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Whitehead groups of certain hyperbolic manifolds

  • A. J. Nicas (a1) and C. W. Stark (a2)

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.

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[1] L. A. Best . On torsion-free discrete subgroups of PSL (2, C) with compact orbit space. Canad. J. Math. 23 (1971), 451460.

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

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

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

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

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

[10] C. Weber and H. Seifert . Die beiden Dodekaederräume. Math. Z. 37 (1933), 237253.

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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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