Formality of function spaces - Mathematics > Algebraic Topology

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Abstract: Let $X$ be a nilpotent space such that there exists $p\geq 1$ with$H^p(X,\mathbb Q) e 0$ and $H^n(X,\mathbb Q)=0$ if $n>p$. Let $Y$ be am-connected space with $m\geq p+1$ and $H^*(Y,\mathbb Q)$ is finitely generatedas algebra. We assume that $X$ is formal and there exists $p$ odd such that$H^p(X,\mathbb Q) e 0$. We prove that if the space $\mathcal F(X,Y)$ ofcontinuous maps from $X$ to $Y$ is formal, then $Y$ has the rational homotopytype of a product of Eilenberg Mac Lane spaces. At the opposite, we exhibit anexample of a formal space $\mathcal F(S^2,Y)$ where $Y$ is not rationallyequivalent to a product of Eilenberg Mac Lane spaces.

Autor: Micheline Vigué-Poirrier (Paris 13)

Fuente: https://arxiv.org/