Advances in Two-Dimensional Homotopy and Combinatorial Group Theory

192 Pages·2018·1.948 MB·English

by  Wolfgang Metzler, Stephan Rosebrock

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat
http://www.cambridge.org/mathematics
332 Handbookoftiltingtheory,L.ANGELERIHÜGEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndEdition),A.KOCK
334 TheNavier–Stokesequations,N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability,A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules,I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory,J.MF.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory,N.YOUNG(ed)
339 GroupsStAndrews2005I,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory,J.B.CONREY,D.W.FARMER,F.MEZZADRI
&N.C.SNAITH(eds)
342 Ellipticcohomology,H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI,J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII,J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry,A.NEEMAN
346 Surveysincombinatorics2007,A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics,N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis,P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY
&A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY
&A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas,A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials,J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis,J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis,K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence,J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations,D.DELBOURGO
357 Algebraictheoryofdifferentialequations,M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory,M.R.BRIDSON,P.H.KROPHOLLER
&I.J.LEARY(eds)
359 Modulispacesandvectorbundles,L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA
&S.RAMANAN(eds)
360 Zariskigeometries,B.ZILBER
361 Words:Notesonverbalwidthingroups,D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories,R.BAUTISTA,L.SALMERÓN&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008,F.CUCKER,A.PINKUS&M.J.TODD(eds)
364 Partialdifferentialequationsandfluidmechanics,J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009,S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems,B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena,G.BLOWER
368 GeometryofRiemannsurfaces,F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks,M.DRAIEF&L.MASSOULIÉ
370 Theoryofp-adicdistributions,S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH
371 Conformalfractals,F.PRZYTYCKI&M.URBAN´SKI
372 Moonshine:Thefirstquartercenturyandbeyond,J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds)
373 Smoothness,regularityandcompleteintersection,J.MAJADAS&A.G.RODICIO
374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction,S.ALINHAC
375 Triangulatedcategories,T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds)
376 Permutationpatterns,S.LINTON,N.RUŠKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY
378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 Realandcomplexsingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
381 Symmetriesandintegrabilityofdifferenceequations,D.LEVI,P.OLVER,Z.THOMOVA&
P.WINTERNITZ(eds)
382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJÍCˇEK
383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems,B.MARCUS,K.PETERSEN&
T.WEISSMAN(eds)
386 Independence-friendlylogic,A.L.MANN,G.SANDU&M.SEVENSTER
387 GroupsStAndrews2009inBathI,C.M.CAMPBELLetal(eds)
388 GroupsStAndrews2009inBathII,C.M.CAMPBELLetal(eds)
389 Randomfieldsonthesphere,D.MARINUCCI&G.PECCATI
390 Localizationinperiodicpotentials,D.E.PELINOVSKY
391 Fusionsystemsinalgebraandtopology,M.ASCHBACHER,R.KESSAR&B.OLIVER
392 Surveysincombinatorics2011,R.CHAPMAN(ed)
393 Non-abelianfundamentalgroupsandIwasawatheory,J.COATESetal(eds)
394 Variationalproblemsindifferentialgeometry,R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds)
395 Howgroupsgrow,A.MANN
396 Arithmeticdifferentialoperatorsoverthep-adicintegers,C.C.RALPH&S.R.SIMANCA
397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology,J.BOLTE&F.STEINER(eds)
398 Mathematicalmodelsincontactmechanics,M.SOFONEA&A.MATEI
399 Circuitdoublecoverofgraphs,C.-Q.ZHANG
400 Densespherepackings:ablueprintforformalproofs,T.HALES
401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory,B.DENG,J.DU&Q.FU
402 Mathematicalaspectsoffluidmechanics,J.C.ROBINSON,JL.RODRIGO&W.SADOWSKI(eds)
403 Foundationsofcomputationalmathematics,Budapest2011,F.CUCKER,T.KRICK,A.PINKUS&
A.SZANTO(eds)
404 Operatormethodsforboundaryvalueproblems,S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds)
405 Torsors,étalehomotopyandapplicationstorationalpoints,AN.SKOROBOGATOV(ed)
406 Appalachiansettheory,J.CUMMINGS&E.SCHIMMERLING(eds)
407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups,J.N.BRAY,D.F.HOLT&
C.M.RONEY-DOUGAL
408 Complexityscience:theWarwickmaster’scourse,R.BALL,V.KOLOKOLTSOV&R.S.MACKAY(eds)
409 Surveysincombinatorics2013,S.R.BLACKBURN,S.GERKE&M.WILDON(eds)
410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups,
T.CECCHERINI-SILBERSTEIN,F.SCARABOTTI&F.TOLLI
411 Modulispaces,L.BRAMBILA-PAZ,O.GARCÍA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds)
412 Automorphismsandequivalencerelationsintopologicaldynamics,D.B.ELLIS&R.ELLIS
413 Optimaltransportation,Y.OLLIVIER,H.PAJOT&C.VILLANI(eds)
414 AutomorphicformsandGaloisrepresentationsI,F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
415 AutomorphicformsandGaloisrepresentationsII,F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
416 Reversibilityindynamicsandgrouptheory,A.G.O’FARRELL&I.SHORT
417 Recentadvancesinalgebraicgeometry,C.D.HACON,M.MUSTAT¸A˘&M.POPA(eds)
418 TheBloch–KatoconjecturefortheRiemannzetafunction,J.COATES,A.RAGHURAM,A.SAIKIA&
R.SUJATHA(eds)
419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations,J.C.MEYER&
D.J.NEEDHAM
420 Arithmeticandgeometry,L.DIEULEFAITetal(eds)
421 O-minimalityandDiophantinegeometry,G.O.JONES&A.J.WILKIE(eds)
422 GroupsStAndrews2013,C.M.CAMPBELLetal(eds)
423 Inequalitiesforgrapheigenvalues,Z.STANIC
424 Surveysincombinatorics2015,A.CZUMAJetal(eds)
425 Geometry,topologyanddynamicsinnegativecurvature,C.S.ARAVINDA,F.T.FARRELL&
J.-F.LAFONT(eds)
426 Lecturesonthetheoryofwaterwaves,T.BRIDGES,M.GROVES&D.NICHOLLS(eds)
427 RecentadvancesinHodgetheory,M.KERR&G.PEARLSTEIN(eds)
428 GeometryinaFréchetcontext,C.T.J.DODSON,G.GALANIS&E.VASSILIOU
429 Sheavesandfunctionsmodulop,L.TAELMAN
430 RecentprogressinthetheoryoftheEulerandNavier-Stokesequations,J.C.ROBINSON,J.L.RODRIGO,
W.SADOWSKI&A.VIDAL-LÓPEZ(eds)
431 Harmonicandsubharmonicfunctiontheoryontherealhyperbolicball,M.STOLL
432 Topicsingraphautomorphismsandreconstruction(2ndEdition),J.LAURI&R.SCAPELLATO
433 RegularandirregularholonomicD-modules,M.KASHIWARA&P.SCHAPIRA
434 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems(2ndEdition),K.TAIRA
435 GradedringsandgradedGrothendieckgroups,R.HAZRAT
436 Groups,graphsandrandomwalks,T.CECCHERINI-SILBERSTEIN,M.SALVATORI&
E.SAVA-HUSS(eds)
437 Dynamicsandanalyticnumbertheory,D.BADZIAHIN,A.GORODNIK&N.PEYERIMHOFF(eds)
438 Randomwalksandheatkernelsongraphs,M.T.BARLOW
439 Evolutionequations,K.AMMARI&S.GERBI(eds)
440 Surveysincombinatorics2017,A.CLAESSONetal(eds)
441 Polynomialsandthemod2SteenrodalgebraI,G.WALKER&RM.W.WOOD
442 Polynomialsandthemod2SteenrodalgebraII,G.WALKER&RM.W.WOOD
443 Asymptoticanalysisingeneralrelativity,T.DAUDÉ,D.HÄFNER&J.-P.NICOLAS(eds)
444 Geometricandcohomologicalgrouptheory,P.H.KROPHOLLER,I.J.LEARY,C.MARTÍNEZ-PÉREZ&
B.E.A.NUCINKIS(eds)
445 Introductiontohiddensemi-Markovmodels,J.VANDERHOEK&R.J.ELLIOTT
446 Advancesintwo-dimensionalhomotopyandcombinatorialgrouptheory,W.METZLER&S.
ROSEBROCK(eds)

