Stratified Spaces and Tangential Struc
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130 5. Degrees, Intersection Numbers, and the Euler Characteristic. 2. Intersection Numbers and the Euler Characteristic M c: Jr+1 Exercises 8. Let be a compad lHlimensionaJ submanifold without boundary. Two points x, y E R"+' - M are separated by M if and only if lkl(x,Y}.MI * O. lSee 1. A complex polynomial of degree n defines a map of the Riemann sphere to itself Exercise 7.) of degree n. What is the degree of the map defined by a rational function p(z)!q(z)? 9. The Hopfinvariant ofa map f:5' ~ 5' is defined to be the linking number Hlf) ~ 2. (a) Let M, N, P be compact connected oriented n·manifolds without boundaries Lk(g-l(a),g-l(b)) (see Exercise 7) where 9 is a c~ map homotopic 10 f and a, b are and M .!, N 1. P continuous maps. Then deg(fg) ~ (deg g)(deg f). The same holds distinct regular values of g. The linking number is computed in mod 2 if M, N, P are not oriented. (b) The degree of a homeomorphism or homotopy equivalence is ± 1. f(c) * a, b. *3. Let IDl. be the category whose objects are compact connected n-manifolds and whose (a) H(f) is a well-defined homotopy invariant off which vanishes iff is nuD homo- topic. morphisms are homotopy classes (f] of maps f:M ~ N. For an object M let 7t"(M) (b) If g:5' ~ 5' has degree p then H(fg) ~ pH(f). be the set of homotopy classes M ~ 5". -
New Ideas in Algebraic Topology (K-Theory and Its Applications)
NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems. -
On the Homology of Configuration Spaces
TopologyVol. 28, No. I. pp. I II-123, 1989 Lmo-9383 89 53 al+ 00 Pm&d I” Great Bntam fy 1989 Pcrgamon Press plc ON THE HOMOLOGY OF CONFIGURATION SPACES C.-F. B~DIGHEIMER,~ F. COHEN$ and L. TAYLOR: (Receiued 27 July 1987) 1. INTRODUCTION 1.1 BY THE k-th configuration space of a manifold M we understand the space C”(M) of subsets of M with catdinality k. If c’(M) denotes the space of k-tuples of distinct points in M, i.e. Ck(M) = {(z,, . , zk)eMklzi # zj for i #j>, then C’(M) is the orbit space of C’(M) under the permutation action of the symmetric group Ik, c’(M) -+ c’(M)/E, = Ck(M). Configuration spaces appear in various contexts such as algebraic geometry, knot theory, differential topology or homotopy theory. Although intensively studied their homology is unknown except for special cases, see for example [ 1, 2, 7, 8, 9, 12, 13, 14, 18, 261 where different terminology and notation is used. In this article we study the Betti numbers of Ck(M) for homology with coefficients in a field IF. For IF = IF, the rank of H,(C’(M); IF,) is determined by the IF,-Betti numbers of M. the dimension of M, and k. Similar results were obtained by Ldffler-Milgram [ 171 for closed manifolds. For [F = ff, or a field of characteristic zero the corresponding result holds in the case of odd-dimensional manifolds; it is no longer true for even-dimensional manifolds, not even for surfaces, see [S], [6], or 5.5 here. -
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. -
Combinatorial Topology
Chapter 6 Basics of Combinatorial Topology 6.1 Simplicial and Polyhedral Complexes In order to study and manipulate complex shapes it is convenient to discretize these shapes and to view them as the union of simple building blocks glued together in a “clean fashion”. The building blocks should be simple geometric objects, for example, points, lines segments, triangles, tehrahedra and more generally simplices, or even convex polytopes. We will begin by using simplices as building blocks. The material presented in this chapter consists of the most basic notions of combinatorial topology, going back roughly to the 1900-1930 period and it is covered in nearly every algebraic topology book (certainly the “classics”). A classic text (slightly old fashion especially for the notation and terminology) is Alexandrov [1], Volume 1 and another more “modern” source is Munkres [30]. An excellent treatment from the point of view of computational geometry can be found is Boissonnat and Yvinec [8], especially Chapters 7 and 10. Another fascinating book covering a lot of the basics but devoted mostly to three-dimensional topology and geometry is Thurston [41]. Recall that a simplex is just the convex hull of a finite number of affinely independent points. We also need to define faces, the boundary, and the interior of a simplex. Definition 6.1 Let be any normed affine space, say = Em with its usual Euclidean norm. Given any n+1E affinely independentpoints a ,...,aE in , the n-simplex (or simplex) 0 n E σ defined by a0,...,an is the convex hull of the points a0,...,an,thatis,thesetofallconvex combinations λ a + + λ a ,whereλ + + λ =1andλ 0foralli,0 i n. -
