On the Non-Existence of Elements of Kervaire Invariant
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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. -
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. -
Contemporary Mathematics 220
CONTEMPORARY MATHEMATICS 220 Homotopy Theory via Algebraic Geometry and Group Representations Proceedings of a Conference on Homotopy Theory March 23-27, 1997 Northwestern University Mark Mahowald Stewart Priddy Editors http://dx.doi.org/10.1090/conm/220 Selected Titles in This Series 220 Mark Mahowald and Stewart Priddy, Editors, Homotopy theory via algebraic geometry and group representations, 1998 219 Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov, Editors, Secondary calculus and cohomological physics, 1998 218 Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, Editors, Domain decomposition methods 10, 1998 217 Eric Carlen, Evans M. Harrell, and Michael Loss, Editors, Advances in differential equations and mathematical physics, 1998 216 Akram Aldroubi and EnBing Lin, Editors, Wavelets, multiwavelets, and their applications, 1998 215 M. G. Nerurkar, D.P. Dokken, and D. B. Ellis, Editors, Topological dynamics and applications, 1998 214 Lewis A. Coburn and Marc A. Rieffel, Editors, Perspectives on quantization, 1998 213 Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter, Editors, Studies on composition operators, 1998 212 E. Ramirez de Arellano, N. Salinas, M. V. Shapiro, and N. L. Vasilevski, Editors, Operator theory for complex and hypercomplex analysis, 1998 211 J6zef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers Colloquium, 1997 210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1998 209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential equations, 1997 208 MichelL. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors, Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997 207 Yujiro Kawamata and Vyacheslav V. -
EXOTIC SPHERES and CURVATURE 1. Introduction Exotic
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 45, Number 4, October 2008, Pages 595–616 S 0273-0979(08)01213-5 Article electronically published on July 1, 2008 EXOTIC SPHERES AND CURVATURE M. JOACHIM AND D. J. WRAITH Abstract. Since their discovery by Milnor in 1956, exotic spheres have pro- vided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres. 1. Introduction Exotic spheres are manifolds which are homeomorphic but not diffeomorphic to a standard sphere. In this introduction our aims are twofold: First, to give a brief account of the discovery of exotic spheres and to make some general remarks about the structure of these objects as smooth manifolds. Second, to outline the basics of curvature for Riemannian manifolds which we will need later on. In subsequent sections, we will explore the interaction between topology and geometry for exotic spheres. We will use the term differentiable to mean differentiable of class C∞,and all diffeomorphisms will be assumed to be smooth. As every graduate student knows, a smooth manifold is a topological manifold that is equipped with a smooth (differentiable) structure, that is, a smooth maximal atlas. Recall that an atlas is a collection of charts (homeomorphisms from open neighbourhoods in the manifold onto open subsets of some Euclidean space), the domains of which cover the manifold. Where the chart domains overlap, we impose a smooth compatibility condition for the charts [doC, chapter 0] if we wish our manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps. -
Browder's Theorem and Manifolds with Corners
Browder’s theorem and manifolds with corners Duality and the Wu classes. The “Spanier-Whitehead dual” of a spectrum X is the function spectrum (in which S denotes the sphere spectrum) DX = F (X, S) (which exists by Brown representability). It comes equipped with an “evaluation” pairing DX ∧ X → S. Write H∗(−) for homology with F2 coefficients. If X is finite, then the induced pairing H−∗(DX) ⊗ H∗(X) → F2 is perfect, so there is an isomorphism Hi(X) → Hom(H−i(DX), F2) which may be rewritten using the universal coefficient theorem as i −i Hom(H (X), F2) → H (DX) A Steenrod operation θ induces a contragredient action on the left, which coincides with the action of χθ on the right. Here χ is the Hopf conjugation on the Steenrod algebra. The map χ is an algebra anti-automorphism and an involution, and is characterized on the total Steenod square by the identity of operators χSq = Sq−1 because of the form of the Milnor diagonal. ∞ If M is a closed smooth m manifold and X = Σ M+, then the Thom spectrum M ν of the stable normal bundle (normalized to have formal dimension −m) furnishes the Spanier-Whitehead dual of X. This is “Milnor-Spanier” or “Atiyah” duality. Poincar´eduality is given by the composite isomorphism m−i −∪U −i ν =∼ H (M) −→ H (M ) ←− Hi(M) where U ∈ H−m(M ν) is the Thom class. 