Poincaré duality of the basic intersection cohomology of a Killing foliation

Abstract

We prove that the basic intersection cohomology H * p (M/F ), where F is the singular foliation determined by an isometric action of a Lie group G on the compact manifold M, verifies the Poincaré Duality Property.Cohomology theories are one of the basic tools in the study of invariants of topological and geometrical objects. They provide a good environment for the development of obstruction theories. In the case of regular Riemannian foliations basic cohomology theory proved to be of great importance. In particular, for foliations of compact manifolds, it was possible to define a 1-basic cohomology class κ, the lvarez class, whose vanishing is responsible for tautness. Moreover, the Poincaré duality property holds only in basic twisted cohomology associated to this 1-cohomology class, (see [2] and [3] for the precise statement).In the singular case the situation is even more complicated, for isometric actions the top dimensional basic cohomology can vanish and the Poincaré duality does not hold, [7]. Moreover, the standard procedure for the definition of the tautnes class seems not to work. Perhaps one should approach the problem from a different angle, and consider some other cohomology theory.We introduced the intersection basic cohomology in [10] and the examples and results obtained indicate that this cohomology theory is suitable for the study of topology and geometry of singular Riemannian foliations, [8,9,11,12]. In the present paper we demonstrate that under suitable orientation assumptions the basic intersection cohomology of a Killing foliation satisfies the Poincaré duality property.In the sequel M is a connected, second countable, Haussdorff, without boundary and smooth (of class C ∞ ) manifold. We also write G for a Lie group. *