Persistent homology of products and Gromov-Hausdorff distances between hypercubes and spheres
Date
2023
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Abstract
An exploration in the first half of this dissertation of the relationships among spectral sequences, persistent homology, and products of simplices, including the development of a new concept in categorical product filtration, is followed in the second half by new determinations of a) lower bounds for the Gromov-Hausdorff distance between n-spheres and (n + 1)-hypercubes equipped with the geodesic metric and of b) new lower bounds for the coindexes of the Vietoris-Rips complexes of hypercubes equipped with the Hamming metric. In their paper, "Spectral Sequences, Exact Couples, and Persistent Homology of Filtrations", Basu and Parida worked on building an n-derived exact couple from an increasing filtration X of simplicial complexes, C(n)(X) = {D(n)(X), E(n)(X), i(n), j(n), ∂(n)}. The terms E(n)∗,∗ (X) are the bigraded vector spaces of a spectral sequence that has differentials d(r)(X), and the terms D(n)∗,∗ (X) are the persistent homology groups H∗,∗∗ (X). They proved that there exists a long exact sequence whose groups are H∗,∗ ∗ (X) and whose bigraded vector spaces are (E∗∗, ∗(X), d∗(X)). We establish in Section 3 of this dissertation a new, similar theorem in the case of the categorical product filtration X × Y that states that there exists a long exact sequence consisting of ⊕(l+j=n) H∗,∗ l (X) ⊗ H∗,∗j (Y) and of the bigraded vector spaces E∗ ∗,∗(X × Y) of (E∗ ∗,∗(X × Y ),d∗(X × Y)), and prove it in part using Künneth formulas on homology. The emphasis on product spaces continues in Section 5, where we establish new lower bounds for the Gromov-Hausdorff distance between n-spheres and (n+1)-hypercubes, I(n+1), when both are equipped with the geodesic distance. From these lower bounds, we conjecture new lower bounds for the coindices of the Vietoris-Rips complexes of hypercubes when equipped with the Hamming metric. We then determine new lower bounds for the coindices of the Vietoris-Rips complexes of hypercubes, a) by producing a map between spheres and the geometric realizations of Vietoris-Rips complexes of hypercubes using abstract convex combination and balanced sets, and b) by decomposing hollow n-cubes (homotopically equivalent to the above-mentioned spheres) into simplices of smaller dimension and smaller diameter.
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Subject
Hamming metric
persistent homology
Vietoris-Rips
hypercube
Gromov-Hausdorff distance
spectral sequence