Equivariant basic cohomology of Riemannian foliations

Abstract

Abstract The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.