DOCSLIB.ORG

Lecture Notes on Basic Differential Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes on Basic Differential Topology LECTURE NOTES ON BASIC DIFFERENTIAL TOPOLOGY ANTONIO LERARIO These are notes for the course \Advanced Geometry 2" for the Master Diploma in Mathematics at the University of Trieste and at SISSA. These notes are by no means complete: excellent references for the subject are the books [3,5,7,8,9], from which in fact many proofs are taken or adapted. The notes contain some exercises, which the reader is warmly encouraged to solve (sometimes part of a proof is left as an exercise). I apologize in advance for the many mistakes and imprecisions that the reader might find: I will greatly appreciate if he/she could point out any of them. Date: November 4, 2020. 1 2 ANTONIO LERARIO 1. Differentiable manifolds Definition 1 (m{dimensional Ck manifold). Let m N and k N ;! : A manifold M of dimension m and class Ck is a paracompact, Hausdorff2 topological2 space,[ f1 withg countably many connected components1 and such that: (1) for every point x M there exists a neighborhood U of x and a continuous function 2 : U Rm which is a homeomorphism onto an open subset of Rm (the pair (U; ) is called a!chart); (2) for every pairs of charts (U ; ) and (U ; ) such that U U = the map 1 1 2 2 1 \ 2 6 ; 1 m 2 1 U− U : 1(U1 U2) R ◦ j 1\ 2 \ ! is a Ck map (for k = 0 we obtain topological manifolds, for k 1 differentiable manifolds, for k = smooth manifolds and for k = ! analytic manifolds).≥ 1 S k A collection of charts (Uj; j) j J as above such that j J Uj = M is called a C atlas. f g 2 2 Remark 2. In the second condition above, given two charts (U1; 1) and (U2; 2) such that 1 k U U = it is not enough to check that the map from − is C , but also that its 1 2 2 1 U1 U2 \ 6 k; ◦ j \ 3 inverse is C . For instance, let M = R with the two charts (U1; 1) = (R; x) and (U2; 2) = (R; x ): 1 3 ! 1 1=3 0 Then − = x is C but − = x is only C . 2 ◦ 1 1 ◦ 2 k Remark 3 (Atlases and differential structures). Two C atlases A = (Uα; α) α A and B = f kg 2 k (Vβ;'β) β B for M, are said to be equivalent if their union A B is still a C atlas. A C - f g 2 [ differential structure on M is the choice of an equivalence class of Ck atlases. If we take the union of all atlases belonging to a Ck-differential structure, we obtain a maximal Ck atlas. This atlas contains every chart that is compatible with the chosen differentiable structure. (There is a natural one-to-one correspondence between differentiable structures and maximal differentiable atlases.) From now on we will assume that the atlas we work with is maximal, so that we will have all possible charts available. A simple way to enrich a given atlas is as follows. Given a chart : U Rm around a point x M (as in point (1) of Definition1), and given a neighborhood V !U of x we can easily 2 m ⊂ construct a chart ' : V R by simply taking ' = V . Note that in this way we can construct ! j 1 m a chart (V; ') around any point with V contractible: it is enough to take V = − (BR ( (x); )) for > 0 small enough. Example 4. If ' : M Rn is a homeomorphism, then M is an analytic manifold. In fact one ! 1 m m can cover M with the single chart (M; '), and ' '− = id m : R R is analytic. ◦ R ! Example 5. Given an open set U Rm, the single chart given by the inclusion U, Rm turns it into a smooth manifold. Why is⊆ the condition \with countably many connected components"! in the above definition satisfied by U? (Try to prove it directly: an open set in Rm can have only countably many connected components). Example 6 (Product manifolds). If M and N are smooth manifolds with respective atlases (Uα; α) α A and (Vβ;'β) β B, then M N is naturally a smooth manifold with the atlas f g 2 f g 2 × (Uα Vβ; α 'β) (α,β) A B. f × × g 2 × n 2 2 n+1 Example 7 (Spheres). The unit sphere S = x0 + + xn = 1 R can be endowed with the structure of a smooth manifold as follows.f Consider··· the pointge ⊂= (1; 0;:::; 0) Sn and the 0 2 1Recall that a paracompact space is a topological space X for which every open cover has a locally finite refinement. More precisely: given an open cover U = fUαgα2A fo X, there exists another open cover V = fVβ gβ2B such that (i) for every β 2 B there exists α(β) 2 A such that Vβ ⊂ Uα(β) (i.e. V refines U); (ii) for every x 2 X