Introduction (Lecture 1)
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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. -
Introduction to Gauge Theory Arxiv:1910.10436V1 [Math.DG] 23
Introduction to Gauge Theory Andriy Haydys 23rd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles . .4 2.1.1 Basic notions . .4 2.1.2 Operations on vector bundles . .5 2.1.3 Sections . .6 2.1.4 Covariant derivatives . .6 2.1.5 The curvature . .8 2.1.6 The gauge group . 10 2.2 Principal bundles . 11 2.2.1 The frame bundle and the structure group . 11 2.2.2 The associated vector bundle . 14 2.2.3 Connections on principal bundles . 16 2.2.4 The curvature of a connection on a principal bundle . 19 arXiv:1910.10436v1 [math.DG] 23 Oct 2019 2.2.5 The gauge group . 21 2.3 The Levi–Civita connection . 22 2.4 Classification of U(1) and SU(2) bundles . 23 2.4.1 Complex line bundles . 24 2.4.2 Quaternionic line bundles . 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory . 26 3.1.1 The Chern classes . 28 3.2 The Chern–Simons functional . 30 3.3 The modui space of flat connections . 32 3.3.1 Parallel transport and holonomy . -
PIECEWISE LINEAR TOPOLOGY Contents 1. Introduction 2 2. Basic
PIECEWISE LINEAR TOPOLOGY J. L. BRYANT Contents 1. Introduction 2 2. Basic Definitions and Terminology. 2 3. Regular Neighborhoods 9 4. General Position 16 5. Embeddings, Engulfing 19 6. Handle Theory 24 7. Isotopies, Unknotting 30 8. Approximations, Controlled Isotopies 31 9. Triangulations of Manifolds 33 References 35 1 2 J. L. BRYANT 1. Introduction The piecewise linear category offers a rich structural setting in which to study many of the problems that arise in geometric topology. The first systematic ac- counts of the subject may be found in [2] and [63]. Whitehead’s important paper [63] contains the foundation of the geometric and algebraic theory of simplicial com- plexes that we use today. More recent sources, such as [30], [50], and [66], together with [17] and [37], provide a fairly complete development of PL theory up through the early 1970’s. This chapter will present an overview of the subject, drawing heavily upon these sources as well as others with the goal of unifying various topics found there as well as in other parts of the literature. We shall try to give enough in the way of proofs to provide the reader with a flavor of some of the techniques of the subject, while deferring the more intricate details to the literature. Our discussion will generally avoid problems associated with embedding and isotopy in codimension 2. The reader is referred to [12] for a survey of results in this very important area. 2. Basic Definitions and Terminology. Simplexes. A simplex of dimension p (a p-simplex) σ is the convex closure of a n set of (p+1) geometrically independent points {v0, . -
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). -
Daniel Irvine June 20, 2014 Lecture 6: Topological Manifolds 1. Local
Daniel Irvine June 20, 2014 Lecture 6: Topological Manifolds 1. Local Topological Properties Exercise 7 of the problem set asked you to understand the notion of path compo- nents. We assume that knowledge here. We have done a little work in understanding the notions of connectedness, path connectedness, and compactness. It turns out that even if a space doesn't have these properties globally, they might still hold locally. Definition. A space X is locally path connected at x if for every neighborhood U of x, there is a path connected neighborhood V of x contained in U. If X is locally path connected at all of its points, then it is said to be locally path connected. Lemma 1.1. A space X is locally path connected if and only if for every open set V of X, each path component of V is open in X. Proof. Suppose that X is locally path connected. Let V be an open set in X; let C be a path component of V . If x is a point of V , we can (by definition) choose a path connected neighborhood W of x such that W ⊂ V . Since W is path connected, it must lie entirely within the path component C. Therefore C is open. Conversely, suppose that the path components of open sets of X are themselves open. Given a point x of X and a neighborhood V of x, let C be the path component of V containing x. Now C is path connected, and since it is open in X by hypothesis, we now have that X is locally path connected at x. -
Math 600 Day 6: Abstract Smooth Manifolds
Math 600 Day 6: Abstract Smooth Manifolds Ryan Blair University of Pennsylvania Tuesday September 28, 2010 Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 1 / 21 Outline 1 Transition to abstract smooth manifolds Partitions of unity on differentiable manifolds. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 2 / 21 A Word About Last Time Theorem (Sard’s Theorem) The set of critical values of a smooth map always has measure zero in the receiving space. Theorem Let A ⊂ Rn be open and let f : A → Rp be a smooth function whose derivative f ′(x) has maximal rank p whenever f (x) = 0. Then f −1(0) is a (n − p)-dimensional manifold in Rn. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 3 / 21 Transition to abstract smooth manifolds Transition to abstract smooth manifolds Up to this point, we have viewed smooth manifolds as subsets of Euclidean spaces, and in that setting have defined: 1 coordinate systems 2 manifolds-with-boundary 3 tangent spaces 4 differentiable maps and stated the Chain Rule and Inverse Function Theorem. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 4 / 21 Transition to abstract smooth manifolds Now we want to define differentiable (= smooth) manifolds in an abstract setting, without assuming they are subsets of some Euclidean space. Many differentiable manifolds come to our attention this way. Here are just a few examples: 1 tangent and cotangent bundles over smooth manifolds 2 Grassmann manifolds 3 manifolds built by ”surgery” from other manifolds Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 5 / 21 Transition to abstract smooth manifolds Our starting point is the definition of a topological manifold of dimension n as a topological space Mn in which each point has an open neighborhood homeomorphic to Rn (locally Euclidean property), and which, in addition, is Hausdorff and second countable. -
