A rational model for the evaluation map

Let $X$ be an $l$-connected space and $U$ a connected CW complex with $\dim U \leq l$. Let ${\mathcal F}(U, X)$ be the space of continuous maps from $U$ to $X$. In this paper, an algebraic model for the evaluation map ${\mathcal F}(U, X) \times U \to X$ is considered in terms of the model for the function space due to Brown and Szczarba [B-S]. It turns out that the Brown and Szczarba model for the function space coincides with Haefliger's model.

[B-S] Brown Jr, E. H. and Szczarba, R. H.: On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349(1997), 4931-4951.