Rational formality of mapping spaces

TL;DR

This paper characterizes when the mapping space between certain finite nilpotent CW complexes is rationally formal, showing it occurs precisely when the target has a specific rational homotopy type.

Contribution

It generalizes previous results by providing a complete characterization of the rational formality of mapping spaces under specified conditions.

Findings

Mapping space is rationally formal iff Y has the rational homotopy type of a product of odd spheres.

Extends results of Vigué-Poirrier and Yamaguchi to broader classes of complexes.

Provides a necessary and sufficient condition for rational formality in this context.

Abstract

Let X and Y be finite nilpotent CW complexes with dimension of X less than the connectivity of Y. Generalizing results of Vigu\'e-Poirrier and Yamaguchi, we prove that the mapping space Map(X,Y) is rationally formal if and only if Y has the rational homotopy type of a finite product of odd dimensional spheres.