KA he CALcc

Suppose U C 1122 Is OPEN Aro f: U- 1122 IS A Smooth Function.

I: WRITE f- = ( f, ,fz) .

Is THEME A Smooth F : U → 112 wit't

dot,

-

-f,Are III. = fe ?

NOTE THAT Since such AN F SATISFIES dg?y,

= df#g×,

,A Necessary Condition IS THAT d¥z = 0¥

,

Is This Also Sufficient ?- Xz

Io : Consider f : 1122- fo} - IR ' f (x,Hu) = ( ft i ) It's Easy To CHECK THE Conditions

Bui THEME Is No Smooth F : 1122-903 → IR WITH O= G Amo dg=fz .

THI: LEF Hc 1122 BE A Star-SHAPED OPEN SET Ams Suppose f- : U - 1122 SATISFIES THE Condition . THEN

There IS A Smooth F: 1122→ IR Such THAT d,,

= -f,Aab = f-2

.

Proof : Assume WLOG THAT Xo -- O C- U IS A POINT SUCH THAT THE LINE SEGMENT From Xo To × LIES It

U For Every XE U . DEFINE F: U → 112 By

F-(x , ,Xu) = fo't x. f. Ctx. , txdtxzfzltxz.tw) dtTHEN d,lx, ex. ) -- fjff.lt, ,tx) t tx, 0¥

,

Ctx, .tn) + tuff

,

Ctx, ,txd)dt

Am ¥ tf, Ctx, ,txd= f. Ctx, ,tx4t tx, 0¥,

Ctx, , tx.lt txz II. Ctx , ,tx. )

⇒ o (x, .xi= fjfodtttfiltxi.txittxr.to#.ltx..txD - 8¥ Ctx. .tw ) )dte-

= O

= fjfdytf , Ctx, ,tx.)) dt = tf , Ctx, .tw It!! = f. ( x . ,x . )

A Similar Calculation Shows I fr.

, ,

with MIKES this WEEK? THE Totoro Of U Ensures THAT THE INTRGanz Is Dearer,So It MAKES SENSE

TO TRY TO UNDKustom Corbit, ons ON U Critica GUARANTEE Solutions TO THESE Sonnets Of QUESTIONS.

DIE: IE U c 1122 IS OPEN,

CA( U,112k ) I§ THE SET OF Smooth Functions f : U → 112k

.This IS A Vector SPACE .

DEFINE THE FOLLOWING OPERATIONS

grad : CTU,ie) - CTU

,IR' ) grad 141 -- ( ¥. .

0¥)

curl : CTU,oh - CHU

,R ) curl Lee, .ch) -- DIT

.

- off,

NOTE THAT Curto grad = O

Dier: H' ( ul -- kerlcurlllimlgradlNOTE THAT H'( u) --O IF U c 1122 Is Stan - SHAPED

.However

,H' (1122-904) to AsTHE First Example Shows

.

DRI : H -( u) = kerlgradlWE Can. Extreme Titis To OPEN Sees U c 112k i grad (ft = ( off. , 7¥, . . .

, daffy)Ttm; U EIR

"Is Connected ⇒ Ho ( u) = 112

Pryor : Suppose grad(f)= O .THEN f- Is locally CONSTANT : EACH Xo EU Has A NBits V(Xo) with f- (x) -- flxo)

For ALL X C- V(Xo) . It U Is Conniecries ,THEN EVEN Lucaccy Constant Function IS CONSTANT ( It Xo E U

,THE SET

{xeu / f- (x) = ffxo)}= f-' fffxo)) IS CLOSER SINCE f- Is Continuous,Aae OPEN S ,NCE f locum Constant ⇒ It Is ACO" U )

⇒ Howl -- IR .

CONVERSELY,Ik U IS NOT CONNECTEp

,THE.x THEME IS A SMOOTH SURJECTION f -. U → 90115

.THIS Is LocaLa, 2

Constant t So grad(H -- O ⇒ dim Hola) > 1. ,

=

LET's EXTEND THIS To 3 VARIABLES.

LET U E 1123 BE OPEN.WE HAVE 3 operators :

grad : Cocu, uh - Cola,

R' ) grad IH -

- ( 9¥, ,do¥ ,

7¥.)

curl : CTU,1124 → 0141123) arlff.fr,fsl=f¥x. - offs ,

'¥3- 5¥,9¥ - Itu)

divi CTU,IR' )- CTU

,IR) divlfi.fi

,f) = dot

,tdtd

NOTE : Curtograd -- O A.us diuocurl -- O:

o - c-( u .ir/EidC9u,iR ' ) CTU,IR' ) s CTU.IR/s0

IS A Complex.

