# Magnitude homology of geodesic metric spaces with an upper curvature bound

## TL;DR

This paper investigates the magnitude homology of geodesic metric spaces with an upper curvature bound, revealing vanishing results for small parameters and linking closed geodesics to non-trivial homology.

## Contribution

It provides new vanishing theorems for magnitude homology in curvature-bounded spaces and computes homology for classical spaces like spheres and hyperbolic spaces.

## Key findings

- Magnitude homology vanishes for small l and n > 0 in CAT(κ) spaces
- Explicit computation of magnitude homology for spheres, Euclidean, hyperbolic, and projective spaces
- Presence of closed geodesics implies non-trivial magnitude homology

## Abstract

In this article, we study the magnitude homology of geodesic metric spaces of curvature $\leq \kappa$, especially ${\rm CAT}(\kappa)$ spaces. We will show that the magnitude homology $MH^{l}_{n}(X)$ 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.