Homotopy type of space of maps into a $K(G,n)$

摘要

Let $X$ be a connected CW complex and let $K(G,n)$ be an Eilenberg-Mac Lane CW complex where $G$ is abelian. As $K(G,n)$ may be taken to be an abelian monoid, the weak homotopy type of the space of continuous functions $X o K(G,n)$ depends only upon the homology groups of $X$. The purpose of this note is to prove that this is true for the actual homotopy type. Precisely, the space $mathrm{map}_* ig(X, K(G,n)ig)$ of pointed continuous maps $X o K(G,n)$ is shown to be homotopy equivalent to the Cartesian product[ prod_{i leq n} mathrm{map}_* ig(M_i, K(G,n)ig). ]Here, $M_i$ is a Moore complex of type $Mig(H_i(X), iig)$. The spaces of functions are equipped with the compact open topology.