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Browsing by Subject "Topology"

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  • Hallamaa, Luukas (2019)
    The purpose of this thesis is to present some dimension theory of separable metric spaces, and with the theory developed, prove Brouwer’s Theorem on the Invariance of Domain. This theorem states, that if we embed a subset of the n-dimensional Euclidean space into the aforementioned space, this embedding is an open map. We begin by revising some elementary theory of point-set topology, that should be familiar to any graduate student in mathematics. Drawing from these rudiments, we move on to the concept of dimension. The dimension theory presented is based on the notion of the small inductive dimension. We define this dimension function for regular spaces and state and prove various results that hold for this function. Although this dimension function is defined on regular spaces, we mainly focus on separable metric spaces. Among other things, we prove that the small inductive dimension of the Euclidean n-space is exactly n. This proof makes use of the famous Brouwer Fixed-Point Theorem, which we naturally also prove. We give a combinatorial proof of the Fixed-Point Theorem, which relies on Sperner’s lemma. We move on to develop some theory regarding the extensions of functions. These various results on extensions allow us to finally prove the theorem that lent its name to this thesis: Brouwer’s Theorem on the Invariance of Domain.
  • Kelomäki, Tuomas (2020)
    This thesis provides a proof and some applications for the famous result in topology called the Borsuk-Ulam theorem. The standard formulation of the Borsuk-Ulam theorem states that for every continuous map from an n-sphere to n-dimensional Euclidean space there are antipodal points that map on top of each other. Even though the claim is quite elementary, the Borsuk-Ulam theorem is surprisingly difficult to prove. There are many different kinds of proofs to the Borsuk-Ulam theorem and nowadays the standard method of proof uses heavy algebraic topology. In this thesis a more elementary, geometric proof is presented. Some fairly fundamental geometric objects are presented at the start. The basics of affine and convex sets, simplices and simplicial complexes are introduced. After that we construct a specific simplicial complex and present a method, iterated barycentric subdivision, to make it finer. In addition to simplicial complexes, the theory we are using revolves around general positioning and perturbations. Both of these subjects are covered briefly. A major part in our proof of the Borsuk-Ulam theorem is to show that a certain homotopy function F from a specific n + 1-manifold to the n-dimensional Euclidean space can be by approximated another map G. Moreover this approximation can be done in a way so that the kernel of G is a symmetric 1-manifold. The foundation for approximating F is laid with iterated barycentric subdivision. The approximation function G is obtained by perturbating F on the vertices of the simplicial complex and by extending it locally affinely. The perturbation is done in a way so that the image of vertices is in a general position. After proving the Borsuk-Ulam theorem, we present a few applications of it. These examples show quite nicely how versatile the Borsuk-Ulam theorem is. We prove two formulations of the Ham Sandwich theorem. We also deduce the Lusternik-Schnirelmann theorem from the Borsuk- Ulam theorem and with that we calculate the chromatic numbers of the Kneser graphs. The final application we prove is the Topological Radon theorem.
  • Vuorenmaa, Elmo (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.