Cohomological Tautness of Singular Riemannian Foliations

For a Riemannian foliation F on a compact manifold M , J. A. 'Alvarez L'opez proved that the geometrical tautness of F , that is, the existence of a Riemannian metric making all the leaves minimal submanifolds of M, can be characterized by the vanishing of a basic cohomology class (the 'Alvarez class). In this work we generalize this result to the case of a singular Riemannian foliation K on a compact manifold X. In the singular case, no bundle-like metric on X can make all the leaves of K minimal. In this work, we prove that the 'Alvarez classes of the strata can be glued in a unique global 'Alvarez class. As a corollary, if X is simply connected, then the restriction of K to each stratum is geometrically taut, thus generalizing a celebrated result of E. Ghys for the regular case.