标题： 欧几里得集的幅度同调等价 显示英文标题

标题： Magnitude homology equivalence of Euclidean sets

摘要： 幅度同调是度量空间的$\mathbf{R}^+$-分次同调理论，它捕捉了测地线复杂性的信息。 在这里，我们解决这样一个问题：在存在反向映射并在同调中诱导互为逆映射的情况下，两个度量空间何时具有幅度同调等价性？ 我们在闭合欧几里得集的情况下给出了一个具体的几何必要且充分条件。 在此过程中，我们引入了内边界和核心的凸几何概念，并证明了卡拉泰奥多里经典定理在闭合凸集情况下的加强版本。 显示英文摘要

摘要： Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense that there exist back-and-forth maps inducing mutually inverse maps in homology? We give a concrete geometric necessary and sufficient condition in the case of closed Euclidean sets. Along the way, we introduce the convex-geometric concepts of inner boundary and core, and prove a strengthening for closed convex sets of the classical theorem of Carath\'eodory.