0 The rather boring collapse map M+ → S dualizes to a much more interesting map ι : S0 → M ν, which in cohomology induces the map ι∗ : x ∪ U 7→ hx, [M]i k By Poincar´eduality, for each k there is a unique class vk ∈ H (M) m−k k such that for any x ∈ H (M), hSq x, [M]i = hxvk, [M]i. -
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. -
On the Work of Michel Kervaire in Surgery and Knot Theory
1 ON MICHEL KERVAIRE'S WORK IN SURGERY AND KNOT THEORY Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/eaar/slides/kervaire.pdf Geneva, 11th and 12th February, 2009 2 1927 { 2007 3 Highlights I Major contributions to the topology of manifolds of dimension > 5. I Main theme: connection between stable trivializations of vector bundles and quadratic refinements of symmetric forms. `Division by 2'. I 1956 Curvatura integra of an m-dimensional framed manifold = Kervaire semicharacteristic + Hopf invariant. I 1960 The Kervaire invariant of a (4k + 2)-dimensional framed manifold. I 1960 The 10-dimensional Kervaire manifold without differentiable structure. I 1963 The Kervaire-Milnor classification of exotic spheres in dimensions > 4 : the birth of surgery theory. n n+2 I 1965 The foundation of high dimensional knot theory, for S S ⊂ with n > 2. 4 MATHEMATICAL REVIEWS + 1 Kervaire was the author of 66 papers listed (1954 { 2007) +1 unlisted : Non-parallelizability of the n-sphere for n > 7, Proc. Nat. Acad. Sci. 44, 280{283 (1958) 619 matches for "Kervaire" anywhere, of which 84 in title. 18,600 Google hits for "Kervaire". MR0102809 (21() #1595) Kervaire,, Michel A. An interpretation of G. Whitehead's generalizationg of H. Hopf's invariant. Ann. of Math. (2)()69 1959 345--365. (Reviewer: E. H. Brown) 55.00 MR0102806 (21() #1592) Kervaire,, Michel A. On the Pontryagin classes of certain ${\rm SO}(n)$-bundles over manifolds. Amer. J. Math. 80 1958 632--638. (Reviewer: W. S. Massey) 55.00 MR0094828 (20 #1337) Kervaire, Michel A. Sur les formules d'intégration de l'analyse vectorielle. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
Journal of Scientometric Research
Vol 2 | Issue 1 | Jan-April 2013 ISSN :2321-6654 www.jscires.org Journal of Scientometric Research An Official PublicationAn Official of SCIBIOLMED.ORG Publication of Pharmacognosy Network Worldwide [Phcog.Net] www.jscires.org CONTENTS Review Article Text mining for science and technology: A re‑ view – Part II‑citation and discovery Research Articles Dark energy: A scientometric mapping of publications On the growth behavior of yearly citations of cumula‑ tive papers of individual authors Cahit Arf: Exploring his scientific influence using social network analysis, author co‑citation maps and single publication h index many more... Research in Progress Understanding trends and changes in media coverage of nanotechnology in India Perspective Paper Analysis of Iranian and British university websites by world wide web consortium Scientific Correspondence r‑index: Quantifying the quality of an individual’s scientific research output www.jscires.org Editor-in-Chief : Dr. Sujit Bhattacharya Published by [email protected] New Delhi, INDIA www.scibiolmed.org Managing Editor : Dr. Ahmed, M, [email protected] Bangalore, INDIA JSCIRES RESEARCH ARTICLE Cahit Arf: Exploring his scientific influence using social network analysis, author co‑citation maps and single publication h index1 Yaşar Tonta*, A. Esra Özkan Çelik1 Department of Information Management, Faculty of Letters, Hacettepe University, 06800 Beytepe, Ankara, Turkey, 1Registrar’s Office, Hacettepe University, 06800 Beytepe, Ankara, Turkey 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. -
Milnor, John W. Groups of Homotopy Spheres
