Rational formality of function spaces

Abstract

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

10 pages