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

  • Andrew J. Nicas (a1)

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 Whj1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.

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[1] S. E. Cappell . Manifolds with fundamental group a generalized free product. I. Bull. Amer. Math. Soc. 80 (1974), 11931198.

[2] S. E. Cappell . Unitary nilpotent groups and Hermitian K-theory. Bull. Amer. Math. Soc. 80 (1974), 11171122.

[4] J. Hempel . Orientation reversing involutions and the first Betti number for finite coverings of 3-manifolds. Invent. Math. 67 (1982), 133142.

[5] J. Hempel . Homology of coverings. Pacific J. Math. 112 (1984), 83113.

[8] A. J. Nicas and C. W. Stark . Higher Whitehead groups of certain bundles over Seifert manifolds. Proc. Amer. Math. Soc. 91 (1984), 15.

[13] R. G. Swan . K-theory of finite Groups and Orders. Lecture Notes in Math.149, Springer-Verlag, 1970.

[15] W. Thurston . Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357381.

[16] F. Waldhausen . Algebraic K-theory of generalized free products. Ann. of Math. 108 (1978), 135256.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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