Extensions of maps from suspensions of finite projective spaces (Q909292)

H. Miller's proof of the Sullivan Conjecture shows that if X is a nilpotent space and \(H^*(X;{\mathbb{Z}}/2{\mathbb{Z}})\) is bounded, then \([\Sigma^ tRP^{\infty},X]=0\). We prove a finite version of this theorem. Specifically, we give a number k such that the restriction map \([\Sigma^ tRP^{2^ k},X]\to [\Sigma^ tRP^ n,X]\) is the zero map. The number k depends on n, t, and the connectivity and dimension of \(\bar H^*(X;{\mathbb{Z}}/2{\mathbb{Z}})\). The proof factors the restriction map through another group of homotopy classes of maps, \([\Sigma^{t-1}T_ 1(2^ k),X]\), and studies the filtrations in the Bousfield-Kan spectral sequences for \([\Sigma^{t-1}T_ 1(2^ k),X]\) and \([\Sigma^ tRP^ n,X]\).