An introduction to Characteristic Classes

136 Pages·2021·0.951 MB·English

by  Jean-Paul Brasselet

PANTONESolidCoated313C
An introduction to
Characteristic Classes
C C
M M
Y Y
CM CM
MY MY
CY CY
CMY CMY
K Jean-Paul Brasselet K
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a impa
ISBN 978-65-89124-15-3
impa
Institutode
Matemática
PuraeAplicada 9 786589 124153
Jean-Paul Brasselet
An introduction to
Characteristic Classes
AnintroductiontoCharacteristicClasses
Primeiraimpressão,julhode2021
Copyright©2021Jean-PaulBrasselet.
PublicadonoBrasil/PublishedinBrazil.
ISBN 978-65-89124-15-3
MSC(2020)Primary: 14C17,Secondary:55R40,57R20,57R25,58K45
CoordenaçãoGeral CarolinaAraujo
Produção BooksinBytes Capa IzabellaFreitas&JackSalvador
RealizaçãodaEditoradoIMPA
IMPA www.impa.br
EstradaDonaCastorina,110 [email protected]
JardimBotânico
22460-320RiodeJaneiroRJ
Contents
Preface vi
1 Introduction 1
1.1 Manifoldsandpseudomanifolds . . . . . . . . . . . . . . . . . . . . 1
1.2 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Orientationofpseudomanifolds . . . . . . . . . . . . . . . . 6
1.2.2 Orientationofmanifolds . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Orienteddoublecovering . . . . . . . . . . . . . . . . . . . 9
1.3 Poincaréisomorphism(manifolds) . . . . . . . . . . . . . . . . . . 10
1.4 Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 TheGaußmap(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 GeneralisationoftheGaußmap . . . . . . . . . . . . . . . . 16
1.6 Fibrebundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6.1 Fibrebundles . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6.2 Vectorbundles . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.3 Examplesoffibrebundles–realcase . . . . . . . . . . . . . 18
1.6.4 Examplesoffibrebundles–complexcase . . . . . . . . . . 20
2 The“Euler–Poincaré”characteristic 24
2.1 TheGreekperiod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Maurolico–Descartes–Euler . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Maurolico(1494–1575) . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Descartes(1596–1650) . . . . . . . . . . . . . . . . . . . . 26
2.2.3 Euler(1707–1783) . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.4 Descartes’TheoremisequivalenttoEulerFormula . . . . . . 29
2.2.5 ProofsofEulerFormula . . . . . . . . . . . . . . . . . . . . 29
2.2.6 Thegeneralization: Euler–Poincarécharacteristic . . . . . . . 29
2.3 Poincaré–HopfTheorem . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Theindexofavectorfield. . . . . . . . . . . . . . . . . . . . 31
2.3.2 RelationwiththeGaußmap . . . . . . . . . . . . . . . . . . 36
2.4 ProofofPoincaré–HopfTheorem . . . . . . . . . . . . . . . . . . . 40
2.4.1 Thesmoothcasewithoutboundary . . . . . . . . . . . . . . 40
2.4.2 Thesmoothcasewithboundary . . . . . . . . . . . . . . . . 44
3 Characteristicclasses: smoothcase 47
3.1 Generalobstructiontheory . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 Indexofanr-frame . . . . . . . . . . . . . . . . . . . . . . 50
3.1.2 Generalobstructiontheory . . . . . . . . . . . . . . . . . . . 52
3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.1 Stiefel–Whitneyclasses . . . . . . . . . . . . . . . . . . . . 56
3.2.2 Chernclasses . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Singularvarieties 64
4.1 Stratifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Whitneystratifications . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 FundamentalpropertiesofWhitneystratifications . . . . . . . . . . 69
4.5 Poincaréhomomorphism . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Alexanderisomorphism . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6.1 Cellulartubes . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Poincaré–HopfTheorem(singularvarieties) 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Whytheradialvectorfields? . . . . . . . . . . . . . . . . . . . . . 76
5.3 Whythedualcellsdecomposition? . . . . . . . . . . . . . . . . . . 80
5.4 Radialvectorfields . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.1 Radialvectorfields–Localconstruction . . . . . . . . . . . 81
5.4.2 Radialvectorfields–Globalconstruction . . . . . . . . . . . 84
