Manifolds and Poincaré Complexes
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Commentary on the Kervaire–Milnor Correspondence 1958–1961
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 603–609 http://dx.doi.org/10.1090/bull/1508 Article electronically published on July 1, 2015 COMMENTARY ON THE KERVAIRE–MILNOR CORRESPONDENCE 1958–1961 ANDREW RANICKI AND CLAUDE WEBER Abstract. The extant letters exchanged between Kervaire and Milnor during their collaboration from 1958–1961 concerned their work on the classification of exotic spheres, culminating in their 1963 Annals of Mathematics paper. Michel Kervaire died in 2007; for an account of his life, see the obituary by Shalom Eliahou, Pierre de la Harpe, Jean-Claude Hausmann, and Claude We- ber in the September 2008 issue of the Notices of the American Mathematical Society. The letters were made public at the 2009 Kervaire Memorial Confer- ence in Geneva. Their publication in this issue of the Bulletin of the American Mathematical Society is preceded by our commentary on these letters, provid- ing some historical background. Letter 1. From Milnor, 22 August 1958 Kervaire and Milnor both attended the International Congress of Mathemati- cians held in Edinburgh, 14–21 August 1958. Milnor gave an invited half-hour talk on Bernoulli numbers, homotopy groups, and a theorem of Rohlin,andKer- vaire gave a talk in the short communications section on Non-parallelizability of the n-sphere for n>7 (see [2]). In this letter written immediately after the Congress, Milnor invites Kervaire to join him in writing up the lecture he gave at the Con- gress. The joint paper appeared in the Proceedings of the ICM as [10]. Milnor’s name is listed first (contrary to the tradition in mathematics) since it was he who was invited to deliver a talk. -
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 . -
Whitehead Torsion, Part II (Lecture 4)
Whitehead Torsion, Part II (Lecture 4) September 9, 2014 In this lecture, we will continue our discussion of the Whitehead torsion of a homotopy equivalence f : X ! Y between finite CW complexes. In the previous lecture, we gave a definition in the special case where f is cellular. To remove this hypothesis, we need the following: Proposition 1. Let X and Y be connected finite CW complexes and suppose we are given cellular homotopy equivalences f; g : X ! Y . If f and g are homotopic, then τ(f) = τ(g) 2 Wh(π1X). Lemma 2. Suppose we are given quasi-isomorphisms f :(X∗; d) ! (Y∗; d) and g :(Y∗; d) ! (Z∗; d) between finite based complexes with χ(X∗; d) = χ(Y∗; d) = χ(Z∗; d). Then τ(g ◦ f) = τ(g)τ(f) in Ke1(R). Proof. We define a based chain complex (W∗; d) by the formula W∗ = X∗−1 ⊕ Y∗ ⊕ Y∗−1 ⊕ Z∗ d(x; y; y0; z) = (−dx; f(x) + dy + y0; −dy0; g(y0) + dz): Then (W∗; d) contains (C(f)∗; d) as a based subcomplex with quotient (C(g)∗; d), so an Exercise from the previous lecture gives τ(W∗; d) = τ(g)τ(f). We now choose a new basis for each W∗ by replacing each basis element of y 2 Y∗ by (0; y; 0; g(y)); this is an upper triangular change of coordinates and therefore does not 0 0 affect the torsion τ(W∗; d). Now the construction (y ; y) 7! (0; y; y ; g(y)) identifies C(idY )∗ with a based subcomplex of W∗ having quotient C(−g ◦ f)∗. -
Algebraic Equations for Nonsmoothable 8-Manifolds
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. NICHOLAAS H. KUIPER Algebraic equations for nonsmoothable 8-manifolds Publications mathématiques de l’I.H.É.S., tome 33 (1967), p. 139-155 <http://www.numdam.org/item?id=PMIHES_1967__33__139_0> © Publications mathématiques de l’I.H.É.S., 1967, 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/ ALGEBRAIC EQUATIONS FOR NONSMOOTHABLE 8-MANIFOLDS by NICOLAAS H. KUIPER (1) SUMMARY The singularities of Brieskorn and Hirzebruch are used in order to obtain examples of algebraic varieties of complex dimension four in P^C), which are homeo- morphic to closed combinatorial 8-manifolds, but not homeomorphic to any differentiable manifold. Analogous nonorientable real algebraic varieties of dimension 8 in P^R) are also given. The main theorem states that every closed combinatorial 8-manifbld is homeomorphic to a Nash-component with at most one singularity of some real algebraic variety. This generalizes the theorem of Nash for differentiable manifolds. § i. Introduction. The theorem of Wall. From the smoothing theory of Thorn [i], Munkres [2] and others and the knowledge of the groups of differential structures on spheres due to Kervaire, Milnor [3], Smale and Gerf [4] follows a.o. -
