NDLI: Feuilletages en surfaces, cycles évanouissants et variétés de Poisson

Content Provider	Springer Nature Link
Author	Cuesta, F. Alcalde Hector, G.
Copyright Year	1997
Abstract	Surface foliations, vanishing cycles and Poisson manifolds. Afoliated cylinder of a foliated manifold (M,F) is a path of integral loopsc t forF. Such a cylinder defines anon-trivial vanishing cycle c 0 ifc t is null-homotopic in its supportF t for eacht>0, butc 0 is not null-homotopic in its supportF 0. Vanishing cycles were introduced by S. P. Novikov to study qualitative aspects of codimension one foliations. In this paper we apply this notion to the study of higher codimensional foliations.The first aim is to show the influence that the triviality of vanishing cycles exerts on the topology of thehomotopy groupoid ofF. It is natural to try to reduce the study of triviality to more regular vanishing cycles, as well as to obtain a nice criterion of triviality. In this way, we introduce the notion ofregular vanishing cycle as the “orthogonal” version (for a riemannian metric onM) of the classical notion of immersed vanishing cycle and the notion ofcoherent vanishing cycle, i.e. an integral discD 1 with boundaryc 1 extends to a global foliated homotopyD t such thatc t is the boundary ofD t for eacht>0. We also prove that the triviality of these vanishing cycles implies the triviality of all vanishing cycles. For compact foliated manifolds, we obtain the following criterion: a regular coherent vanishing cycle is non-trivial if and only if the area of the discsD t converges to infinity.Finally, we give two applications of these results to surface foliations: we generalize the Reeb stability theorem to higher codimensions and we resolve the problem of the symplectic realization of Poisson structures supported by surface foliations.
Ending Page	213
Page Count	23
Starting Page	191
File Format	PDF
ISSN	00269255
e-ISSN	14365081
Journal	Monatshefte für Mathematik
Issue Number	3
Volume Number	124
Language	French
Publisher	Springer-Verlag
Publisher Date	1997-01-01
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	cycles évanouissants Mathematics variétés de Poisson Foliations; geometric theory Feuilletages
Content Type	Text
Resource Type	Article
Subject	Mathematics

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