Persistent homology of products and Gromov-Hausdorff distances between hypercubes and spheres - Vargas-Rosario, Daniel, author; Adams, Henry, advisor; Hulpke, Alexander, committee member; Duflot-Miranda, Jeanne, committee member; Bacon, Joel, committee member (Colorado State University. Libraries, 2023)
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.
Keywords: Hamming metric; persistent homology; Vietoris-Rips; hypercube; Gromov-Hausdorff distance; spectral sequence