DOCSLIB.ORG

206 APPENDIX a GLOSSARY FLOW CHARTS Category Theory & Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
206 APPENDIX a GLOSSARY FLOW CHARTS Category Theory & Set Theory 206 APPENDIX A GLOSSARY FLOW CHARTS category theory & set theory injective union surjective quotient map intersection bijective inverse map set element equivalence relation image pre-image object map category 207 point-set topology compactification homeomorphism complete quotient space Hausdor↵ imbedding metric completion gluing paracompact isometry cutting interior continuous cauchy boundary subspace topology compact metric topology convergent limit point connected limit closure topology open metric sequence closed category theory & set theory 208 numbers and algebra modular arithmetic vector space torsion real numbers, R group action module Z/nZ group integers, Z ring rational numbers, Q field binary operation identity element point-set topology cardinality inverse operation metric completion natural numbers, Z+ commutative cauchy, convergence associative metric, sequence linear category theory & set theory 209 linear algebra linear isomorphisms determinants Cramer’s rule multilinear maps tensor products exterior products biinear maps symmetric skew-symmetric definite the plane, R2 linear maps 3-space, R3 bases ordered bases rank inner products n space, Rn dimension orientation − nullity invariance of domain numbers and algebra real numbers, R module, vector space ring, field 210 differential vector calculus critical values the inverse function theorem the regular value theorem the implicit function theorem the constant rank level set theorem higher order mixed partial derivatives Rm n smooth maps A Jacobian matrix in ⇥ represents the m n m immersions and submersions total derivative Dfa of f : R R at a R . ! 2 di↵eomorphisms and embeddings partial derivatives @ @xi a m n m of f : R R at a R ! 2 linear algebra linear maps f : Rn Rm point-set topology ! multilinear map, exterior product continuity, limit linear isomorphisms, determinants 211 manifolds transition functions circles and spheres, submanifolds oriented manifolds S0,S1,S2,S3,Sn,and knots and links in S3 smooth manifolds projective spaces, RPn spanning surfaces in S3 Riemannian manifolds topological manifolds (with or without boundary) coordinate charts atlases of charts linear algebra point-set topology n-space,Rn differential vector calculus continuity, quotient space multilinear map, exterior product higher order mixed partial derivatives cutting, gluing linear isomorphisms, determinants smooth maps Hausdor↵, paracompact inner products Jacobian matrices compact, connected symmetric bilinear mappings interior, boundary, closure orientation 212 covering spaces and fundamental groups structure theorem universal covering spaces the fundamental group path connectedness concatenation of paths local path connectedness the lifting correspondence semilocal simply-connectedness homotopy paths covering spaces homotopy of paths example: R S1 ! homotopy of maps point-set topology continuity, quotient space algebra cutting, gluing groups Hausdor↵, paracompact group actions compact, connected interior, boundary, closure 213 APPENDIX B HOMOLOGY AND KNOTS B.1 Overview Working in the smooth (or piecewise linear) category ensures that every link L S3 ⇢ is tame, meaning that its embedding : n S1 , L S3 extends to an embedding r=1 ! ⇢ of disjoint, compact, solid tori : n SF1 D2 , S3, where (x,0)= (x) for each r=1 ⇥ ! ⇣ ⌘ x n S1. Call the image of a closedF regular neighborhood, or thickening of L, and 2 r=1 F denote this image by ⌫L. Also let ⌫Ls and ⌫Ls respectively denote the closed and open 1 regular neighborhoods of any component Ls = r=s S of L, and let s, s denote the ⇣ ⌘ 3 restrictions = 1 , = . The openF 3-manifold S L is called the link s r=s S s (S1 D2) | | r=s ⇥ \ F complement; the compact 3-manifoldF with boundary S3 ⌫L is called the link exterior. \ Link exteriors provide an (almost too) intuitive setting in which to introduce the basic machinery of singular homology; by contrast, the traditional approach to the sub- ject traces its historical development, following the axiomatic constructions of simplicial homology and singular homology, proving their equivalence, and then deriving proper- ties. This traditional approach is necessary in order to build a solid, logical foundation for homology theory; flowcharts in the Appendices reflect these logical foundations. For ex- pository purposes, however, homology theory for 3-manifolds is all about submanifolds. If M is a 3-manifold, then we consider the following eight homology groups: