Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Takefumi Nosaka
Quandles and
Topological Pairs
Symmetry, Knots,
and Cohomology
123
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Takefumi Nosaka
Quandles and Topological
Pairs
Symmetry, Knots, and Cohomology
123
Takefumi Nosaka
Department ofMathematics
Tokyo Institute of Technology
Tokyo
Japan
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Preface
Thisbooksurveysquandletheory,startingfrombasicmotivationandpassingonto
introduce recent developments and the results of my own research. Topological
applications and related topics are presented throughout. While the topological
aspectsarethefocus,Ialsowouldliketorepresenthowquandletheoryisevaluated
in mathematics. Thus, this book is a crash course in quandle theory.
The first four chapters contain what I consider to be the basics of quandles and
aim to express topological objects in terms of quandles. The remaining chapters
focusonstudyingtheobjectsindetailfromtheviewpointsofhomotopytheoryand
group cohomology. These chapters reflect my own research interests and applica-
tions. For the most part, as prerequisites for reading this book, the reader needs
elements of algebra (groups, rings, modules, and homological algebra) and of
algebraic topology (C1-manifold, fundamental group, covering spaces,
CW-complexes, and (co)-homology). However, concerning low-dimensional
topology, I give a list of elementary notation and facts in Appendix A without
proof; the reader can simply take these facts on faith (for the details, see the
references at the end of this book).
Igiveanumberofexercises,someofwhichimplyarbitraryabbreviationstocut
alongproofshort.SoIonlycitedreferencesfortheanswers.Ithinkthatthedetails
are not so essential in this book, or the details involve tedious computations.
In closing, I sincerely express my thanks to Toshiyuki Akita, J. Scott Carter,
Katsumi Ishikawa, Ye Liu, Hirofumi Niibo, Józef H. Przytycki, Masahico Saito,
SumireSawada,MasayoshiTanno,andSeungYeopYangforcarefulreadingearlier
drafts of this book and for giving me a number of their detailed comments. I also
expressmyappreciationtotherefereesforhisorherusefulcomments.Furthermore,
I gratefully thank Prof. Takeo Ohsawa with whom I have had the opportunity of
workingonthispublication.Moreover,thisbookgrewoutofnotesIwrotefortwo
intensive courses at The University of Tokyo, and the Hokkaido University. I also
thank theattendingstudents and Profs. NariyaKawazumi andTakuya Sakasai.
Tokyo, Japan Takefumi Nosaka
July 2017
v
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Basics of Quandles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Definitions and Examples of Quandles. . . . . . . . . . . . . . . . . . . . . 5
2.2 Characterization Theorem of Quandles from Groups. . . . . . . . . . . 9
2.3 Quandles and Centrally Extended Groups . . . . . . . . . . . . . . . . . . 12
2.4 Link Quandles and Their Properties. . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Appendix: Historical and Topological Comments on Quandles . . . 16
3 X-Colorings of Links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Characterization of X-colorings . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Example 1; Parabolic SL -Representations. . . . . . . . . . . . . 25
2
3.2.2 Example 2; Core Quandle and Double Branched
Covering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.3 Example 3; First Cohomology with Local Coefficients. . . . 28
3.3 Appendix: Colorings and Braid Actions. . . . . . . . . . . . . . . . . . . . 30
4 Some of Quandle Cocycle Invariants of Links . . . . . . . . . . . . . . . . . 33
4.1 Origin: The Homotopy Invariant from the Rack Space . . . . . . . . . 33
4.2 Shadow Cocycle Invariant Associated with X-sets . . . . . . . . . . . . 36
4.3 Cocycle Invariant with Non-Abelian Coefficients . . . . . . . . . . . . . 39
4.4 Binary and Trinary Cocycle Invariants. . . . . . . . . . . . . . . . . . . . . 41
5 Topology of the Rack Space and the 2-Cocycle Invariant. . . . . . . . . 45
5.1 Rack and Quandle Homology with Properties . . . . . . . . . . . . . . . 45