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DOI:10.1017/9781316555798
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Contents
Editors’Preface pageix
AddressesofAuthors xi
1 ASurveyofrecentProgressonsomeProblemsin2-dimensional
Topology
JensHarlander 1
1.1 Introduction 1
1.2 Basicdefinitions 2
1.3 Basicquestions 7
1.4 TheD(2)-problemandtheEilenberg-Ganeaconjecture 9
1.5 Dunwoody’scounterexampletorelationlifting 11
1.6 Solvingtherealizationproblemforsomegroups 13
1.7 Exoticpresentationsofthetrefoilgroup 15
1.8 ExoticalmostpresentationsfortheKleinbottlegroup 17
1.9 Appendix: Geometric Realization for Algebraic 2-
ComplexeswithFiniteFundamentalGroup 19
ByF.RudolfBeylandJensHarlander
2 FurtherResultsconcerningtheAndrews-Curtis-Conjectureand
itsGeneralizations
CynthiaHog-AngeloniandWolfgangMetzler 27
2.1 Introduction 27
2.2 Projectionsontotestgroups 28
2.3 Fixedsubcomplexes 30
2.4 (AC)forspinesof3-manifolds 33
2.5 Considerationsonlength 34
2.6 OnthefutureoftheAndrews-Curtisproblem 35
vi Contents
3 AspectsofTQFTandComputationalAlgebra
HolgerKadenandSimonKing 36
3.1 Introduction 36
3.2 The Concept of Topological Quantum Field theory
(TQFT) 37
3.3 TQFTon2-complexes 40
3.4 TQFTons-move3-cells 52
3.5 Ageneralconstructionofstatesuminvariants 58
3.6 Matveev-Piergallinicalculus 68
4 LabelledOrientedTreesandtheWhitehead-Conjecture
StephanRosebrock 72
4.1 Introduction 72
4.2 Thefiniteandtheinfinitecase 73
4.3 Sphericaldiagrams 74
4.4 ClassesofasphericalLOTs 77
4.5 TheasphericityofinjectiveLOTs 86
4.6 Twoexamples 95
4.7 VirtualKnotsandLOTs 96
4.8 L2-homologyandWhitehead’sasphericityconjecture 98
ByJensHarlander
5 2-Complexesand3-Manifolds
JaninaGlock,CynthiaHog-AngeloniandSergeiMatveev 103
5.1 Introduction 103
5.2 ExistenceandUniqueness 104
5.3 k-ConnectivityofWhiteheadGraphs 106
5.4 M.H.A.Newman’sDiamondLemmaandanewversion
ofit 114
5.5 UniversalScheme 117
5.6 Kneser-MilnorTheorem 118
5.7 PrimedecompositionsofknotsinF×I 120
6 TheRelationGapProblem
JensHarlander 128
6.1 Introduction 128
6.2 TheRelationGapproblem 129
6.3 Examples 132
6.4 AnInfiniteRelationGap 133
6.5 Efficiency 137
6.6 PresentationswithCyclicRelationModules 142
6.7 FiniteGroups 143
Contents vii
7 OntheRelationGapProblemforFreeProducts
CynthiaHog-AngeloniandWolfgangMetzler 149
7.1 Introduction 149
7.2 Semisplit presentations, flower relations and integral
flowerchains 150
7.3 ApplicationstotheEpsteinGroups 158
7.4 Ideasforfurtherwork 159
7.5 Flower chains and Splitting with respect to field
coefficients 161
7.6 ProofoftheTheoremonsemisplitQ∗∗-transformations 165
References 167
Index 177
Erratum 180