DEFINE Hola ) Ann H'

( ul As Before Aa . SET H2 (ul = kerldiv) (in (curl )

THI: IF U E 1123 Is Stan -Shapes,THEN How = H

'

cul = H'(at = O .

PI : THE 2 -D Proofs Still work For Ho Ann H'.

Suppose U IS Stan -SHARES with RESPECT To 0 . Suppose

F- Us 1123 Has div f -- o . Define G : Us 112'By GCE) -- Sj ( Fl tx) xtxldt THEN It's Easy To CHECK

THAT c-rt ( Fl tx) x tx) = It ( t' Ff txt) . Titus, curl Kix)) = Sift ( t' Atx)) dt = Ft) . "IK U E 1123 Is Hoi Star - Sharks

,It's Possible To Have H' la)# o Ams H' (u) ¥0 .

ez ; LET S -- f (x, ,xz, xD c- 1123 I xftxi -- t t X3=0 } Ans LET U -- 1123-5.DE,ewE f : U → 1123 By

flex.nl -- t.IT#T.ii.xi7I.iIIiTEixsIIIIII'7 )Direct (Accurate, SHOWS curl f -- O ⇒ { f) E H' ( u ) . Consider THE Curve 81 E) = (Est

,O,sintl

,

- ite teh

Noire Thar im 8C U .

Suppose grad F -- f ou U

.THEN

I I! Fcrcthdt = FINI-et) - FCK - ite) ) - o Ase → o#

But Tite Chea .ae DUE Gives

¥ Flott)) -- f. (htt) . t ! Htt f. HEH) -Kit, t f, Kitt .rs'

HI

⇒ s. ÷:*. III. no,.co.ma..ro.. I s:i,

Titus,[ f) ¥0 IN H

'

lat .* L

--

WE CAN Externs Titis Tb SUBSETS U E IR"To Construct Vector SPACES

IP ( ul - f DIFFERENTIABLE P - Forms ON U }

= { Shinooth wi U→ APIR" }

ANY DEANE THE EXTER.io#DERluAuEd : R" cut → RP"( u ) WHICH AGREES with dir, grad, curl WHEN IT

lldaxths SENSE . We Have d ? O Amo So we Have A Complex

Os routs n'cut d-Acute's . . .- rn

- '

cut d- rncul→ 0

An" Sir Hilal = Y÷tp'

= cwsiop-fon.msExact p - Forms

THESE Ark HIGHEn - Dimensional Versions of THE PROBLEM OR FINDING Anti Derivatives : H " ( u) Measures How MANY

( loses p - Forums And Not Exact .

Poincare (banana : If U IS A Star -SHAPES OPEN SET THEN Hoke) -- IR Ann HP ( ul --O,p> 0. 3

--

PRI: KAIRRONS THE low- Dimensions Caskey,

p - Foran) Axe Built From ( 0cm Information To ASSEMBLE GLOBAL Information About THE SET U . WE CAN

EXTEND Forms Tb MANIFOLDS Ams Bump A Connes Pours ,.NG Theory . THIS ESSENTIALLY GETS Ar THE QUESTION:

Can WE Solve PDEs one MAN#ones ?

A Cookie Foran WE RT(U ) SATISFIES dw=O , WHICH IS ESSENTIALLY A COLLECTION OF HOMOGENEOUS DifferentiateEQUATIONS

. Finn,.ie n with da = w G wks ( local Solutions To THIS SYSTEM.THIS IsAn' Our QUESTION,

Certainly PREDATING HOMOLOG THEORY.

ALLO R THIS works Onley ON MAN IF ours , THOUGH, Anis wk Ark INTERESTEs IN OTHER SPACES. So THE

QUESTION IS i Can WE BUILD A THEORY Of functions ON CW - COMPLEXES,SAY,THAT IS ANALOGOUS To THIS LOCAL

OBSTRUCTION PROBLEM ? THE Answer IS YES ,

SUPPOSE WE Have A CHAIN Complex OF FREE ABELIAN Groups :

(• : - - - → Cnt , d- Cnts ( n-i → -- -

(Ei G BR Aa. ABELIAN Grove . Apply Homz, I - , G) To C . :

- - - → Hom ( Cn-i. C) Is Hom ( Cn , C) Is Hom (Cnn , G)→ --.

