Proof of the Jordan Curve Theorem
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Geometric Algorithms
Geometric Algorithms primitive operations convex hull closest pair voronoi diagram References: Algorithms in C (2nd edition), Chapters 24-25 http://www.cs.princeton.edu/introalgsds/71primitives http://www.cs.princeton.edu/introalgsds/72hull 1 Geometric Algorithms Applications. • Data mining. • VLSI design. • Computer vision. • Mathematical models. • Astronomical simulation. • Geographic information systems. airflow around an aircraft wing • Computer graphics (movies, games, virtual reality). • Models of physical world (maps, architecture, medical imaging). Reference: http://www.ics.uci.edu/~eppstein/geom.html History. • Ancient mathematical foundations. • Most geometric algorithms less than 25 years old. 2 primitive operations convex hull closest pair voronoi diagram 3 Geometric Primitives Point: two numbers (x, y). any line not through origin Line: two numbers a and b [ax + by = 1] Line segment: two points. Polygon: sequence of points. Primitive operations. • Is a point inside a polygon? • Compare slopes of two lines. • Distance between two points. • Do two line segments intersect? Given three points p , p , p , is p -p -p a counterclockwise turn? • 1 2 3 1 2 3 Other geometric shapes. • Triangle, rectangle, circle, sphere, cone, … • 3D and higher dimensions sometimes more complicated. 4 Intuition Warning: intuition may be misleading. • Humans have spatial intuition in 2D and 3D. • Computers do not. • Neither has good intuition in higher dimensions! Is a given polygon simple? no crossings 1 6 5 8 7 2 7 8 6 4 2 1 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 18 4 18 4 19 4 19 4 20 3 20 3 20 1 10 3 7 2 8 8 3 4 6 5 15 1 11 3 14 2 16 we think of this algorithm sees this 5 Polygon Inside, Outside Jordan curve theorem. -
INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C. -
Lecture 1-2: Connectivity, Planar Graphs and the Jordan Curve Theorem
Topological Graph Theory∗ Lecture 1-2: Connectivity, Planar Graphs and the Jordan Curve Theorem Notes taken by Karin Arikushi Burnaby, 2006 Summary: These notes cover the first and second weeks of lectures. Topics included are basic notation, connectivity of graphs, planar graphs, and the Jordan Curve Theorem. 1 Basic Notation graph: G denotes a graph, with no loops or multiple edges. When loops and multiple edges are present, we call G a multigraph. vertex set of G: denoted V (G) or V . The order of G is n = |V (G)|, or |G|. edge set of G: denoted E(G) or E. q = |E(G)| or kGk. subgraphs: H is a spanning subraph of G if V (H) = V (G) and E(H) ⊆ E(G). H is an induced subgraph of G if V (H) ⊆ V (G), and E(H) is all edges of G with both ends in V (H). paths: Pn denotes a path with n vertices; also called an n-path. cycles: Cn denotes a cycle with n vertices; also called an n-cycle. union of graph: G ∪ H has vertex set V (G) ∪ V (H) and edge set E(G) ∪ E(H). intersection of graph: G ∩ H has vertex set V (G) ∩ V (H) and edge set E(G) ∩ E(H). 2 Connectivity We begin by covering some basic terminology. k-connected: A graph G is k-connected if |G| ≥ k + 1 and for all S ⊆ V (G) with |S| < k, the graph G − S is connected. S separates G if G − S is not connected. cutvertex: v ∈ V (G) is a cutvertex if G − {v} is not connected. -
How Twisted Can a Jordan Curve Be?
