Higher-Dimensional Knots According to Michel Kervaire

146 Pages·2017·1.17 MB·English

by  Françoise Michel

Series of Lectures in Mathematics
Françoise Michel F
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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
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This book is written to provide graduate students with the basic concepts -
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necessary to read texts in higher-dimensional knot theory and its relations i to Michel Kervaire
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with singularities. The first chapters are devoted to a presentation of e
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Pontrjagin’s construction, surgery and the work of Kervaire and Milnor on s
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homotopy spheres. We pursue with Kervaire’s fundamental work on the group n
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of a knot, knot modules and knot cobordism. We add developments due to l

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Levine. Tools (like open books, handlebodies, plumbings, …) often used but
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www.ems-ph.org
Michel/Weber | Rotis Sans | Pantone 287, Pantone 116 | 170 x 240 mm | RB: 7.2 mm

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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.
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of the copyright owner must be obtained.
© 2017 European Mathematical Society
Contact address:
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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