Knots and Primes: An Introduction to Arithmetic Topology

204 Pages·2011·3.89 MB·English

by  Masanori Morishita

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Masanori Morishita
Knots and Primes
An Introduction to Arithmetic Topology
MasanoriMorishita
GraduateSchoolofMathematics
KyushuUniversity
Fukuoka819-0395
Japan
[email protected]
TheEnglishlanguageeditionisbasedontheJapaneseoriginaledition:
MusubimetoSosubyMasanoriMorishita
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To thememoryofmymother,
ChiekoMorishita

Preface
The theme of the present book is the analogy between knot theory and number
theory,basedonthehomotopicalanalogiesbetweenknotsandprimes,3-manifolds
and number rings. Thus, the purpose of this book is to discuss and present, in a
parallelandsystematicmanner,theanalogiesbetweenthefundamentalnotionsand
theoriesofknottheoryandnumbertheory.Forthesakeofreaders,basicmaterials
fromeachfieldarerecollectedinChap.2.
If we look back over the history of knot theory and number theory, an origin
ofthemoderndevelopmentofbothfieldsmaybefoundintheworkofC.F.Gauss
(1777–1855).Theaimofthisbookmayberephrasedasbridgingthetwowaysthat
branchedoutafterGaussandprovidingafoundationofarithmetictopology.
This volume is an English translation of my Japanese book [M12] with some
thingsadded.IthankProfessorY.MatsumotoforrecommendingthatIshouldwrite
a book on this subject. The contents of this book grew out of my intensive lec-
tures at some universities in Japan (Kyushu, Kyoto, Tohoku and Tokyo) on vari-
ousoccasionsduring2002–2007andattheUniversityofHeidelbergin thefall of
2008.ItakethisopportunitytoacknowledgemygratitudetoY.Taguchi,K.Kato,
A.Yukie,T.Yamazaki,T.OdaandD.Vogelforinvitingmetogivelecturesonarith-
metictopology,andIthankY.Terashimaforusefulcommunicationandjointwork
onChap.14.IamthankfultoH.Hida,M.Kaneko,M.Kato,M.Kurihara,Y.Mizu-
sawa and S. Ohtani for useful communication in the course of writing this book,
and to F. Amano, H. Nibo and Y. Takakura for pointing out some misprints in the
Japaneseversion.IalsothanktherefereesfortheirusefulcommentsandL.Stoney,
D.AkmanavicˇiusandM.Nakamurafortheirhelpwiththeproductionofthistext.
I would like to thank C. Deninger, M. Kapranov, T. Kohno, B. Mazur, J. Morava
and T. Ono for their encouragements and interests in my work. Finally, I express
myheartythankstoJ.HillmanandK.Murasugiforansweringpatientlymyques-
tionsonknot/linktheoryovertheyears,especiallytoJ.Hillmanforhisuseful(both
linguisticandmathematical)commentsonthemanuscriptofthisEnglishversion.
Fukuoka,Japan MasanoriMorishita
vii

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Two Ways that Branched out from C.F. Gauss—Quadratic
ResiduesandLinkingNumbers . . . . . . . . . . . . . . . . . . . 1
1.2 GeometrizationofNumberTheory . . . . . . . . . . . . . . . . . 4
1.3 TheOutlineofThisBook . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminaries—FundamentalGroupsandGaloisGroups. . . . . . . 9
2.1 TheCaseofTopologicalSpaces . . . . . . . . . . . . . . . . . . . 9
2.2 TheCaseofArithmeticRings . . . . . . . . . . . . . . . . . . . . 24
2.3 ClassFieldTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 KnotsandPrimes,3-ManifoldsandNumberRings . . . . . . . . . . 49
4 LinkingNumbersandLegendreSymbols . . . . . . . . . . . . . . . 55
4.1 LinkingNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 LegendreSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 DecompositionsofKnotsandPrimes . . . . . . . . . . . . . . . . . . 61
5.1 DecompositionofaKnot . . . . . . . . . . . . . . . . . . . . . . 61
5.2 DecompositionofaPrime . . . . . . . . . . . . . . . . . . . . . . 64
6 HomologyGroupsandIdealClassGroupsI—GenusTheory . . . . 69
6.1 HomologyGroupsandIdealClassGroups . . . . . . . . . . . . . 69
6.2 GenusTheoryforaLink . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 GenusTheoryforPrimeNumbers . . . . . . . . . . . . . . . . . . 73
7 LinkGroupsandGaloisGroupswithRestrictedRamification . . . . 77
7.1 LinkGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Pro-l GaloisGroupswithRestrictedRamification . . . . . . . . . 80
8 MilnorInvariantsandMultipleResidueSymbols . . . . . . . . . . . 85
8.1 FoxFreeDifferentialCalculus. . . . . . . . . . . . . . . . . . . . 85
8.2 MilnorInvariants. . . . . . . . . . . . . . . . . . . . . . . . . . . 93
ix