# Cohomological Tautness of Singular Riemannian Foliations

## TL;DR

This paper extends the concept of geometrical tautness characterized by basic cohomology classes from regular to singular Riemannian foliations on compact manifolds, establishing a global Alvarez class and implications for simply connected spaces.

## Contribution

It generalizes Alvarez López's result to singular foliations by defining a unique global Alvarez class from strata and shows tautness in simply connected cases.

## Key findings

- Existence of a unique global Alvarez class for singular Riemannian foliations.
- In simply connected manifolds, the foliation's restriction to each stratum is geometrically taut.
- Generalization of Ghys's result to singular foliations.

## Abstract

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.