Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T01:53:50.474Z Has data issue: false hasContentIssue false

Whitehead groups of certain hyperbolic manifolds

Published online by Cambridge University Press:  24 October 2008

A. J. Nicas
Affiliation:
University of Toronto
C. W. Stark
Affiliation:
Brandeis University

Extract

An aspherical manifold is a connected manifold whose universal cover is contractible. It has been conjectured that the Whitehead groups Whj (π1 M) (including the projective class group, the original Whitehead group of π1M, and the higher Whitehead groups of [9]) vanish for any compact aspherical manifold M. The present paper considers this conjecture for twelve hyperbolic 3-manifolds constructed from regular hyperbolic polyhedra. Hyperbolic manifolds are of special interest in this regard since so much is known about their topology and geometry and very little is known about the algebraic K-theory of hyperbolic manifolds whose fundamental groups are not generalized free products.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Best, L. A.. On torsion-free discrete subgroups of PSL (2, C) with compact orbit space. Canad. J. Math. 23 (1971), 451460.CrossRefGoogle Scholar
[2]Cappell, S. E.. Unitary nilpotent groups and Hermitian K-theory. Bull. Amer. Math. Soc. 80 (1974), 11171122.CrossRefGoogle Scholar
[3]Cappell, S. E.. Manifolds with fundamental group a generalized free product. I. Bull. Amer. Math. Soc. 80 (1974), 11931198.CrossRefGoogle Scholar
[4]Hempel, J.. Orientation reversing involutions and the first Betti number for finite coverings of 3-manifolds. Invent. Math. 67 (1982), 133142.CrossRefGoogle Scholar
[5]Hempel, J.. Homology of coverings. Pacific J. Math., (in the Press).Google Scholar
[6]Nicas, A. J.. Induction theorems for groups of homotopy manifold structures. Mem. Amer. Math. Soc. 267 (1982).Google Scholar
[7]Nicas, A. J. and Stark, C. W.. Higher Whitehead groups of certain bundles over Seifert manifolds. (In the Press.)Google Scholar
[8]Thurston, W.. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357381.CrossRefGoogle Scholar
[9]Waldhausen, F.. Algebraic K-theory of generalized free products. Ann. of Math. 108 (1978), 135256.CrossRefGoogle Scholar
[10]Weber, C. and Seifert, H.. Die beiden Dodekaederräume. Math. Z. 37 (1933), 237253.CrossRefGoogle Scholar