Braids in Pau -- an Introduction
ANNALES MATHÉMATIQUES BLAISE PASCAL Enrique Artal Bartolo and Vincent Florens Braids in Pau – An Introduction Volume 18, no 1 (2011), p. 1-14. <http://ambp.cedram.org/item?id=AMBP_2011__18_1_1_0> © Annales mathématiques Blaise Pascal, 2011, tous droits réservés. L’accès aux articles de la revue « Annales mathématiques Blaise Pas- cal » (http://ambp.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://ambp.cedram.org/legal/). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Publication éditée par le laboratoire de mathématiques de l’université Blaise-Pascal, UMR 6620 du CNRS Clermont-Ferrand — France cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales mathématiques Blaise Pascal 18, 1-14 (2011) Braids in Pau – An Introduction Enrique Artal Bartolo Vincent Florens Abstract In this work, we describe the historic links between the study of 3-dimensional manifolds (specially knot theory) and the study of the topology of complex plane curves with a particular attention to the role of braid groups and Alexander-like invariants (torsions, different instances of Alexander polynomials). We finish with detailed computations in an example. Tresses à Pau – une introduction Résumé Dans ce travail, nous décrivons les liaisons historiques entre l’étude de variétés de dimension 3 (notamment, la théorie de nœuds) et l’étude de la topologie des courbes planes complexes, dont l’accent est posé sur le rôle des groupes de tresses et des invariantes du type Alexander (torsions, différents incarnations des polynômes d’Alexander). -
An Introduction to Differentiable Manifolds
Mathematics Letters 2016; 2(5): 32-35 http://www.sciencepublishinggroup.com/j/ml doi: 10.11648/j.ml.20160205.11 Conference Paper An Introduction to Differentiable Manifolds Kande Dickson Kinyua Department of Mathematics, Moi University, Eldoret, Kenya Email address: [email protected] To cite this article: Kande Dickson Kinyua. An Introduction to Differentiable Manifolds. Mathematics Letters. Vol. 2, No. 5, 2016, pp. 32-35. doi: 10.11648/j.ml.20160205.11 Received : September 7, 2016; Accepted : November 1, 2016; Published : November 23, 2016 Abstract: A manifold is a Hausdorff topological space with some neighborhood of a point that looks like an open set in a Euclidean space. The concept of Euclidean space to a topological space is extended via suitable choice of coordinates. Manifolds are important objects in mathematics, physics and control theory, because they allow more complicated structures to be expressed and understood in terms of the well–understood properties of simpler Euclidean spaces. A differentiable manifold is defined either as a set of points with neighborhoods homeomorphic with Euclidean space, Rn with coordinates in overlapping neighborhoods being related by a differentiable transformation or as a subset of R, defined near each point by expressing some of the coordinates in terms of the others by differentiable functions. This paper aims at making a step by step introduction to differential manifolds. Keywords: Submanifold, Differentiable Manifold, Morphism, Topological Space manifolds with the additional condition that the transition 1. Introduction maps are analytic. In other words, a differentiable (or, smooth) The concept of manifolds is central to many parts of manifold is a topological manifold with a globally defined geometry and modern mathematical physics because it differentiable (or, smooth) structure, [1], [3], [4]. -