HOW TWISTED CAN A JORDAN CURVE BE? Manfred Sauter Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless num- ber of possible exceptions. — Felix Klein (1849–1925) We start by fixing notation. A closed curve is a continuous map γ : [0, 1] → 2 R with γ(0) = γ(1), and it is a closed Jordan curve if in addition γ|[0,1) is injective. Similarly one defines a (Jordan) arc, a polygonal (Jordan) arc and a closed polygonal (Jordan) curve. It is well-known, of course, that a closed Jordan curve γ partitions the plane into three connected components: the bounded open interior int γ, the unbounded open exterior ext γ and the compact (image of the) curve itself, which is the common boundary of the first two components. For a captivating historical overview of the struggle to understand curves and toplological dimension we refer to [Cri99], where most of the classical topological results that appear in this note are featured. For a closed polygonal Jordan curve γ there is a simple criterion to check whether a point x ∈ R2 \ γ is in its interior. This criterion is algorithmic in nature and useful in computational geometry, see for example [O’R98, Section 7.4]. Consider a ray emerging from x towards infinity and count the number of intersections with γ. If the ray is chosen such that it intersects the segments of γ only ouside of their end points (which can be easily arranged for a polygonal Jordan curve) or if intersections with the end points are counted appropriately, the point x is in the interior of γ if and only if the number of intersections is odd. -
Knots, 3-Manifolds and the Lickorish Wallace Theorem
KNOTS, 3-MANIFOLDS AND THE LICKORISH WALLACE THEOREM A REPORT submitted in partial fulfillment of the requirements for the award of the dual degree of Bachelor of Science-Master of Science in MATHEMATICS by M V AJAY KUMAR NAIR (13076) DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH BHOPAL BHOPAL - 462066 April 2018 i CERTIFICATE This is to certify that M V Ajay Kumar Nair, BS-MS (Dual Degree) student in Department of Mathematics has completed bonafide work on the dissertation entitled `Knots, 3-manifolds and the Lickorish Wallace theorem' under my supervision and guidance. April 2018 Dr. Kashyap Rajeevsarathy IISER Bhopal Committee Member Signature Date ii ACADEMIC INTEGRITY AND COPYRIGHT DISCLAIMER I hereby declare that this project is my own work and, to the best of my knowledge, it contains no materials previously published or written by another person, or substantial proportions of material which have been ac- cepted for the award of any other degree or diploma at IISER Bhopal or any other educational institution, except where due acknowledgement is made in the document. I certify that all copyrighted material incorporated into this document is in compliance with the Indian Copyright (Amendment) Act (2012) and that I have received written permission from the copyright owners for my use of their work, which is beyond the scope of the law. I agree to indemnify and saveguard IISER Bhopal from any claims that may be arise from any copyright violation. April 2018 M V Ajay Kumar Nair IISER Bhopal iii ACKNOWLEDGEMENT I would like to extend my heartfelt gratitude to my thesis supervisor Dr Kashyap Rajeevsarathy for the persistent inspiration and guidance. -
A Note on the Homotopy Type of the Alexander Dual
A NOTE ON THE HOMOTOPY TYPE OF THE ALEXANDER DUAL EL´IAS GABRIEL MINIAN AND JORGE TOMAS´ RODR´IGUEZ Abstract. We investigate the homotopy type of the Alexander dual of a simplicial complex. In general the homotopy type of K does not determine the homotopy type of its dual K∗. Moreover, one can construct for each finitely presented group G, a simply ∗ connected simplicial complex K such that π1(K ) = G. We study sufficient conditions on K for K∗ to have the homotopy type of a sphere. We also extend the simplicial Alexander duality to the context of reduced lattices. 1. Introduction Let A be a compact and locally contractible proper subspace of Sn. The classical n Alexander duality theorem asserts that the reduced homology groups Hi(S − A) are isomorphic to the reduced cohomology groups Hn−i−1(A) (see for example [6, Thm. 3.44]). The combinatorial (or simplicial) Alexander duality is a special case of the classical duality: if K is a finite simplicial complex and K∗ is the Alexander dual with respect to a ground set V ⊇ K0, then for any i ∼ n−i−3 ∗ Hi(K) = H (K ): Here K0 denotes the set of vertices (i.e. 0-simplices) of K and n is the size of V .A very nice and simple proof of the combinatorial Alexander duality can be found in [4]. An alternative proof of this combinatorial duality can be found in [3]. In these notes we relate the homotopy type of K with that of K∗. We show first that, even though the homology of K determines the homology of K∗ (and vice versa), the homotopy type of K does not determine the homotopy type of K∗. -
