Topology Seminar: Finiteness Obstructions for G-Spaces up to hG-equivalence

Two G-spaces X and Y are said to be hG-equivalent if their Borel constructions are equivalent over BG. In this talk we introduce an obstruction which determines when a G-sphere X is hG-equivalent to a finite G-sphere Y, at a prime p dividing the order of G. We will also discuss how to generalize this to a global finiteness obstruction, and, in the case of G-spheres, relate the finiteness obstruction to the dimension function of a G-sphere.