Variations on a Theme of Borel: An Essay on the Role of the Fundamental Group in Rigidity

365 Pages·2022·2.642 MB·English

by  Shmuel Weinberger

CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
J. BERTOIN, B. BOLLOBÁS, W. FULTON, B. KRA,
I. MOERDIJK, C. PRAEGER, P. SARNAK,
B. SIMON, B. TOTARO
213VariationsonaThemeofBorel
blih d li b C bid i i
CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
J. BERTOIN, B. BOLLOBÁS, W. FULTON, B. KRA, I. MOERDIJK,
C. PRAEGER, P. SARNAK, B. SIMON, B. TOTARO
Acompletelistofbooksintheseriescanbefoundatwww.cambridge.org/mathematics
Recenttitlesincludethefollowing:
196. TheTheoryofHardy’sZ-Function.ByA.IVIC´
197. InducedRepresentationsofLocallyCompactGroups.ByE.KANIUTHandK.F.TAYLOR
198. TopicsinCriticalPointTheory.ByK.PERERAandM.SCHECHTER
199. CombinatoricsofMinusculeRepresentations.ByR.M.GREEN
200. SingularitiesoftheMinimalModelProgram.ByJ.KOLLÁR
201. CoherenceinThree-DimensionalCategoryTheory.ByN.GURSKI
202. Canonical Ramsey Theory on Polish Spaces. By V. KANOVEI, M. SABOK, and
J.ZAPLETAL
203. APrimerontheDirichletSpace.ByO.EL-FALLAH,K.KELLAY,J.MASHREGHI,and
T.RANSFORD
204. GroupCohomologyandAlgebraicCycles.ByB.TOTARO
205. RidgeFunctions.ByA.PINKUS
206. ProbabilityonRealLieAlgebras.ByU.FRANZandN.PRIVAULT
207. AuxiliaryPolynomialsinNumberTheory.ByD.MASSER
208. RepresentationsofElementaryAbelianp-GroupsandVectorBundles.ByD.J.BENSON
209. Non-HomogeneousRandomWalks.ByM.MENSHIKOV,S.POPOVandA.WADE
210. FourierIntegralsinClassicalAnalysis(SecondEdition).ByC.D.SOGGE
211. Eigenvalues,MultiplicitiesandGraphs.ByC.R.JOHNSONandC.M.SAIAGO
212. Applications of Diophantine Approximation to Integral Points and Transcendence. By
P.CORVAJAandU.ZANNIER
213. VariationsonaThemeofBorel.ByS.WEINBERGER
214. TheMathieuGroups.ByA.A.IVANOV
215. SlendernessI:Foundations.ByR.DIMITRIC
216. JustificationLogic.ByS.ARTEMOVandM.FITTING
217. DefocusingNonlinearSchrödingerEquations.ByB.DODSON
218. TheRandomMatrixTheoryoftheClassicalCompactGroups.ByE.S.MECKES
219. OperatorAnalysis.ByJ.AGLER,J.E.MCCARTHY,andN.J.YOUNG
220. Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory. By
C.WENDL
221. MatrixPositivity.ByC.R.JOHNSON,R.L.SMITH,andM.J.TSATSOMEROS
222. AssouadDimensionandFractalGeometry.ByJ.M.FRASER
223. CoarseGeometryofTopologicalGroups.ByC.ROSENDAL
224. AttractorsofHamiltonianNonlinearPartialDifferentialEquations.ByA.KOMECHandE.
KOPYLOVA
225. NoncommutativeFunction-TheoreticOperatorFunctionandApplications.ByJ.A.BALL
andV.BOLOTNIKOV
226. TheMordellConjecture.ByA.MORIWAKI,H.IKOMA,andS.KAWAGUCHI
227. TranscendenceandLinearRelationsof1-Periods.ByA.HUBERandG.WÜSTHOLZ
228. Point-CountingandtheZilber–PinkConjecture.ByJ.PILA
229. LargeDeviationsforMarkovChains.ByA.D.DEACOSTA
230. FractionalSobolevSpacesandInequalities.ByD.E.EDMUNDSandW.D.EVANS
blih d li b C bid i i
Variations on a Theme of Borel
An Essay on the Role of the Fundamental
Group in Rigidity
SHMUEL WEINBERGER
UniversityofChicago
blih d li b C bid i i
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,
NewDelhi–110025,India
103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107142596
DOI:10.1017/9781316529645
©ShmuelWeinberger2023
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2023
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Weinberger,Shmuel,author.
Title:VariationsonathemeofBorel/ShmuelWeinberger,UniversityofChicago.
Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversityPress,
2021.|Series:Cambridgetractsinmathematics;213|Includesbibliographicalreferences.
Identifiers:LCCN2022024936(print)|LCCN2022024937(ebook)|ISBN
9781107142596(hardback)|ISBN9781316507490(ebook)
Subjects:LCSH:Rigidity(Geometry)|Manifolds(Mathematics)|Three-manifolds
(Topology)|Graphtheory.|BISAC:MATHEMATICS/Geometry/General|
MATHEMATICS/Geometry/General
Classification:LCCQA640.77.W452021(print)|LCCQA640.77(ebook)|DDC
514/.34–dc23/eng20220823
LCrecordavailableathttps://lccn.loc.gov/2022024936