Z F Hk(M; ) and Hk(M; 2) for k =0,1,2,3, taking coefficients in the integers or in the two- F Z Z Z F element field 2 = /2 . Each element of each Hk(M; ) (resp. Hk(M; 2)) can be seen 214 Figure B.1: Left: a longitude λ and meridian µ on the torus boundary @⌫L of the thickened link ⌫L, in case L is the lh trefoil. Right: a Seifert surface F S3 ⌫L with @F = λ. ⇢ \ as an equivalence class [s] of oriented (resp. unoriented) embedded submanifolds s M ⇢ of dimension n (this fails when M has higher dimension). The equivalence is straightfor- ward in the case of disjoint, connected submanifolds: taking coefficients in the integers Z, a disjoint pair of connected, n-dimensional submanifolds s ,s M are equivalent, 1 2 ⇢ [s ]=[s ], i↵ s s bounds an oriented (n+1)-dimensional submanifold S M, such that 1 2 1 [ 2 ⇢ the induced orientation on @S = s s preserves the orientation on s and reverses that 1 [ 2 1 on s2. In particular, [s1] = 0 and s1 is called nullhomologous if s1 bounds an oriented submanifold S M. The equivalence works the same with coefficients in Z/2Z, omitting ⇢ everything about orientation. Let M = S3 ⌫L be a link exterior, and let M = S3-cut-along-F be a link exterior 1 \ 2 Z Z cut along a spanning surface. In the case of such Mi, i =1,2, both H0(Mi; )= and F F Z F H0(Mi; 2)= 2, since both Mi are path connected, and both H3(Mi; )=0=Hx(Mi, 2), since both Mi, although compact and oriented, have nonempty boundary. Each primitive 215 Z F element of H1(Mi; ) (resp. H1(Mi; 2)) can be seen as a simple closed curve that bounds Z F no oriented (resp. unoriented) surface in Mi, and elements of H2(Mi; ) and H2(Mi; 2) can be seen as embedded surfaces F M (compact and without boundary) that do not ⇢ i bound. The first homology groups, H1, will be the most useful of these groups, because of the linking pairing lk : H (X;Z) H (Y;Z) Z for any disjoint subsets X,Y S3, which 1 ⌦ 1 ! ⇢ is determined by the following property. If ↵ = f (S1) X, β = g(S1) Y are simple closed ⇢ ⇢ curves, then their linking number lk(↵,β) is the degree of a map h defined as follows. Let z S3 (X Y) be any point, let ' : S3 z R3 be stereographic projection, and let 2 \ [ \{ } ! h : S1 S1 S2 be given by ⇥ ! ' g(y) ' f (x) h :(x,y) ◦ − ◦ . (B.1) 7! ' g(y) ' f (x) ◦ − ◦ 3 The degree of this map can be seen, by choosing an “upward” direction in R , as the number of times ↵ passes directly above β in a right-handed way, minus the number of times ↵ passes directly above β in a left-handed way: lk(↵,β)= β β . (B.2) α − α This gives: L L Lr lk(L ,L )= r s L . (B.3) r s − s B.2 Longitudes, meridia, and boundary slopes Given a thickening ⌫L of a link L, first homology groups coming from each link component Ls distinguish two isotopy classes of curves on @⌫Ls, the longitude λs and 216 the meridian µs, which in turn give rise to the extremely useful notion of boundary slopes. A meridian µs is any simple closed curve on @⌫Ls that represents a generator of H (S3 ⌫L ;Z) and is nullhomologous in H (⌫L ;Z). For example, p @D2 is 1 \ s 1 s r=s{ } ⇥ ⇣ ⌘ a meridian for any point p S1.Alongitude λ is any simple closed curveF on @⌫L that 2 s s is nullhomologous in H (S3 ⌫L ;Z) and is a generator of H (⌫L ;Z). Thus, µ bounds 1 \ s 1 s s an oriented surface (in fact a disk) in ⌫Ls but not in @⌫Ls, and a longitude λs bounds an oriented surface in S3 ⌫L but not in @⌫L (c.f. Figure B.1). The Mayer-Vietoris sequence \ s s 3 for S -cut-along @⌫Ls proves that any two meridia on @⌫Ls are isotopic in @⌫Ls, as are any two longitudes (c.f. the Appendix). Recall, given a spanning surface F for a link L, that we isotope F to intersect each component ⌫L of the thickening ⌫L in an annulus. Also assume that each pair (L ,F s s \ @⌫L ) is co-oriented, i.e. that these curves are oriented so that [L ]=[F @⌫L ] in H (⌫L ). s s \ s 1 s The sum of linking numbers n lk(L ,F @⌫L ) is called the net slope of F, and the n- s=1 s \ s tuple (lk(L ,F @⌫L ))n is calledP the boundary slope of F. The latter characterizes the s \ s s=1 isotopy class of F @⌫L on @⌫L: each F @⌫L is homologous on @⌫L to [λ ] + lk(L ,F \ \ s s s s \ @⌫Ls)[µs] when µs is oriented so that lk(Ls,µs) = +1, and simple closed curves on the torus are determined by their homology class. 3 1 3 Explicitly, take an embedding ⇠ : F˙ , S , with ⇠ ˙ = : S , S (and = ! |@F s ! th ˙ 1F ˙ s s, where each s is the restriction of to the s copy Ls of S , and each s(Ls)=Ls isF the sth component of L), and let ' : S3 z R3 be stereographic projection as before, \{ } ! ˙ where z S3 ⇠(F). Also take L to be a pusho↵ of L˙ into F˙, such that the restriction 2 \ s s ˙ ˙ ˙ s := ⇠ ˙ maps Ls to F @⌫Ls =:bLs.
