# Cohomologie des variétés feuilletées

Abstract : We associate to each foliated Morse function f on a measured foliation a longitudinal complex which computes the longitudinal cohomology introduced by A. Connes, as we prove. The q index space of this complex is given by the field $E^q=(l^2(C^q \cap L))_L$ , where C^q denotes the manifold of the q index longitudinal critical points of the generalized Morse function, and where L is a generic leaf. The differential $\delta^q:E^q \rightarrow E^{q+1}$ is defined by studying how the orientation of the unstable manifold is carried along a trajectory linking to a q index critical point and a q+1 index critical point. To prove that this complex computes the longitudinal cohomology, we show that it can be seen as the limit, when $\tau \rightarrow \infty$, of a foliated complex $(W^q_{\tau,L},d^q_{\tau,L})$ considered by A. Connes and T. Fack.
This extends to the foliated case a result of \emph{B. Helffer} et \emph{J. Sjörstrand}.
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Cited literature [71 references]

https://tel.archives-ouvertes.fr/tel-00358710
Contributor : Christophe Jaloux <>
Submitted on : Wednesday, February 4, 2009 - 10:47:45 AM
Last modification on : Wednesday, July 8, 2020 - 12:43:14 PM
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• HAL Id : tel-00358710, version 1

### Citation

Christophe Jaloux. Cohomologie des variétés feuilletées. Mathématiques [math]. Université Claude Bernard - Lyon I, 2008. Français. ⟨tel-00358710⟩

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