CW towers and mapping spaces

Let …→Z3→Z2→Z1…→Z3→Z2→Z1 be a tower of Hurewicz fibrations where each ZiZi is homotopy equivalent to a CW complex. An important question is when also the inverse limit space is homotopy equivalent to a CW complex; we present a sufficient condition that is necessary up to a looping of the tower. When the homotopy groups of the ZiZi vanish above a fixed dimension, we obtain simple necessary and sufficient conditions. The main application is map(X,Y)map(X,Y), the space of continuous maps from a countable CW complex X to a CW complex Y. Our results are sharpest when Y   is countable with finitely many nontrivial homotopy groups. Then, the path component of f∈map(X,Y)f∈map(X,Y) is homotopy equivalent to a CW complex if and only if all homotopy groups πk(map(X,Y),f)πk(map(X,Y),f) are countable and f is not a phantom map. Let Y   be nilpotent and let l⁎:Y→Y(P)l⁎:Y→Y(P) be localization at the set of primes P  . We show that if Y(P)Y(P) is a Postnikov section and map(X,Y)map(X,Y) has CW homotopy type, the induced mapping l⁎:map(X,Y)→map(X,Y(P))l⁎:map(X,Y)→map(X,Y(P)) is localization at P on each path component, thereby extending a classical result from localization theory.