$L_{i}nfty$ rational homotopy of mapping spaces

Abstract

In this paper we describe explicit $L_{\infty }$ algebras modeling the rational homotopy type of any component of the spaces $\map(X,Y)$ and $\map^*(X,Y)$ of free and pointed maps between the finite nilpotent CW-complex $X$ and the finite type nilpotent CW-complex $Y$. When $X$ is of finite type, non necessarily finite, we also show that the algebraic covers of these $L_{\infty }$ algebras model the universal covers of the corresponding mapping spaces.

19 pages