A QUOTIENT CRITERION FOR SYZYGIES IN EQUIVARIANT COHOMOLOGY

Let X be a manifold with an action of a torus T such that all isotropy groups are connected and satisfying some other mild hypotheses. We provide a necessary and sufficient criterion for the equivariant cohomology H (T) (*) (X) with real coefficients to be a certain syzygy as module over H*(BT). It turns out that, possibly after blowing up the non-free part of the action, this only depends on the orbit space X/T together with its stratification by orbit type. Our criterion unifies and generalizes results of many authors about the freeness and torsion-freeness of equivariant cohomology for various classes of T-manifolds.