Postnikov Towers and Gottlieb Groups of Orbit Spaces

Let X be a 1-connected space with the homotopy type of a CW -space and H a finite group acting freely on X by homeomorphisms homotopic to the identity. We prove that lkη∗Gk(X) ⊆ Gk(X/H) for all k > 1 and some estimated positive integer lk which depends on k, where Gk is the k′th Gottlieb group and η : X → X/H is the quotient map to the orbit space X/H. We show that lk is independent of k for X with the homotopy type of a finite CW -space. We also obtain that lπk(X) ⊆ Gk(X) for some positive integer l (independent on k) provided some restrictions are placed on the space X and the integer k > 1. Moreover, η∗Gk(X)p = Gk(X/H)p for the p-primary components, where p is a prime not dividing the order |H| of the group H.