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Higher-Dimensional Knots According to Michel Kervaire Higher-Dimensional Techniques in This Fascinating Area

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Higher-Dimensional Knots According to Michel Kervaire Higher-Dimensional Techniques in This Fascinating Area Series of Lectures in Mathematics Françoise Michel Michel and Claude Weber Françoise Claude Weber Françoise Michel Higher-Dimensional Knots Claude Weber According to Michel Kervaire Michel Kervaire wrote six papers on higher-dimensional knots which can be considered fundamental to the development of the theory. They are not only of historical interest but naturally introduce to some of the essential to Michel Kervaire According Knots Higher-Dimensional Higher-Dimensional techniques in this fascinating area. This book is written to provide graduate students with the basic concepts Knots According necessary to read texts in higher-dimensional knot theory and its relations with singularities. The first chapters are devoted to a presentation of Pontrjagin’s construction, surgery and the work of Kervaire and Milnor on to Michel Kervaire homotopy spheres. We pursue with Kervaire’s fundamental work on the group of a knot, knot modules and knot cobordism. We add developments due to Levine. Tools (like open books, handlebodies, plumbings, …) often used but hard to find in original articles are presented in appendices. We conclude with a description of the Kervaire invariant and the consequences of the Hill– Hopkins–Ravenel results in knot theory. ISBN 978-3-03719-180-4 www.ems-ph.org Michel/Weber | Rotis Sans | Pantone 287, Pantone 116 | 170 x 240 mm | RB: 7.2 mm EMS Series of Lectures in Mathematics Edited by Ari Laptev (Imperial College, London, UK) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series (for a complete listing see our homepage at www.ems-ph.org): Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration Koen Thas, A Course on Elation Quadrangles Benoît Grébert and Thomas Kappeler, The Defocusing NLS Equation and Its Normal Form Armen Sergeev, Lectures on Universal Teichmüller Space Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, 3-Manifold Groups Hans Triebel, Tempered Homogeneous Function Spaces Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink and Jürgen Sander, Four Faces of Number Theory Alberto Cavicchioli, Friedrich Hegenbarth and Dušan Repovš, Higher-Dimensional Generalized Manifolds: Surgery and Constructions Davide Barilari, Ugo Boscain and Mario Sigalotti, Geometry, Analysis and Dynamics on sub- Riemannian Manifolds, Volume I Davide Barilari, Ugo Boscain and Mario Sigalotti, Geometry, Analysis and Dynamics on sub- Riemannian Manifolds, Volume II Dynamics Done with Your Bare Hands, Françoise Dal’Bo, François Ledrappier and Amie Wilkinson, (Eds.) Hans Triebel, PDE Models for Chemotaxis and Hydrodynamics in Supercritical Function Spaces Françoise Michel Claude Weber Higher-Dimensional Knots According to Michel Kervaire Authors: Françoise Michel Claude Weber Institut de Mathématiques de Toulouse Section de Mathématiques Université Paul Sabatier Université de Genève 118 route de Narbonne 2-4 rue de Lièvre, C.P. 64 31062 Toulouse Cedex 9 1211 Genève 4 France Switzerland E-mail: [email protected] E-mail: [email protected] 2010 Mathematics Subject Classification: 57Q45, 57R65, 32S55 Key words: Knots in high dimensions, homotopy spheres, complex singularities ISBN 978-3-03719-180-4 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, broadcasting, 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. © 2017 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich Switzerland Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: Alison Durham, Manchester, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Preface The aim of this book is to present Kervaire’s work on differentiable knots in higher dimensions in codimension q = 2. As explained in our introduction (see Section 1.7), this book is written to make the reading of papers by Michel Kervaire and Jerome Levine easier. Thus we hope to communicate our enthusiasm for higher-dimensional knot theory. In order to appreciate the importance of Kervaire’s contribution, we describe in Chapters2 to4 what was, at the time, the situation in differential topology and in knot theory in codimension q ≥ 3. In Chapter5, we expose Kervaire’s characterization of the fundamental group of a knot complement. In Chapter6, we explain Kervaire and Levine’s work on knot modules. In Chapter7, we detail Kervaire’s construction of the “simple knots” classified by Jerome Levine. Chapter8 summarizes Kervaire and Levine’s results on knot cobordism. In Chapter9, we apply higher-dimensional knot theory to singularities of complex hypersurfaces. AppendixesA toD are devoted to a discussion of some basic concepts, known to the experts: signs, Seifert hypersurfaces, open book decompositions and handlebodies. In AppendixE, we conclude with an exposition of the results of Hill–Hopkins– Ravenel on the Kervaire invariant problem and its consequences for knot theory in codimension 2. The point of view adopted here is somewhat pseudo-historical. When we make explicit Kervaire’s work we try to follow him closely, in order to retain some of the flavor of the original texts. When necessary we add further contributions, often due to Levine. We also propose developments that occurred later. We thank Peter Landweber for patiently correcting our English. The figures in AppendixF are due to Cam Van Quach. We thank her for her friendly collaboration. Françoise Michel Claude Weber Geneva, September 2016 Contents Preface...................................... v 1 Introduction.................................. 1 1.1 Kervaire’s six papers on knot theory.................1 1.2 A brief description of the content of the book............2 1.3 What is a knot?............................3 1.4 Knots in the early 1960s.......................4 1.5 Higher-dimensional knots and homotopy spheres..........4 1.6 Links and singularities........................5 1.7 Final remarks.............................6 1.8 Conventions and notation.......................7 2 Some tools of differential topology...................... 9 2.1 Surgery from an elementary point of view..............9 2.2 Vector bundles and parallelizability................. 10 2.3 The Pontrjagin method and the J-homomorphism.......... 11 3 The Kervaire–Milnor study of homotopy spheres.............. 15 3.1 Homotopy spheres.......................... 15 3.2 The groups Θn and bPn+1 ....................... 15 3.3 The Kervaire invariant........................ 20 3.4 The groups Pn+1 ........................... 22 3.5 The Kervaire manifold........................ 24 4 Differentiable knots in codimension ≥3................... 27 4.1 On the isotopy of knots and links in any codimension........ 27 4.2 Embeddings and isotopies in the stable and metastable ranges.... 28 4.3 Below the metastable range...................... 29 5 The fundamental group of a knot complement................ 33 5.1 Homotopy n-spheres embedded in Sn+2 ............... 33 5.2 Necessary conditions to be the fundamental group of a knot complement 35 5.3 Sufficiency of the conditions if n ≥ 3 ................. 37 5.4 The Kervaire conjecture........................ 40 5.5 Groups that satisfy the Kervaire conditions.............. 41 viii Contents 6 Knot modules................................. 43 6.1 The knot exterior........................... 43 6.2 Some algebraic properties of knot modules.............. 45 6.3 The qth knot module when q < n/2 ................. 48 6.4 Seifert hypersurfaces......................... 50 6.5 Odd-dimensional knots and the
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