Magnitude homology, diagonality, and median spaces (Q2240606)

The Künneth formula and the Mayer-Vietoris sequence for the magnitude homology of graphs were proved by by \textit{R. Hepworth} and \textit{S. Willerton} [Homology Homotopy Appl. 19, No. 2, 31--60 (2017; Zbl 1377.05088)]. In this paper, the authors prove that the Künneth formula and the Mayer-Vietoris sequence for the magnitude homology of graphs can be generalized to a metric setting. The authors extend the notion of diagonality of graphs to metric spaces. Then they verify its stability under products, retracts, and filtrations. As an application, the authors show that median spaces are diagonal; in particular any Menger convex median space has vanishing magnitude homology. In Section 1.1, the authors introduce the magnitude homology of median spaces. They prove Proposition 6.3: Median metric spaces are diagonal. In Section 1.2, the authors introduce the Künneth formula and the Mayer-Vietoris sequence. They prove Proposition 4.3 (Künneth theorem -- metric case): If $X$, $Y$ are metric spaces and $X\times Y$ is their $l^1$ product, then there exists a natural ``cross-product'' morphism $MH_\ast(X)\otimes MH_\ast(Y)\xrightarrow{\square} MH_\ast(X\times Y)$ $[f]\otimes [g]\mapsto [f\square g]$, which fits into a natural short exact sequence \[ 0\to MH_\ast(X)\otimes MH_\ast(Y)\xrightarrow{\square} MH_\ast(X\times Y)\to\mathrm{Tor}(MH_{\ast-1}(X),MH_\ast(Y))\to 0. \] and Theorem 4.14 (Mayer-Vietoris -- metric case): If $X=Y\cup Z$ is a gated decomposition of $X$ and $W=Y\cap Z$, then the inclusions $j_Y:W\to Y$, $j_Z:W\to Z$, $i_Y:Y\to X$, $i_Z:Z\to X$ induce a short exact sequence \[ 0\to MH_\ast(W)\xrightarrow{\langle(j_Y)_\ast,-(j_Z)_\ast\rangle} MH_\ast(Y) \oplus MH_\ast(Z)\xrightarrow{(i_Y)_\ast\oplus(i_Z)_\ast} MH_\ast(X)\to 0. \] In Section 2, the authors review the background of metric spaces, graphs, and magnitude homology. In Section 3, the authors study median graphs. In Section 4, the authors prove the Künneth formula and the Mayer-Vietoris sequence. In Section 5, the authors discuss diagonality. In Section 6, the authors prove that median spaces are diagonal.