Load more

Recommended publications
  • Homological Algebra
    Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of
    [Show full text]
  • Basic Properties of Filter Convergence Spaces
    Basic Properties of Filter Convergence Spaces Barbel¨ M. R. Stadlery, Peter F. Stadlery;z;∗ yInstitut fur¨ Theoretische Chemie, Universit¨at Wien, W¨ahringerstraße 17, A-1090 Wien, Austria zThe Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA ∗Address for corresponce Abstract. This technical report summarized facts from the basic theory of filter convergence spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Introduction Mathematical notions such as convergence, continuity, and separation are, at textbook level, usually associated with topological spaces. It is possible, however, to introduce them in a much more abstract way, based on axioms for convergence instead of neighborhood. This approach was explored in seminal work by Choquet [4], Hausdorff [12], Katˇetov [14], Kent [16], and others. Here we give a brief introduction to this line of reasoning. While the material is well known to specialists it does not seem to be easily accessible to non-topologists. In some cases we include proofs of elementary facts for two reasons: (i) The most basic facts are quoted without proofs in research papers, and (ii) the proofs may serve as examples to see the rather abstract formalism at work. 2. Sets and Filters Let X be a set, P(X) its power set, and H ⊆ P(X). The we define H∗ = fA ⊆ Xj(X n A) 2= Hg (1) H# = fA ⊆ Xj8Q 2 H : A \ Q =6 ;g The set systems H∗ and H# are called the conjugate and the grill of H, respectively.
    [Show full text]
  • Basic Topologytaken From
    Notes by Tamal K. Dey, OSU 1 Basic Topology taken from [1] 1 Metric space topology We introduce basic notions from point set topology. These notions are prerequisites for more sophisticated topological ideas—manifolds, homeomorphism, and isotopy—introduced later to study algorithms for topological data analysis. To a layman, the word topology evokes visions of “rubber-sheet topology”: the idea that if you bend and stretch a sheet of rubber, it changes shape but always preserves the underlying structure of how it is connected to itself. Homeomorphisms offer a rigorous way to state that an operation preserves the topology of a domain, and isotopy offers a rigorous way to state that the domain can be deformed into a shape without ever colliding with itself. Topology begins with a set T of points—perhaps the points comprising the d-dimensional Euclidean space Rd, or perhaps the points on the surface of a volume such as a coffee mug. We suppose that there is a metric d(p, q) that specifies the scalar distance between every pair of points p, q ∈ T. In the Euclidean space Rd we choose the Euclidean distance. On the surface of the coffee mug, we could choose the Euclidean distance too; alternatively, we could choose the geodesic distance, namely the length of the shortest path from p to q on the mug’s surface. d Let us briefly review the Euclidean metric. We write points in R as p = (p1, p2,..., pd), d where each pi is a real-valued coordinate. The Euclidean inner product of two points p, q ∈ R is d Rd 1/2 d 2 1/2 hp, qi = Pi=1 piqi.
