Morse and generic contact between foliations
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- by Russell B. Walker PDF
- Trans. Amer. Math. Soc. 254 (1979), 265-281 Request permission
Abstract:
Motivated by the recent work of J. Franks and C. Robinson, the study of the contact between two foliations of equal codimension is begun. Two foliations generically contact each other in certain dimensional sub-manifold complexes. All but a finite number of these contact points are “Morse". In a recent paper by the author, a complete large isotopy “index of contact” is specified for two foliations of ${T^2}$. If contact is restricted to index 0 ("domed contact"), some sharp conclusions are made as to the topology of the manifold and isotopy classes of the two foliations. It is hoped that this work will lead to the construction of new quasi-Anosov diffeomorphisms and possibly to a new Anosov diffeomorphism.References
-
J. Franks and C. Robinson, A quasi-Anosov diffeomorphism which is not Anosov (to appear).
- Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR 0344555
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5 —, Private communications. B. Lawson, Lectures on the quantitative theory of foliations, Washington University Press, St. Louis, Mo., 1975.
- H. Blaine Lawson Jr., Foliations, Bull. Amer. Math. Soc. 80 (1974), 369–418. MR 343289, DOI 10.1090/S0002-9904-1974-13432-4
- F. Reese Harvey and H. Blaine Lawson Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), no. 2, 223–290. MR 425173, DOI 10.2307/1971032 J. Milnor, Foliations and foliated vector bundles, Princeton Univ., mimeographed notes, 1970. —, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963. G. Reeb, Sur certains properiétés topologiques des variétés feuilletées, Actualités Sci. Indust. No. 1183, Hermann, Paris, 1952.
- Steve Smale, Global analysis and economics. I. Pareto optimum and a generalization of Morse theory, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 531–544. MR 0341535
- Emery Thomas, Vector fields on manifolds, Bull. Amer. Math. Soc. 75 (1969), 643–683. MR 242189, DOI 10.1090/S0002-9904-1969-12240-8
- William Thurston, Noncobordant foliations of $S^{3}$, Bull. Amer. Math. Soc. 78 (1972), 511–514. MR 298692, DOI 10.1090/S0002-9904-1972-12975-6 R. Walker, An index of contact on ${T^2}$; toral combinatorics, J. Combinatorics (to appear).
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 254 (1979), 265-281
- MSC: Primary 58F15; Secondary 57R30, 58F18
- DOI: https://doi.org/10.1090/S0002-9947-1979-0539918-1
- MathSciNet review: 539918