KEiH FoA 12

Suppose X t Y Are Sprees.

Les p ,: Xx Y - X Aus pc : XxY →Y BE THE Protection Maes

.

THE Crosse Product IS THE Diemen MAP

H't

( Xi RI x H't(Tir)- HTXxYiR )

x x f - pitta) upElp)Since This Is Bilinear we CAN REPLACE X BT ④r - THIS Mar IS A RING Homomorphism Ik UkSto ( a④ b) ( exo d) = f)"

" "Cac ④ bd )

.

THI : Tine Cross Prome H*(Xi R) ④r HYY.IR/eH*lXxYi,R/IsA-IsomoneinsmorRi.iosIt X - Y Arie Cw - Complexes t H'

' (Yi R) ISA Finitely Generates Free R-Normie For Au K

(ez R A FIELD) .

PR : we win Prove A Monk Gkn Enn Bks-a- (Attn. "⑧

-

CoHomology Of SP HENES Ams AN AP Pdx Cariou- - --

Rican Tita H't ( Shi 2) = ZG)) @" ) 12km. This Is Also Eaux To THE Exterior ALGRBNA Az,

Ik n Is ODD WE PMEREN 12,92) t Ik n IS Kuku Wh PREEBe 219231£) because or Titek NNE.it Formula , WHERE DEGREES AN AFTER .

So, HH s

" '

x Smx . -- x 5421 ) = Az Ca . . .. , de) ,hit -- ni

,Anni ooo ,

b-i, eg

H't 15×53 ; 2) = 2631*2) ④ Az, ,Ifl -- 3 a auf -- C- if"pua -- Sua e H5( 5×53,2) .

⇒ H't ( Tn ; 2) = H'T '

iz)-

- Az Ca. . .. . an ) , Kit -- I

.

Also, H 't flip"x IR Phi I Zeca .pl/( It'

,pm) .

DRI : A D - Atoka Structure on 112"

Is A Promo 112^+112" - IR"

S- ca THF

a ( btc) = ab tac Ams Ik a to,THEME IS A- X E IR

"wit't ax =L

,WittenE l IS A Distinguishes

ElkanEnt Satisfy , no l - X -- X . I =x For Au X E 1124.( Noth : THS Product NKR- No.- BE Commutative on

Kuku Associative. )

Eg : n = I i 112 WITH USUAL MOLT , PLL CAPON

n 2 : ¢ I 1122 WITH COMP etx MULTIPLICATION

None Examples : Tite Cross Produce ON 1123. THEne IS No Unit , Nor Do Enemies Have Inverses .

Titan: Ix 112"

Has THE Structure 0k A Division ALGEBRA outer 112,THE- n Must BE A Power Of 2 .

Pilar , THE Mars Xtsax AND X th Xa Anh Li- Kan Isomorphisms Fox Any a #O . It Follows THAT

THE www.pucm-iov MAP 112%112" → 112" Ih' Ducks A MAP h.. IRP" "x 112pm

- '

→ IRP" '

,what Is A

HOMEOMORPHISM With RESTRICTED To EACH SUBSPACE 112pm- '

x Sys on 1×4×1128""

.

h Is Continuous

Snuck It IS THE Quotient Of A BillaEnn Mae .

Consider h 't : H't ( IRP" " ; 212)- H't ( IRP"

-'

x'RP" "

i a) 13

245M€ ) - Zala . .at/Cai,ai )

WHA Is htt t)? It Must Earn I. 4 , t Ikeda ,Kane kit Zz . THE Incursion 112pm "- 112pm yep

" '

INTO THE First Factor SENDS d , To 2 AND 22 Tb 0 IN Colton Loot .AM Since hh RESTRICTS TO AHomeomorphism ON THE First Factor

,k,to . Similarly

,ke -1-0 . THUS

,h 't (2) = d

, taz .

Since 24--0, WK Must Have (a. the)n= O ⇒ €,

(1) Lk -0 ⇒ (%) m.DZ For O - Ken .But

Titis Harriers Orry For n --24 Soused.

, ,

No : Tite Same Proof works Fon Cl INSTEAD Of 112 ⇒ IK Er IS A Division Ac -6k Bra Outen Q,THEN

(1) = 0 Du Z For Oaken ⇒ n=l.