WEIGHT FILTRATIONS in ALGEBRAIC K-THEORY Daniel R
November 13, 1992 WEIGHT FILTRATIONS IN ALGEBRAIC K-THEORY Daniel R. Grayson University of Illinois at Urbana-Champaign Abstract. We survey briefly some of the K-theoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher algebraic K-theory developed by Quillen in 1972. It is the purpose of this article to explain the sense in which the previous statement is true, and to explain how it is thought that the motivic cohomology groups with rational coefficients arise from K-theory through the intervention of the Adams operations. We give a basic description of algebraic K-theory and explain how Quillen’s idea [42] that the Atiyah-Hirzebruch spectral sequence of topology may have an algebraic analogue guides the search for motivic cohomology. There are other useful survey articles about algebraic K-theory: [50], [46], [23], [56], and [40]. I thank W. Dwyer, E. Friedlander, S. Lichtenbaum, R. McCarthy, S. Mitchell, C. Soul´e and R. Thomason for frequent and useful consultations. 2. Constructing topological spaces. In this section we explain the considerations from combinatorial topology which give rise to the higher algebraic K-groups. The first principle is simple enough to state, but hard to implement: when given an interesting group (such as the Grothendieck group of a ring) arising from some algebraic situation, try to realize it as a low-dimensional homotopy group (especially π0 or π1) of a space X constructed in some combinatorial way from the same algebraic inputs, and study the homotopy type of the resulting space. -
Combinatorial Topology and Applications to Quantum Field Theory
Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Chair Professor Ian Agol Professor Constantin Teleman Professor Joel Moore Fall 2018 Abstract Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vivek Shende, Chair Topology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and dis- crete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists. 1 To my loving parents, kind friends, and patient teachers. i Contents I Discrete Topology Toolbox1 1 Basics4 1.1 Discrete Spaces..........................4 1.1.1 Cellular Maps and Cellular Approximation.......6 1.1.2 Triangulations and Barycentric Subdivision......6 1.1.3 PL-Manifolds and Combinatorial Duality........8 1.1.4 Discrete Morse Flows...................9 1.2 Chains, Cycles, Cochains, Cocycles............... 13 1.2.1 Chains, Cycles, and Homology.............. 13 1.2.2 Pushforward of Chains.................. 15 1.2.3 Cochains, Cocycles, and Cohomology......... -
A Survey of the Foundations of Four-Manifold Theory in the Topological Category
A SURVEY OF THE FOUNDATIONS OF FOUR-MANIFOLD THEORY IN THE TOPOLOGICAL CATEGORY STEFAN FRIEDL, MATTHIAS NAGEL, PATRICK ORSON, AND MARK POWELL Abstract. The goal of this survey is to state some foundational theorems on 4-manifolds, especially in the topological category, give precise references, and provide indications of the strategies employed in the proofs. Where appropriate we give statements for manifolds of all dimensions. 1. Introduction Here and throughout the paper \manifold" refers to what is often called a \topological manifold"; see Section 2 for a precise definition. Here are some of the statements discussed in this article. (1) Existence and uniqueness of collar neighbourhoods (Theorem 2.5). (2) The Isotopy Extension theorem (Theorem 2.10). (3) Existence of CW structures (Theorem 4.5). (4) Multiplicativity of the Euler characteristic under finite covers (Corollary 4.8). (5) The Annulus theorem 5.1 and the Stable Homeomorphism Theorem 5.3. (6) Connected sum of two oriented connected 4-manifolds is well-defined (Theorem 5.11). (7) The intersection form of the connected sum of two 4-manifolds is the sum of the intersection forms of the summands (Proposition 5.15). (8) Existence and uniqueness of tubular neighbourhoods of submanifolds (Theorems 6.8 and 6.9). (9) Noncompact connected 4-manifolds admit a smooth structure (Theorem 8.1). (10) When the Kirby-Siebenmann invariant of a 4-manifold vanishes, both connected sum with copies of S2 S2 and taking the product with R yield smoothable manifolds (Theorem 8.6). × (11) Transversality for submanifolds and for maps (Theorems 10.3 and 10.8). -
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. -
Differential Topology from the Point of View of Simple Homotopy Theory