( ll l l l

Cn-' C

"(htt

WHEN S IS THE DUAL Co Bouman, Map : Ik ft Hou ( Cn,G) THEN f( f) (2) = f- ( da )

→2 C- ( n * i

SINCE 0-0=0, we Have SS -- O Ann we CAN DEFINE COHOMOLOGY Groves

H' ( C .,G) e Ker 8 :C

"

- C" '

c- Coomes-

Tu f : "

→(nc- COBOUNDARIES

ez : C. : O - 2521 - Z ⑤ 2152130 Ho -- 21,It,-- 2 I 21+0212 , Hz -- O

, 11-3=21I 1- ( 0,22)

Cz Cz C , Co

° 0

TAKE 6=21 : Os 21 → 21+021 - Z → Z- O C'-- Hom ( 2021,21) = 21+021 f

,la

,b) = 9

, fala, b) = b(O

c' CZ C

' f,fr BASIS

f (f) ( n ) = f. (On ) -- f , 10,41=0 ⇒ f : 21+021 - 21 Is ( a.blts 2b ⇒ Ker S -- { laid} IZ

fl (n ) = felon) = f. ( 0,41=2n⇒ Ho -- 21 , H

'-- Z

,H'= Ilias = 212

,

H'=D

0TAKE 6=212 : O → 2120-212+024 → 212524-70 ⇒ Hole.,2121=212

Carb ) 1- (O,2b)

-- o H

' ( (•,214=212+021,

H2 ( C.,21,1=212

HBK. , 2121=212

Q : CAN WE Determine HTC..G) IN Terms ok It * (C.) Ann G ? 4

A : First NOTE THAT There IS A Natuna Mar h : H"

( C.,G) → Home, ( Hn ( C .), G)

DEFINES As Follows.

It' f : Cn - G Is A Coc > core ( Sf -- O),THEN f- d -- O Ann So f

VANISHES ON Bn E Cn.Titus

,f IH Ducks A Mar fo : Zn (Bn→ Gi i.e.

,AN Recrement OR

Homz ( Hn (C .),G)

.Ite f Lites IK im S

,f-- 8g = GO ,

THEN f- Is O on Zn ⇒ fo -- o .So we

Dae, .ie hh By h ( CH) = [fol e Home ( Hulce)

,G)

.

(La: h IS Sunt ECT NE.

Pilot : Consider Tine Stout Exact Sewanee O - Zn - (n → Bn - i → 0 . This Spurs Since

Bn - , IS FREE AND So There Is A Protection p : Cn - Zn Restrictive Tho id ou Zn.

So If WE Have A MAP Cfo : Zn→G , we Have y = Hoop : (n → G EXTENDING Cfo .

In Particular,

Titis Externs Homomorphisms En- G VANISHING ON Bn Tb Mars Cu → G THAR STILL VANISH

ON Bn ; THAT IS, It Externs Mars Hn ( Ca) → G Tb Enemies Of Kers Ams Passino Tb

Quotients wie Gier A Map Hom ( Hn (Ce ),

G) - H" ( Ca

,G )

.IF WE Follow THIS By h WE Get

THE IDENTITY ON Homz ( Hn ( C. ) , G) t HENCE h Is Sunseri vie Ano WR Goe A Sp Short

Euro SEQUENCE O- Kesh - Hnk. , G) Is Homz, ( Hull. ) , G) → o

←

Want Is beer h ? DetailsAne D- THE Book,But Hku 's THE PUNCH link :

kwh = Ext ( Hn. . ( C . ) , G)Withe THIS IS DEE, was As Follows

.

LET H BE A FINITE- y Giv k Ratko Abeyta Grove. Titter

1 . Ext ( Hiott ',G) e Ext ( H

,e) to Ext ( H '

,G)

2. Ext ( H

,G) = o Ik H Is FREE

.

3. Ext ( In

,G) I GING

e.g : Ext ( In ,Im) E Zm/nZm I Zlgcdln , m)

es : Consider Previous Exam .ae : C. = O - 2321 - 21+021-92-30 Ho -- 21,H,=D 212

,He -

- O, Hz --21

I 1- ( 10,2)

⇒ HTC. ,2tHom( Ho , 217=-21 ; H' (C. , 2) I Houtz ① 21,2) to Ext (¥1,21 I Howl 2. 211=-2H'( Ce , 2) I Hom ( O , 2) to Ext (210212,21 ) I 212 ; HSCC

.,2) E Hom ( 21 , 2) to Ext 10,211 I 21

-

Ho(C . , 212) I Howl#Zz ) I 2h ,

' H' ( Ca,Dc) = Hour (2+0212,26) to Ext (21,2k) I 212+0212 (⑦ O)

H'( Ce,4) I Hom ( o , 2h) to Ext (Z ⑤ 2h , 2h) I 2h ; H'(C ., 2h ) I Hom (21,212) to Ext 10,2k) I 2h

UNIVERSAL COEFFICIENT THI : Tithe Is A Split Short EXACT SEQUENCE- -

O- Ext ( Hn .. (CD, C ) - Hnk..G) → Howl Hale. )

,G) → 0

THIS SEQUENCE IS Not Naturae Scene CE Tine SPLITTING IS NI CANONICAL.

