Persistent homology of the sum metric

Abstract

Given finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, we investigate the persistent homology $P{H}_{\ast }(X\times Y)$ of the Cartesian product $X\times Y$ equipped with the sum metric $d_X+d_Y$. Interpreting persistent homology as a module over a polynomial ring, one might expect the usual Künneth short exact sequence to hold. We prove that it holds for $P{H}_0$ and $P{H}_1$, and we illustrate with the Hamming cube $\{0,1{\}}^k$ that it fails for $P{H}_n,n\ge 2$. For $n=2$, the prediction for $P{H}_2(X\times Y)$ from the expected Künneth short exact sequence has a natural surjection onto $P{H}_2(X\times Y)$. We compute the nontrivial kernel of this surjection for the splitting of Hamming cubes $\{0,1{\}}^k=\{0,1{\}}^{k-1}\times \{0,1\}$. For all $n\ge 0$, the interleaving distance between the prediction for $P{H}_n(X\times Y)$ and the true persistent homology is bounded above by the minimum of the diameters of $X$ and $Y$. As preliminary results of independent interest, we establish an algebraic Künneth formula for simplicial modules over the ring $\kappa [{\mathbb{R}}_{+}]$ of polynomials with coefficients in a field $\kappa$ and exponents in ${\mathbb{R}}_{+}=[0,\infty )$, as well as a Künneth formula for the persistent homology of ${\mathbb{R}}_{+}$-filtered simplicial sets -- both of these Künneth formulas hold in all homological dimensions $n\ge 0$.

To appear in Journal of Pure and Applied Algebra