This is a combined Topology/Algebraic Topology seminar. For a group G, we define a notion of dimension in terms of dimension change of the of the top homology between a free G space X and it's quotient X/G.  We show that this is well defined and calculate this "Action-dimension" for certain groups, including finitely generated solvable groups, free groups, finite groups, and connected Lie groups.  As a consequence we give a positive answer to a conjecture of J Kollar: Let f: R^n ---> T^n be the universal covering and M a closed submanifold in T^n = (S^1)^n  such that f^{-1}(M) has the homotopy type of a finite complex, then M = T^n.