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

## Jaka Smrekar

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 XK(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 map(X, K(G,n)) of pointed continuous maps XK(G,n) is shown to be homotopy equivalent to the Cartesian product
 ∏ i ≤ n map∗(Mi, K(G,n)).
Here, Mi is a Moore complex of type M(Hi(X), i). The spaces of functions are equipped with the compact open topology.

Homology, Homotopy and Applications, Vol. 15 (2013), No. 1, pp.137-149.

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