Daniel Irvine June 20, 2014 Lecture 6: Topological Manifolds 1. Local
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A Guide to Topology
i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. -
Open 3-Manifolds Which Are Simply Connected at Infinity
OPEN 3-MANIFOLDS WHICH ARE SIMPLY CONNECTED AT INFINITY C H. EDWARDS, JR.1 A triangulated open manifold M will be called l-connected at infin- ity il each compact subset C of M is contained in a compact poly- hedron P in M such that M—P is connected and simply connected. Stallings has shown that, if M is a contractible open combinatorial manifold which is l-connected at infinity and is of dimension ra^5, then M is piecewise-linearly homeomorphic to Euclidean re-space E" [5]. Theorem 1. Let M be a contractible open 3-manifold, each of whose compact subsets can be imbedded in £3. If M is l-connected at infinity, then M is homeomorphic to £3. Notice that, in order to prove the 3-dimensional Poincaré conjec- ture, it would suffice to prove Theorem 1 without the hypothesis that each compact subset of M can be imbedded in £3. For, if M is a simply connected closed 3-manifold and p is a point of M, then M —p is a contractible open 3-manifold which is clearly l-connected at infinity. Conversely, if the 3-dimensional Poincaré conjecture were known, then the hypothesis that each compact subset of M can be imbedded in £3 would be unnecessary. All spaces and mappings in this paper are considered in the poly- hedral or piecewise-linear sense, unless otherwise stated. As usual, by an open re-manifold is meant a noncompact connected space triangu- lated by a countable simplicial complex without boundary, such that the link of each vertex is piecewise-linearly homeomorphic to the usual (re—1)-sphere. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
DIFFERENTIAL GEOMETRY COURSE NOTES 1.1. Review of Topology. Definition 1.1. a Topological Space Is a Pair (X,T ) Consisting of A
DIFFERENTIAL GEOMETRY COURSE NOTES KO HONDA 1. REVIEW OF TOPOLOGY AND LINEAR ALGEBRA 1.1. Review of topology. Definition 1.1. A topological space is a pair (X; T ) consisting of a set X and a collection T = fUαg of subsets of X, satisfying the following: (1) ;;X 2 T , (2) if Uα;Uβ 2 T , then Uα \ Uβ 2 T , (3) if Uα 2 T for all α 2 I, then [α2I Uα 2 T . (Here I is an indexing set, and is not necessarily finite.) T is called a topology for X and Uα 2 T is called an open set of X. n Example 1: R = R × R × · · · × R (n times) = f(x1; : : : ; xn) j xi 2 R; i = 1; : : : ; ng, called real n-dimensional space. How to define a topology T on Rn? We would at least like to include open balls of radius r about y 2 Rn: n Br(y) = fx 2 R j jx − yj < rg; where p 2 2 jx − yj = (x1 − y1) + ··· + (xn − yn) : n n Question: Is T0 = fBr(y) j y 2 R ; r 2 (0; 1)g a valid topology for R ? n No, so you must add more open sets to T0 to get a valid topology for R . T = fU j 8y 2 U; 9Br(y) ⊂ Ug: Example 2A: S1 = f(x; y) 2 R2 j x2 + y2 = 1g. A reasonable topology on S1 is the topology induced by the inclusion S1 ⊂ R2. Definition 1.2. Let (X; T ) be a topological space and let f : Y ! X. -
Lecture Notes C Sarah Rasmussen, 2019
Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Introduction to knots and embeddings 13 Lecture 3: Link diagrams and Alexander polynomial skein relations 17 Lecture 4: Handle decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Morse functions from handle decompositions 24 Lecture 6: Handle-bodies and Heegaard diagrams 28 Lecture 7: Fundamental group presentations from Heegaard diagrams 36 Lecture 8: Alexander polynomials from fundamental groups 39 Lecture 9: Fox calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston classification for mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn surgery 64 Lecture 15: 3-manifolds from Dehn surgery 68 Lecture 16: Seifert fibered spaces 72 Lecture 17: Hyperbolic manifolds 76 Lecture 18: Embedded surface representatives 80 Lecture 19: Incompressible and essential surfaces 83 Lecture 20: Connected sum 86 Lecture 21: JSJ decomposition and geometrization 89 Lecture 22: Turaev torsion and knot decompositions 92 Lecture 23: Foliations 96 Lecture 24. Taut Foliations 98 Errata: Catalogue of errors/changes/addenda 102 References 106 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . ◦ ν(X) { the interior of ν(X). This notation is somewhat redundant, but emphasises openness. -