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 699–710 http://dx.doi.org/10.1090/bull/1506 Article electronically published on June 12, 2015 SELECTED MATHEMATICAL REVIEWS related to the paper in the previous section by JOHN MILNOR MR0148075 (26 #5584) 57.10 Kervaire, Michel A.; Milnor, John W. Groups of homotopy spheres. I. Annals of Mathematics. Second Series 77 (1963), 504–537. The authors aim to study the set of h-cobordism classes of smooth homotopy n-spheres; they call this set Θn. They remark that for n =3 , 4thesetΘn can also be described as the set of diffeomorphism classes of differentiable structures on Sn; but this observation rests on the “higher-dimensional Poincar´e conjecture” plus work of Smale [Amer. J. Math. 84 (1962), 387–399], and it does not really form part of the logical structure of the paper. The authors show (Theorem 1.1) that Θn is an abelian group under the connected sum operation. (In § 2, the authors give a careful treatment of the connected sum and of the lemmas necessary to prove Theorem 1.1.) The main task of the present paper, Part I, is to set up methods for use in Part II, and to prove that for n = 3 the group Θn is finite (Theorem 1.2). (For n =3the authors’ methods break down; but the Poincar´e conjecture for n =3wouldimply that Θ3 = 0.) We are promised more detailed information about the groups Θn in Part II. The authors’ method depends on introducing a subgroup bPn+1 ⊂ Θn;asmooth homotopy n-sphere qualifies for bPn+1 if it is the boundary of a parallelizable man- ifold. -
Fighting for Tenure the Jenny Harrison Case Opens Pandora's
Calendar of AMS Meetings and Conferences This calendar lists all meetings and conferences approved prior to the date this issue insofar as is possible. Instructions for submission of abstracts can be found in the went to press. The summer and annual meetings are joint meetings with the Mathe January 1993 issue of the Notices on page 46. Abstracts of papers to be presented at matical Association of America. the meeting must be received at the headquarters of the Society in Providence, Rhode Abstracts of papers presented at a meeting of the Society are published in the Island, on or before the deadline given below for the meeting. Note that the deadline for journal Abstracts of papers presented to the American Mathematical Society in the abstracts for consideration for presentation at special sessions is usually three weeks issue corresponding to that of the Notices which contains the program of the meeting, earlier than that specified below. Meetings Abstract Program Meeting# Date Place Deadline Issue 890 t March 18-19, 1994 Lexington, Kentucky Expired March 891 t March 25-26, 1994 Manhattan, Kansas Expired March 892 t April8-10, 1994 Brooklyn, New York Expired April 893 t June 16-18, 1994 Eugene, Oregon April4 May-June 894 • August 15-17, 1994 (96th Summer Meeting) Minneapolis, Minnesota May 17 July-August 895 • October 28-29, 1994 Stillwater, Oklahoma August 3 October 896 • November 11-13, 1994 Richmond, Virginia August 3 October 897 • January 4-7, 1995 (101st Annual Meeting) San Francisco, California October 1 December March 4-5, 1995 -
REMEMBERING MARK MAHOWALD 1931–2013 Mark Mahowald Was a Dominant Figure in Algebraic Topology for Decades. After Being Trained
REMEMBERING MARK MAHOWALD 1931–2013 DOUGLAS C. RAVENEL WITH CONTRIBUTIONS BY MARTIN C. TANGORA, STEWART B. PRIDDY, DONALD M. DAVIS, MARK J. BEHRENS AND WEN-HSIUNG LIN UNABRIDGED VERSION OF A SIMILAR ARTICLE THAT APPEARED IN THE JULY 2016 ISSUE OF THE NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY, PHOTOS NOT INCLUDED Mark Mahowald was a dominant figure in algebraic topology for decades. After being trained as an analyst he became a self trained homotopy theorist in the early 1960s. His 1967 AMS Memoir The metastable homotopy of S n, with its dozens of charts packed with arcane information, established him as a master of computations related to the homotopy groups of spheres. It came to be known in the field simply as “the red book.” It was the unanimous opinion of experts that he knew this subject far better than anyone else. His insight and intuition were legendary. Countless coworkers benefited from his ideas and advice. He was born on December 1, 1931, in Albany, Minnesota, a very small and mostly German Catholic community. He was one of seven children. His father was a family physician who ran his practice out of his home. There was an enyclopedia in the house which Mark would consult whenever he was curious about something. He was largely self taught. He also spent a lot of time on the nearby farm that his father owned. Among his six siblings, two would become academics, one a medical doctor and one a Jesuit priest. Mark first attended college at Carnegie Tech (now Carnegie Mellon University) in Pitts- burgh and planned to major in chemical engineering.