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Metric thickenings and complexes on metric spaces

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Title: Metric thickenings and complexes on metric spaces
Author(s): Vuorenmaa, Elmo
Contributor: University of Helsinki, Faculty of Science
Degree program: Master's Programme in Mathematics and Statistics
Specialisation: Mathematics
Language: English
Acceptance year: 2021
In topology, one often wishes to find ways to extract new spaces out of existing spaces. For example, the suspension of a space is a fundamental technique in homotopy theory. However, in recent years there has been a growing interest in extracting topological information out of discrete structures. In the field of topological data-analysis one often considers point clouds, which are finite sets of points embedded in some R^m. The topology of these sets is trivial, however, often these sets have more structure. For example, one might consider a uniformly randomly sampled set of points from a circle S1. Clearly, the resulting set of points has some geometry associated to it, namely the geometry of S1. The use of certain types of topological spaces called Vietoris-Rips and Cech complexes allows one to study the "underlying topology" of point clouds by standard topological means. This in turn enables the application of tools from algebraic topology, such as homology and cohomology, to be applied to point clouds. Vietoris-Rips and Cech complexes are often not metrizable, even though they are defined on metric spaces. The purpose of this thesis is to introduce a homotopy result of Adams and Mirth concerning Vietoris-Rips metric thickenings. In the first chapter, we introduce the necessary measure theory for the main result of the thesis. We construct the 1-Wasserstein distance, and prove that it defines a metric on Polish spaces. We also note, that the 1-Wasserstein distance is a metric on general metric spaces. In the sequel, we introduce various complexes on spaces. We study simplicial complexes on R^n and introduce the concept of a realization. We then prove a theorem on the metrizability of a realization of a simplicial complex. We generalize simplicial complexes to abstract simplicial complexes and study the geometric realization of some complexes. We prove a theorem on the existence of geometric realizations for abstract simplicial complexes. Finally, we define Vietoris-Rips and Cech complexes, which are complexes that are formed on metric spaces. We introduce the nerve lemma for Cech complexes, and prove a version of it for finite CW-complexes. The third chapter introduces the concept of reach, which in a way measures the curvature of the boundary of a subset of R^n. We prove a theorem that characterizes convex, closed sets of R^n by their reach. We also introduce the nearest point projection map π, and prove its continuity. In the final chapter, we present some more measure theory, which leads to the definitions of Vietoris-Rips and Cech metric thickenings. The chapter culminates in constructing an explicit homotopy equivalence between a metric space X of positive reach and its Vietoris-Rips metric thickening.
Keyword(s): Topology Vietoris-Rips complex simplicial complex metric thickening 1-Wasserstein metric

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