Rational formality of function spaces
Micheline Vigue-Poirrier

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 be a nilpotent space such that there exists with and if . Let be a m-connected space with and is finitely generated as algebra. We assume that the odd part of the rational Hurewicz homomorphism: is non-zero. We prove that if the space of continuous maps from to is rationally formal, then 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 where is not rationally equivalent to a product of Eilenberg Mac Lane spaces.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Advanced Topology and Set Theory