LCebookrecordavailableathttps://lccn.loc.gov/2022024937
ISBN978-1-107-14259-6Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
blih d li b C bid i i
Contents
Preface pageix
1 Introduction 1
1.1 IntroductiontoGeometricRigidity 1
1.2 TheBorelConjecture 4
1.3 Notes 8
2 ExamplesofAsphericalManifolds 10
2.1 Low-DimensionalExamples 10
2.2 ConstructionsofLattices 14
2.3 SomeMoreExoticAsphericalManifolds 20
2.4 Notes 27
3 FirstContact:theProperCategory 31
3.1 Overview 31
3.2 K\G(cid:2)andItsLarge-ScaleGeometry 33
3.3 Surgery 39
3.4 StrongApproximation 45
3.5 Property(T) 48
Appendix:Property(T)andExpanders 53
3.6 CohomologyofLattices 56
3.7 MixingtheIngredients 65
3.8 Morals 70
3.9 Notes 71
4 HowCanItBeTrue? 79
4.1 Introduction 79
4.2 TheHirzebruchSignatureTheorem 81
4.3 TheNovikovConjecture 84
4.4 FirstPositiveResults 85
v
blih d li b C bid i i
vi Contents
4.5 Novikov’sTheorem 92
4.6 Curvature,Tangentiality,andControlledTopology 94
4.7 Surgery,Revisited 99
4.8 ControlledTopology,Revisited 106
4.9 ThePrincipleofDescent 110
4.10 SecondaryInvariants 114
4.11 Notes 119
5 PlayingtheNovikovGame 129
5.1 Overview 129
5.2 AnteingUp:IntroductiontoIndexTheory 134
Appendix:AGlimpsethroughtheLookingGlass 139
5.3 PlayingtheGame:WhatHappensinParticularCases? 147
5.4 TheMoral 160
5.5 PlayingtheBorelGame 163
5.6 Notes 175
6 EquivariantBorelConjecture 186
6.1 Motivation 186
6.2 Trifles 189
6.3 h-Cobordisms 199
6.4 Cappell’sUNilGroups 202
6.5 TheSimplestNontrivialExamples 206
6.6 GeneralitiesaboutStratifiedSpaces 212
6.7 TheEquivariantNovikovConjecture 220
6.8 TheFarrell–JonesConjecture 232
6.9 ConnectiontoEmbeddingTheory 235
6.10 EmbeddingTheory 242
6.11 Notes 249
7 ExistentialProblems 258
7.1 SomeQuestions 258
7.2 The“WallConjecture”andVariants 261
7.3 TheNielsenProblemandtheConner–RaymondConjecture 267
7.4 Products:OntheDifferencethataGroupActionMakes 272
7.5 Fibering 277
7.6 ManifoldswithExcessiveSymmetry 283
7.7 Notes 286
8 Epilogue:ASurveyofSomeTechniques 291
8.1 Codimension-1Methods 291
8.2 InductionandControl 294
blih d li b C bid i i
Contents vii
8.3 DynamicsandFoliatedControl 296
8.4 TensorSquareTrick 300
8.5 TheBaum–ConnesConjecture 302
8.6 A-T-menability,UniformEmbeddability,andExpanders 306
References 311
SubjectIndex 340
blih d li b C bid i i
blih d li b C bid i i
Preface
Thisessayisaworkofhistoricalfiction–the“WhatifEleanorRooseveltcould
fly?”kind.1TheBorelconjectureisacentralproblemintopology:itassertsthe
topological rigidity of aspherical manifolds (definitions below!). Borel made
his conjecture in a letter to Serre some 65 years ago,2 after learning of some
workofMostowontherigidityofsolvmanifolds.
We shall reimagine Borel’s conjecture as being made after Mostow had
proved the more famous rigidity theorem that bears his name – the rigidity
ofhyperbolicmanifoldsofdimensionatleastthree–asthegeometricrigidity
ofhyperbolicmanifoldsisstrongerthanwhatistrueofsolvmanifolds,andthe
geometricpictureisclearer.
I will consider various related problems in a completely ahistorical order.
Mymotiveinallthisistohighlightandexplainvariousideas,especiallyrecur-
ringideas,thatilluminateour(oratleastmyown)currentunderstandingofthis
area.
Based on the analogy between geometry and topology imagined by Borel,
onecanmakemanyotherconjectures:variationsonBorel’stheme.Many,but
perhapsnotall,ofthesevariantsarefalseandonecannotblamethemonBorel.
(Onseveraloccasionshedescribedfeelingluckythatheduckedthebulletand
hadnotconjecturedsmoothrigidity–aphenomenonindistinguishabletothe
mathematicsofthetimefromthestatementthathedidconjecture.)
However, even the false variants are false for good reasons, and study-
ing these can be quite fun (and edifying); all of the problems we consider
enrichourunderstandingofthegeometricandanalyticpropertiesofmanifolds.
Verumexerroris.
1 SeeSaturdayNightLive,season4,episode4.
2 May2,1953.
ix
h d i 0 0 9 8 3 6 2964 00 blih d li b C bid i i