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Rational formality of function spaces

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Micheline Vigué-Poirrier

Let $X$ be a nilpotent space such that there exists $N\geq 1$ 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 $m\geq N+1$ and $H^*(Y,\mathbb Q)$ is finitely
generated as algebra. We assume that the odd part of the rational Hurewicz
homomorphism: $\pi _{odd}(X)\otimes \mathbb Q\rightarrow H_{odd}(X,\mathbb Q)$
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.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 1, pp. 99-108