Algebraic Hopf invariants and rational models for mapping spaces

In this paper we will define an invariant $m{c}_{\mathrm{\infty }}(f)$ of maps $f:X \rightarrow Y_{$\mathbb{Q}$}$betweenafiniteCW-complexandarationalspace$Y_{$\mathbb{Q}$}$\mathrm{.}Weprovethatthisinvariantiscomplete,i\mathrm{.}e\mathrm{.}$mc_{\infty}(f)=mc_{\infty}(g)$ifanonlyif$f$and$g$ are homotopic. We will also construct an $L_{\mathrm{\infty }}$-model for the based mapping space $Ma{p}_{\ast }(X,Y_{\mathbb{Q}})$ from a $C_{\mathrm{\infty }}$-coalgebra and an $L_{\mathrm{\infty }}$-algebra.