there exists a neighborhood Vx of x which intersects only finitely many elements of V (i.e. V is locally finite). It is worth noticing that if a Hausdorff space is locally Euclidean (i.e. if it satisfies condition (1) of Definition 1) and connected, then this space is paracompact if and only if it is second countable (i.e. its topology has a countable basis), see the discussion you can find at this webpage https://math.stackexchange.com/questions/ 527642/the-equivalence-between-paracompactness-and-second-countablity-in-a-locally-eucl. There exist locally Euclidean spaces which are Hausdorff, paracompact but not second countable (for example R with the discrete topology); that is why we also add the condition that a manifold should have countably many connected components. LECTURE NOTES ON BASIC DIFFERENTIAL TOPOLOGY 3 e0 bc Sn b p1 n R b b ψ1(p2) ψ1(p1) b p2 n n Figure 1. The stereographic projection 1 : S e0 R . nf g ! n n two open sets of the the sphere defined by U1 = S e0 and U2 = S e0 . We produce explicit nf ng \{− ng and nice homeomorphisms (i.e. charts) 1 : U1 R and 2 : U2 R , called stereographic projections (see Figure1), as follows: ! ! 1 1 (x ; : : : ; x ) = (x ; : : : ; x ) and (x ; : : : ; x ) = (x ; : : : ; x ): 1 0 n 1 x 1 n 2 0 n 1 + x 1 n − 0 0 1 n n It is easy to verify that − : R S is given by: 1 ! 1 1 2 − (y ; : : : ; y ) = y 1; 2y ;:::; 2y : 1 1 n 1 + y 2 k k − 1 n k k 1 n n This in particular allows one to write the explicit expression for 2 1 U− U : R 0 R 0 : ◦ j 1\ 2 nf g ! nf g 1 y − (y) = ; 2 ◦ 1 y 2 k k n which is a smooth map. Hence (U1; 1); (U2; 2) is a smooth atlas for S and turns it into a smooth manifold. This is called thef standard differentialg structure on Sn. Example 8 (Real projective spaces). The real projective space RPn can be endowed with the n n+1 structure of smooth manifold as follows. Recall that RP = (R 0 )= , where p1 p2 if and nf g ∼ ∼ only if there exists λ = 0 such that p1 = λp2. We denote by [x0; : : : ; xn] the equivalence class n+16 of (x0; : : : ; xn) R 0 (the xj are called homogeneous coordinates). For every j = 0; : : : ; n 2 nf g n consider the open set Uj RP defined by: ⊂ U = [x ; : : : ; x ] such that x = 0 ; j f 0 n j 6 g n together with the homeomorphism j : Uj R given by: ! x0 cxj xn j([x0; : : : ; xn]) = ;:::; ;:::; xj xj xj (here the \hat" symbol denotes that this element has been removed from the list). The inverse 1 n n − : R RP is given by: j ! 1 j− (y0;:::; ybj; : : : ; yn) = [y0;:::; 1; : : : yn]; where the \1" is in position j. As a consequence, for every i = j we have: 6 1 y0 cyi 1 yn i j− (y0;:::; ybj; : : : ; yn) = ;:::; ;:::; ;::: ; ◦ yi yi yi yj which is a diffeomorphism of Rn 0 to itself. nf g Exercise 9. Prove that RP1 and S1 are homeomorphic. Example 10 (real Grassmannians). The real Grassmannian G(k; n) consists of the set of all k-dimensional vector subspaces of Rn, endowed with the quotient topology of the map: n k q : M R × such that rk(M) = k G(k; n); q(M) = span columns of M : f 2 g ! f g 4 ANTONIO LERARIO In other words, G(k; n) (as a topological space) can be considered as the quotient of the set of n k n k real matrices of rank k (viewed as a subset of R × ) under the equivalence relation: × k M1 M2 there exists L GL(R ) such that M1 = M2L: ∼ () 2 n 1 Observe that G(1; n) = RP − and that the above definition mimics the equivalence relation v1 v2 if and only if there exists λ GL(R) = R 0 such that v1 = λv2: ∼ 2 nf g We want to endow G(k; n) with the structure of a smooth manifold. For every multi-index J = (j ; : : : ; j ) n we denote by M the k k submatrix of M obtained by selecting the rows 1 k 2 k jJ × j1; : : : ; jk (in this way M J c denotes the (n k) k submatrix of M obtained by selecting the complementary rows). Forj every such multi-index− ×J we define the open set: U = [M] G(k; n) such that det(M ) = 0 : J f 2 jJ 6 g (Note that this set is well defined.) Mimicking again the definition for projective spaces, we define (n k) k the manifold charts J : UJ R − × by: ! 1 ([M]) = (MM − ) c : J J jJ The expression of the inverse of a matrix in terms of its determinant and its cofactor sshows that n 1 for every pair of indices J ;J the map − is smooth.
Load more

Recommended publications
  • Hirschdx.Pdf
    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".