Arxiv:1303.6028V2 [Math.DG] 29 Dec 2014 B Ahrglrlvlhprufc Scle an Called Is Hypersurface Level Regular Each E Scle the Called Is Set E.[T3 )
ISOPARAMETRIC FUNCTIONS ON EXOTIC SPHERES CHAO QIAN AND ZIZHOU TANG Abstract. This paper extends widely the work in [GT13]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy n-sphere (n > 4) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [St96] ( only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres ). As an application, we improve a beautiful result of B´erard-Bergery [BB77] ( see also pp.234-235 of [Be78] ). 1. Introduction Let N be a connected complete Riemannian manifold. A non-constant smooth function f on N is called transnormal, if there exists a smooth function b : R R such that f 2 = → |∇ | b( f ), where f is the gradient of f . If in addition, there exists a continuous function a : ∇ R R so that f = a( f ), where f is the Laplacian of f , then f is called isoparametric. → △ △ Each regular level hypersurface is called an isoparametric hypersurface and the singular level set is called the focal set. The two equations of the function f mean that the regular level hypersurfaces of f are parallel and have constant mean curvatures, which may be regarded as a geometric generalization of cohomogeneity one actions in the theory of transformation groups ( ref. [GT13] ). Owing to E. Cartan and H. F. M¨unzner [M¨u80], the classification of isoparametric hy- persurfaces in a unit sphere has been one of the most challenging problems in submanifold geometry. -
Graphs and Patterns in Mathematics and Theoretical Physics, Volume 73
http://dx.doi.org/10.1090/pspum/073 Graphs and Patterns in Mathematics and Theoretical Physics This page intentionally left blank Proceedings of Symposia in PURE MATHEMATICS Volume 73 Graphs and Patterns in Mathematics and Theoretical Physics Proceedings of the Conference on Graphs and Patterns in Mathematics and Theoretical Physics Dedicated to Dennis Sullivan's 60th birthday June 14-21, 2001 Stony Brook University, Stony Brook, New York Mikhail Lyubich Leon Takhtajan Editors Proceedings of the conference on Graphs and Patterns in Mathematics and Theoretical Physics held at Stony Brook University, Stony Brook, New York, June 14-21, 2001. 2000 Mathematics Subject Classification. Primary 81Txx, 57-XX 18-XX 53Dxx 55-XX 37-XX 17Bxx. Library of Congress Cataloging-in-Publication Data Stony Brook Conference on Graphs and Patterns in Mathematics and Theoretical Physics (2001 : Stony Brook University) Graphs and Patterns in mathematics and theoretical physics : proceedings of the Stony Brook Conference on Graphs and Patterns in Mathematics and Theoretical Physics, June 14-21, 2001, Stony Brook University, Stony Brook, NY / Mikhail Lyubich, Leon Takhtajan, editors. p. cm. — (Proceedings of symposia in pure mathematics ; v. 73) Includes bibliographical references. ISBN 0-8218-3666-8 (alk. paper) 1. Graph Theory. 2. Mathematics-Graphic methods. 3. Physics-Graphic methods. 4. Man• ifolds (Mathematics). I. Lyubich, Mikhail, 1959- II. Takhtadzhyan, L. A. (Leon Armenovich) III. Title. IV. Series. QA166.S79 2001 511/.5-dc22 2004062363 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given. -
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, . -
Mapping Class Group of a Handlebody
