Twisted cohomology theories and the single obstruction to lifting

Abstract

p o h = f L. A map g: K-> E such that p ° g — f and g I L = h we call a lifting of f rel h. In this paper single obstruction Γ(f) e Hf (K, L, f if) is defined, ί? is a so-called Z?-spectrum, and H* ( if) is cohomology in that spectrum. If a lifting of / rel h exists, Γ{f) = 0; this condition is also sufficient if the fiber of p is /^-connected and dim (K/L) ^ 2k + 1. If g0 and #i are liftings of / rel h, a single obstruction <5(#o, #i; h)eH(K, L, f: if) is also defined; if # 0 and #i are connected by a homotopy of liftings of / rel h δ(g 0, g^ h)=0; this condition is, also sufficient if p is /^-connected and dim (K/L) ^ 2k. In § 4, a spectral sequence is constructed for cohomology in a i?-spectrum, based on the Postnikov tower of that spectrum, and the relationship between the single obstruction and the classical obstructions is defined.