Note on the rational cohomology of the function space of based maps

Abstract

In this paper, for a formal, path connected, finite-dimensional CW-complex $X$ of finite type and a $q$-connected space $Y$ of finite type with $q \geq \rm{dim} X$, we determine the necessary and sufficient condition for the rational cohomology algebra $H^*(\mathcal{F}_*(X,Y);\mathbb{Q})$ of the function space $\mathcal{F}_*(X,Y)$ of based maps to be free.