WHAT Ark Tithe Division ALGEBRAS ( IF THEY Exist) ?

D= 0 112

D= I 1C No.TANK

D= 2 Ifl = Quaternions 112 " with Basis I,i, 5, k Arin Quartermaine Multiplication

the Fact,WE HAVE QUATERNION, a Prospective Space HIP

"

: It Consists Of"

QUATERNION ,c links " IN1H""

-

- 112" " " "

. Help"Is a Quotient Of S

" +3C 1124" " WITH F.Ben Sb And WE CAN INDucsw.ee,4BUILD Http

"

From LHP"

the Attache, .→ A 4h - CEU Via THE Mars . Note IHP

'= S Ans we Have

4THE Quotient S?- S Causes THE Hore ANAP ( J

- -

UST AS WITH THE Quoi , But S?→ S? Gp ' )

d =3 ① =

"

Octonions ' ' On CATLEY NUMBERS. 1128 WITH A Noni Associative PROD - 5

Basis : { lo,e, yer , es, Ey, er, eco, e> } lie ; = - fi; eo t Eiji, eh , Eiji, =)

For ijk-- 123,145,176,246,257,347, 365

2 IS Antisymmetriccoli -- lieu --fi , eoeo -- eo

THIS IS A Lukins OBJET,But IT Works .

WE GET THE Conus spore , no PROTECTIVE SPACES ①P? A Quotientog strut? c 1128Mt! ①" t

.

'①p

'-58 WITH S' § THE Connes pond.ro Hoek Map .

d > 3 ? Suppose f : S""- S

"

Is A-y Continuous harm. Choose Generous SE HU-

'

( s"" ) Are

-1E H"

( S " ) Ans (Er y BE A Coerce thermosEntail 4 .Since 204=0, yuy IS A CoBoundary :

yuy = Su ( Simonian Cotton.noroot ) . Also,Since f

't

(7)E- HMS"" ' ) =D ( n 7,2) Titone Is An (n- D -Coats .-

x on Sh- '

s- ca Tite f#(g)= Sx

.Note That xuf

#ly ) a f

#tu) ane Boat (2n- e) - (⑥ chanson'S

"!

Now S ( Xu f # ly) - f#(u) ) = 8 ( x u 8×1 - f# ( yay ) = Sxosx - f"(y ) of

#41=0 ( 21 - Coerces )

So THE Coitosnucoa, Ccxss OF THIS Coc -ice E IS A Multiple m f of THE GE,- Enna Sf H2" - ' ( §" )

.

THIS INTKG.sn Is thirteens.ge OR THE Choices Mark t Iss (Auto Tite Hoff IHVAnt Ok f . THIS IS AHomann the variant t So THE ASSIGNMENT V: Tzu. . ( S

" ) → 21 8ft) -- ml f) IS A Homomorphism.

FALI i.n ODD ⇒ 8=0

2 . n Even ⇒ 2 C- Turf

3 . n -- 2,4, 8 t f Tine Hoek MAP ⇒ mlf) =L .

THI: UP To Homotopy Tmz on, Mars Of Hoek Invasion / Are The 3A B- ve. Hence The Orry

Division AwkB.ms Occur In Dimensions 1,2,4,8.,,

14MANIFOLDS-

Def; A NAAN irons Of D ,mission n IS A HausDomek Spack IN IN WHICH EACH Point Has A

NEIGHBORHOOD ltoomfconaaepitc To IR"

.A COMPACT MANIFOLD IS CALLED Cios.

Note : It XE Ah,Tina Hi (M

,an-9×3 ; 2) I Hi (Irn, 1127903; Z ) (excision)

I Iti. .( 112^-903

.

- Z )

I Ain ( S".

- 2)Are So THE Loca Homo Kooy Greeves VAN is it For it n

.

eg : 5,112pm

,Eph

,#Pn← dim -- 4h

① P" : As A REAL MAN Ifans Titis Hm,Dimension 2n

.

INDEED,Tak Stas

Ui = { Ezo: .. . : E) E Eph l Zi = , } Coven Ep" AND Exit Ui k E

"

a 1124

ORIENTATION OF MANIFOLDS- - -

You HAVE Some Experience with THE IDEA OK ①nitration. Dutoit way,

AN Ontario MANIFOLD Hrs AN" Inside " AND AN "

ours , DE"

eg n

. µ ..