5.4.3 Poincaré–HopfTheoremforsingularvarieties. . . . . . . . . 86
6 Schwartzclasses 88
6.1 Radialextensionofframes. . . . . . . . . . . . . . . . . . . . . . . 89
6.1.1 Localradialextensionofr-frames . . . . . . . . . . . . . . . 89
6.1.2 Globalradialextensionofr-frames . . . . . . . . . . . . . . 90
6.2 Obstructioncocyclesandclasses . . . . . . . . . . . . . . . . . . . 92
7 MacPhersonclasses 93
7.1 Nashtransformation . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 localEulerobstruction . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3 Constructiblesetsandfunctions . . . . . . . . . . . . . . . . . . . . 97
7.4 Matherclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.5 MacPhersonclasses . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8 Developmentsandperspectives 103
8.1 Remarksandcomplements. . . . . . . . . . . . . . . . . . . . . . . 103
8.2 AboutthefundamentalChernarticle. . . . . . . . . . . . . . . . . . 104
8.3 ThepolarvarietiesandMatherclasses. . . . . . . . . . . . . . . . . 106
8.4 MoredevelopmentsofChernclassesforsingularvarieties . . . . . . 106
8.4.1 Bivariantclasses . . . . . . . . . . . . . . . . . . . . . . . . 106
8.4.2 Othergeneralizationsofclassesinthesingularcase. . . . . . 107
8.4.3 Hirzebruchformalism . . . . . . . . . . . . . . . . . . . . . 108
8.5 TheEulerlocalobstruction . . . . . . . . . . . . . . . . . . . . . . 108
8.6 Someapplicationsinothermathematicaldomainsandintheoretical
physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Bibliography 110
IndexofAuthors 117
Index 120
List of Figures
1.1 Triangulationsofthesphere. . . . . . . . . . . . . . . . . . . . . 2
1.2 TheWhitneyumbrella. . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thepinchedtorusandthesuspensionofthetorus . . . . . . . . . 6
1.4 Compatibleorientations. . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Thedoublecone. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Dualcells1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Dualcells2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 ThePythagoraspolyhedra. . . . . . . . . . . . . . . . . . . . . . 25
2.2 TheTheaetetuspolyedra . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Hexahedra,(6faces)and5,6,7ou8vertices . . . . . . . . . . . . 28
2.4 ThemapSn 1 Rn 0 . . . . . . . . . . . . . . . . . . . . . . 33
(cid:0)
! nf g
2.5 Vectorfieldsindimension2. . . . . . . . . . . . . . . . . . . . . 34
2.6 Singularityofsecondtype(n 2). . . . . . . . . . . . . . . . . . 35
D
2.7 The"-neighbourhoodofM . . . . . . . . . . . . . . . . . . . . . 38
2.8 Thevectorfieldw.x/. . . . . . . . . . . . . . . . . . . . . . . . . 39
2.9 Vectorfieldsonthesphereandthetorus. . . . . . . . . . . . . . . 41
2.10 BarycentersandbarycentricsubdivisionK . . . . . . . . . . . . . 42
0
2.11 TheHopfvectorfieldH inatriangle(cid:140)(cid:27)0;(cid:27)1;(cid:27)2(cid:141): . . . . . . . . 42
2.12 Poincaré–HopfTheoremfortherealpr(cid:128)ojec(cid:128)tive(cid:128)plane. . . . . . . . 43
3.1 Themappr v.r/ Sk 1 V .Kn/.. . . . . . . . . . . . . . . 50
2 (cid:0) r
(cid:14) W !
4.1 Thepinchedtorusandthesuspensionofthetorus . . . . . . . . . 65
4.2 Astratification. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Whitneycondition(a) . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Whitneycondition(b) . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Stratificationofthecone. . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Stratificationsofthe(realpartofthe)varietyV. . . . . . . . . . . 69
4.7 Distinguishedneighbourhood. . . . . . . . . . . . . . . . . . . . 71
5.1 M.-H.Schwartz’scounterexample. . . . . . . . . . . . . . . . . . 77
5.2 Vectorfieldsonthepinchedtorus . . . . . . . . . . . . . . . . . . 79
5.3 Parallelextension. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 transversevectorfield. . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 A“tapered”neighbourhood. . . . . . . . . . . . . . . . . . . . . 83
5.6 Localradialextensionofavectorfield . . . . . . . . . . . . . . . 84
5.7 Theradialvectorfield . . . . . . . . . . . . . . . . . . . . . . . . 86
7.1 TheNashtransformation. . . . . . . . . . . . . . . . . . . . . . . 96
Preface
Definingthebirthofcharacteristicclassesisnotclear.
