University of Southern Maine USM Digital Commons

Thinking Matters Symposium Archive Student Scholarship

Spring 4-2015

Topological Manifolds

Grant Wilson University of Southern Maine

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Recommended Citation Wilson, Grant, "Topological Manifolds" (2015). Thinking Matters Symposium Archive. 39. https://digitalcommons.usm.maine.edu/thinking_matters/39

This Poster Session is brought to you for free and open access by the Student Scholarship at USM Digital Commons. It has been accepted for inclusion in Thinking Matters Symposium Archive by an authorized administrator of USM Digital Commons. For more information, please contact [email protected]. Topological Manifolds

Presenter: Grant Wilson Faculty Mentor: Laurie Woodman Affiliation: University of Southern Maine Department: Mathematics and Statistics ABSTRACT Example 2: 2-Sphere Topological Manifolds are abstract spaces that locally resemble Euclidean 푆 ≔ 푥, 푦, 푧 : 푥2 + 푦2 + 푧2 = 1 Why Hausdorff? space. For example, consider a round globe and a flat map. The map is 2 This is the unit sphere in ℝ2. a 2-dimensional representation of a 3-dimensional space. Given any The Hausdorff property is required to prevent unusual It is an example of a 2-manifold point on the globe we can find a corresponding position on the map, spaces qualifying as manifolds. and vice versa. This correspondence is called a chart. With a sufficient Some examples of Atlases on the unit sphere are: number of charts, we can describe the whole space. Such a collection Consider the topological space: of charts is called an Atlas. It is possible to construct different Atlases Orthographic Projection: (X , T). ;where X = ℝ ∪ 푝 where 푝 is an extra point for the same space, allowing us to move from one chart, to the space, 푈 = 푂 , 푂 푒 푒 1 2 not in ℝ . And 풯 = 푎, 푏 : 푎 < 푏 ∪ { 푐, 0 ∪ 푝 ∪ to another chart. This process is called a transition map. 푒 푂1 푥, 푦, 푧 = 푥, 푦 ; 푧 > 0 0, 푑 : 푐 < 0, 푑 > 0 The areas of focus for this project include several examples of manifolds 푂 푥, 푦, 푧 = 푥, 푦 ; 푧 ≤ 0 such as curves, n-spheres, and the torus. We explore and illustrate 2 different approaches to charts on these manifolds, the properties of a manifold, examples of spaces that fail to meet these requirements, and Example 3: N-Sphere the derivation of transition maps. The n-sphere is a collection of points that Defn. A Real n-dimensional topological manifold is a extends the idea of a sphere to higher This space is locally homeomorphic to ℝ 푎푛푑is 2nd Hausdorff, Second Countable, topological space which is dimensions. Although we cannot graphically countable, but it is not Haudorff. locally homeomorphic to n-dimensional Euclidean space, i.e. represent this space we can still talk about R. its charts and atlases. Consider points 푝 amd 0. Can we find two distinct 푆 ≔ {풙 = 푥 , 푥 , … , 푥 : | 풙| = 1 } 푒 Defn. 푛 1 2 푛 open sets U, V such that 푝 ∈ 푈 푎푛푑 0 ∈ 푉? Locally Homeomorphic – Small open sets on a manifold M, 푒 Similar to the previous examples in 1, and 2 All neighborhoods of 0 are of the form (a,b) where a < resemble Euclidean space. i.e. there exists a space, we can form an atlas through ? 0 and b > 0. homeomorphism f:M  V where V is a subset of R. This orthographic projection. All neighborhoods of 푝 are of the form 푐, 0 ∪ 푝 ∪ function f is called a chart. 푒 푒 푈 = 휑1, 휑2 ; 푤ℎ푒푟푒 0, 푑 . 휑1 풙 = 푥1, 푥2, … , 푥푛−1 ; 푥푛 > 0 A collection of charts that span the entire space M, is called 휑2 풙 = 푥1, 푥2, … , 푥푛−1 ; 푥푛 ≤ 0 an Atlas. Example 1: 1-Sphere Example 4: Torus Is a 2- manifold. 1 1 Transition Map 2 2 Defn. 푆 × 푆 푤𝑖푡ℎ 푐ℎ푎푟푡푠 {휑푆1, 휑푆1 } 푆1 ≔ 푥, 푦 : 푥 + 푦 = 1 휃1 This is the familiar definition of a the TORUS AS CARTESIAN PRODUCT Clearly every configuration of U and V will have a non unit circle centered at the origin. 휑1 empty intersection. Atlas 1: (orthographic projection) Therefore (X,풯) 𝑖푠 푛표푡 푎 푚푎푛𝑖푓표푙푑. 푈 = 휑1, 휑2 휑1 푥, 푦 = 푦; 푦 > 0 휑2 푥, 푦 = 푦; 푦 ≤ 0 Atlas 2: References 푦 nd 휃 푥, 푦 = 푡푎푛−1 ; 푥 ≠ −1 -Foundations of Topology: C. Wayne Patty; 2 edition 1 푥 + 1 -Geometry and Toplogy Marco Guarltieri −1 푦 휃2 푥, 푦 = 푡푎푛 ; 푥 ≠ −1 -Introduction to Toplogy: Pure and Applied: Adams 1 − 푥 Franzosa