CW towers and mapping spaces

摘要

Let ... -> Z(3) -> Z(2) -> Z(1) be a tower of Hurewicz fibrations where each Z(i) 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 Z(i) vanish above a fixed dimension, we obtain simple necessary and sufficient conditions. The main application is map(X, Y), the space of continuous maps from a cou