Browsing by Subject "simplicial complex"
Now showing items 1-2 of 2
-
(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.
-
(2019)We study a subcategory of topological spaces called polyhedrons. In particular, the work focuses on simplicial complexes out of which polyhedrons are constructed. With simplicial complexes we can calculate the homology groups of polyhedrons. These are computationally easier to handle compared to singular homology groups. We start by introducing simplicial complexes and simplicial maps. We show how polyhedrons and simplicial complexes are related. Simplicial maps are certain maps between simplicial complexes. These can be transformed to piecewise linear maps between polyhedrons. We prove the simplicial approximation theorem which states that for any continuous function between polyhedrons we can find a piecewise linear map which is homotopic to the continuous function. In section 4 we study simplicial homology groups. We prove that on polyhedrons the simplicial homology groups coincide with singular homology groups. Next we give an algorithm for calculating the homology groups from matrix presentations of boundary homomorphisms. Also examples of these calculations are given for some polyhedrons. In the last section, we assign an integer called the Lefschetz number for continuous maps from a polyhedron to itself. It is calculated using the induced map between homology groups of the polyhedron. With the help of Hopf trace theorem we can also calculate the Lefschetz number using the induced map between chain complexes of the polyhedron. We prove the Lefschetz fixed-point theorem which states that if the Lefschetz number is not zero, then the continuous function has a fixed-point.
Now showing items 1-2 of 2