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Transactions of the American Mathematical Society

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



The stability of foliations
of orientable 3-manifolds
covered by a product

Author: Sandra L. Shields
Journal: Trans. Amer. Math. Soc. 348 (1996), 4653-4671
MSC (1991): Primary 57M12, 57M20, 57N10, 57R30, 58F10
MathSciNet review: 1355076
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Abstract: We examine the relationship between codimension one foliations that are covered by a trivial product of hyperplanes and the branched surfaces that can be constructed from them. We present a sufficient condition on a branched surface constructed from a foliation to guarantee that all $C^1$ perturbations of the foliation are covered by a trivial product of hyperplanes. We also show that a branched surface admits a strictly positive weight system if and only if it can be constructed from a fibration over $S^1$.

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  • 1. J. Christy and S. Goodman, Branched surfaces transverse to codimension one foliations, working paper, Emory Univ., Atlanta, GA, and University of North Carolina, Chapel Hill, NC, n.d.
  • 2. J. Christy, Branched surfaces and attractors. I: Dynamic branched surfaces, Trans. Amer. Math. Soc. 336 (1993), 759--784. MR 93f:58135
  • 3. W. Floyd and U. Oertel, Incompressible surfaces via branched surfaces, Topology 23 (1984), 117--125. MR 85a:57007
  • 4. J. Franks and R. F. Williams, Anomolous Anosov flows, Common Theory of Dynamical Systems, Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 158--174. MR 82e:58078
  • 5. J. Hempel, 3-manifolds Ann. of Math. Studies, vol. 86, Princeton Univ. Press, Princeton, NJ, 1976. MR 54:3702
  • 6. Morris W. Hirsch, Stability of compact leaves of foliations, Dynamical Systems (Proc. Sympos., Salvador, Brazil, 1971; M. Peixoto, ed.), Academic Press, New York, 1973, pp. 135--153. MR 48:12555
  • 7. R. Langevin and H. Rosenberg, On stability of compact leaves and fibrations, Topology 16 (1977), 107--112. MR 57:1508
  • 8. S. P. Novikov, Topology of foliations, Trudy Moskov. Mat. Obschch. 14 (1965), 248--277; English transl., Trans. Moscow Math. Soc. 14 (1977), 268--303. MR 34:824
  • 9. C. Palmeira, Open manifolds foliated by planes, Ann. of Math. (2) 107 (1978), 109--131. MR 58:18490
  • 10. J. Plante, Stability of codimension one foliations by compact leaves, Topology 22 (1983), 173--177. MR 84g:57022
  • 11. ------, Anosov flows, Amer. J. Math. 94 (1972), 729--754. MR 51:14099
  • 12. G. Reeb, Sur certaines propriétés topologiques des variétées feuilletées, Actualités Sci. Indust., no. 1183, Hermann, Paris, 1952. MR 14:1113a
  • 13. S. Shields, Branched surfaces and the stability of compact leaves, Thesis, University of North Carolina, Chapel Hill, NC, 1991.
  • 14. ------, Branched surfaces and the simplest foliations of 3-manifolds, preprint.
  • 15. A. Verjovsky, Codimension one Anosov flows, Bol. Soc. Math. Mexicana 19 (1974), 49--77. MR 55:4282
  • 16. R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. No. 43 (1973), 169--203. MR 50:1289

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Additional Information

Sandra L. Shields
Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424

Keywords: Branched surface, foliation, leaf space, holonomy map, topological equivalency
Received by editor(s): February 5, 1993
Received by editor(s) in revised form: October 9, 1995
Article copyright: © Copyright 1996 American Mathematical Society