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Topology from the Differentiable Viewpoint
    TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT By John W. Milnor Princeton University Based on notes by David W. Weaver The University Press of Virginia Charlottesville PREFACE THESE lectures were delivered at the University of Virginia in December 1963 under the sponsorship of the Page-Barbour Lecture Foundation. They present some topics from the beginnings of topology, centering about L. E. J. Brouwer’s definition, in 1912, of the degree of a mapping. The methods used, however, are those of differential topology, rather than the combinatorial methods of Brouwer. The concept of regular value and the theorem of Sard and Brown, which asserts that every smooth mapping has regular values, play a central role. To simplify the presentation, all manifolds are taken to be infinitely differentiable and to be explicitly embedded in euclidean space. A small amount of point-set topology and of real variable theory is taken for granted. I would like here to express my gratitude to David Weaver, whose untimely death has saddened us all. His excellent set of notes made this manuscript possible. J. W. M. Princeton, New Jersey March 1965 vii CONTENTS Preface v11 1. Smooth manifolds and smooth maps 1 Tangent spaces and derivatives 2 Regular values 7 The fundamental theorem of algebra 8 2. The theorem of Sard and Brown 10 Manifolds with boundary 12 The Brouwer fixed point theorem 13 I 3. Proof of Sard’s theorem 16 4. The degree modulo 2 of a mapping 20 J Smooth homotopy and smooth isotopy 20 5. Oriented manifolds 26 The Brouwer degree 27 6. Vector fields and the Euler number 32 7.
    [Show full text]
  • Differential Topology Forty-Six Years Later
    Differential Topology Forty-six Years Later John Milnor n the 1965 Hedrick Lectures,1 I described its uniqueness up to a PL-isomorphism that is the state of differential topology, a field that topologically isotopic to the identity. In particular, was then young but growing very rapidly. they proved the following. During the intervening years, many problems Theorem 1. If a topological manifold Mn without in differential and geometric topology that boundary satisfies Ihad seemed totally impossible have been solved, 3 n 4 n often using drastically new tools. The following is H (M ; Z/2) = H (M ; Z/2) = 0 with n ≥ 5 , a brief survey, describing some of the highlights then it possesses a PL-manifold structure that is of these many developments. unique up to PL-isomorphism. Major Developments (For manifolds with boundary one needs n > 5.) The corresponding theorem for all manifolds of The first big breakthrough, by Kirby and Sieben- dimension n ≤ 3 had been proved much earlier mann [1969, 1969a, 1977], was an obstruction by Moise [1952]. However, we will see that the theory for the problem of triangulating a given corresponding statement in dimension 4 is false. topological manifold as a PL (= piecewise-linear) An analogous obstruction theory for the prob- manifold. (This was a sharpening of earlier work lem of passing from a PL-structure to a smooth by Casson and Sullivan and by Lashof and Rothen- structure had previously been introduced by berg. See [Ranicki, 1996].) If B and B are the Top PL Munkres [1960, 1964a, 1964b] and Hirsch [1963].
    [Show full text]
  • Spaces of Polygons in the Plane and Morse Theory
    Swarthmore College Works Mathematics & Statistics Faculty Works Mathematics & Statistics 4-1-2005 Spaces Of Polygons In The Plane And Morse Theory Don H. Shimamoto Swarthmore College, [email protected] C. Vanderwaart Follow this and additional works at: https://works.swarthmore.edu/fac-math-stat Part of the Mathematics Commons Let us know how access to these works benefits ouy Recommended Citation Don H. Shimamoto and C. Vanderwaart. (2005). "Spaces Of Polygons In The Plane And Morse Theory". American Mathematical Monthly. Volume 112, Issue 4. 289-310. DOI: 10.2307/30037466 https://works.swarthmore.edu/fac-math-stat/30 This work is brought to you for free by Swarthmore College Libraries' Works. It has been accepted for inclusion in Mathematics & Statistics Faculty Works by an authorized administrator of Works. For more information, please contact [email protected]. Spaces of Polygons in the Plane and Morse Theory Author(s): Don Shimamoto and Catherine Vanderwaart Source: The American Mathematical Monthly, Vol. 112, No. 4 (Apr., 2005), pp. 289-310 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/30037466 Accessed: 19-10-2016 15:46 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly This content downloaded from 130.58.65.20 on Wed, 19 Oct 2016 15:46:08 UTC All use subject to http://about.jstor.org/terms Spaces of Polygons in the Plane and Morse Theory Don Shimamoto and Catherine Vanderwaart 1.
