Localization of basic characteristic classes

Töben, Dirk

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 537-570.

Nous introduisons des classes et des nombres caractéristiques basiques d’un feuilletage riemannien. Si la variété riemannienne est complète, simplement connexe (ou plus généralement si le feuilletage est un feuilletage de Killing transversalement orientable) et si l’espace des ahérences des feuilles est compact, alors les nombres caractéristiques basiques sont déterminés par la dynamique infinitésimale du feuilletage en l’union des adhérences des feuilles fermées. En effet, ils peuvent être calculés avec un théorème de localisation de type Atiyah-Bott-Berline-Vergne pour la cohomologie équivariante basique.

We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold $M$ is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type localization theorem for equivariant basic cohomology.

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