Free and Locally Free Actions

This chapter addresses free and locally free actions. It uses the Cartan model to compute the equivariant cohomology of a circle action, so equivariant cohomology is taken with real coefficients. An action is said to be free if the stabilizer of every point consists only of the identity element. It turns out that the equivariant cohomology of a free circle action is always u-torsion. More generally, an action of a topological group G on a topological space X is locally free if the stabilizer Stab(x) of every point is discrete. The chapter then proves that the equivariant cohomology of a locally free circle action on a manifold is also u-torsion.