Abstract
A foliation with all leaves compact (compact foliation) is called locally stable if every leaf has a basis of neighborhoods which are unions of leaves. We study the relationship between the first real cohomology group of leaves and the local stability of compact foliations. We show by example that the topology of the typical leaves (i.e. leaves with zero holonomy) has no influence on the local stability of the foliation while — at least for small codimensions — (less than 4 in general or less than 5 for foliations on compact minifolds) — a locally unstable foliation has a leaf F with infinite holonomy and a finite covering F' of F such that H1(F'; IR) ≠ O. We also prove a related structural stability result for fibre bundles.
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Bibliography
Borel, A.: Introduction aux groupes arithmétiques. Act. Sci. et Ind. N° 1341, Paris: Hermann 1969
Borel, A.: Some finiteness properties of Adele groups over number fields. Publ. Math. IHES16, 1–30 (1963)
Conze, J.-P., Guivarc'h, Y.: Remarque sur la distalité dans les espaces vectoriels. C.R. Acad. Sc. Paris278, 1083–1086 (1974)
Edwards, R.D., Millett, K.C., Sullivan, D.: Foliations with all leaves compact. Topology16, 13–32 (1977)
Epstein, D.B.A.: Periodic flows on 3-manifolds. Annals of Math.95, 68–82 (1972)
Epstein, D.B.A.: Foliations with all leaves compact. Ann. Inst. Fourier, Grenoble26, 265–282 (1976)
Epstein, D.B.A.: A topology for the space of foliations. Geometry and Topology, Proceedings, Rio de Janeiro 1976. Lecture Notes in Mathematics597, 132–150. Berlin-Heidelberg-New York: Springer 1977
Epstein, D.B.A., Rosenberg, H.: Stability of compact foliations. Geometry and Topology, Proceedings, Rio de Janeiro 1976. Lecture Notes in Mathematics597, 151–160. Berlin-Heidelberg-New York: Springer 1977.
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics583. Berlin-Heidelberg-New York: Springer 1977
Hu, S.-T.: Theory of retracts. Detroit: Wayne State University Press 1965
Kazhdan, D.A.: Connection of the dual space of a group with the structure of its closed subgroups. Funktsional'nyi Analiz i Ego Prilozheniya1, 71–74 (1967)
Kneser, M.: Erzeugende und Relationen verallgemeinerter Einheitengruppen. Journal f.d. reine und angewandte Math. 214/15, 345–349 (1964)
Langevin, R., Rosenberg, H.: On stability of compact leaves and fibrations. Topology16, 107–112 (1977)
Margulis, G.A.: Finiteness of quotient groups of discrete subgroups. Punktsional'nyi Analiz i Ego Prilozheniya13, No. 3, 178–187 (1979)
Menger, K.: Über die Dimension von Punktmengen II. Monatsh. für Math. und Phys.34, 137–161 (1926)
Montgomery, D.: Pointwise periodic homeomorphisms. Amer. J. Math.59, 118–120 (1937)
Platonov, V.P.: The problem of strong approximation and the Kneser-Tits conjecture. Izv. Akad. Nauk, ser. Math.33, 1211–1220 (1969); appendix in Izv. Akad. Nauk, Ser. Math.34, 775–777 (1970)
Platonov, V.P., Zalesskii, A.E.: On a problem of Auerbach. Dokl. Akad. Nauk BSSR10, 5–6 (1966)
Reeb, G.: Sur certaines propriétés topologiques des variétés feuilletées. Act. Sci. et Ind. 1183. Paris: Hermann 1952
Schmidt, K.: Amenability, Kazhdan's property T, strong ergodicity, and invariant means for ergodic group actions. Preprint. Univ. of Warwick 1980
Sullivan, D.: A new flow. Bull. A.M.S.82, 331–332 (1976)
Sullivan, D.: A counterexample to the periodic orbit conjecture. Publ. Math. IHES46, 5–14 (1976)
Thurston, W.: A generalization of the Reeb stability theorem. Topology13, 347–352 (1974)
Vogt, E.: Foliations of codimension 2 with all leaves compact. manuscripts math.18, 187–212 (1976)
Vogt, E.: A periodic flow with infinite Epstein hierarchy. manuscripta math.22, 403–412 (1977)
Wang, S.P.: The dual space of semi-simple Lie groups. Amer. J. Math.91, 921–937 (1969)
Wang, S.P.: On density properties of S-subgroups of locally compact groups. Ann. of Math.94, 325–329 (1971)
Weaver, N.: Pointwise periodic homeomorphisms of continua. Ann. of Math.95, 83–85 (1972)
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Vogt, E. The first cohomology group of leaves and local stability of compact foliations. Manuscripta Math 37, 229–267 (1982). https://doi.org/10.1007/BF01168511
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DOI: https://doi.org/10.1007/BF01168511