Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T14:16:38.548Z Has data issue: false hasContentIssue false
  • English
  • Français

An infinite family of non-Haken hyperbolic 3-manifolds with vanishing Whitehead groups

Published online by Cambridge University Press:  24 October 2008

Andrew J. Nicas
Andrew J. Nicas
Affiliation:
Department of Mathematics, University of Toronto, Toronto M5S 1A1, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]:

Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism,

and its analogue in algebraic K-theory:

Conjecture B. The Whitehead groups Whj(π1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 2 , March 1986 , pp. 239 - 246
Copyright
Copyright © Cambridge Philosophical Society 1986

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]Cappell, S. E.. Manifolds with fundamental group a generalized free product. I. Bull. Amer. Math. Soc. 80 (1974), 1193–1198.CrossRefGoogle Scholar
[2]Cappell, S. E.. Unitary nilpotent groups and Hermitian K-theory. Bull. Amer. Math. Soc. 80 (1974), 1117–1122.CrossRefGoogle Scholar
[3]Farrell, F. T. and Hsiang, W. C.. On the rational homotopy groups of the diffeomorphism groups of discs, spheres, and aspherical manifolds. Proc. Sympos. Pure Math. 32 (1978), 325–338.CrossRefGoogle Scholar
[4]Hempel, J.. Orientation reversing involutions and the first Betti number for finite coverings of 3-manifolds. Invent. Math. 67 (1982), 133–142.CrossRefGoogle Scholar
[5]Hempel, J.. Homology of coverings. Pacific J. Math. 112 (1984), 83–113.CrossRefGoogle Scholar
[6]Nicas, A. J.. Induction theorems for groups of homotopy manifold structures. Mem. Amer. Math. Math. Soc. 267 (1982).Google Scholar
[7]Nicas, A. J.. Induction theorems for higher Whitehead groups (preprint).Google Scholar
[8]Nicas, A. J. and Stark, C. W.. Higher Whitehead groups of certain bundles over Seifert manifolds. Proc. Amer. Math. Soc. 91 (1984), 1–5.CrossRefGoogle Scholar
[9]Nicas, A. J. and Stark, C. W.. Whitehead groups of certain hyperbolic manifolds. Math. Proc. Cambridge Philos. Soc. 95 (1984), 299–308.CrossRefGoogle Scholar
[10]Nicas, A. J. and Stark, C. W.. Whitehead groups of certain hyperbolic manifolds. II (preprint).Google Scholar
[11]Richardson, J. S. and Rubinstein, J. H.. Hyperbolic manifolds from regular polyhedra (preprint).Google Scholar
[12]Riley, R.. A quadratic parabolic group. Math. Proc. Cambridge Philos. Soc. 77 (1975), 281–288.CrossRefGoogle Scholar
[13]Swan, R. G.. K-theory of finite Groups and Orders. Lecture Notes in Math. 149, Springer-Verlag, 1970.CrossRefGoogle Scholar
[14]Thurston, W.. The geometry and topology of 3-manifolds. Princeton University notes, 1980.Google Scholar
[15]Thurston, W.. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357–381.CrossRefGoogle Scholar
[16]Waldhausen, F.. Algebraic K-theory of generalized free products. Ann. of Math. 108 (1978), 135–256.CrossRefGoogle Scholar