Notation and Basic Facts in Knot Theory
Appendix A Notation and Basic Facts in Knot Theory In this appendix, our aim is to provide a quick review of basic terminology and some facts in knot theory, which we use or need in this book. This appendix lists them item by item (without proof); so we do no attempt to give a full rigorous treatments; Instead, we work somewhat intuitively. The reader with interest in the details could consult the textbooks [BZ, R, Lic, KawBook]. • First, we fix notation on the circle S1 := {(x, y) ∈ R2 | x2 + y2 = 1}, and consider a finite disjoint union S1. A link is a C∞-embedding of L :S1 → S3 in the 3-sphere. We denote often by #L the number of the disjoint union, and denote the image Im(L) by only L for short. If #L is 1, L is usually called a knot, and is written K instead. This book discusses embeddings together with orientation. For an oriented link L, we denote by −L the link with its orientation reversed, and by L∗ the mirror image of L. • (Notations of link components). Given a link L :S1 → S3, let us fix an open tubular neighborhood νL ⊂ S3. Throughout this book, we denote the complement S3 \ νL by S3 \ L for short. Since we mainly discuss isotopy classes of S3 \ L,we may ignore the choice of νL. 2 • For example, for integers s, t ∈ Z , the torus link Ts,t (of type (s, t)) is defined by 3 2 s t 2 2 S Ts,t := (z,w)∈ C z + w = 0, |z| +|w| = 1 . -
Riemann's Mapping Theorem
4. Del Riemann’s mapping theorem version 0:21 | Tuesday, October 25, 2016 6:47:46 PM very preliminary version| more under way. One dares say that the Riemann mapping theorem is one of more famous theorem in the whole science of mathematics. Together with its generalization to Riemann surfaces, the so called Uniformisation Theorem, it is with out doubt the corner-stone of function theory. The theorem classifies all simply connected Riemann-surfaces uo to biholomopisms; and list is astonishingly short. There are just three: The unit disk D, the complex plane C and the Riemann sphere C^! Riemann announced the mapping theorem in his inaugural dissertation1 which he defended in G¨ottingenin . His version a was weaker than full version of today, in that he seems only to treat domains bounded by piecewise differentiable Jordan curves. His proof was not waterproof either, lacking arguments for why the Dirichlet problem has solutions. The fault was later repaired by several people, so his method is sound (of course!). In the modern version there is no further restrictions on the domain than being simply connected. William Fogg Osgood was the first to give a complete proof of the theorem in that form (in ), but improvements of the proof continued to come during the first quarter of the th century. We present Carath´eodory's version of the proof by Lip´otFej´erand Frigyes Riesz, like Reinholdt Remmert does in his book [?], and we shall closely follow the presentation there. This chapter starts with the legendary lemma of Schwarz' and a study of the biho- lomorphic automorphisms of the unit disk. -
Arxiv:1607.08163V2 [Math.GT] 24 Sep 2016
LECTURES ON THE TRIANGULATION CONJECTURE CIPRIAN MANOLESCU Abstract. We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pinp2q symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology. 1. Introduction The triangulation conjecture stated that every topological manifold can be triangulated. The work of Casson [AM90] in the 1980's provided counterexamples in dimension 4. The main purpose of these notes is to describe the proof of the following theorem. Theorem 1.1 ([Man13]). There exist non-triangulable n-dimensional topological manifolds for every n ¥ 5. The proof relies on previous work by Galewski-Stern [GS80] and Matumoto [Mat78], who reduced this problem to a different one, about the homology cobordism group in three dimensions. Homology cobordism can be explored using the techniques of gauge theory, as was done, for example, by Fintushel and Stern [FS85, FS90], Furuta [Fur90], and Frøyshov [Fro10]. In [Man13], Pinp2q-equivariant Seiberg-Witten Floer homology is used to construct three new invariants of homology cobordism, called α, β and γ. The properties of β suffice to answer the question raised by Galewski-Stern and Matumoto, and thus prove Theorem 1.1. The paper is organized as follows. Section 2 contains background material about triangulating manifolds. In particular, we sketch the arguments of Galewski-Stern and Matumoto that reduced Theorem 1.1 to a problem about homology cobordism. In Section 3 we introduce the Seiberg-Witten equations, finite dimensional approxima- tion, and the Conley index. -
