Series of Lectures in Mathematics Françoise Michel F r a n Claude Weber ç o is e M ich Françoise Michel e Higher-Dimensional Knots l a n Claude Weber d C According to Michel Kervaire la u d e W e b e r Michel Kervaire wrote six papers on higher-dimensional knots which can Higher-Dimensional be considered fundamental to the development of the theory. They are not only of historical interest but naturally introduce to some of the essential H techniques in this fascinating area. ig Knots According h e r This book is written to provide graduate students with the basic concepts - D necessary to read texts in higher-dimensional knot theory and its relations i to Michel Kervaire m with singularities. The first chapters are devoted to a presentation of e n Pontrjagin’s construction, surgery and the work of Kervaire and Milnor on s i o homotopy spheres. We pursue with Kervaire’s fundamental work on the group n a of a knot, knot modules and knot cobordism. We add developments due to l K Levine. Tools (like open books, handlebodies, plumbings, …) often used but n hard to find in original articles are presented in appendices. We conclude o t s with a description of the Kervaire invariant and the consequences of the Hill– A Hopkins–Ravenel results in knot theory. c c o r d i n g t o M i c h e l K e r v a ISBN 978-3-03719-180-4 ir e 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 TheaimofthisbookistopresentKervaire’sworkondifferentiableknotsinhigher dimensionsincodimensionq =2. Asexplainedinourintroduction(seeSection1.7), this book is written to make the reading of papers by Michel Kervaire and Jerome Levineeasier. Thuswehopetocommunicateourenthusiasmforhigher-dimensional knot theory. In order to appreciate the importance of Kervaire’s contribution, we describeinChapters2to4whatwas,atthetime,thesituationindifferentialtopology and in knot theory in codimension q ≥ 3. In Chapter 5, we expose Kervaire’s characterization of the fundamental group of a knot complement. In Chapter 6, we explain Kervaire and Levine’s work on knot modules. In Chapter 7, we detail Kervaire’sconstructionofthe“simpleknots”classifiedbyJeromeLevine. Chapter8 summarizes Kervaire and Levine’s results on knot cobordism. In Chapter 9, we apply higher-dimensional knot theory to singularities of complex hypersurfaces. AppendixesAtoDaredevotedtoadiscussionofsomebasicconcepts,knowntothe experts: signs, Seifert hypersurfaces, open book decompositions and handlebodies. In Appendix E, we conclude with an exposition of the results of Hill–Hopkins– Ravenel on the Kervaire invariant problem and its consequences for knot theory in codimension2. The point of view adopted here is somewhat pseudo-historical. When we make explicitKervaire’sworkwetrytofollowhimclosely, inordertoretainsomeofthe flavoroftheoriginaltexts. Whennecessaryweaddfurthercontributions,oftendue toLevine. Wealsoproposedevelopmentsthatoccurredlater. We thank Peter Landweber for patiently correcting our English. The figures in AppendixFareduetoCamVanQuach. Wethankherforherfriendlycollaboration. FrançoiseMichel ClaudeWeber Geneva,September2016 Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Kervaire’ssixpapersonknottheory . . . . . . . . . . . . . . . . . 1 1.2 Abriefdescriptionofthecontentofthebook . . . . . . . . . . . . 2 1.3 Whatisaknot? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Knotsintheearly1960s . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Higher-dimensionalknotsandhomotopyspheres . . . . . . . . . . 4 1.6 Linksandsingularities . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Finalremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.8 Conventionsandnotation . . . . . . . . . . . . . . . . . . . . . . . 7 2 Sometoolsofdifferentialtopology. . . . . . . . . . . . . . . . . . . . . . 9 2.1 Surgeryfromanelementarypointofview . . . . . . . . . . . . . . 9 2.2 Vectorbundlesandparallelizability . . . . . . . . . . . . . . . . . 10 2.3 ThePontrjaginmethodandthe J-homomorphism . . . . . . . . . . 11 3 TheKervaire–Milnorstudyofhomotopyspheres. . . . . . . . . . . . . . 15 3.1 Homotopyspheres . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 ThegroupsΘn andbPn+1 . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 TheKervaireinvariant . