Suppose G is a finite group, and f is a map from a CW complex F to the fixed point of a G-CW complex Y. Is it possible to extend F to a finite G-CW complex X satisfying XG = F, and extend f to a G-map g: X→ Y, such that g is a homotopy equivalence after forgetting the G-action?
We extend the classical results of Lowell Jones and Robert Oliver to the general setting. For semi-free action, we encounter a finiteness type obstruction. For general action by group of not prime power order, the obstruction is the Euler characteristics over components of YG. We calculate such obstructions for various examples.
This is a joint work with Sylvain Cappell and Shmuel Weinberger.