Volume 21, issue 2 (2021)

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Magnitude homology of geodesic metric spaces with an upper curvature bound

Yasuhiko Asao

Algebraic & Geometric Topology 21 (2021) 647–664
Abstract

We study the magnitude homology of geodesic metric spaces of curvature $\le \kappa$, especially $CAT\left(\kappa \right)$ spaces. We will show that the magnitude homology ${MH}_{n}^{l}\left(X\right)$ of such a metric space $X$ vanishes for small $l$ and all $n>0$. Consequently, we can compute magnitude homology in small length gradings for spheres ${\mathbb{𝕊}}^{n}$, the Euclidean spaces ${\mathbb{𝔼}}^{n}$, the hyperbolic spaces ${ℍ}^{n}$ and real projective spaces ${ℝℙ}^{n}$ with the standard metric. We also show that the existence of a closed geodesic in a metric space guarantees the nontriviality of magnitude homology.

Keywords
magnitude homology, geodesics, CAT(k) spaces
Mathematical Subject Classification 2010
Primary: 51F99, 55N35