WhoofPythagoras,Plato,Maurolico,Descartes,Euler,Poincaré,Hopf... can
beconsideredasthecreatorofthecharacteristicclasses?
ThisisthereasonwhyIinviteyouduringthecourseforacruiseinwhichwe
willmeetthesepeopleandothers... whowillsharetheircontributionwithus.
Our cruise starts on the island of Samos, Greece in 570 BC where we meet
Pythagorasplayingwithrepresentationsofthetetrahedron,thehexahedron(cube)
and the octahedron. Thales comes from Miletus not far from the island to play
with us. We leave the island for Athens where we meet Theaetetus, less known
thanPlato,althoughheisthefinderofthe5platonicpolyhedra.
After playing with the polyhedra, through the Mediterranean Sea we leave
GreeceforSicilywherewehaveawalkontheSyracusebeachwithArchimedes
in230BC.StillinSicily,inMessina,muchlater,inDecember26,1537,wemeet
an Italian priest, Francesco Maurolico who, apparently does not care of the war
between Charles V and the Pope against the Turks and prefers to spend his time
describingtheplanarrepresentationsoftheplatonicpolyhedra. Maurolicotellsus
thatheobservedthatthe5platonicpolyhedrasatisfytheformula:
#vertices #edges #faces 2:
(cid:0) C D C
The boat takes us to Stockholm, in January 1650, where Descartes is invited
by the Queen Christina of Sweden. Descartes is very ill. He entrusts us with his
manuscripts, containing among other things a “nice theorem”. When Descartes
dies,fewdayslater,wetaketheboatwhichtransportshismanuscriptstoParisin
asafe. ArrivinginParis,theboatsinks(doyouknowhowtoswim?). Fortunately,
after3daysintheriverSeine,thesafeisrecovered,allowinglaterLeibniztocopy
Descartes’manuscriptsandtotakethesecopiestoHanoverinGermany.
Still in Germany, in 1750 we go to the Jean-Sebastien Bach’s funerals. Then
in Berlin, on November 14, we meet Euler who just sent a letter to his friend
Goldbachsayingthathediscoveredtheformulathatnowbearshisname,thisfor
allconvexpolyhedrainR3. Theformulaappearsnowinfactasacorollaryofthe
theoremthatDescartesshowedus.
We return in Paris, in 1885, for the Victor Hugo’s funerals. Poincaré is there
and he says us that he has been able to generalize the Euler characteristic for all
dimensions. We stay in Paris to follow the construction of the Eiffel Tower and
ofthefirstmetroline. WemeetagainPoincaréin1899whotellsusthatthechar-
acteristic, now called Euler–Poincaré, is the obstruction to the construction of a
vectorfieldtangenttoacompactsmoothsurface.
BackinBerlin,in1927,wegotothecinematoseethenewsilentfilm“Metropo-
lis”byFritzLang. WearesittingnexttoHopfwhoinvitesustoknowhowhegen-
eralizesthePoincaréresultforalldimensions,obtainingthenowcalledPoincaré–
Hopftheorem.
HopfadviseshisstudentStiefeltostudytheobstructiontoconstructanr-frame
tangenttoasmoothmanifold. Forus,wecontinueourcruise,thistimeontheliner
“Normandie”, on May 29, 1935 for its inaugural crossing to USA. In a festive
atmosphere,hewinsthe“blueribbon”. WereachWeston,whereWhitneyshows
usaroundhismarveloushouse. HetakesustoclimbthepeaksofMassachusetts
(don’tyoufeeldizzy?). Whitneytellsusthathehasasimilarconstructiontothe
Stiefel’sone. TheStiefel–Whitneyclassesareborn.
WestayintheUnitedStatesduringWWIIandCherninvitesustoPrincetonin
1946toreadushispoemsandtellushowheconstructs,inthecomplexsetting,the
“Chern classes” in so many ways, not just using the obstruction theory but also,
amongothers,bydecompositionofGrassmannianmanifoldsintoSchubertcycles
and by differential forms. Chern gets us so excited that we forget our cruise. He
suggestsustofollowevolutionofthetheories: Chern–Gauß–Bonnet,Chern–Weil,
Chern–Simons.
BackinFrance,in1965,wetakethetrainfromParistoLilleinwhichwemeet
a woman making strange drawings on sheets of paper. Marie-Hélène Schwartz
explains to us that she understood why, in general, the Poincaré–Hopf Theorem
does not work for singular varieties and, radiant, she explains to us that we must