Magnitude homology of geodesic metric spaces with an upper curvature bound

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