Topology - Wikipedia, the Free Encyclopedia Page 1 of 7
Topology - Wikipedia, the free encyclopedia Page 1 of 7 Topology From Wikipedia, the free encyclopedia Topology (from the Greek τόπος , “place”, and λόγος , “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct A Möbius strip, an object with only one discipline developed, which was referred to in Latin as the surface and one edge. Such shapes are an geometria situs (“geometry of place”) or analysis situs object of study in topology. (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20 th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms , which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line. Topology includes many subfields. The most basic and traditional division within topology is point-set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology , which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology , which primarily studies manifolds and their embeddings (placements) in other manifolds. -
L -Betti Numbers of R-Spaces and the Integral Foliated Simplicial Volume
Marco Schmidt L2-Betti Numbers of R-Spaces and the Integral Foliated Simplicial Volume 2005 Mathematik L2-Betti Numbers of R-Spaces and the Integral Foliated Simplicial Volume Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakult¨at der Westf¨alischen Wilhelms-Universit¨at M¨unster vorgelegt von Marco Schmidt aus Berlin – 2005 – Dekan: Prof. Dr. Klaus Hinrichs Erster Gutachter: Prof. Dr. Wolfgang L¨uck Zweiter Gutachter: Prof. Dr. Anand Dessai Tag der m¨undlichen Pr¨ufung: 30. Mai 2005 Tag der Promotion: 13. Juli 2005 Introduction The origin of this thesis is the following conjecture of Gromov [26, Section 8A, 2 (2) p. 232] revealing a connection between the L -Betti numbers bk (M)and the sim- plicial volume M of a closed oriented connected aspherical manifold M. Conjecture. Let M be a closed oriented connected aspherical manifold with M = 0. Then (2) ≥ bk (M)=0 for all k 0. The first definition of L2-Betti numbers for cocompact free proper G-manifolds with G-invariant Riemannian metric (due to Atiyah [2]) is given in terms of the heat kernel. We will briefly recall this original definition at the beginning of Chap- ter 1. Today, there is an algebraic and more general definition of L2-Betti numbers which works for arbitrary G-spaces. Analogously to ordinary Betti numbers, they are given as the “rank” of certain homology modules. More precisely, the k-th L2- (2) N Betti number bk (Z; G) of a G-space Z is the von Neumann dimension of the k-th twisted singular homology group of Z with coefficients in the group von Neumann algebra N G. -
LECTURE 1. Differentiable Manifolds, Differentiable Maps
LECTURE 1. Differentiable manifolds, differentiable maps Def: topological m-manifold. Locally euclidean, Hausdorff, second-countable space. At each point we have a local chart. (U; h), where h : U ! Rm is a topological embedding (homeomorphism onto its image) and h(U) is open in Rm. Def: differentiable structure. An atlas of class Cr (r ≥ 1 or r = 1 ) for a topological m-manifold M is a collection U of local charts (U; h) satisfying: (i) The domains U of the charts in U define an open cover of M; (ii) If two domains of charts (U; h); (V; k) in U overlap (U \V 6= ;) , then the transition map: k ◦ h−1 : h(U \ V ) ! k(U \ V ) is a Cr diffeomorphism of open sets in Rm. (iii) The atlas U is maximal for property (ii). Def: differentiable map between differentiable manifolds. f : M m ! N n (continuous) is differentiable at p if for any local charts (U; h); (V; k) at p, resp. f(p) with f(U) ⊂ V the map k ◦ f ◦ h−1 = F : h(U) ! k(V ) is m differentiable at x0 = h(p) (as a map from an open subset of R to an open subset of Rn.) f : M ! N (continuous) is of class Cr if for any charts (U; h); (V; k) on M resp. N with f(U) ⊂ V , the composition k ◦ f ◦ h−1 : h(U) ! k(V ) is of class Cr (between open subsets of Rm, resp. Rn.) f : M ! N of class Cr is an immersion if, for any local charts (U; h); (V; k) as above (for M, resp. -
CONNECTEDNESS-Notes Def. a Topological Space X Is
