Scalable spaces
Geometric Topology
2022-09-16 v3 Algebraic Topology
Differential Geometry
Metric Geometry
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.
Cite
@article{arxiv.1912.00590,
title = {Scalable spaces},
author = {Aleksandr Berdnikov and Fedor Manin},
journal= {arXiv preprint arXiv:1912.00590},
year = {2022}
}
Comments
35 pages, 3 figures; minor changes, accepted for publication in Inventiones