Persistent homology of the sum metric
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Given finite metric spaces (X, d_X) and (Y, d_Y), we investigate the persistent homology PH_*(X × Y) of the Cartesian product X × 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 PH_0 and PH_1, and we illustrate with the Hamming cube {0,1}^k that it fails for PH_n, n ≥ 2. For n = 2, the prediction for PH_2(X × Y) from the expected Künneth short exact sequence has a natural surjection onto PH_2(X × Y). We compute the nontrivial kernel of this surjection for the splitting of Hamming cubes {0,1}^k = {0,1}^k-1×{0,1}. For all n ≥ 0, the interleaving distance between the prediction for PH_n(X × 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 κ[R_+] of polynomials with coefficients in a field κ and exponents in R_+ = [0,∞), as well as a Künneth formula for the persistent homology of R_+-filtered simplicial sets -- both of these Künneth formulas hold in all homological dimensions n ≥ 0.
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