Künneth Formulae in Persistent Homology

Abstract

The classical Künneth formula in algebraic topology describes the homology of a product space in terms of that of its factors. In this paper, we prove Künneth-type theorems for the persistent homology of the categorical and tensor product of filtered spaces. That is, we describe the persistent homology of these product filtrations in terms of that of the filtered components. In addition to comparing the two products, we present two applications in the setting of Vietoris-Rips complexes: one towards more efficient algorithms for product metric spaces with the maximum metric, and the other recovering persistent homology calculations on the n-torus. Contents 1. Introduction 1.1. Sketch of proof 1.2. Organization of the paper 1.3. Related work 2. Preliminaries 2.1. Persistent homology 2.2. The Classical Künneth Theorems 2.3. Homotopy Colimits 3. Products of Diagrams of Spaces 3.1. A comparison theorem 4. A Persistent Künneth formula for the categorical product 5. A Persistent Künneth formula for the generalized tensor product 5.1. The tensor product and Tor of graded modules 5.

Operations on Metric Thickenings

“…Major questions include determining the homotopy type of the Vietoris-Rips complex of a given space at all scale parameters r (see in particular [2] which determines all Vietoris-Rips complexes of the circle), and of determining the topology of the Vietoris-Rips complex of a product, wedge sum, or other gluing of spaces whose individual Vietoris-Rips complexes are known. Metric gluings were studied extensively in [4], and products in [8,15]. Here we study similar questions, not about the Vietoris-Rips simplicial complex but the Vietoris-Rips metric thickening.…”

“…Vietoris-Rips and Čech simplicial complexes preserve certain homotopy properties under products and wedge sums. Indeed, the case of (L ∞ ) products is given in [2, Proposition 10.2], [15], [25] and the case of wedge sums is given in [4,5,9,24].…”

Sliding Window Persistence of Quasiperiodic Functions

“…Our third contribution leverages the aforementioned approximation strategy, parameters choices and a recent persistent Künneth formula [15], to establish bounds for the cardinality and persistence of strong toroidal features in dgm R j pSW d,τ f q, for 1 ď j ď N . We prove the following (Section 6):…”

“…The second result needed to describe dgm R j pT N p F , } ¨}8 q is a Künneth formula for Rips persistent homology and the maximum metric [15,Corollary 4.6]. Proposition 6.2 ([15]).…”

On Vietoris–Rips complexes of ellipses

“…Our Theorem 1 furthermore describes the 3-dimensional homology of VR(Q n ; r) that appears at scale r = 2. See [28,44] and [2,Proposition 10.2] for versions of the Künneth formula for Vietoris-Rips complexes which hold when the metric on X × Y is instead the supremum metric.…”