Arf Invariant
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Obstructions to Approximating Maps of N-Manifolds Into R2n by Embeddings
Topology and its Applications 123 (2002) 3–14 Obstructions to approximating maps of n-manifolds into R2n by embeddings P.M. Akhmetiev a,1,D.Repovšb,∗,A.B.Skopenkov c,1 a IZMIRAN, Moscow Region, Troitsk, Russia 142092 b Institute for Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box 2964, Ljubljana, Slovenia 1001 c Department of Mathematics, Kolmogorov College, 11 Kremenchugskaya, Moscow, Russia 121357 Received 11 November 1998; received in revised form 19 August 1999 Abstract We prove a criterion for approximability by embeddings in R2n of a general position map − f : K → R2n 1 from a closed n-manifold (for n 3). This approximability turns out to be equivalent to the property that f is a projected embedding, i.e., there is an embedding f¯: K → R2n such − that f = π ◦ f¯,whereπ : R2n → R2n 1 is the canonical projection. We prove that for n = 2, the obstruction modulo 2 to the existence of such a map f¯ is a product of Arf-invariants of certain quadratic forms. 2002 Elsevier Science B.V. All rights reserved. AMS classification: Primary 57R40; 57Q35, Secondary 54C25; 55S15; 57M20; 57R42 Keywords: Projected embedding; Approximability by embedding; Resolvable and nonresolvable triple point; Immersion; Arf-invariant 1. Introduction Throughout this paper we shall work in the smooth category. A map f : K → Rm is said to be approximable by embeddings if for each ε>0 there is an embedding φ : K → Rm, which is ε-close to f . This notion appeared in studies of embeddability of compacta in Euclidean spaces—for a recent survey see [15, §9] (see also [2], [8, §4], [14], [16, * Corresponding author. -
February 19, 2016
February 19, 2016 Professor HELGE HOLDEN SECRETARY OF THE INTERNATIONAL MATHEMATICAL UNION Dear Professor Holden, The Turkish Mathematical Society (TMD) as the Adhering Organization, applies to promote Turkey from Group I to Group II as a member of IMU. We attach an overview of the developments of Mathematics in Turkey during the last 10 years (2005- 2015) preceded by a historical account. With best regards, Betül Tanbay President of the Turkish Mathematical Society Report on the state of mathematics in Turkey (2005-2015) This is an overview of the status of Mathematics in Turkey, prepared for the IMU for promotion from Group I to Group II by the adhering organization, the Turkish Mathematical Society (TMD). 1-HISTORICAL BACKGROUND 2-SOCIETIES AND CENTERS RELATED TO MATHEMATICAL SCIENCES 3-NUMBER OF PUBLICATIONS BY SUBJECT CATEGORIES 4-NATIONAL CONFERENCES AND WORKSHOPS HELD IN TURKEY BETWEEN 2013-2015 5-INTERNATIONAL CONFERENCES AND WORKSHOPS HELD IN TURKEY BETWEEN 2013-2015 6-NUMBERS OF STUDENTS AND TEACHING STAFF IN MATHEMATICAL SCIENCES IN TURKEY FOR THE 2013-2014 ACADEMIC YEAR AND THE 2014-2015 ACADEMIC YEAR 7- RANKING AND DOCUMENTS OF TURKEY IN MATHEMATICAL SCIENCES 8- PERIODICALS AND PUBLICATIONS 1-HISTORICAL BACKGROUND Two universities, the Istanbul University and the Istanbul Technical University have been influential in creating a mathematical community in Turkey. The Royal School of Naval Engineering, "Muhendishane-i Bahr-i Humayun", was established in 1773 with the responsibility to educate chart masters and ship builders. Gaining university status in 1928, the Engineering Academy continued to provide education in the fields of engineering and architecture and, in 1946, Istanbul Technical University became an autonomous university which included the Faculties of Architecture, Civil Engineering, Mechanical Engineering, and Electrical and Electronic Engineering. -
Cahit Arf: Exploring His Scientific Influence Using Social Network Analysis and Author Co-Citation Maps
