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

Abstract

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 $X \to K(G,n)$ 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 $\mathrm{map}_* \big(X, K(G,n)\big)$ of pointed continuous maps $X \to K(G,n)$ is shown to be homotopy equivalent to the Cartesian product \[ \prod_{i \leq n} \mathrm{map}_* \big(M_i, K(G,n)\big). \] Here, $M_i$ is a Moore complex of type $M\big(H_i(X), i\big)$. The spaces of functions are equipped with the compact open topology.