A condition for the stability of ℝ-covered on foliations of 3-manifolds

Goodman, Sue; Shields, Sandi

doi:10.1090/S0002-9947-00-02391-6

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