The Ways I Know How to Define the Alexander Polynomial
ALL THE WAYS I KNOW HOW TO DEFINE THE ALEXANDER POLYNOMIAL KYLE A. MILLER Abstract. These began as notes for a talk given at the Student 3-manifold seminar, Spring 2019. There seems to be many ways to define the Alexander polynomial, all of which are somehow interrelated, but sometimes there is not an obvious path between any two given definitions. As the title suggests, this is an exploration of all the ways I know how to define this knot invariant. While we will touch on a number of facts and properties, these notes are not meant to be a complete survey of the Alexander polynomial. Contents 1. Introduction2 2. Alexander’s definition2 2.1. The Dehn presentation2 2.2. Abelianization4 2.3. The associated matrix4 3. The Alexander modules6 3.1. Orders7 3.2. Elementary ideals8 3.3. The Fox calculus 10 3.4. Torus knots 13 3.5. The Wirtinger presentation 13 3.6. Generalization to links 15 3.7. Fibered knots 15 3.8. Seifert presentation 16 3.9. Duality 18 4. The Conway potential 18 4.1. Alexander’s three-term relation 20 5. The HOMFLY-PT polynomial 24 6. Kauffman state sum 26 6.1. Another state sum 29 7. The Burau representation 29 8. Vassiliev invariants 29 9. The Alexander quandle 29 9.1. Projective Alexander quandles 33 10. Reidemeister torsion 33 11. Knot Floer homology 33 References 33 Date: May 3, 2019. 1 DRAFT 2020/11/14 18:57:28 2 KYLE A. MILLER 1. Introduction Recall that a link is an embedded closed 1-manifold in S3, and a knot is a 1-component link. -
Positive Alexander Duality for Pursuit and Evasion
POSITIVE ALEXANDER DUALITY FOR PURSUIT AND EVASION ROBERT GHRIST AND SANJEEVI KRISHNAN Abstract. Considered is a class of pursuit-evasion games, in which an evader tries to avoid detection. Such games can be formulated as the search for sections to the com- plement of a coverage region in a Euclidean space over a timeline. Prior results give homological criteria for evasion in the general case that are not necessary and sufficient. This paper provides a necessary and sufficient positive cohomological criterion for evasion in a general case. The principal tools are (1) a refinement of the Cechˇ cohomology of a coverage region with a positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore homology of the coverage gaps with a positive cone encoding time orienta- tion, and (3) a positive variant of Alexander Duality. Positive cohomology decomposes as the global sections of a sheaf of local positive cohomology over the time axis; we show how this decomposition makes positive cohomology computable as a linear program. 1. Introduction The motivation for this paper comes from a type of pursuit-evasion game. In such games, two classes of agents, pursuers and evaders move in a fixed geometric domain over time. The goal of a pursuer is to capture an evader (by, e.g., physical proximity or line-of-sight). The goal of an evader is to move in such a manner so as to avoid capture by any pursuer. This paper solves a feasibility problem of whether an evader can win in a particular setting under certain constraints. We specialize to the setting of pursuers-as-sensors, in which, at each time, a certain region of space is \sensed" and any evader in this region is considered captured. -
Arxiv:1610.07421V4 [Math.AT] 3 Feb 2017 7 Need Also “Commutative Cubes” 12
Modelling and computing homotopy types: I I Ronald Brown School of Computer Science, Bangor University Abstract The aim of this article is to explain a philosophy for applying the 1-dimensional and higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to n-cubes of pointed spaces. Key words: homotopy types, groupoids, higher groupoids, Seifert-van Kampen theorems 2000 MSC: 55P15,55U99,18D05 Contents Introduction 2 1 Statement of the philosophy3 2 Broad and Narrow Algebraic Data4 3 Six Anomalies in Algebraic Topology6 4 The origins of algebraic topology7 5 Enter groupoids 7 6 Higher dimensions? 10 arXiv:1610.07421v4 [math.AT] 3 Feb 2017 7 Need also “Commutative cubes” 12 8 Cubical sets in algebraic topology 15 ITo aqppear in a special issue of Idagationes Math. in honor of L.E.J. Brouwer to appear in 2017. This article, has developed from presentations given at Liverpool, Durham, Kansas SU, Chicago, IHP, Galway, Aveiro, whom I thank for their hospitality. Preprint submitted to Elsevier February 6, 2017 9 Strict homotopy double groupoids? 15 10 Crossed Modules 16 11 Groupoids in higher homotopy theory? 17 12 Filtered spaces 19 13 Crossed complexes and their classifying space 20 14 Why filtered spaces? 22 Bibliography 24 Introduction The usual algebraic topology approach to homotopy types is to aim first to calculate specific invariants and then look for some way of amalgamating these to determine a homotopy type.