-REc : ALL OK THIS works Outen PRETTY GENERIC 121 -Vos R Ans Coo Is A COMPLEX OF R -Monks.S

WE Oney NEED SUBMODULE> Of Frick Mobutu Tb BE FREE,WHICH HAPPENS

, e.g . With R IS A PID .

In Particular,Ik C. Is A CHA ,- Complex Of Uk cion SPACES OVEN A Fiber k

,Then It Is A FACT

THAR Extra (Hn. . ( C. ),V) -- O For ¥ Viktor Smee V

. Titus,

H" ( Coo

,V ) I Homie ( Hunk.) , V )

18 , Cons . Dean THE COMPLEX OF QI - VK.cm SpreesCz Cz C , Co

O → Q SQ → ↳⑦ ④ → ④ → O

L -(a , -4)

(a. f) tsatf

THE.no Ho -- O, H ,-- O

,Ha -- O , Hz -- Q

A- Pray Hom al - , Q) ⇒ Ho--o, H

'--Ela .at/R&as4=o:

o - fi- → EI -00%-01+2=0, it

'-

- a2 1- (2 , x )

( a ,B)↳ a -A

Calculator or 8 : co-- Home!QQ ) Has Basis id :& → a .

So 5-(id) (u , u ) = id OLUND= id ( Utv)

Titus,Solid) IS THE Mae Gto Q→ On TAKING Chiu) Tho Utv

.This Is

= u + ✓

THE Sum OF THE Basis Buenas p ,: Q to Q → IQ And Pz : K too→ Ox

4.H ten I. u ) 1- u

So,Unseen Tine Iso s co = Hom ( 6×01 I #

,id vs I Am C

'= HomCato

,d) E Q Q p, ↳ ( 1. o)

,pztsco.is

S> go : ④ → Of ①Q Is a ↳ G.a ) .

S '

p , (a) =p ,lout =p , ( u ,

- u) -- u i S'

polo) -- pcldv) -- Pa (v ,- v) = - v

. So S ' (a ,p ) = 2 -f .

I

SINGulam COHOMOLOGY--

(KT X BE A Tfpoeoo em Space AND Lhs Cn (X) Denote The Group Orsini overran - CHA ,#s .DEFINE

C" (Xi 6) = Homzccncxl , G)

So : A CoCH YE C"

(Xie) Assures To Any ri on- X A Vause Cllr) E G .

S : C"

(Xia - C" " ( x .

- GI

Seco ) = 4106) = El- ' lice ( Heu. . . . .. .. . .vn's)ir : ont '→ X

we Have 82=0 Ann So we DEFINE The nth CoHooa Gray OI Xw Coefficients II.GTf De Tine n th Cortona ology Of The CoonPLEX

on CTX .

- G) → c'Cx.

- Gies . . . - [Hide -- n

El (x) -- Ker 8"= n - Cocyc# s Ams B

"

( x) = in S" -'

= n - CubanDANES

* UCT "

O → Ext (Hu. . CX ) , e ) - Hn (Xi G)→ Hom( Hix), G) → 0 Is Split Exact .

et Ksn ⇒ Hits ; 2) =/ Z i-- on

0 else

GI : If WE Cann JUST Compute Coitomorooy P- RELY ALGEBRAICALLY VIA UCT, WHY Bother ? 6

I : Co Homo Loot ACTUALLY HASA RAUCH Richter STRUCTURE.

So LET's Force A item.

=

HolliesCo (x) =Z{Points Of X} ⇒ A CoCHAIN U E co (XiG) IS AN Arbitrary function Y '

- X→ G

(Not NECESSARILY Continuous).Suppose 86 = 0

.Tithe Ik o '

-Evo

,U ,)- X Is A l - Sanpete

Ulu , - Vo) -O ⇒ flu, )=6Cuo) ⇒ y Is Constant ON PATH Components Of X .

Titus,

HO(Xi G) = kerf -- f functions from PATH Components 04 X To 6$= Homa ( Ho ( x)

,G)

It Particular,Ho (Xi Z) Hss Rane Kaun To THE Number of Pettit Components OK X .

Note : IF F IS A Fikes,THEN It

" (Xi F) E Home ( Hn Nif) , F).

GmFAREDUCERS COHOMOLOGY- -

Duterte THE AUGMENTED CHA,a Complex .

.. . - Cz Cx ) - C, 1×1 → Co (x ) 52130

⇒ In (Xi G) = Hnk .

- G) Form > o.

Wars About IIOCXIG) ?