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. BARRY MAZUR Differential topology from the point of view of simple homotopy theory Publications mathématiques de l’I.H.É.S., tome 15 (1963), p. 5-93 <http://www.numdam.org/item?id=PMIHES_1963__15__5_0> © Publications mathématiques de l’I.H.É.S., 1963, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CHAPTER 1 INTRODUCTION It is striking (but not uncharacteristic) that the « first » question asked about higher dimensional geometry is yet unsolved: Is every simply connected 3-manifold homeomorphic with S3 ? (Its original wording is slightly more general than this, and is false: H. Poincare, Analysis Situs (1895).) The difficulty of this problem (in fact of most three-dimensional problems) led mathematicians to veer away from higher dimensional geometric homeomorphism-classificational questions. Except for Whitney's foundational theory of differentiable manifolds and imbed- dings (1936) and Morse's theory of Calculus of Variations in the Large (1934) and, in particular, his analysis of the homology structure of a differentiable manifold by studying critical points of 0°° functions defined on the manifold, there were no classificational results about high dimensional manifolds until the era of Thorn's Cobordisme Theory (1954), the beginning of" modern 9? differential topology. -
Topics in Low Dimensional Computational Topology
THÈSE DE DOCTORAT présentée et soutenue publiquement le 7 juillet 2014 en vue de l’obtention du grade de Docteur de l’École normale supérieure Spécialité : Informatique par ARNAUD DE MESMAY Topics in Low-Dimensional Computational Topology Membres du jury : M. Frédéric CHAZAL (INRIA Saclay – Île de France ) rapporteur M. Éric COLIN DE VERDIÈRE (ENS Paris et CNRS) directeur de thèse M. Jeff ERICKSON (University of Illinois at Urbana-Champaign) rapporteur M. Cyril GAVOILLE (Université de Bordeaux) examinateur M. Pierre PANSU (Université Paris-Sud) examinateur M. Jorge RAMÍREZ-ALFONSÍN (Université Montpellier 2) examinateur Mme Monique TEILLAUD (INRIA Sophia-Antipolis – Méditerranée) examinatrice Autre rapporteur : M. Eric SEDGWICK (DePaul University) Unité mixte de recherche 8548 : Département d’Informatique de l’École normale supérieure École doctorale 386 : Sciences mathématiques de Paris Centre Numéro identifiant de la thèse : 70791 À Monsieur Lagarde, qui m’a donné l’envie d’apprendre. Résumé La topologie, c’est-à-dire l’étude qualitative des formes et des espaces, constitue un domaine classique des mathématiques depuis plus d’un siècle, mais il n’est apparu que récemment que pour de nombreuses applications, il est important de pouvoir calculer in- formatiquement les propriétés topologiques d’un objet. Ce point de vue est la base de la topologie algorithmique, un domaine très actif à l’interface des mathématiques et de l’in- formatique auquel ce travail se rattache. Les trois contributions de cette thèse concernent le développement et l’étude d’algorithmes topologiques pour calculer des décompositions et des déformations d’objets de basse dimension, comme des graphes, des surfaces ou des 3-variétés. -
On the Topology of Discrete Strategies∗
The final version of this paper was published in The International Journal of Robotics Research, 29(7), June 2010, pp. 855–896, by SAGE Publications Ltd, http://online.sagepub.com. c Michael Erdmann On the Topology of Discrete Strategies∗ Michael Erdmann School of Computer Science Carnegie Mellon University [email protected] December 2009 Abstract This paper explores a topological perspective of planning in the presence of uncertainty, focusing on tasks specified by goal states in discrete spaces. The paper introduces strategy complexes. A strategy complex is the collection of all plans for attaining all goals in a given space. Plans are like jigsaw pieces. Understanding how the pieces fit together in a strategy complex reveals structure. That structure characterizes the inherent capabilities of an uncertain system. By adjusting the jigsaw pieces in a design loop, one can build systems with desired competencies. The paper draws on representations from combinatorial topology, Markov chains, and polyhedral cones. Triangulating between these three perspectives produces a topological language for describing concisely the capabilities of uncertain systems, analogous to concepts of reachability and controllability in other disciplines. The major nouns in this language are topological spaces. Three key theorems (numbered 1, 11, 20 in the paper) illustrate the sentences in this language: (a) Goal Attainability: There exists a strategy for attaining a particular goal from anywhere in a system if and only if the strategy complex of a slightly modified system is homotopic to a sphere. (b) Full Controllability: A system can move between any two states despite control uncertainty precisely when its strategy complex is homotopic to a sphere of dimension two less than the number of states.