Formal Geometry and Bordism Operations

421 Pages·2019·5.125 MB·English

by  Eric Peterson

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 177
EditorialBoard
B. BOLLOBA´S, W. FULTON, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
FORMALGEOMETRYANDBORDISMOPERATIONS
Thistextorganizesarangeofresultsinchromatichomotopytheory,runningasingle
thread through theorems in bordism and a detailed understanding of the moduli of
formal groups. It emphasizes the naturally occurring algebro-geometric models that
presage the topological results,taking the reader through a pedagogical development
of the field. In addition to forming the backbone of the stable homotopy category,
these ideas have found application in other fields: the daughter subject “elliptic
cohomology” abuts mathematical physics, manifold geometry, topological analysis,
andtherepresentationtheoryofloopgroups.Thecommonlanguageemployedwhen
discussingthesesubjectsshowcasestheirunityandguidesthereaderbreezilyfromone
domain to the next, ultimately culminating in the construction of Witten’s genus for
Stringmanifolds.
This text is an expansion of a set of lecture notes for a topics course delivered at
HarvardUniversityduringthespringtermof2016.
Eric Peterson works in quantum compilation for near-term supremacy hardware at
RigettiComputinginBerkeley,California.HewaspreviouslyaBenjaminPeirceFellow
atHarvardUniversity.
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
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B.Bolloba´s,W.Fulton,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
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Formal Geometry
and Bordism Operations
ERIC PETERSON
HarvardUniversity,Massachusetts
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DOI:10.1017/9781108552165
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Title:Formalgeometryandbordismoperations/EricPeterson(RigettiComputing,
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Description:Cambridge;NewYork,NY:CambridgeUniversityPress,[2019]|
Series:Cambridgestudiesinadvancedmathematics;177
Identifiers:LCCN2018034169|ISBN9781108428033(hardback:alk.paper)
Subjects:LCSH:Topologicalmanifolds.|Manifolds(Mathematics)|Geometry,
Algebraic.|Boundaryvalueproblems.
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Letusbegladwedon’tworkinalgebraicgeometry.
J.F.Adams[Ada78b,Section2.1]

Contents
ForewordbyMatthewAndo pageix
Preface xi
Introduction 1
Conventions 7
1 UnorientedBordism 9
1.1 ThomSpectraandtheThomIsomorphism 10
1.2 CohomologyRingsandAffineSchemes 16
1.3 TheSteenrodAlgebra 22
1.4 HopfAlgebraCohomology 31
1.5 TheUnorientedBordismRing 39
2 ComplexBordism 49
2.1 CalculusonFormalVarieties 51
2.2 DivisorsonFormalCurves 60
2.3 LineBundlesAssociatedtoThomSpectra 65
2.4 PowerOperationsforComplexBordism 73
2.5 ExplicitlyStabilizingCyclic MU–PowerOperations 83
2.6 TheComplexBordismRing 91
3 FiniteSpectra 97
3.1 DescentandtheContextofaSpectrum 99
3.2 TheStructureofM I:TheAffineCover 111
fg
3.3 TheStructureofM II:LargeScales 121
fg
3.4 TheStructureofM III:SmallScales 132
fg
3.5 NilpotenceandPeriodicityinFiniteSpectra 140
3.6 ChromaticFractureandConvergence 154
vii
viii Contents
4 UnstableCooperations 165
4.1 UnstableContextsandtheSteenrodAlgebra 167
4.2 AlgebraicMixedUnstableCooperations 178
4.3 UnstableCooperationsforComplexBordism 188
4.4 DieudonnéModules 194
4.5 OrdinaryCooperationsforLandweberFlatTheories 203
4.6 CooperationsamongGeometricPointsonM 211
fg
5 Theσ-Orientation 219
5.1 CoalgebraicFormalSchemes 220
5.2 SpecialDivisorsandtheSpecialSplittingPrinciple 226
5.3 ChromaticAnalysisofBU[6,∞) 238
5.4 AnalysisofBU[6,∞)atInfiniteHeight 247
5.5 ModularFormsand MU[6,∞)-Manifolds 256
5.6 ChromaticSpinandStringOrientations 270
AppendixA PowerOperations 283
A.1 RationalChromaticPhenomena(NathanielStapleton) 285
A.2 OrientationsandPowerOperations 307
A.3 TheSpectrumofModularForms 322
A.4 OrientationsbyE Maps 333
∞
AppendixB LooseEnds 349
B.1 HistoricalRetrospective(MichaelHopkins) 349
B.2 TheRoadAhead 352
References 379
Index 401
Preface
Thisbookowesanincredibleamounttoanumberofincrediblepeople.Under-
standingtheresearchprogramsummarizedherehasbeenoneoftheprimary
pursuits–ifnottheprimarypursuit–ofmyyoungacademiccareer.Ithasbeen
anunmistakableandenormousprivilegenotonlytobegrantedthetimeand
freedomtolearnaboutthesebeautifulideas,butalsotolearndirectlyfromtheir
progenitors.IoweverylargedebtstoMatthewAndo,MichaelHopkins,and
NeilStrickland,foreachhavingshownmesuchindividualattentionandcare,
aswellasforhavingworkedoutthetailofthislongstory.Matt,inparticular,is
thepersonwhogotmeintohighermathematics,andIfeelthatforadecadenow
Ihavebeenpayingforwardthegoodwillanddeepfriendlinessheshowedme
duringmytimeatUrbana–Champaign.MikeandNeilarenotfarbehind.Mike
has been my supervisor in one sense or another for years running, in which
rolehehasbeencontinuallyencouragingandgiving.Amongotherthings,Neil
sharedwithmeanoteofhisthateventuallyblossomedintomythesisproblem,
whichisanawfullynicegifttohavegiven.
Lessdirectlyinideasbutnolessdirectlyinstewardship,Ialsooweavery
largedebttomyPh.D.adviser,ConstantinTeleman.BythetimeIarrivedatUC
Berkeley,Iwasalreadytoosoakedthroughwithhomotopytheorytodevelopa
flavorforhissortofmathematicalphysics,andhenonethelessendeavoredtomeet
mewhereIwas.ItwasConstantinwhoencouragedmetofocusspecialattention
onmakingtheseideasaccessible,speakingunderstandably,andhighlighting
theconnectionswithnearbyfields.Heemphasizedthatmathematicsdonein
isolation,ratherthaninmaximalconnectiontootherpeople,ismathematics
wasted.Hehasverymuchcontributedtomypassionforcommunicationand
xi