NOTE THAT H ° (Xi G ) IS THE Confection Of Funtctous THAR And Constant ON PATH Composers die X.

E : Co (x) → Z SENS Etch r : OSX Fo I C- 21.So Ik e : Z → G IS A Homomorphism

E't (e) G)= e (Ek)) = Ceci )

THIS IS A Constant Function XIG ⇒ im E't= f constant Functions }

This,Ttocxi G) = Hoc XiG) ( { Constant Functions on x}

RE# we ↳ Homo Lott- -

TAKE A Pain (XIA ) too Duteize THE Exact SEQUENCE O → Cn (A)Is Cn Cx ) Is Cn ( X.A) → O :

O- C"

(X. Aitc) t Cncxic) C"

(Aidoo

THIS IS EXACT : it Restricts A Cochran, ou X To A CoCHAN ON A .

It f--Cutt) → G

,Then f

CAN BE EXTRUDES Tb Aa or Cn ( X) By ASSIGNING THE VA core 0 To Au Sian pucks No- IN A ⇒[ ¥ IS SURJECTIVE

. THE KERNEL Of Itt IS ALL Co chains TAK,.io THE VALUE On Cn (A).

THESE COMESPonsTo Mans Cn (X. A) = Cn WICCA) →6 ⇒ Keri't -- Hom (Cn CX. A ) , 6) = im#

. j # Is Cheaney Insecure.

THE Routine CoBoundary 8 : C"

(X.Ai G)→ It' ( X. Aib) IS THE Restriction Of Tite AbsoluteCo BOUNDARY t So THE Groups H

"

(X. A ,

- G) Are DEFINES.I# t j't Commute WITH f Since

i t j Commune with d- WE THEMEFork Have A Lo no Exact Co Homo Looy SR Quince

- - - s H'(X.A .

- G) HYXIG) H" (Aids H "

' (X.Aid- n . - A HNCAIGI IH " "(x. A .

-G )th D th

Hom (Hula), e) Homlltntilx.at,G)

Et : ( Dn,Sn" ) : . . - Hn

-

:(D" ) - Hn- '

( Sh- '

y § Hn (D",S" " ) - H

"

(Dn )- - - 7I 11

O B-i UCT 0 By UCT

VARIANCE-

(o Homero# IS A CONTRAVARIANT functor : Ik f:( X.A) u (Y , B) Iss Continuous , WE Gta AMnr f 't : HYY, B ; G) → H' ( X.AIG)

.

A'Deeks,f#: C

" (Y,BIG)- C

" ( X.Aid Is Done To

f# : Ca ( X, Al - Cn CY , B ) Ann Smick Of# = f# 0 we Have f#8=8 f

't

.

We Also Have A

Commutative Dito nnnn O - Ext ( Hn-' ( X.Al , G)→ H"

( X.Ai G) h- Hom ( Hnl X.A) , c)→ o

tf * I 't Ff't THTO → text ( Andy

,B),G)→ HTY.BR) → Howl Haly, D. G) so

H0nY CE Ix f , g : ( X. At → H. B) Are Homework , THEN f#g't: H

" ( Y,is,

- G) → H' (X.A.

-G).

If : 1) ✓ALEE THE Honnocooy Proof. ,,

fxcis.org : ZCACX, I aint A .

Titan i : ( X - Z ,A - Z ) - ( X.A ) INDUCES Isis

i 't : H't

IX. Ai G) → It#

(X- Z,A-Zico)

PI : U CT t FIVE LEMMA .On DUAL TEE THE Homo Loot Proof

. ,,

SINCE SINGULAR Corto nnorocy IS DIKE, c - a- To Compute, we WANT TO USE OTHERTHEORIES .

Simple Cohomology

(E- X BE A D- Complex,Ac X A Subcompact . Duterte On ( X.A) To GET 0

"

( X,AIG)-- Hom (Onlx.A),G)⇒ HI ( x.Ai 6) E Hn ( X. AIG)

(IELLULAR COHOMOLOGY- -

LET X BE A Cw - Computer.Duautre Hn (X

"

,X" " ) To Get { It " ( X " , X

" "

; Gl } .

ANA-like - Viktor is--

Ik X= Au B,Duarte O - Cn( An B) → Cn CA) to Cut B)→ Cn ( Xl → O Tb 6kt THE low Exact Sea:

- . .- Hn (Xi G) - H

"

(AIG ) to H " ( B.

- 6) - H"

Larbi G)→ Hn"

( X .

- A- → - - -

=

(up Products- -

Contra ✓AMAN ok IMA kiss It Possible To DER , we EXTRA Structure ON CO 'tonnouooy.

