Magnitude homology of geodesic metric spaces with an upper curvature bound

Abstract

In this article, we study the magnitude homology of geodesic metric spaces of curvature $\le \kappa$, especially $\mathrm{CAT}(\kappa )$ spaces. We will show that the magnitude homology $M{H}_n^l(X)$ of such a meric space $X$ vanishes for small $l$ and all $n>0$. Conseqently, we can compute a total $\mathbb{Z}$-degree magnitude homology for small $l$ for the shperes ${\mathbb{S}}^n$, the Euclid spaces ${\mathbb{E}}^n$, the hyperbolic spaces ${\mathbb{H}}^n$, and real projective spaces ${\mathbb{RP}}^n$ with the standard metric. We also show that an existence of closed geodesic in a metric space guarantees the non-triviality of magnitude homology.

16pages

Cited by in corpus (3)

• Magnitude homology of enriched categories and metric spaces
• Magnitude homology equivalence of Euclidean sets
• Magnitude homology of geodesic space