5.2 The Rack Space and its Properties. . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Topology of the Non-Abelian Cocycle Invariants . . . . . . . . . . . . . 52
5.4 Examples; Second Quandle Homology and 2-Cocycles. . . . . . . . . 55
vii
viii Contents
6 Topology on the Quandle Homotopy Invariant. . . . . . . . . . . . . . . . . 59
6.1 Monoid Structure on the Universal Covering . . . . . . . . . . . . . . . . 59
6.2 On the Classifying Map of the Rack Space . . . . . . . . . . . . . . . . . 61
6.3 Rack Spaces of the Knot Quandles . . . . . . . . . . . . . . . . . . . . . . . 64
6.4 Topological Interpretation of the Homotopy Invariant. . . . . . . . . . 65
6.5 Application; Computation of Third Quandle Homology . . . . . . . . 68
7 Relative Group Homology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.1 Review; Group Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.2 Relative Group Homology as a Mapping Cone . . . . . . . . . . . . . . 74
7.3 Hochschild Relative Group Homology. . . . . . . . . . . . . . . . . . . . . 76
7.4 Examples of (Relative) Group (Co)cycles . . . . . . . . . . . . . . . . . . 79
7.4.1 Example 1; Group 3-Cocycles of Cyclic Groups . . . . . . . . 79
7.4.2 Example 2; the Chern-Simons 3-Class . . . . . . . . . . . . . . . 80
7.4.3 Example 3; Presented Groups and Knot Groups . . . . . . . . 81
8 Inoue–Kabaya Chain Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.1 Original Definition of Inoue–Kabaya Chain Map . . . . . . . . . . . . . 83
8.2 Guideline to Applications and Fundamental Classes . . . . . . . . . . . 85
8.3 Relation to the Chern-Simons Invariants of Hyperbolic
Links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.4 Lifts of Inoue–Kabaya Chain Map. . . . . . . . . . . . . . . . . . . . . . . . 89
8.4.1 For Some Quandle Operations on Groups. . . . . . . . . . . . . 89
8.4.2 Recovery of All Mochizuki’s 3-Cocycles . . . . . . . . . . . . . 92
8.4.3 Quandle Cocycles from Invariant Theory . . . . . . . . . . . . . 94
8.4.4 Binary Cocycle Invariants Versus Cohomology
Pairings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Appendix A: Notation and Basic Facts in Knot Theory. .... ..... .... 99
Appendix B: Automorphism Groups from Quandles... .... ..... .... 107
Appendix C: Small Quandles, and Some Quandle Homology..... .... 119
References.... .... .... .... ..... .... .... .... .... .... ..... .... 127
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 135
Notational Conventions
In this book, group actions are generally understood to be right actions. Given
groups H (cid:2)G, we will usually write HnG instead of G=H for the right coset Ha.
Furthermore, conjugation by h2G implies a map G!G; g7!h(cid:3)1gh. Similarly,
G-modules are understood to be right G-modules.
Furthermore, given groups G and K with a homomorphism /:G!AutðKÞ;
g7!/ , the semi-direct product, KoG, is the group on K(cid:4)G with operation
g
ðk;gÞIðk0;g0Þ:¼ð/ ðkÞ(cid:5)k0; gg0Þ; for k;k0 2K; g;g0 2G:
g0
Weareofteninterestedinsometorsionpartsofabeliangroups.Inthissituation,
a homomorphism f :A!B between abelian groups is said to be a ½1=N(cid:6)-iso-
morphism and is denoted by f :Affi½1=N(cid:6) B, if the restriction of f on the ‘-torsion
parts of A and B is an isomorphism for any prime ‘ that does not divide N.
Next, here is a list of the abbreviated notation that will be used many times:
pL p1ðS3nLÞ, i.e., The fundamental group of a link complement S3nL.
G0 The commutator subgroup of a group G.
G The abelianization of G, that is, G ¼G=G0.
ab ab
StabðxÞ The stabilizer subgroup of x2X, when G acts on a set X.
Z(cid:2)Q The ring consisting of integers and the rational number field.
R(cid:2)C The real number field and the complex number field.
F The finite field of order q.
q
SL ðFÞ The special linear group of n(cid:4)n matrices with determinant 1 over a
n
field F.
H ðY;MÞ The cellular homology of a CW-complex Y, where we sometimes
n
consider M as a local coefficient module.
p ðYÞ The n-th homotopy group of a connected CW-complex Y.
n
XY Theloopspace.Thatis,XY ofapointedtopologicalspaceY isdefined
to be the space of based maps from the circle S1 to Y with the
compact-open topology
ix
Chapter 1
Introduction
A quandle is an algebraic system that will be defined shortly in Definition 2.1.