    [Show full text]
  • Algebraic Topology
    Algebraic Topology Vanessa Robins Department of Applied Mathematics Research School of Physics and Engineering The Australian National University Canberra ACT 0200, Australia. email: [email protected] September 11, 2013 Abstract This manuscript will be published as Chapter 5 in Wiley's textbook Mathe- matical Tools for Physicists, 2nd edition, edited by Michael Grinfeld from the University of Strathclyde. The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology. arXiv:1304.7846v2 [math-ph] 10 Sep 2013 1 Contents 1 Introduction 3 2 Homotopy Theory 4 2.1 Homotopy of paths . 4 2.2 The fundamental group . 5 2.3 Homotopy of spaces . 7 2.4 Examples . 7 2.5 Covering spaces . 9 2.6 Extensions and applications . 9 3 Homology 11 3.1 Simplicial complexes . 12 3.2 Simplicial homology groups . 12 3.3 Basic properties of homology groups . 14 3.4 Homological algebra . 16 3.5 Other homology theories . 18 4 Cohomology 18 4.1 De Rham cohomology . 20 5 Morse theory 21 5.1 Basic results . 21 5.2 Extensions and applications . 23 5.3 Forman's discrete Morse theory . 24 6 Computational topology 25 6.1 The fundamental group of a simplicial complex . 26 6.2 Smith normal form for homology . 27 6.3 Persistent homology . 28 6.4 Cell complexes from data . 29 2 1 Introduction Topology is the study of those aspects of shape and structure that do not de- pend on precise knowledge of an object's geometry.
    [Show full text]
  • 16 Continuous Functions Definition 16.1. Let F
    Math 320, December 09, 2018 16 Continuous functions Definition 16.1. Let f : D ! R and let c 2 D. We say that f is continuous at c, if for every > 0 there exists δ>0 such that jf(x)−f(c)j<, whenever jx−cj<δ and x2D. If f is continuous at every point of a subset S ⊂D, then f is said to be continuous on S. If f is continuous on its domain D, then f is said to be continuous. Notice that in the above definition, unlike the definition of limits of functions, c is not required to be a limit point of D. It is clear from the definition that if c is an isolated point of the domain D, then f must be continuous at c. The following statement gives the interpretation of continuity in terms of neighborhoods and sequences. Theorem 16.2. Let f :D!R and let c2D. Then the following three conditions are equivalent. (a) f is continuous at c (b) If (xn) is any sequence in D such that xn !c, then limf(xn)=f(c) (c) For every neighborhood V of f(c) there exists a neighborhood U of c, such that f(U \D)⊂V . If c is a limit point of D, then the above three statements are equivalent to (d) f has a limit at c and limf(x)=f(c). x!c From the sequential interpretation, one gets the following criterion for discontinuity. Theorem 16.3. Let f :D !R and c2D. Then f is discontinuous at c iff there exists a sequence (xn)2D, such that (xn) converges to c, but the sequence f(xn) does not converge to f(c).
    [Show full text]
  • A TEXTBOOK of TOPOLOGY Lltld
    SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1.
    [Show full text]
  • 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.
    [Show full text]
  • Introduction to Homology
    Introduction to Homology Matthew Lerner-Brecher and Koh Yamakawa March 28, 2019 Contents 1 Homology 1 1.1 Simplices: a Review . .2 1.2 ∆ Simplices: not a Review . .2 1.3 Boundary Operator . .3 1.4 Simplicial Homology: DEF not a Review . .4 1.5 Singular Homology . .5 2 Higher Homotopy Groups and Hurweicz Theorem 5 3 Exact Sequences 5 3.1 Key Definitions . .5 3.2 Recreating Groups From Exact Sequences . .6 4 Long Exact Homology Sequences 7 4.1 Exact Sequences of Chain Complexes . .7 4.2 Relative Homology Groups . .8 4.3 The Excision Theorems . .8 4.4 Mayer-Vietoris Sequence . .9 4.5 Application . .9 1 Homology What is Homology? To put it simply, we use Homology to count the number of n dimensional holes in a topological space! In general, our approach will be to add a structure on a space or object ( and thus a topology ) and figure out what subsets of the space are cycles, then sort through those subsets that are holes. Of course, as many properties we care about in topology, this property is invariant under homotopy equivalence. This is the slightly weaker than homeomorphism which we before said gave us the same fundamental group. 1 Figure 1: Hatcher p.100 Just for reference to you, I will simply define the nth Homology of a topological space X. Hn(X) = ker @n=Im@n−1 which, as we have said before, is the group of n-holes. 1.1 Simplices: a Review k+1 Just for your sake, we review what standard K simplices are, as embedded inside ( or living in ) R ( n ) k X X ∆ = [v0; : : : ; vk] = xivi such that xk = 1 i=0 For example, the 0 simplex is a point, the 1 simplex is a line, the 2 simplex is a triangle, the 3 simplex is a tetrahedron.
