A condition for the stability of $\mathbb {R}$-covered on foliations of 3-manifolds
HTML articles powered by AMS MathViewer
- by Sue Goodman and Sandi Shields PDF
- Trans. Amer. Math. Soc. 352 (2000), 4051-4065 Request permission
Abstract:
We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be $\mathbb {R}$-covered. This condition can be readily verified for many examples. Further, if an $\mathbb {R}$-covered foliation has a compact leaf $L$, then any transverse loop meeting $L$ lifts to a copy of the leaf space, and the ambient manifold fibers over $S^1$ with $L$ as fiber.References
- T. Barbot: Geometrie transverse des flots d’Anovox, Thesis, Ecole Norm. Sup., Lyon, 1992.
- Thierry Barbot, Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles, Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247–270 (French, with English summary). MR 1332403, DOI 10.1017/S0143385700008361
- J. Christy and S. Goodman: Branched surfaces transverse to codimension one foliations, preprint.
- Sérgio R. Fenley, Anosov flows in $3$-manifolds, Ann. of Math. (2) 139 (1994), no. 1, 79–115. MR 1259365, DOI 10.2307/2946628
- S. Fenley: Continuous extension of Anosov foliations, preprint.
- David Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
- David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
- David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
- Étienne Ghys, Flots d’Anosov sur les $3$-variétés fibrées en cercles, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 67–80 (French, with English summary). MR 758894, DOI 10.1017/S0143385700002273
- Sue E. Goodman, Closed leaves in foliations of codimension one, Comment. Math. Helv. 50 (1975), no. 3, 383–388. MR 423371, DOI 10.1007/BF02565757
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
- Morris W. Hirsch, Stability of compact leaves of foliations, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 135–153. MR 0334236
- S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian). MR 0200938
- F. Palmeira: Open manifolds foliated by planes, Annals of Math. 107 (1978), 109-131.
- J. F. Plante, Anosov flows, transversely affine foliations, and a conjecture of Verjovsky, J. London Math. Soc. (2) 23 (1981), no. 2, 359–362. MR 609116, DOI 10.1112/jlms/s2-23.2.359
- J. F. Plante, Solvable groups acting on the line, Trans. Amer. Math. Soc. 278 (1983), no. 1, 401–414. MR 697084, DOI 10.1090/S0002-9947-1983-0697084-7
- J. F. Plante, Diffeomorphisms without periodic points, Proc. Amer. Math. Soc. 88 (1983), no. 4, 716–718. MR 702306, DOI 10.1090/S0002-9939-1983-0702306-5
- Harold Rosenberg, Foliations by planes, Topology 7 (1968), 131–138. MR 228011, DOI 10.1016/0040-9383(68)90021-9
- R. Roussarie, Plongements dans les variétés feuilletées et classification de feuilletages sans holonomie, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 101–141 (French). MR 358809
- Paul A. Schweitzer, Existence of codimension one foliations with minimal leaves, Ann. Global Anal. Geom. 9 (1991), no. 1, 77–81. MR 1116633, DOI 10.1007/BF02411357
- S. Shields: Branched surfaces and the stability of compact leaves, thesis, University of North Carolina at Chapel Hill (1991).
- Sandra L. Shields, Stability for a class of foliations covered by a product, Michigan Math. J. 44 (1997), no. 1, 3–20. MR 1439665, DOI 10.1307/mmj/1029005617
- V. Solodov: On the universal cover of Anosov flows, Research announcement.
- Dennis Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979), no. 2, 218–223. MR 535056, DOI 10.1007/BF02566269
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088
- Alberto Verjovsky, Codimension one Anosov flows, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 2, 49–77. MR 431281
- R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203. MR 348794
Additional Information
- Sue Goodman
- Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3902
- Email: seg@math.unc.edu
- Sandi Shields
- Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424
- Email: shields@math.cofc.edu
- Received by editor(s): September 3, 1996
- Received by editor(s) in revised form: April 18, 1998
- Published electronically: May 12, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 4051-4065
- MSC (2000): Primary 57M12, 57M20, 57N10, 57R30
- DOI: https://doi.org/10.1090/S0002-9947-00-02391-6
- MathSciNet review: 1624178