Scalable spaces

TL;DR

Scalable spaces are a class of formal, simply connected manifolds with a special cohomology embedding property, impacting rational homotopy theory and providing counterexamples to Gromov's conjecture on higher homotopy distortion.

Contribution

The paper introduces the concept of scalable spaces, linking formality with a metric property, and demonstrates their implications in rational homotopy theory and Gromov's conjecture.

Findings

Scalable spaces are formal and have cohomology algebra embeddings.

Spaces that are formal but not scalable challenge Gromov's conjecture.

Scalability relates to quantitative homotopy theory properties.

Abstract

\emph{Scalable spaces} are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality. They are also characterized by particularly nice behavior from the point of view of quantitative homotopy theory. Among other results, we show that spaces which are formal but not scalable provide counterexamples to Gromov's long-standing conjecture on distortion in higher homotopy groups.