CONNECTEDNESS-Notes Def. A topological space X is disconnected if it admits a non-trivial splitting: X = A [ B; A \ B = ;; A; B open in X; and non-empty. (We'll abbreviate `disjoint union' of two subsets A and B {meaning A\B = ;{ by A t B.) X is connected if no such splitting exists. A subset C ⊂ X is connected if it is a connected topological space, when endowed with the induced topology (the open subsets of C are the intersections of C with open subsets of X.) Note that X is connected if and only if the only subsets of X that are simultaneously open and closed are ; and X. Example. The real line R is connected. The connected subsets of R are exactly the intervals. (See [Fleming] for a proof.) Def. A topological space X is path-connected if any two points p; q 2 X can be joined by a continuous path in X, the image of a continuous map γ : [0; 1] ! X, γ(0) = p; γ(1) = q. Path-connected implies connected: If X = AtB is a non-trivial splitting, taking p 2 A; q 2 B and a path γ in X from p to q would lead to a non- trivial splitting [0; 1] = γ−1(A) t γ−1(B) (by continuity of γ), contradicting the connectedness of [0; 1]. Example: Any normed vector space is path-connected (connect two vec- tors v; w by the line segment t 7! tw + (1 − t)v; t 2 [0; 1]), hence connected. The continuous image of a connected space is connected: Let f : X ! Y be continuous, with X connected. -
Maximal Atlases and Holomorphic Maps of Riemann Surfaces
Supplementary Notes, Exercises and Suggested Reading for Lecture 3: Maximal Atlases and Holomorphic Maps of Riemann Surfaces An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T. E. Venkata Balaji August 16, 2012 An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T. E. Venkata Balaji Supplementary Notes, Exercises and Suggested Reading for Lecture 3:Maximal Atlases and Holomorphic Maps of Riemann Surfaces 1 Some Defintions and Results from Topology. Read up the following topics from a standard textbook on Topology. You may consult for example the book by John L. Kelley titled General Topology and the book by George F. Simmons titled An Introduction to Topology and Modern Analysis. a) Regular and Normal Spaces. Recall that a topological space is called Hausdorff if any two distinct points can be separated by disjoint open neighborhoods. Hausdorffness is also denoted by T2 and is stronger than T1 for which every point is a closed subset. A topological space is called regular if given a point and a closed subset not containing that point, there are open disjoint subsets, one containing the given point and the other the closed subset. In other words, a point and a closed subset not containing that point can be separated by disjoint open neighborhoods. A topological space is called T3 if it is T1 and regular. A topological space is called normal if any two disjoint closed subsets can be separated by disjoint open neighborhoods. A topological space is called T4 if it is T1 and normal. -
On Digital Simply Connected Spaces and Manifolds: a Digital Simply Connected 3- Manifold Is the Digital 3-Sphere
On digital simply connected spaces and manifolds: a digital simply connected 3- manifold is the digital 3-sphere Alexander V. Evako Npk Novotek, Laboratory of Digital Technologies. Moscow, Russia e-mail: [email protected]. Abstract In the framework of digital topology, we study structural and topological properties of digital n- dimensional manifolds. We introduce the notion of simple connectedness of a digital space and prove that if M and N are homotopy equivalent digital spaces and M is simply connected, then so is N. We show that a simply connected digital 2-manifold is the digital 2-sphere and a simply connected digital 3-manifold is the digital 3-sphere. This property can be considered as a digital form of the Poincaré conjecture for continuous three-manifolds. Key words: Graph; Dimension; Digital manifold; Simply connected space; Sphere 1. Introduction A digital approach to geometry and topology plays an important role in analyzing n-dimensional digitized images arising in computer graphics as well as in many areas of science including neuroscience, medical imaging, industrial inspection, geoscience and fluid dynamics. Concepts and results of the digital approach are used to specify and justify some important low-level image processing algorithms, including algorithms for thinning, boundary extraction, object counting, and contour filling. Usually, a digital object is equipped with a graph structure based on the local adjacency relations of digital points [5]. In papers [6-7], a digital n-surface was defined as a simple undirected graph and basic properties of n-surfaces were studied. Paper [6] analyzes a local structure of the digital space Zn.