Cahit Arf: Exploring His Scientific Influence Using Social Network Analysis and Author Co-citation Maps 1 2 Yaşar Tonta and A. Esra Özkan Çelik 1 [email protected] Department of Information Management, Hacettepe University, 06800 Beytepe, Ankara, TR 2 [email protected] Registrar's Office, Hacettepe University, 06800 Beytepe, Ankara, TR Abstract Cahit Arf (1910-1997), a famous Turkish scientist whose picture is depicted in one of the Turkish banknotes, is a well known figure in mathematics with his discoveries named after him (e.g., Arf invariant, Arf rings, the Hasse- Arf theorem). Although Arf may not be considered as a prolific scientist in terms of number of papers (he authored a total of 23 papers), his influence on mathematics and related disciplines was profound. As he was active before, during and after the World War II, Arf’s contributions were not properly listed in citation indexes and thus did not generate that many citations even though several papers with “Arf” in their titles appeared in the literature. This paper traces the influence of Arf in scientific world using citation analysis techniques first. It reviews the scientific impact of Arf by analyzing both the papers authored by Arf and papers whose titles or keywords contain various combinations of “Arf rings”, “Arf invariant”, and so on. The paper then goes on to study Arf’s contributions using social network analysis and author co-citation analysis techniques. CiteSpace and pennant diagrams are used to explore the scientific impact of Arf by mapping his cited references derived from Thomson Reuters’ Web of Science (WoS) database. -
The Kervaire Invariant in Homotopy Theory
The Kervaire invariant in homotopy theory Mark Mahowald and Paul Goerss∗ June 20, 2011 Abstract In this note we discuss how the first author came upon the Kervaire invariant question while analyzing the image of the J-homomorphism in the EHP sequence. One of the central projects of algebraic topology is to calculate the homotopy classes of maps between two finite CW complexes. Even in the case of spheres – the smallest non-trivial CW complexes – this project has a long and rich history. Let Sn denote the n-sphere. If k < n, then all continuous maps Sk → Sn are null-homotopic, and if k = n, the homotopy class of a map Sn → Sn is detected by its degree. Even these basic facts require relatively deep results: if k = n = 1, we need covering space theory, and if n > 1, we need the Hurewicz theorem, which says that the first non-trivial homotopy group of a simply-connected space is isomorphic to the first non-vanishing homology group of positive degree. The classical proof of the Hurewicz theorem as found in, for example, [28] is quite delicate; more conceptual proofs use the Serre spectral sequence. n Let us write πiS for the ith homotopy group of the n-sphere; we may also n write πk+nS to emphasize that the complexity of the problem grows with k. n n ∼ Thus we have πn+kS = 0 if k < 0 and πnS = Z. Given the Hurewicz theorem and knowledge of the homology of Eilenberg-MacLane spaces it is relatively simple to compute that 0, n = 1; n ∼ πn+1S = Z, n = 2; Z/2Z, n ≥ 3. -
The Arf and Sato Link Concordance Invariants
transactions of the american mathematical society Volume 322, Number 2, December 1990 THE ARF AND SATO LINK CONCORDANCE INVARIANTS RACHEL STURM BEISS Abstract. The Kervaire-Arf invariant is a Z/2 valued concordance invari- ant of knots and proper links. The ß invariant (or Sato's invariant) is a Z valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants {/?,} associated with a two com- ponent link of linking number zero where each ßi is a Z valued concordance invariant and ß0 = ß . In this paper we demonstrate a formula for the Arf invariant of a two component link L = X U Y of linking number zero in terms of the ß invariant of the link: arf(X U Y) = arf(X) + arf(Y) + ß(X U Y) (mod 2). This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link's components. We then estab- lish a formula for \ß\ in terms of the link's Alexander polynomial A(x, y) = (x- \)(y-\)f(x,y): \ß(L)\ = \f(\, 1)|. Finally we find a relationship between the ß{ invariants and linking numbers of lifts of X and y in a Z/2 cover of the compliment of X u Y . 1. Introduction The Kervaire-Arf invariant [KM, R] is a Z/2 valued concordance invariant of knots and proper links. The ß invariant (or Sato's invariant) is a Z valued concordance invariant of two component links of linking number zero discov- ered by Levine (unpublished) and studied by Sato [S], Cochran [C], and Daniel Ruberman. -
The Arf-Kervaire Invariant Problem in Algebraic Topology: Introduction
