The singular cohomology of the inverse limit of a Postnikov tower is representable

Abstract

Let X 1 ← X 2 ← ⋯ {X_1} \leftarrow {X_2} \leftarrow \cdots be an inverse sequence of spaces and maps satisfying (i) each X n {X_n} has the homotopy type of a CW complex, (ii) each f n {f_n} is a Hurewicz fibration, and (iii) the connectivity of the fiber of f n {f_n} goes to ∞ \infty with n n . Let X ^ \hat X be the inverse limit of the sequence. It is shown that the natural homomorphism H ˇ k ( X ^ , G ) → H k ( X ^ , G ) \check {H}^k(\hat {X},G) \to H^k(\hat {X}, G) (from Čech cohomology to singular cohomology, with ordinary coefficient module G G ) is an isomorphism for all k k . It follows that lim → n [ X n , K ( G , k ) ] ≅ [ X ^ , K ( G , k ) ] {\lim _{ \to n}}[{X_n},K(G,k)] \cong [\hat X,K(G,k)] for any Eilenberg- Mac Lane space K ( G , k ) K(G,k) . It is also shown that, except in trivial cases, X X does not have the homotopy type of a CW complex.