( ASIDE : Titanic Is A Pioneer Structure ON Hx ( X),(Acc Eos Tine Pontian PRo ,

But It's Not As Natuna.)

(Er RBR A Rino ( usually 2,24,Q) .

Lkr YE C " (X.

- R) t te CMX.

-R).

THE Cup Projet Oke - tIS THE CocHa ,n Q u t C- Cnt" ( Xi R ) Dee, Nico By

de ut ) ( r ) = 4 ( oleo,.. ..vn/t(4cun.....vn+m3)

/ trFRONT n - FACE

G: Ohm-Xo, •

BACK m - FACEOK T

We , Vries Tf tow This Inn-Us A PROD- ⇐ ON CO Homo root. 8

Lkxan : S ( y ut ) -- ⇐g)ut + C- 15608 t ) YECNCXIR),tc-cmfxiitl.pro: Liu r : on '-→ '

→ X.

Then

⇐ ceutlcrl -- EEE't letter. . . . .,oi . ... .vn them. . .

-mmmm )

C- 1146 - SHH-

- i nice Lola..... .ms ) than.

..

.' un .-mail

ADDING THESE, THE LASTTerm or The First Sonn CANCELS THE first Term Oro Tine Secours Are

THE REST IS EXACTLY 8 ( Clut) G) = ( fu t) Cdr) .#

Cory : THE Cup Pro out OF Two Co CYCLES ISA Coc yccE.

PI : Suppose 84--0=8 t. Then 814- t ) = Scent ± yo St = o - O = o.o

Also,THE↳ Pronoun of A Coc-Kok Ano A Co boundary Is A CoBourn Any

.

It Fod lows THAR

we Have An Indoors (re Promis u : H" ( Xie) x Hml Xi RI - Hntmlx

.

- R)(IN Fact, U Is Buin Enn Am So WE CAN REPLACE X By ④ r) It Is Associative AroDistributive Since It IS ON THE CHAIN LEVEL

.IK R HAS AN IDK .

- try In,THEN THE CLASS

I E HOI Xi R ) ( ICH = Ir,re Co (Xi) IS THE ID.ee#ii-- For THE Cup PROD- 5

.It Follows

Tita H" ( Xi R) =

.

HiNik) ISA Graney Ri .

We CAN EXTEND Tb RELATIVE Co Hoon Owo > :

H "CX.ir/xHmlX.AiR)-HntmCxX.AiR1HnlXiAiRlxHmCXiR) Is It" + " (X.Air )

HYXAIR) x Hml x.Air ) is H"" (X. AIR)

Prof : Ix f:X- Y Is Continuous,Then f

't

: H" ( Y

,

- R ) - H#

(X.' R ) ISA Rinko Honnonnonritism.

Prove : A ye C" ( Yi RI,tf CKY.

- R )

( f #y u f#t) G) = f#

e ( oleo....us/f#tl4sun....un+ms )

= y ( frlsu. . .. . uns) t ( folau. .. . unni )

= ( yw t ) ( fr )

= f # Lieut ) Gr) . ,THIN : Srrrosie RIS Commutative

,a E H

" ( X.Air), fc. Hml X.Air ) . Then a - f- f)

" "

Bua .In

Particular,Ix n IS ODD , THEN 2cL-23=0 c- Ht" (X.AIR) . So I' H'" (X. AIR) Has No 2- Torsion, a wa --o .

PRI: Horrify , -16 CALCULATION (look IT UP) .It IH - owes SHUFFLING THE Onnen OF THE

✓knacks OR A Simplex or Are THE SHOWING THE Iho Docter CHA , ,v MAP Iss HomoTopic To id. y,

E±Ae a 9>:/.

( Rt X BETHE Torus 51×5' WITH O - Computer Structure : I0

,:

we know It"

(x ; WE Hom ( H, 1*1,21) E Hom (21021,211 . A Basis or r n me:

"i :*: ÷::

"

:c :: i://a.in#v .i

j '

wie Naka To REPRESENT a Ans f Dy Coc -locks .

Define L B !-- - - - - -

- - - - - - -- - - -

;- -

Cf : C , ( x) - 21 By 461=1 ⇒ Vertical Arc Crosses e t ,

cel -- o else #aul : C

,(x)→ 21 By t (e)= , ⇒ Horizontal Arc Crosses e-

+let -- O else

Ne , f Cf (r) = O Ik r Does Not Contain a or THE TWO RIGHT DIAGONAL Arcs i ie. 84 ( Ty ) -- O

Syco ,) = y (dr. ) = 4cal - ellen) to = l - 1=0 LET eij Be The Encore ADJACENTSacral = 410%1=4 Cen) + cells) - 46231=1 to -1=0 p r , + r,84 Irs ) = 4 (ers ) - 4cal SEO -- I - I - o = 0

So 6 Is A Coercive Ams y (a) =L ⇒ [ e)=L .

