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Knots and Primes: An Introduction to Arithmetic Topology PDF

204 Pages·2012·1.32 MB·English
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Universitext Universitext SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA VincenzoCapasso UniversitàdegliStudidiMilano,Milan,Italy CarlesCasacuberta UniversitatdeBarcelona,Barcelona,Spain AngusJ.MacIntyre QueenMary,UniversityofLondon,London,UK KennethRibet UniversityofCalifornia,Berkeley,Berkeley,CA,USA ClaudeSabbah CNRS,ÉcolePolytechnique,Palaiscau,France EndreSüli UniversityofOxford,Oxford,UK WojborA.Woyczynski CaseWesternReserveUniversity,Cleveland,OH,USA Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolutionofteachingcurricula,intoverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Forfurthervolumes: www.springer.com/series/223 Masanori Morishita Knots and Primes An Introduction to Arithmetic Topology MasanoriMorishita GraduateSchoolofMathematics KyushuUniversity Fukuoka819-0395 Japan [email protected] TheEnglishlanguageeditionisbasedontheJapaneseoriginaledition: MusubimetoSosubyMasanoriMorishita Copyright©SpringerJapan2009 AllRightsReserved ISSN0172-5939 e-ISSN2191-6675 Universitext ISBN978-1-4471-2157-2 e-ISBN978-1-4471-2158-9 DOI10.1007/978-1-4471-2158-9 SpringerLondonDordrechtHeidelbergNewYork BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressControlNumber:2011940954 MathematicsSubjectClassification: 11Rxx,11Sxx,57Mxx ©Springer-VerlagLondonLimited2012 Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,asper- mittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish- ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissuedbythe CopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesentto thepublishers. Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree forgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions thatmaybemade. Coverdesign:VTeXUAB,Lithuania Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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

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This is a foundation for arithmetic topology - a new branch of mathematics which is focused upon the analogy between knot theory and number theory. Starting with an informative introduction to its origins, namely Gauss, this text provides a background on knots, three manifolds and number fields. Com
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