THE ARF-KERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY: INTRODUCTION MICHAEL A. HILL, MICHAEL J. HOPKINS, AND DOUGLAS C. RAVENEL ABSTRACT. This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper in this volume, The Arf-Kervaire invariant problem in algebraic topology: Sketch of the proof. A rigorous account can be found in our preprint The non-existence of elements of Kervaire invariant one on the arXiv and on the third author’s home page. The latter also has numerous links to related papers and talks we have given on the subject since announcing our result in April, 2009. CONTENTS 1. Background and history 3 1.1. Pontryagin’s early work on homotopy groups of spheres 3 1.2. Our main result 8 1.3. The manifold formulation 8 1.4. The unstable formulation 12 1.5. Questions raised by our theorem 14 2. Our strategy 14 2.1. Ingredients of the proof 14 2.2. The spectrum Ω 15 2.3. How we construct Ω 15 3. Some classical algebraic topology. 15 3.1. Fibrations 15 3.2. Cofibrations 18 3.3. Eilenberg-Mac Lane spaces and cohomology operations 18 3.4. The Steenrod algebra. 19 3.5. Milnor’s formulation 20 3.6. Serre’s method of computing homotopy groups 21 3.7. The Adams spectral sequence 21 4. Spectra and equivariant spectra 23 4.1. -
Durham Research Online
Durham Research Online Deposited in DRO: 25 April 2019 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Cha, Jae Choon and Orr, Kent and Powell, Mark (2020) 'Whitney towers and abelian invariants of knots.', Mathematische Zeitschrift., 294 (1-2). pp. 519-553. Further information on publisher's website: https://doi.org/10.1007/s00209-019-02293-x Publisher's copyright statement: c The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk Mathematische Zeitschrift https://doi.org/10.1007/s00209-019-02293-x Mathematische Zeitschrift Whitney towers and abelian invariants of knots Jae Choon Cha1,2 · Kent E. -
A Topologist's View of Symmetric and Quadratic
1 A TOPOLOGIST'S VIEW OF SYMMETRIC AND QUADRATIC FORMS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/eaar Patterson 60++, G¨ottingen,27 July 2009 2 The mathematical ancestors of S.J.Patterson Augustus Edward Hough Love Eidgenössische Technische Hochschule Zürich G. H. (Godfrey Harold) Hardy University of Cambridge Mary Lucy Cartwright University of Oxford (1930) Walter Kurt Hayman Alan Frank Beardon Samuel James Patterson University of Cambridge (1975) 3 The 35 students and 11 grandstudents of S.J.Patterson Schubert, Volcker (Vlotho) Do Stünkel, Matthias (Göttingen) Di Möhring, Leonhard (Hannover) Di,Do Bruns, Hans-Jürgen (Oldenburg?) Di Bauer, Friedrich Wolfgang (Frankfurt) Di,Do Hopf, Christof () Di Cromm, Oliver ( ) Di Klose, Joachim (Bonn) Do Talom, Fossi (Montreal) Do Kellner, Berndt (Göttingen) Di Martial Hille (St. Andrews) Do Matthews, Charles (Cambridge) Do (JWS Casels) Stratmann, Bernd O. (St. Andrews) Di,Do Falk, Kurt (Maynooth ) Di Kern, Thomas () M.Sc. (USA) Mirgel, Christa (Frankfurt?) Di Thirase, Jan (Göttingen) Di,Do Autenrieth, Michael (Hannover) Di, Do Karaschewski, Horst (Hamburg) Do Wellhausen, Gunther (Hannover) Di,Do Giovannopolous, Fotios (Göttingen) Do (ongoing) S.J.Patterson Mandouvalos, Nikolaos (Thessaloniki) Do Thiel, Björn (Göttingen(?)) Di,Do Louvel, Benoit (Lausanne) Di (Rennes), Do Wright, David (Oklahoma State) Do (B. Mazur) Widera, Manuela (Hannover) Di Krämer, Stefan (Göttingen) Di (Burmann) Hill, Richard (UC London) Do Monnerjahn, Thomas ( ) St.Ex. (Kriete) Propach, Ralf ( ) Di Beyerstedt, Bernd -
Remarks on the Green-Schwarz Terms of Six-Dimensional Supergravity
Remarks on the Green-Schwarz terms of six-dimensional supergravity theories Samuel Monnier1, Gregory W. Moore2 1 Section de Mathématiques, Université de Genève 2-4 rue du Lièvre, 1211 Genève 4, Switzerland [email protected] 2NHETC and Department of Physics and Astronomy Rutgers University Piscataway, NJ 08855, USA [email protected] Abstract We construct the Green-Schwarz terms of six-dimensional supergravity theories on space- times with non-trivial topology and gauge bundle. We prove the cancellation of all global gauge and gravitational anomalies for theories with gauge groups given by products of Upnq, SUpnq and Sppnq factors, as well as for E8. For other gauge groups, anomaly cancellation is equivalent to the triviality of a certain 7-dimensional spin topological field theory. We show in the case of a finite Abelian gauge group that there are residual global anomalies imposing constraints on the 6d supergravity. These constraints are compatible with the known F-theory models. Interest- ingly, our construction requires that the gravitational anomaly coefficient of the 6d supergravity theory is a characteristic element of the lattice of string charges, a fact true in six-dimensional F-theory compactifications but that until now was lacking a low-energy explanation. We also arXiv:1808.01334v2 [hep-th] 25 Nov 2018 discover a new anomaly coefficient associated with a torsion characteristic class in theories with a disconnected gauge group. 1 Contents 1 Introduction and summary 3 2 Preliminaries 8 2.1 Six-dimensional supergravity theories . ........... 8 2.2 Some facts about anomalies . 12 2.3 Anomalies of six-dimensional supergravities . -
