Let $X$ be a compact topological space and $T$ a complex torus acting on $X$ without fixed points. What, in general, can be said about the topology of $X$ (ex. rational (co-)homology)? I know one can show $H_{T}^*(X)$ to be a torsion module over the eqivariant cohomology of a point, but perhaps there are more sweeping statements one can make. I would appreciate any and all references and suggestions.