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 ThegroupsPn+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 TheKervairemanifold . . . . . . . . . . . . . . . . . . . . . . . . 24 4 Differentiableknotsincodimension≥3. . . . . . . . . . . . . . . . . . . 27 4.1 Ontheisotopyofknotsandlinksinanycodimension . . . . . . . . 27 4.2 Embeddingsandisotopiesinthestableandmetastableranges . . . . 28 4.3 Belowthemetastablerange . . . . . . . . . . . . . . . . . . . . . . 29 5 Thefundamentalgroupofaknotcomplement. . . . . . . . . . . . . . . . 33 5.1 Homotopyn-spheresembeddedinSn+2 . . . . . . . . . . . . . . . 33 5.2 Necessaryconditionstobethefundamentalgroupofaknotcomplement 35 5.3 Sufficiencyoftheconditionsifn ≥ 3 . . . . . . . . . . . . . . . . . 37 5.4 TheKervaireconjecture. . . . . . . . . . . . . . . . . . . . . . . . 40 5.5 GroupsthatsatisfytheKervaireconditions. . . . . . . . . . . . . . 41 viii Contents 6 Knotmodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.1 Theknotexterior . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Somealgebraicpropertiesofknotmodules. . . . . . . . . . . . . . 45 6.3 Theqthknotmodulewhenq < n/2 . . . . . . . . . . . . . . . . . 48 6.4 Seiferthypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.5 Odd-dimensionalknotsandtheSeifertform . . . . . . . . . . . . . 52 6.6 Even-dimensionalknotsandthetorsionSeifertform. . . . . . . . . 56 7 Odd-dimensionalsimplelinks. . . . . . . . . . . . . . . . . . . . . . . . 61 7.1 q-handlebodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.2 TherealizationtheoremforSeifertmatrices . . . . . . . . . . . . . 62 7.3 Levine’sclassificationofembeddingsofhandlebodiesincodimension1 65 7.4 Levine’sclassificationofsimpleodd-dimensionalknots . . . . . . . 67 8 Knotcobordism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.2 Theeven-dimensionalcase . . . . . . . . . . . . . . . . . . . . . . 69 8.3 Theodd-dimensionalcase . . . . . . . . . . . . . . . . . . . . . . 71 9 Singularitiesofcomplexhypersurfaces. . . . . . . . . . . . . . . . . . . 75 9.1 ThetheoryofMilnor . . . . . . . . . . . . . . . . . . . . . . . . . 75 9.2 AlgebraiclinksandSeifertforms . . . . . . . . . . . . . . . . . . . 78 9.3 Cobordismofalgebraiclinks . . . . . . . . . . . . . . . . . . . . . 79 9.4 ExamplesofSeifertmatricesassociatedtoalgebraiclinks . . . . . . 80 9.5 Mumfordandtopologicaltriviality . . . . . . . . . . . . . . . . . . 81 9.6 JoinsandPham–Brieskornsingularities . . . . . . . . . . . . . . . 83 9.7 ThepaperbyKauffman[56] . . . . . . . . . . . . . . . . . . . . . 86 9.8 ThepaperbyKauffmanandNeumann[57] . . . . . . . . . . . . . 87 9.9 Kauffman–Neumann’sconstructionwhenbothlinksarethebinding ofanopenbookdecomposition . . . . . . . . . . . . . . . . . . . . 88 A Linkingnumbersandsigns. . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.1 Theboundaryofanorientedmanifold . . . . . . . . . . . . . . . . 91 A.2 Linkingnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 B ExistenceofSeiferthypersurfaces. . . . . . . . . . . . . . . . . . . . . . 93 C Openbookdecompositions. . . . . . . . . . . . . . . . . . . . . . . . . . 95 C.1 Openbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 C.2 Browder’slemma2 . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Contents ix D Handlebodiesandplumbings. . . . . . . . . . . . . . . . . . . . . . . . . 99 D.1 Bouquetsofspheresandhandlebodies . . . . . . . . . . . . . . . . 99 D.2 Parallelizablehandlebodies . . . . . . . . . . . . . . . . . . . . . . 101 D.3 m-dimensionalsphericallinksinS2m+1 . . . . . . . . . . . . . . . 103 D.4 Plumbing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 E Homotopy spheres embedded in codimension 2 and the Kervaire–Arf–- Robertello–Levineinvariant. . . . . . . . . . . . . . . . . . . . . . . . . 109 E.1 Whichhomotopyspherescanbeembeddedincodimension2? . . . 109 E.2 TheKervaire–Arf–Robertello–Levineinvariant . . . . . . . . . . . 111 E.3 The Hill–Hopkins–Ravenel result and its influence on the KARL invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 F Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 IndexofNotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 IndexofQuotedTheorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 IndexofTerminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133