Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Finite covers of $ 3$-manifolds containing essential tori

Author: John Luecke
Journal: Trans. Amer. Math. Soc. 310 (1988), 381-391
MSC: Primary 57N10; Secondary 57M10
MathSciNet review: 965759
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown in this paper that if a Haken $ 3$-manifold contains an incompressible torus that is not boundary-parallel then either it has a finite cover that is a torus-bundle over the circle or it has finite covers with arbitrarily large first Betti number.

References [Enhancements On Off] (What's this?)

  • [A-M] M. F. Atiyah and I. G. McDonald, Introduction to commutative algebra, Addison-Wesley, 1969. MR 0242802 (39:4129)
  • [A-S] Roger C. Alperin and Peter B. Shalen, Linear groups of finite cohomological dimension, preprint. MR 609046 (82c:20087)
  • [Ga] David Gabai, On $ 3$-manifolds finitely covered by surface-bundles, Low-dimensional Topology and Kleinian Groups, London Math. Soc. Lecture Notes, no. 112, Cambridge Univ. Press, Cambridge, England, 1986. MR 903863 (88m:57021)
  • [He1] John Hempel, $ 3$-manifolds, Ann. of Math. Studies, no. 86, Princeton Univ. Press, Princeton, N.J., 1976. MR 0415619 (54:3702)
  • [He2] -, Homology of coverings, Pacific J. Math. 112 (1984), 83-114. MR 739142 (85e:57001)
  • [He3] -, Orientation reversing involutions and the first Betti number for finite coverings of $ 3$-manifolds, Invent. Math. 67 (1982), 133-143. MR 664329 (83m:57010)
  • [He4] -, Virtually Haken manifolds, preprint.
  • [Hi] Graham Higman, The units of group rings, Proc. London Math. Soc. 46 (1940), 231-248. MR 0002137 (2:5b)
  • [J] William Jaco, Lectures on $ 3$-manifold topology, CBMS Regional Conf. Ser. in Math., 43, Amer. Math. Soc., Providence, R.I., 1979.
  • [K-L] S. Kojima and D. D. Long, Virtual Betti numbers of some hyperbolic $ 3$-manifolds, preprint.
  • [M] Wilhelm Magnus, Noneuclidean tesselations and their groups, Academic Press, New York, 1974. MR 0352287 (50:4774)
  • [S] P. Samuel, À propos du théorème des unités, Bull. Sci. Math. (2) 90 (1966), 89-96. MR 0204454 (34:4296)
  • [Th1] William Thurston, The geometry and topology of $ 3$-manifolds, lecture notes.
  • [Th2] -, Three dimensional manifolds, Kleinian groups, and hyperbolic geometry, Proc. Sympos. Pure Math., vol 39, Amer. Math. Soc., Providence, R.I., 1983, pp. 87-111.
  • [Wa] F. Waldhausen, Eine Klasse von $ 3$-dimensionalen Mannigfaltigkeiten I, Invent. Math. 3 (1967), 308-333; II, Invent. Math. 4 (1967), 87-117. MR 0235576 (38:3880)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N10, 57M10

Retrieve articles in all journals with MSC: 57N10, 57M10

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society