Obstructions for constructing equivariant fibrations

Let G be a finite group and H be a family of subgroups of G which is closed under conjugation and taking subgroups. Let B be a G-CW-complex whose isotropy subgroups are in H and let F = {F H} H e{open}H be a compatible family of H -spaces. A G -fibration over B with the fiber type F = {F H} H e{open}H is a G -equivariant fibration p: E → B where p -1(b) is G b -homotopy equivalent to F Gb for each b e{open} B. In this paper, we develop an obstruction theory for constructing G-fibrations with the fiber type F over a given G -CW-complex B. Constructing G -fibrations with a prescribed fiber type F is an important step in the construction of free G -actions on finite CW-complexes which are homotopy equivalent to a product of spheres.