Algebraic Footprints of Geometric Features in Homology

Suppose we are given a topological space, either as an abstract mathematical object or as the underlying shape of some data. The most important topological properties of interest are the holes: one-dimensional holes appearing as bottlenecks in the contraction of loops, two-dimensional holes showing up as voids, etc. Such holes are formalized with the concept of homology, whose study has occupied algebraic topology and other areas of mathematics for more than a century. The past two decades have witnessed the development of a variant called persistent homology. Among its motivations are the desires to measure sizes of holes and to compute homology from finite subsets. A common workflow in this research is the following: given a finite subset of some underlying shape, construct a combinatorial approximation via a Vietoris-Rips or a Cech complex, and use this complex to infer homological information about the shape. Indeed, it is possible to obtain the homology of a shape using parameters of appropriately small scale. The motivating idea of this proposal is the observation that the mentioned complexes contain more information than just the topology of the shape. In particular, for intermediate scales we can interpret high-dimensional homology as reflections of geometric features. For example, we can detect interesting closed and contractible geodesics from two- or three-dimensional homology. The proposal outlines a general theory for the retrieval of geometric information from persistent homology. The approach contains aspects of combinatorics and discrete geometry (configurations of points on spheres), of topology (Vietoris-Rips nerve theorem), and of geometry (local contractibility properties). Concepts in this theory provide connections to other areas within mathematics, such as the contractibility of metric graphs, the length and Laplacian spectra of manifolds, and the existence of 1-Lipschitz cohomology representatives.

In a metric space, there is a well defined non-negative distance between any two of its points, and these distances satisfy a few natural conditions, including symmetry: the distance from A to B is that same as that from B to A. For example, a closed surface in which the distance is the length of the shortest curve that connects two points in the surface is a metric space. The starting point of this project was the insight that subtle geometric properties, such as the length of a locally shortest non-contractible loop, is reflected in the homology of a nested sequence of infinite complexes constructed from neighborhoods whose size is controlled by a real parameter. For example, the Vietoris--Rips complex that connects all points at distance smaller than or equal to r > 0 developes a filling of this loop when r reaches one third of its length. The goal of this project is to shed light on this connection and to move it closer to applications. The infinite complexes can be viewed as limits of the complexes obtained from finite samplings, so the difference between their topologies may be interpreted as measuring the lack of information caused by finite sampling. A particularly useful direction to get closer to applications is the generalization of the setting to quasi-metric spaces, in which the distances are allowed to violate the symmetry condition: the distance from A to B may be different from that from B to A. We investigate how this affects the homology of the corresponding complexes, and how this difference can be used to better understand non-symmetric geo-spatial data, such as the mobility of Austrians during different phases of the COVID-19 pandemic.