Samnium Cream-ou Shows [ t) -- B .

None : @ ut) (r . ) = Celal 46,41=0

( yo t) (re ) = 4lb) then ) -- o ⇒ Out TAKES VALUE / on C = oytrz -rz -5 ,

( pot) (rs ) = 9k, tlezy )-- I Do Oc -- o ⇒ Cc) Generates Hz ( x)

( yo t ) Cry ) = 4lb) tle -41=0 So Ce ut RinksErrs THE Generator 8eH4Xi2)

TH-s, auf -- V .

Also,Etsy to CHECK THAT tu TAKES VALUE - I on. c ⇒ [ tu 6)= -8 .

A Simi LAN CAL curation Applies To A Cluster Surface Nlg of G Ew us g :

I' d, if, , arise , . - - , as,fg Generate H'( Mg ,- Z) t 8 Generates H2 (Mg

.

- 211,THEN

Li ups = { 8 5=5 di was-- o

O its fi up; = O

2.LEI XI IRP? Since It

'( Xi 2) = Hour ( Hick . 21) = Hou ( 2h ,

211=0 ANDHYX.IR/=ExtC2zpt-ZzTltErkAneN/oINTknESti.v6Cup Products Inc H * ( X ,

- Z ).

But,with Zz- Coefficients

HYX ; 21,1=212 , H' (Xi = Howl 212,214=212 , HYX ,-214 = Ext (212,2121=212 Ann Sobre Mian

Hauk A Nontrivial Probe Ui H' (X .

- K) x H' (Xi 4)→ HEX.

- 2h ) .#

b O → Cz - C,→ Co- o O -214,23- 2k{a. b. c3→ video

WIU to axbtc a → wt u Ze Lat L)

¥1. . fu: ÷÷÷o : it:÷÷.Durr (OCHA , .ee To [at b) : Salut - Celan) -- 4( ath te ) = Itt to--o

- - - -- - - - - - -

¥- - - Sell) = 6104-- ylatbtc) -- O

y Ula) -- I614=1

⑥ uce ) cut 6161414=0 ⇒(6 - o) Dum

b U(b)=0 To Cath)v-7W (you) (4--414614=1 ⇒ sauce) Gen!

it Co"0GR6 10

H*( Xi R) = It

" (Xi R ) IS A GRaDABx Grove AND THE Cue Product MAKES This

A Rime ; the Fact Asu R - ALGEBRA SINCE WK CAN adultery By Elements Dr RE Ho (Xi R) .

None Tuns of = fill" "Mfa Winkie lat -- i Ix at Hill

.

-r).

e.g : H*( IRP} 2h ) = 212525/43) ,

• c. it' ( IRP' ,

- 2h) .

Extknion ALGEBRAS--

A# [a , , .. . , du) = FREE R - 'MODULE with Basis di,

. - - hi , ,i,c izc --- C in

,SUBJECT TO did ; = -a ,- ai t

21=0Arla . . .. , 2n) -

-

,

Akela, . -- , ant Is The Exterior

Acoersnaeg: X -- Toms

,H't

(Xix ) -- Aqsa , ,a Bt H'(Xia .

Facts : I.H't

( II Kin ) ITH"

Hai Rl

t. It * ( YX.ir/=aTH*tXaiR )

es : Gp2 # slug "

Notre : H# (GP2,- Z ) '

.Z o Z O Z

⇒ Same Is True For Corto urology

Hut ( s-us " ; 2e ) : Z O Z O 21

Be WHAT About THE Rows Structure ?

IITs' ust ; IE Tt 'T s!2) x #→ ( 54,-211 = Zfx.it/Cx2=o,yEo,xy=o)lxH--2,lyl--4

Htt ( Eph; 2) I 2123/43) ( To Be Proven Below) . THESE R-s Ane Noe Isonaorritec.

Ttm : H * ( IRPhi 2h) I Recall@" t ' ) t H 't ( IRP?- 2h) I Zach,lat -- l

H't ( Eph ; 2) = ZENKanti) t HHEpa; 2) E Z Ca) , 121=2 .

PI : THIS Actually Say, H"( IRP" i Ze) Is GENErater By 2k

,where d IS A Generator or H'(IRP?-4)

This Is True Triune> For IRP'-- S

'

,But Direct (Accountor Shower It Is True For IRP ?

