Rational formality of function spaces

TL;DR

This paper investigates the conditions under which the space of continuous maps between certain rational spaces is formal, showing it implies the target space has a specific rational homotopy type, with a counterexample illustrating limitations.

Contribution

It establishes a criterion linking the formality of function spaces to the rational homotopy type of the target space, and provides a counterexample to the generality of this relationship.

Findings

If (X,Y) is rationally formal, then Y has the rational homotopy type of a finite product of Eilenberg-Mac Lane spaces.

An example of a formal function space (S^2,Y) where Y is not a product of Eilenberg-Mac Lane spaces.

The odd part of the rational Hurewicz homomorphism is non-zero under the given conditions.

Abstract

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: $\pi_{odd}(X)\otimes \mathbb Q\to H_{odd}(X,\mathbb Q)$ 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.