Formality of function spaces

Abstract

Let $X$ be a nilpotent space such that there exists $p\ge 1$ with $H^p(X,\mathbb{Q})\ne 0$ and $H^n(X,\mathbb{Q})=0$ if $n>p$. Let $Y$ be a m-connected space with $m\ge p+1$ and $H^{\ast }(Y,\mathbb{Q})$ is finitely generated as algebra. We assume that $X$ is formal and there exists $p$ odd such that $H^p(X,\mathbb{Q})\ne 0$. We prove that if the space $\mathcal{F}(X,Y)$ of continuous maps from $X$ to $Y$ is formal, then $Y$ has the rational homotopy type of a product of Eilenberg Mac Lane spaces. At the opposite, we exhibit an example of a formal space $\mathcal{F}(S^2,Y)$ where $Y$ is not rationally equivalent to a product of Eilenberg Mac Lane spaces.

8 pages