Assume Inductively THA It * ( IR Pii 2h) I 22cL) / @ it ' ) Fon ich .Notre Tita The Ductus .N

u : IRP"- IRP

"

Iho - c.es An. Isomorphism On Corto nnocooy u't : Hl( lip" ; 2h)→ HUMP",-21€ ) , lekTitus

,It Suffices The Snow THA Ik di E H

"( IRP" ; 2h) Are a ; E H

5 ( LRP "; 212 ) Are Generations,

it ,- =n,

THEN di U2 ; to Do It" ( IRP

"

; 2h) .

We know 1121075/21, .

Lir Si -- { Go, --

, Xi ,o.- id l EXE -- IS a S

"

A,→ 5 -- {Ko. ..,o , xn-s , .. . as

"

Since its -_ u, Xi

-- Xu- j

⇒ Sin SJ = { 10 , -- so, It , O, -- , o) } .

Passive To Quotients we GE- IRP :c IRP" tIRPJCIRP " w# * 112pin IRPJ -- Gpb = {fo : -- - :O : ho .. :o)}

.

None TH# S,-⇐ 112PM. IRP " -'

wish,IRP'- sp}

DKRormatioothksrn-c.rs To IRP "' '

.

WE Also Have DK DixDJ ⇒ Di -- { (Xo : -- - ixi : o -o:3 },DI {so : .. :O : xn-j : ..-

n

. xD}⇒ Dn = { fxo : .. - n

. Xin , :L : Xin : --- n

. Xn)} In Homogeneous Coordinates.

(on Sroka Title FOLLOWING Commutative DING Nam ( ALL COHOMOLOGY WITH Zz- Coffees)y ,

Hiller)x H''

(iron)#Hulme" )T T

yiqppynzpn-ippgxHTIRPYIRPHRPY-ynuppn.ippn-g.is)t t

H''

( Dn , Dn- DJ) x H'T Dn,D"-Di) °_ It

"

(Dn,Dn- 905 )

WE WANT TO SHOW THA Tite Tor Horizonrn that Is Sort.EC true.

Cn: Alec Ukraine Arrows Are Isomorphisms.

Pryor : THE Bottom RIGHT Mnr H"

( IRP",RP"- Gp3) - HKD" , D

"-fuk) IS An Isomorphism By Excision

.

THE TOP RIGHT MAP A're Rns I- The lows Exact Shao,e~ae Of THE Pain ( 112pm,IRP"-Spy) = 412pm

,112pm) :

. - - - Hn- '

( IRP""

) - Hn- '

( Npn) - Hnlippn,112pm )→ HYMN) - H

" ( lip" " ) → . - -

E T T

BT IteDUCT'VE =3 O # I 0

HYPOTHESIS

For THE LEFT Arrows Cons is,En THE FOLLOWING DIAGRAM←Iso Sauce ALL OTHERS ARE

Hi ( Rpn ) EI Hicken, 'Rpi) Hi ( IRN,iRf- IRP ' ) Hi CD? D'- D' )( Inoue.io/=

"" If → I f II * * )- ⇒

Hi Lippi ) Hicippi,Rpi-y# Hiftp.pi.ippi-gpg) E- Hi (Di

,Di- SR)

LES PA in IRpi-gpgjip.pe-i ( Excision)

⇐ *) Is AN Iso : D" = Dix DJ ⇒ D"-D's (Di -903) xD ' ⇒ Htt icon , Dn -DI) E H'

'

( Di, @i-Sos) xD; )

I Hi (Di,Di- go} )

(x) Is An Iso : WE Cca,.in/Ha-lRPn-lRPJDki-onmatiowThE.-rxa-s To IR pi - ' . IT VE IRP"- IRPJ

,At LEAST

OAK OR THE First i coordinates or v Is Nonzero ⇒ Fl f 90,1 , -- , i -th with Xe ¥0 .LET

ft ( u) -- ft ( Xo : - -- ixn)= (Xo : . - -n

. Xi -i : txi : . - . : txn).THEN Ar t -- I

, f.Cut .- u Ans Ar t-- o, fo (v) C- IRP"!

Claire : THE Bottom Arrow Is Sunt Kot lui ,

PI : Hi ( Dn,Dn- Di ) = H'

'

(DixCD :S"' ) ) Hi ( Di .si "),Greenan. B, EDI) .

HTRTINV .

Similarly,

-HJ ( Dn,D'- DJ ) IS GENK pain By [ DJ)

,Ano Chen- ray TDI) b {Di) = {D) c- HID;D's,'

THUS,Since THE Bottom Arrow IS Sortkctivk

,ALL Horizon in Arrow Ame SURJECTIVE

.

THE Pro. ,c For Cl P"

Is Intention,Except H

"REPLACES It

"kvenywitkn.ie

. ,,

Coe : Hitler; 2h) = 212k) Ano HH Gpa; 2) I 2k ) (dega -- 2) . "