    [Show full text]
  • Configuration Space Integrals and the Topology of Knot And
    Morfismos, Vol. 17, No. 2, 2013, pp. 1{56 Configuration space integrals and the topology of knot and link spaces Ismar Voli´c 1 Abstract This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type invari- ants of classical knots and links. More generally, we also explain the construction of a chain map, given by configuration space integrals, between a certain diagram complex and the deRham complex of the space of knots in dimension four or more. A gen- eralization to spaces of links, homotopy links, and braids is also treated, as are connections to Milnor invariants, manifold calculus of functors, and the rational formality of the little balls operads. 2010 Mathematics Subject Classification: 57Q45, 57M27, 81Q30, 57- R40. Keywords and phrases: configuration space integrals, Bott-Taubes inte- grals, knots, links, homotopy links, braids, finite type invariants, Vas- siliev invariants, Milnor invariants, chord diagrams, weight systems, manifold calculus, embedding calculus, little balls operad, rational for- mality of configuration spaces. Contents 1 Introduction 2 1.1 Organization of the paper 5 2 Preliminaries 6 2.1 Differential forms and integration along the fiber 6 2.2 Space of long knots 9 1The author was supported by the National Science Foundation grant DMS 1205786. 1 2 Ismar Voli´c 2.3 Finite type invariants 10 2.4 Configuration spaces and their compactification 17 3 Configuration space integrals and finite type knot invariants 20 3.1 Motivation: The linking number 20 3.2 \Self-linking" for knots 22 3.3 Finite type two knot invariant 23 3.4 Finite type k knot invariants 30 4 Generalization to Kn, n > 3 33 5 Further generalizations and applications 38 5.1 Spaces of links 38 5.2 Manifold calculus of functors and finite type invariants 43 5.3 Formality of the little balls operad 48 References 52 1 Introduction Configuration space integrals are fascinating objects that lie at the inter- section of physics, combinatorics, topology, and geometry.
    [Show full text]
  • A List of Recommended Books in Topology
    A List of Recommended Books in Topology Allen Hatcher These are books that I personally like for one reason or another, or at least find use- ful. They range from elementary to advanced, but don’t cover absolutely all areas of Topology. The number of Topology books has been increasing rather rapidly in recent years after a long period when there was a real shortage, but there are still some areas that are difficult to learn due to the lack of a good book. The list was made in 2003 and is in need of updating. For the books that were still in print in 2003 I gave the price at that time since this certainly seems like relevant information. One cannot help noticing the wide variation in prices, which shows how ridiculously inflated the prices from some publishers are. When books are available online for free downloading this has been indicated. Unfortunately the number of such books is still small. Here are the main headings for the list: I. Introductory Books II. Algebraic Topology III. Manifold Theory IV. Low-Dimensional Topology V. Miscellaneous I. Introductory Books. General Introductions . Here are two books that give an idea of what topology is about, aimed at a general audience, without much in the way of prerequisites. • V V Prasolov. Intuitive Topology. American Mathematical Society 1995. [$20] • J R Weeks. The Shape of Space. 2nd ed. Marcel Dekker, 2002. [$35] Point-Set Topology. The standard textbook here seems to be the one by Munkres, but I’ve never been able to work up any enthusiasm for this rather pedestrian treatment.
    [Show full text]
  • Topics in Low Dimensional Topology
    Chris Brav, Ash Lightfoot Topics in low dimensional topology The goal of low dimensional topology is to understand manifolds of dimensions three and four, and this seminar will present a selection of techniques, new and old, which have been developed to achieve this. The course will begin from a topological perspective, but turning to recent developments in the subject (namely, Heegaard Floer homology) we will see that algebraic geometry (in particular, character varieties) has a significant role to play. In the topological part, our emphasis will be on constructing examples and building geometric intuition. Note also that, regarding 3-manifold theory as consisting of two halves: knot theory and hyperbolic geometry, we will not touch much on hyperbolic geometry; moreover, our approach to knot theory will be topological, rather than combinatorial. Selected topics • Handlebody decompositions of manifolds, surgery and Morse theory • Heegaard splittings and Heegaard diagrams of 3-manifolds • Dehn surgery and surgery on a link in 3-space • Kirby calculus on 4-manifolds • Representation spaces, character varieties and invariants from Heegaard diagrams • Special topics in 3-manifolds/knot theory (various invariants, cyclic and branched covers, open book decompositions, branched coverings, fibered knots, etc.) • Special topics in 4-manifolds (various invariants, exotic 4-manifolds, trisections of 4-manifolds, knotted surfaces, topological classification, etc.) Prerequisites: All students are welcome. Though familiarity with differential topology (e.g., transversality) and basic algebraic topology (e.g., homology, CW-complexes, fundamental group) will certainly be helpful, students may use this as an opportunity to develop some intuition and motivation for future courses in geometric topology.
    [Show full text]