FUNDAMENTA MATHEMATICAE 158 (1998) Mapping class group of a handlebody by Bronis law W a j n r y b (Haifa) Abstract. Let B be a 3-dimensional handlebody of genus g. Let M be the group of the isotopy classes of orientation preserving homeomorphisms of B. We construct a 2-dimensional simplicial complex X, connected and simply-connected, on which M acts by simplicial transformations and has only a finite number of orbits. From this action we derive an explicit finite presentation of M. We consider a 3-dimensional handlebody B = Bg of genus g > 0. We may think of B as a solid 3-ball with g solid handles attached to it (see Figure 1). Our goal is to determine an explicit presentation of the map- ping class group of B, the group Mg of the isotopy classes of orientation preserving homeomorphisms of B. Every homeomorphism h of B induces a homeomorphism of the boundary S = ∂B of B and we get an embedding of Mg into the mapping class group MCG(S) of the surface S. z-axis α α α α 1 2 i i+1 . . β β 1 2 ε x-axis i δ -2, i Fig. 1. Handlebody An explicit and quite simple presentation of MCG(S) is now known, but it took a lot of time and effort of many people to reach it (see [1], [3], [12], 1991 Mathematics Subject Classification: 20F05, 20F38, 57M05, 57M60. This research was partially supported by the fund for the promotion of research at the Technion. [195] 196 B.Wajnryb [9], [7], [11], [6], [14]). -
[Math.GT] 31 Mar 2004
CORES OF S-COBORDISMS OF 4-MANIFOLDS Frank Quinn March 2004 Abstract. The main result is that an s-cobordism (topological or smooth) of 4- manifolds has a product structure outside a “core” sub s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-complex. The decomposition is highly nonunique so cannot be used to define an invariant, but it shows the topological s-cobordism question reduces to the core case. The simply-connected version of the decomposition (with 1-complex a point) is due to Curtis, Freedman, Hsiang and Stong. Controlled surgery is used to reduce topological triviality of core s-cobordisms to a question about controlled homotopy equivalence of 4-manifolds. There are speculations about further reductions. 1. Introduction The classical s-cobordism theorem asserts that an s-cobordism of n-manifolds (the bordism itself has dimension n + 1) is isomorphic to a product if n ≥ 5. “Isomorphic” means smooth, PL or topological, depending on the structure of the s-cobordism. In dimension 4 it is known that there are smooth s-cobordisms without smooth product structures; existence was demonstrated by Donaldson [3], and spe- cific examples identified by Akbulut [1]. In the topological case product structures follow from disk embedding theorems. The best current results require “small” fun- damental group, Freedman-Teichner [5], Krushkal-Quinn [9] so s-cobordisms with these groups are topologically products. The large fundamental group question is still open. Freedman has developed several link questions equivalent to the 4-dimensional “surgery conjecture” for arbitrary fundamental groups. -
Finite Group Actions on Kervaire Manifolds 3
FINITE GROUP ACTIONS ON KERVAIRE MANIFOLDS DIARMUID CROWLEY AND IAN HAMBLETON M4k+2 Abstract. Let K be the Kervaire manifold: a closed, piecewise linear (PL) mani- fold with Kervaire invariant 1 and the same homology as the product S2k+1 × S2k+1 of M4k+2 spheres. We show that a finite group of odd order acts freely on K if and only if 2k+1 2k+1 it acts freely on S × S . If MK is smoothable, then each smooth structure on M j M4k+2 K admits a free smooth involution. If k 6= 2 − 1, then K does not admit any M30 M62 free TOP involutions. Free “exotic” (PL) involutions are constructed on K , K , and M126 M30 Z Z K . Each smooth structure on K admits a free /2 × /2 action. 1. Introduction One of the main themes in geometric topology is the study of smooth manifolds and their piece-wise linear (PL) triangulations. Shortly after Milnor’s discovery [54] of exotic smooth 7-spheres, Kervaire [39] constructed the first example (in dimension 10) of a PL- manifold with no differentiable structure, and a new exotic smooth 9-sphere Σ9. The construction of Kervaire’s 10-dimensional manifold was generalized to all dimen- sions of the form m ≡ 2 (mod 4), via “plumbing” (see [36, §8]). Let P 4k+2 denote the smooth, parallelizable manifold of dimension 4k+2, k ≥ 0, constructed by plumbing two copies of the the unit tangent disc bundle of S2k+1. The boundary Σ4k+1 = ∂P 4k+2 is a smooth homotopy sphere, now usually called the Kervaire sphere.