Smith theory is a classical theory which reveals the homology information of the fixed point set of a certain space with group action. And the converse question is that, given two spaces F and Y (both without group action), can F be realized as the fixed point set of X(with a group action G), which has the same homotopy type as Y ? So far the best answer is by Robert Oliver, who proves that, for certain type of group, and when Y is a point, the only obstruction of the converse problem is the Euler characteristic.