Editors’ Preface
Ourbookon”Two-dimensionalHomotopyandCombinatorialGroupTheory
(LMS197)”appeared24yearsago.Sincetheturnofthemillenniumwewere
encouragedtopublishinasimilarstylethe(remarkable)progressthathasbeen
obtained in the meantime. On the other hand, none of the fundamental open
questionsliketheAndrews-CurtisConjectureandtheWhitehead-Asphericity
problem have been (fully) solved, and almost no prognosis is possible about
whenitwillfinallyhappen.Wehopethatmathematicians,whoareindanger
of leaving the subject because of this perspective, will be kept active by this
sequelpublication;andnewoneswillbegainedforourimportantfield.
Again,wemixsurveysandoriginalmaterialinthechapters,whichcanbe
readbystudentsandexpertsalike.Ongoingresearchismentionedasworkin
progress. There are ample references to the LMS 197 volume which will be
citedas[I]includingthebibliographyandtheindex,wherenewpublications
andnewnotionsarelisted.
SincethepublicationofLMS197therehavebeenworkshopsoftheauthors
at least every fifth year. We met in Germany, Italy, Russia and the USA, and
thefruitsofthesemeetingsarethebasisofthisbook.Duringameetingclose
toNaumburg(Saale)wemadethedecisiontostartwhatyoucanreadnow.
AmathematicallyenergeticmemberoftheteamofauthorswasNancyWal-
ler.Butwhentheactualwritinghadtostart,sheunexpectedlydied.Herpreprint
[BeWa13]togetherwithF.RudolfBeylcouldnotbefinished,andinparticular
itwasimpossiblewithouthertowriteaplannedchapteronthegeometricreal-
izationproblemforalgebraic2-complexesatthesametime.Hence”Nancy’s
Toy”,whichshehopedthatwouldresultinacounterexample,hasbeenleftto
thefuture.Whatcanbesaidatpresent,iscollectedinanappendixtochapterI
byF.RudolfBeylandJensHarlander.Thisbookcanbeconsideredasserving
forNancyWaller’smemory.
AsfortheLMS197volume,allchaptersarerefereedwithintheteamofau-
x Editors’Preface
thorsandbyanexternalrefereewhohadtheoptionofremaininganonymous.
Wearegratefulfortheirvaluableservice.Thosewhogaveuspermissiontobe
named openly are Laura Biroth (Mainz), Martin Bridson (Oxford), Cameron
Gordon(Austin,TX),PaulLatiolais(Portland,OR),PeterLinnell(Blacksburg,
VA),WajidMannan(Brighton),andTimodeWolff(CollegeStation,TX).We
also thank our former students for their help to prepare this book, some of
whomarementionedinthechaptersandinthebibliography.Aspecialthank
yougoestoWesleyBrowning,whoallowedustoreproducehisnote[Bro76],
whichgrewoutofcorrespondencewithJoanBirmanandwhichshouldnotbe
forgotten.
Last, but not least, we again thank Roger Astley and his team, who gave
encouragementandcontinuousadviceasforLMS197.
July2017
WolfgangMetzlerandStephanRosebrock