3-manifolds, Annals of Mathematics Studies, Princeton University Press, 1976Google Scholar
9.Jaco, W. and Shalen, P., Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc. 220 (1979)Google Scholar
10.Jones, L.E., A paper for F.T. Farrell on his 60'th birthday, High-dimensional manifold topology, 200–260, World Sci. Publ., River Edge, NJ, 2003CrossRefGoogle Scholar
11.Leeb, B., 3-manifolds with(out) metrices of nonpositive curvature, Invent. Math. 122 (1995), 277–289CrossRefGoogle Scholar
12.Poénaru, V., Classification des difféomorphismes des surfaces, Astérisque, 66–67 (1979), 159–180Google Scholar
13.Roushon, S.K., K-theory of virtually poly-surface groups, Algebr. Geom. Topol. 3 (2003), 103–116CrossRefGoogle Scholar
14.Roushon, S.K., The Fibered isomorphism conjecture for complex manifolds, Acta Math. Sin. (Engl. Ser.) 23, no. 4 (2007), 639–658CrossRefGoogle Scholar
15.Roushon, S.K., The Farrell-Jones isomorphism conjecture for 3-manifold groups, J. K- Theory, same issueGoogle Scholar