"'""" re

⇐ ""

"" "" """"""

A- "

liar- Han. .. " i.

Omkar introns 0k A BASIS IN IRB

WE WANT AN ANGEDRAC TOPOLOGY DRRW,Tow OF THIS IDEA.LET's Start with 112? WHATRukn ANN 0MW TAT.on

IS,ItsHou -b BE PRESERVED By A Rotation Avs REVIEWED By A REFLECTION

.

DI : Are 0rtu Ok 112nA .- X Iss A CHOICE OR A GEN Kantor For THE Group Hu ( IR"

,42^-5×4) E Z

.

Nosy Thar Hn ( IR ",112"

-9×3) E Hm, (SI' )

,where SI

'

Is Tite SPHERE Centreman Arx . Since Rotations

Of 112" HAVE DEGNER I Ams Reflections Hauk DEGNER - I

,THIS OBJECT DOES WAAR WE WANT . NOTE Also THAT

IT Y IS An-1 Other Point or 1124,THEN Ha ( ht , 112^-9×4) I Han ( IR" , 112

"- B ) I An ( ith

,112^-9×4 )

,Witten B IS

A BALL (data ,# ix. 6 BOTH X Ano y .

So AN Onlinetailor At × DETERMINES A- OMENTAT ou Ar Y -

So,For A GENEnm Manifold 1h

,A loyal 0riE Ar X IS A CHOICE OR GENEmtorµ×E HnCNN.lk-Ext)

DRI : An Orientation OF AN IS A Function X telex ,WHERE lex IS A Loca orientation A- x , Suat THAT

EACH X EM HAS A Bau B or Finite Radius A- Bout × Sech THA ALL My , Y E B, Are IMAGES OR ASINGLE Generator MBE Hn (ah

,M -B) UN Dien Tine Natuna Mars Hn ( ah

,uh- B)- Harlan

,M - Sys)

.

THIS IS A LOCAL CONSISTENCY CONDITION.

Eg : kN -- S"

.Let p= North Pole Ams q= SOUTH POLE. For p, TAKE THE NB its Sh- {q} z IR

".THIS

Certainly Contains A BALLOi SOME FINITE RADIUS B.

• P

Hn ( 5,5- B) E Hn.,C5' ) a [a]

B

\ = QB[ I f Soo 5 IS ORK- TABLE

.

&

E

Hd 5,5- his)aK

N'R-TATons Exist : Given A→ AMANI Fory NN , CONS in ,E~ I 5

IN = {µ× ( x EAN t µ× Is A LOCAL Orientation Atx }DEE

, ne p : AT → Itn By p(µ×)=x .THIS IS A 2 : I SortEctiov

.

Cu : Twerk IS A Tbpouoog ON AT MAKING THIS A Couronne Mars.

As A Basis,TAKE THE Follow ,no SETS

.

If BCIR" c M ISA BALL Or Finite RADIUS,Ams µBE HIM,

an- B),

(Et UluB) = {µ×E AT ( xe B Ann µ telex } .It's Ent TO CHECK THA- The> works Ans Tita Ulus) B

Prep : ITN IS OMEN TABLE .

Pt : EACH Mx E Tn Has A Camonica Loca Orientation : MIE Itu (In, In- 9u×3) CountsPOND ,is To

Ux unseen Hn ( In, Fn- 9µs) E Hn ( Ulus),UH - End) E Hn ( B

,B- 9×3)

,Ann Br Construction Timex Also

SATISFY THE LOCAL Consistency CONDITION."Z

L

Eg : M -

- IRP. TAKE A Point y C- IRP

,Ams A Sanaa Bru B A Room It . THEN THEY Are Two Copies OR B

LYING Above THIS IN Tn.Since WE U Engrams Tite Cousens or IR p

'

,WE Must Have In 2 5

.

Ppg ; IF HK IS CONNECTED,THEN AN IS OMENTABLE ⇒ RN HAS Two Components

.

In Part , can,M

Is OrrinTABLE IF IT,AN -- O on Alone GENERALLY If IT

, XN Has No Su DGroup OR INDEX 2 .