Floer Homology, Gauge Theory, and Low-Dimensional Topology
Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. -
Triangulations of 3-Manifolds with Essential Edges
TRIANGULATIONS OF 3–MANIFOLDS WITH ESSENTIAL EDGES CRAIG D. HODGSON, J. HYAM RUBINSTEIN, HENRY SEGERMAN AND STEPHAN TILLMANN Dedicated to Michel Boileau for his many contributions and leadership in low dimensional topology. ABSTRACT. We define essential and strongly essential triangulations of 3–manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3–manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold. We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3–manifold to be essential, and a strict angle structure is a sufficient condition for a triangulation to be strongly essential. Moreover, algorithms to test whether a triangulation of a 3–manifold is essential or strongly essential are given. 1. INTRODUCTION Finding combinatorial and topological properties of geometric triangulations of hyper- bolic 3–manifolds gives information which is useful for the reverse process—for exam- ple, solving Thurston’s gluing equations to construct an explicit hyperbolic metric from an ideal triangulation. Moreover, suitable properties of a triangulation can be used to de- duce facts about the variety of representations of the fundamental group of the underlying 3-manifold into PSL(2;C). See for example [32]. In this paper, we focus on one-vertex triangulations of closed manifolds and ideal trian- gulations of the interiors of compact manifolds with boundary. Two important properties of these triangulations are introduced. For one-vertex triangulations in the closed case, the first property is that no edge loop is null-homotopic and the second is that no two edge loops are homotopic, keeping the vertex fixed. -
Experimental Statistics of Veering Triangulations
EXPERIMENTAL STATISTICS OF VEERING TRIANGULATIONS WILLIAM WORDEN Abstract. Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. We obtain experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold. 1. Introduction In 2011, Agol [Ago11] introduced the notion of a layered veering triangulation for certain fibered hyperbolic manifolds. In particular, given a surface Σ (possibly with punctures), and a pseudo-Anosov homeomorphism ' ∶ Σ → Σ, Agol's construction gives a triangulation ○ ○ ○ ○ T for the mapping torus M' with fiber Σ and monodromy ' = 'SΣ○ , where Σ is the surface resulting from puncturing Σ at the singularities of its '-invariant foliations. For Σ a once-punctured torus or 4-punctured sphere, the resulting triangulation is well understood: it is the monodromy (or Floyd{Hatcher) triangulation considered in [FH82] and [Jør03]. In this case T is a geometric triangulation, meaning that tetrahedra in T can be realized as ideal hyperbolic tetrahedra isometrically embedded in M'○ . In fact, these triangulations are geometrically canonical, i.e., dual to the Ford{Voronoi domain of the mapping torus (see [Aki99], [Lac04], and [Gu´e06]). Agol's definition of T was conceived as a generalization of these monodromy triangulations, and so a natural question is whether these generalized monodromy triangulations are also geometric. Hodgson{Issa{Segerman [HIS16] answered this question in the negative, by producing the first examples of non- geometric layered veering triangulations. -
Equivalences to the Triangulation Conjecture 1 Introduction
ISSN online printed Algebraic Geometric Topology Volume ATG Published Decemb er Equivalences to the triangulation conjecture Duane Randall Abstract We utilize the obstruction theory of GalewskiMatumotoStern to derive equivalent formulations of the Triangulation Conjecture For ex n ample every closed top ological manifold M with n can b e simplicially triangulated if and only if the two distinct combinatorial triangulations of 5 RP are simplicially concordant AMS Classication N S Q Keywords Triangulation KirbySieb enmann class Bo ckstein op erator top ological manifold Intro duction The Triangulation Conjecture TC arms that every closed top ological man n ifold M of dimension n admits a simplicial triangulation The vanishing of the KirbySieb enmann class KS M in H M Z is b oth necessary and n sucient for the existence of a combinatorial triangulation of M for n n by A combinatorial triangulation of a closed manifold M is a simplicial triangulation for which the link of every i simplex is a combinatorial sphere of dimension n i Galewski and Stern Theorem and Matumoto n indep endently proved that a closed connected top ological manifold M with n is simplicially triangulable if and only if KS M in H M ker where denotes the Bo ckstein op erator asso ciated to the exact sequence ker Z of ab elian groups Moreover the Triangulation Conjecture is true if and only if this exact sequence splits by or page The Ro chlin invariant morphism is dened on the homology b ordism group of oriented