However,theconceptismotivatedbysomeclassicalmathematics.Precisely,begin-
ningfromancientGreece,manymathematicianshavebeeninterestedinobjects X
withsymmetries,e.g.,reflectionsandthesphere;AsinKlein’sErlangenprogram,
suchsymmetriesareinterpretedassubgroupsinthepermutationgroupofbijections
Bij(X,X),e.g.,CoxetergroupsandSO(n).Thus,inthemathematicalparadigmafter
the19thcentury,thetheoryofgroupsplaysadynamicalrole,andtheyhavedevel-
opedmany areas.Incontrast,quandles appear tobestaticasopposed todynamic,
andtheyreturnourfocustobasicmotivations.Actually,asseeninTheorem2.23of
[Joy, Mat],everyquandleisrepresentedassuchastaticobjectwithsymmetry;the
theoremsometimesimpliesanawkwarddifficultyindealingwithquandles,asold-
fashionedobjects.But,suchadifficultydoesnotalwaysmeansomethingnegative.
Althoughthisbookisprimarilyasurvey,thistextdoesemphasizefreshapproaches
andinterestingperspectivesthatquandlescanbeusefulofstudyingrelativeobjects
withsymmetryandlow-dimensionaltopology(e.g.,knottheory).Theperspectives
canbesummarizedtofouradvantages:Precisely,
©TheAuthor(s)2017 1
T.Nosaka,QuandlesandTopologicalPairs,
SpringerBriefsinMathematics,https://doi.org/10.1007/978-981-10-6793-8_1
2 1 Introduction
(cid:2) (cid:3)
1.ThestudyofGfromrelativeobjectsH\G,e.g.,polyhedra,reflections,sym-
metricspaces,andsoon.Here,wecanprocessthepair H ⊂G asoneobject.
2.Centralextensionsofgroups,e.g.,Spinstructure,K -group,geometricanom-
2
aly.Quandlescansometimesmanagecentralextensionsmoreeasily.
3. Reduction of dimension and non-compactness of G, e.g., cases of
dim(H\G) ≤ dimG, co-compact subgroups H, infinite coverings of mani-
folds.
4. Cubical sets in algebraic topology. Distributive law in the quandle axiom
agreeswithcollectingcubesinasense;see“therackspace”inChaps.5and6.
(cid:4) (cid:5)
Inthisbook,wedescribehowtheseapproachesmaybeaccomplishedandwhat
thebenefitsindoingsoare.
In particular, this book tries to give many examples, in order to see how some
quandles yield interesting phenomena coming from the four perspectives. In the
topologicalchapters,weespeciallytakeup“Alexanderquandles”,whicharesome-
whatanalogoustoabeliangroups(seetheAlexanderizationinSect.B.2).Although
theclassismadeupofthesimplestandmanageableoneslikelinearobjects,weget
someinterestingresults,e.g.,onAlexanderpolynomials,meta-abeliangroups,and
tripleMasseyproducts.
Moreover,thisbookstressestheirapplicationstolow-dimensionaltopology.In3-
dimensionalgeometry,asintheWaldhausen[Wal]andthegeometrizationtheorem,
itisessentiallysignificanttoviewthefundamental groupsofmanifoldswiththeir
boundaryasasingleentity;thisviewisrealizedas“fundamentalquandle”,which
detectsknotsinthe3-sphere;seeChap.2.Here,theboundaryinformationappears
ascentralextensions;seeSect.2.4.Inaddition,thisbookwouldliketodemonstrate
howquandlehomologyissituatedinmathematics.
Furthermore,inthethirdtopic,wewillseethatsomeexistingresultsintopology
canberecoveredfromquandlesand,thus,thattheresultsturnouttobeinterpreted
as2-dimensionaldiagrams(here,somecubicalcuttingsplaykeyroles).Especially,
we succeeded in giving 2-dimensional descriptions of the fundamental homology
3-classandtwistedcohomologypairingsoflinks.Sincesuch3-classesandpairings
have been considered to be speculative and uncomputable, we may conclude that
theseresultsaresignificantachievementsofquandletheory.
Accordingtothemotivationsmentionedabove,ourapproachtothesubjectcanbe
summarizedasfollows:WebegininChap.2bydefiningquandlesandgivingmany
examples,fromtherelativeviewpointofthehomogeneoussetH\G.Thetopological
motivation will always be kept in sight, we will immediately discuss perspectives
forthetopologicalapplicationsofquandles.
Chapters3and4concernlow-dimensionaltopologicalparts.Precisely,quandles
arecompatiblewith(relative)fundamentalgroups,andareofusetogetquantitative
information in the groups. Furthermore, we can equip the information with some
gradings,suchascharacteristicclasses.
