Persistent homology and geometry - Lemež, Boštjan (visokošolska dela, 2022)
In thesis we introduce a novel simplicial complex assigned to a decreasing sequence of scales, called the selective Rips complex. It is a generalization of the Vietoris-Rips complex, where a sequence of scales is used instead of a single scale. A simpler version of the selective Rips complexes was previously designed to detect more geometric features than their Rips counterparts. We study the properties of selective Rips complexes and generalize the theory of Vietoris-Rips complexes. We prove the Stability Theorem for selective Rips complexes. The main contributions of this dissertation are various reconstruction results up to the homotopy type using selective Rips complexes. First, we prove that the selective Rips complex of a closed Riemannian manifold ?$X$? is homotopy equivalent to ?$X$? for appropriately small scales. When restricted to Vietoris-Rips complexes, we provide a novel proof of the Hausmann's reconstruction result. Next, we present finite reconstruction results with selective Rips complexes and intrinsic Čech complexes. We prove that if a metric space ?$S$? is close enough to a closed Riemannian manifold ?$X$? in the Gromov-Hausdorff distance, the selective Rips complex (and also the intrinsic Čech complex) of ?$S$? is homotopy equivalent to ?$X$? for appropriately small scales. As a special case, we provide a novel proof of the Latschev's reconstruction result. Finally, we classify the one-dimensional persistence of geodesic spaces arising from selective Rips complexes. We prove that 1-dimensional persistence of Rips and selective Rips complexes are isomorphic up to reparametrization.
Keywords: computational topology; geodesic space; geodetski prostor; homotopska rekonstukcija; homotopy reconstruction; izrek o živcu; matematika; mathematics; nerve theorem; računska topologija; Riemannian manifolds; Riemannove mnogoterosti; selective Rips complex; selektivni Ripsov kompleks; Vietoris-Rips complex; Vietoris-Ripsov kompleks