Cohomology of singular Riemannian foliations

Abstract

Associated to a smooth foliation ( M , F ) are defined the basic and the foliated cohomologies. These cohomologies are related to the de Rham cohomology by the de Rham spectral sequence of F , E r , d R s , t ( M ) , constructed by filtering the de Rham complex of the manifold. For a Riemannian foliation on a compact manifold the second term of this spectral sequence, E 2 , d R s , t ( M ) , is finite dimensional and a topological invariant. In this Note we prove these two results, fitness dimension and topological invariance, for the cohomology of singular Riemannian foliations. The proof uses the previous theorems for the regular case and the structure of singular Riemannian foliations described by P. Molino. For the basic cohomology these results have been proved by R. Wolak. To cite this article: X.M. Masa, A. Rodríguez-Fernández, C. R. Acad. Sci. Paris, Ser. I 342 (2006).