LAST TIMELx d It It 2 Lieder of LERPCM

along a vector field Xhere Pt U M flow of X

IR

in tlx rPcm rt m

x x X xp i 2 X X Xp i

satisfies 0 Cdf X IL x gap 1 2 apt C17Pa a pp b n

e 1 1 2 0

Cartan's formula Lx 2 d 2 Lx d x

rumrunnerde Rham cohomology

MotivationfromCalc3 if U C 1123 simply connected then curl it o

implies T gradf fersone fIn the language of diff forms2 If da o But if U is simply connected it is an equivalence

fergie desilu

Def The p th deRham cohomology group of a manifold M is the quotientvector Space

Hp µ Ker d cm CP M

imd rm

Ree group here vectorSpaceyR factfurmanifolds eventopologicaloned connectedlocally y pathconnected

H M constant functions If M connected HTM IRdf o m T if M has n connected

e man

timer

H M is a finite dimensional neckspacewithout poof

Def A form Le M is closed if da 0Form a S exact if a dp for some PERP M

go Hp closedP Eexact p horns

closed p form 2 1 7 23 E HP23 22 iff L 2 dp forsome p

Preposition deRham cohomology groups of an n dimensionalmanifold M

havethe following properties

HP M o p n

for AE HPCM b c119 14 there is a bilinear product abe HPT M

satisfying ab CDM ba

If F M N a smoothmap it defies a natural linearmap

F HPCN HP M had commutes withtheproduct

Prod clear since A T't o

if a ED b p3 set ab aA a

IPdosed NdosedNeedto check that snap is closed dkap DLap 1C1Pdadp o

o o

Lap defines a cohomology classneedto checkthat it is well defined suppose we choose another representative for a2 D8 Then a'xp Ledo ap Lap drop in the sa e cobig

clam asLapdcrapchanging p p peds is lady

f 23 F 2 e F 2 closed F'a defiedined'ecMV a

spam L 2 D8 FIT IF 2L

so theoperator Ltd FFF Ef 453 is well defined

F snap IF xp EE'SaFfpF 2 IF'Mso F respectsthe product

is

asmooth

Wesaythat F MxEa b N is smooth if it is a restrictionof F MxEae b e N

or re for some E o

1 generally F N U3MxEabT 7Thereon Let F MxEg N be thmap.FI Fax tandconsider the inducedmap on de Rha cohomology F HP N HP MThen i F'T Ft

Prod Let a 123 e HPCN F d ERPCMXLo.isT infact MxCE HCclosed him on N

c F 2 p t dt ay splittingT T

Pfo no M p i fer on M

Splithy tt is clear in a word system depending on t dependingont

n III.es IEItt m a.s itgneraksaueoHhecdX ft

closed F a closed o d pedtardmptdtx dtadmr

dm exteriorderivative i thednt Ffa Ffa M

J IdtEffFt 2 dmIdt8 triGIF 2 Fid plt f

doer to N

closed ferns Fit FIL delter by an exact bM

Fia to a

Corollary The deRham cohomology groups of M IR are zero her p o

Prod Set F lRnxEo D IR F idCx t tX Fo maps 112 to 0 C IR

constant napFi HRRn 1HPC1R Dfo vanishesD forany seer 0412 Fo

FF HPCR HER P o

zeromapby Proposition F Fo HPCR O P o

IR is connected Hock locallyconstantfunctions Rn

IR

By a similar argument

Poincare if U C.IR a star shaped regionopen Ux EU tea D tx EU

tfthen HP a R if p o

0 if p o

HP Mx IR HP M for anymanifold M

FtCa x Ca tx deformation retractionof M IR onto M

Examplei H'CS

IEEE.IEsgtaagucee Ide i.EE s'k

pudefudbgHyf.la nU5lEIYdfe0 obviously leg herdegree reason where Cee it t

petclosed and Ues Yawhere 4C6,2T

assume p df S compact f must havea win and a macFlat function onS df mustvanish somewhere

but fu is www.skg fu is netexactso tics o

and containsthenonzero class I µ

Let 2 C sics anyhorn want a dh gee h'Cee

doe solution hat JdsgCs Copereda function of 4

this solutionis pernodc if Itsgcs 0

generally gcn fatdsgcs Ice with gCoe hide

So meanvalueofgo S 9th go t.cat dsgce

Sei 2 go pet dh H S Span µ Rw wda exactform

Theory For n o Hp gm112 if p o or h

o otherwise

P Let n i caseone wasdiscussedabove Let Icpc n

Let 2Esedosed Shen R Hy du since HER o

YI stereographiccharts 21µs do for some uerPEUV ons uesir CH

HER on UAV aHun Hun du du du v

u v c R sea UIdi.EERxSn1HPClRxSn Hr Sn o for pen

byinductionu u du were 4mV

U_Eu Belo hiksurrentLet 4 bumpfunction on UN s t 4 1 on U n Vd

vi qj Belo 4 w global Cp 2 form on 5extended

synerodef.ve

pu on U

globalCpi hoon restrictions to U n V agreeUtd Yu on V

then dp L x is exact HPCS 7 0

for Icp n

If p f U V E ridged UNL a constantfunction uingthat UN is connected for a 1

pa o U

global function dp LVtc on V agree on overlap

If P n u u defnes a day in HP UN e HP S _IRadnate

Let H 5 Spent T UN S xReroded 5 tf

Sn 1So u v No wat Dw If 4 0 then we do as aboveandPradaglobal NtseeXEIR tempsf 2 dpf is lnear a 2

andindependent ofthechoice of u V slittingthem by an exact ten can beabsorbedat wdentics EL

4 bumpfunction on IRNeedto fuel with nonzero Q Set D 4 dta o EN Sn

edyhsiabyrereuh.gg uextendedbyzero

1461dg u err uoutsideSupp4 4

sit du L

v 1014olds o err v u v 1 1614 0 0 on Un VTo a

a