Higher-Order Intersections in Low-Dimensional Topology
Higher-order intersections in SPECIAL FEATURE low-dimensional topology Jim Conanta, Rob Schneidermanb, and Peter Teichnerc,d,1 aDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996-1300; bDepartment of Mathematics and Computer Science, Lehman College, City University of New York, 250 Bedford Park Boulevard West, Bronx, NY 10468; cDepartment of Mathematics, University of California, Berkeley, CA 94720-3840; and dMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved February 24, 2011 (received for review December 22, 2010) We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants W of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with – previously known link invariants such as Milnor, Sato Levine, and Fig. 1. (Left) A canceling pair of transverse intersections between two local Arf invariants. We also define higher-order Sato–Levine and Arf sheets of surfaces in a 3-dimensional slice of 4-space. The horizontal sheet invariants and show that these invariants detect the obstructions appears entirely in the “present,” and the red sheet appears as an arc that to framing a twisted Whitney tower. Together with Milnor invar- is assumed to extend into the “past” and the “future.” (Center) A Whitney iants, these higher-order invariants are shown to classify the exis- disk W pairing the intersections. (Right) A Whitney move guided by W tence of (twisted) Whitney towers of increasing order in the 4-ball. -
Brown-Kervaire Invariants
Brown-Kervaire Invariants Dissertation zur Erlangung des Grades ”Doktor der Naturwissenschaften” am Fachbereich Mathematik der Johannes Gutenberg-Universit¨at in Mainz Stephan Klaus geb. am 5.10.1963 in Saarbr¨ucken Mainz, Februar 1995 Contents Introduction 3 Conventions 8 1 Brown-Kervaire Invariants 9 1.1 The Construction ................................. 9 1.2 Some Properties .................................. 14 1.3 The Product Formula of Brown ......................... 16 1.4 Connection to the Adams Spectral Sequence .................. 23 Part I: Brown-Kervaire Invariants of Spin-Manifolds 27 2 Spin-Manifolds 28 2.1 Spin-Bordism ................................... 28 2.2 Brown-Kervaire Invariants and Generalized Kervaire Invariants ........ 35 2.3 Brown-Peterson-Kervaire Invariants ....................... 37 2.4 Generalized Wu Classes ............................. 40 3 The Ochanine k-Invariant 43 3.1 The Ochanine Signature Theorem and k-Invariant ............... 43 3.2 The Ochanine Elliptic Genus β ......................... 45 3.3 Analytic Interpretation .............................. 48 4 HP 2-BundlesandIntegralEllipticHomology 50 4.1 HP 2-Bundles ................................... 50 4.2 Integral Elliptic Homology ............................ 52 4.3 Secondary Operations and HP 2-Bundles .................... 54 5 AHomotopyTheoreticalProductFormula 56 5.1 Secondary Operations .............................. 56 5.2 A Product Formula ................................ 58 5.3 Application to HP 2-Bundles ........................... 62 6 -
Contributions to the History of Number Theory in the 20Th Century Author
Peter Roquette, Oberwolfach, March 2006 Peter Roquette Contributions to the History of Number Theory in the 20th Century Author: Peter Roquette Ruprecht-Karls-Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: [email protected] 2010 Mathematics Subject Classification (primary; secondary): 01-02, 03-03, 11-03, 12-03 , 16-03, 20-03; 01A60, 01A70, 01A75, 11E04, 11E88, 11R18 11R37, 11U10 ISBN 978-3-03719-113-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2013 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 To my friend Günther Frei who introduced me to and kindled my interest in the history of number theory Preface This volume contains my articles on the history of number theory except those which are already included in my “Collected Papers”.