L_∞ models of based mapping spaces

Abstract

In this paper, for any pointed map f : X → Y between finite type nilpotent CW-complexes, we obtain L_∞ and Lie models of map_f^*(X,Y), the pointed space of based maps homotopic to f, in terms of Lie algebras constructed from the Quillen models of X and Y. The main advantage of our approach is to allow X to be an infinite dimensional CW-complex, in which case map_f^*(X,Y) has no longer the homotopy type of a finite type CW-complex.