    [Show full text]
  • Homology Groups of Homeomorphic Topological Spaces
    An Introduction to Homology Prerna Nadathur August 16, 2007 Abstract This paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. It concludes with a proof of the equivalence of simplicial and singular homology groups. Contents 1 Simplices and Simplicial Complexes 1 2 Homology Groups 2 3 Singular Homology 8 4 Chain Complexes, Exact Sequences, and Relative Homology Groups 9 ∆ 5 The Equivalence of H n and Hn 13 1 Simplices and Simplicial Complexes Definition 1.1. The n-simplex, ∆n, is the simplest geometric figure determined by a collection of n n + 1 points in Euclidean space R . Geometrically, it can be thought of as the complete graph on (n + 1) vertices, which is solid in n dimensions. Figure 1: Some simplices Extrapolating from Figure 1, we see that the 3-simplex is a tetrahedron. Note: The n-simplex is topologically equivalent to Dn, the n-ball. Definition 1.2. An n-face of a simplex is a subset of the set of vertices of the simplex with order n + 1. The faces of an n-simplex with dimension less than n are called its proper faces. 1 Two simplices are said to be properly situated if their intersection is either empty or a face of both simplices (i.e., a simplex itself). By \gluing" (identifying) simplices along entire faces, we get what are known as simplicial complexes. More formally: Definition 1.3. A simplicial complex K is a finite set of simplices satisfying the following condi- tions: 1 For all simplices A 2 K with α a face of A, we have α 2 K.
    [Show full text]
  • POINT SET TOPOLOGY Definition 1 a Topological Structure On
    POINT SET TOPOLOGY De¯nition 1 A topological structure on a set X is a family (X) called open sets and satisfying O ½ P (O ) is closed for arbitrary unions 1 O (O ) is closed for ¯nite intersections. 2 O De¯nition 2 A set with a topological structure is a topological space (X; ) O ; = 2;Ei = x : x Eifor some i = [ [i f 2 2 ;g ; so is always open by (O ) ; 1 ; = 2;Ei = x : x Eifor all i = X \ \i f 2 2 ;g so X is always open by (O2). Examples (i) = (X) the discrete topology. O P (ii) ; X the indiscrete of trivial topology. Of; g These coincide when X has one point. (iii) =the rational line. Q =set of unions of open rational intervals O De¯nition 3 Topological spaces X and X 0 are homomorphic if there is an isomorphism of their topological structures i.e. if there is a bijection (1-1 onto map) of X and X 0 which generates a bijection of and . O O e.g. If X and X are discrete spaces a bijection is a homomorphism. (see also Kelley p102 H). De¯nition 4 A base for a topological structure is a family such that B ½ O every o can be expressed as a union of sets of 2 O B Examples (i) for the discrete topological structure x x2X is a base. f g (ii) for the indiscrete topological structure ; X is a base. f; g (iii) For , topologised as before, the set of bounded open intervals is a base.Q 1 (iv) Let X = 0; 1; 2 f g Let = (0; 1); (1; 2); (0; 12) .
    [Show full text]
  • Topology of Homology Manifolds
    BULLETIN(New Series) OF THE AMERICANMATHEMATICAL SOCIETY Volume 28, Number 2, April 1993 TOPOLOGY OF HOMOLOGY MANIFOLDS J. BRYANT, S. FERRY, W. MIO, AND S. WEINBERGER Abstract. We construct examples of nonresolvable generalized «-manifolds, n > 6 , with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed «-manifold. We further investigate the structure of general- ized manifolds and present a program for understanding their topology. By a generalized n-manifold we will mean a finite-dimensional absolute neigh- borhood retract X such that X is a homology «-manifold; that is, for all x e X, Ht(X, X - {x}) = Hi(Rn , R" - {0}). Generalized manifolds arise nat- urally as fixed-point sets of group actions on manifolds, as limits of sequences of manifolds, and as boundaries of negatively curved groups. See [BM, Bo, B, GPW]. Such spaces have most of the homological properties of topological manifolds. In particular, generalized manifolds satisfy Poincaré duality [Bo]. Generalized manifolds also share certain geometric and analytic properties with manifolds. Modern proofs of the topological invariance of rational Pontrja- gin classes show that Pontrjagin classes can be defined for generalized manifolds and (even better!) that the symbol of the signature operator can be defined for these spaces. See [CSW]. In light of this, the following question seems natural: Question 1. Is every generalized manifold X homotopy equivalent to a topo- logical manifold? By [FP], this is true for compact simply connected homology manifolds in all higher dimensions. We shall see below that this is not true in the nonsimply connected case. To continue in this vein, we can consider a strong version of Question 1 that asserts that, for such an X, a manifold M can be chosen coherently for all of its open subsets.
    [Show full text]
  • 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.
    [Show full text]