Higher-dimensional Knots According to Michel Kervaire

146 Pages·2017·0.979 MB·English

by  Francoise Michel, Claude Weber

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