Prior ; Iie M IS CONNECTED,THEN ITN Hass Berthier l or 2 Components

. Ik It Has Two Components,SAY

INI Mo IAR, ,Tita Since pl mo: MMM IS A HOMEOMORPHISM

,AN IS ORIBNTABLK (Since Mo Is) .

Ik

IM IS OMEN TABLE,THEN It HAS Exactly Two Orientations (Since kN Connect;) Ams EACH OR THESE DEFINES A

Component OR IN. "

ITN IS A Bit RESTRICTIVE.

Let 's Burns A BIGGkn Covkn Mz → An :

Mz = { xx / 2x E Huhn, M-Exs) , xena }THIS IS AN EN Fini IME SHEET.sn Coven

,2x 1- X . DEE, me A Sect S : kN- Mz By 5 Cx ) = O E Hn (Ah

,M-69)

.

THE RESTOF Mz IS A Cockcrow Or Copies Of Tin : link,1=1,2, ... , any2 ITN Consists Of k - In

ALTE=E D ow : AN Orient OF AN IS A Section S : AN → Adz Such THAT S Cx) IS A

GENE Raton or Hn CAN,M - 9×4 ) For An x EM .

NOTE THA- THEuk 's Notitia6 SPECIAL About Z.

For Any Commutative R, .ve R with l,AN R-0mkTM ORM

Is Au Assia,vnxEn- X telex E Hn (M , M - Ext i R) I R ,WITH Mx A Generator ( i. e.

,A U In R). WE Hak

Title Cover Mr→AN Aws AN R- Orientation IS A SECTION OF This. Since Hn ( AA

,Ah -Ext

,' R)-7 HIM

,ha-*5)④R

EACH RE R DETERMINES A SUBcovering Mr - AN Consist, .u6 Ok thx ④ r , For Mx A GEN Enron or HIM,

M-9×5,21)

It r Has Onsen Z Iho R , THEN F- - r Amo Mr = M . Otherwise, Mr 2 AT Ars Mr = IMr/Mr=N1,

§ AN ORIENTABLE AMANIFOLD IS R - ORIENTABLE For Every R WHnk A HonoreErraBek AN ISS R- OutwitBok

⇒ R Corra ,#s A Unr or Onsen 2 (⇐ char 12=2) .

So Every MAN,Fars IS Zz- OnitaTABLE.

THI: LEE AN BE A CLOSED,Connects n - Khan , Fours . 16

1.IR th Is R - OMEN.-ABLE, THEN Hn (Ah ; R) - Hn ( M

,M - Ex's

.

- R) Is Ann Isomorphism For Au Xena .

2,Ik M IS NOT ORIENTABLE

,THEN THE Nlm Hy (M ,

- R) → Hu Lan,M- 9*9,72 ) Is thisBct we with IMAGE

{ re R ) 25=03 For Au x fan.

3. Hi (MiR)-- O For is n

Iu Panicum ,Hn Cnn

,

- Z ) -- Z or O Aa Hn Cnn ; Zz) -5 Za.

Dick : An ketamine µ E Hn (Mi R) whose IMAGE µ×E Hallin, M -Ex's ,' R) IS A Generator for Au XEAN IS

CALLED A FuNDAerin Cuts For An.

NITE : Suppose 11h Has A D -Complex Structure.THEN µ Is RezonesEarths By Some {Kiri where THE ri

Are THE n- Simplices of An .Since µ Mms TO A GENEnoon or Hn (M

,M-GS) Fern XE Tut (ri )

, we Must Have

ki = I l Fon Au i.Also

,{Kiri IS A CYCLE Ans So It ri to; Stone A FACE THEN ki Determines ks Am

VICE VERSA.

A Khong DEtanks Analysis Shows Tano A CHOICE OF SIGNS IS Possible ⇐ AN OrientAbcB.Ans It So

{Kiri DEFINES A FUNDaanEvin Class.Outen Ze

,Eri Is ALWAYS A FUNDAMENTAL CLASS

.

-

Homology Of MANIFOLDS-- -

(ET X BE Any SPACE.THE CIA PRor IS A Bunim OPED#ou

n : Culkin) ④ Clcxie) - Ck-e (Xie)Dee

, man Fon Kal BY SETTING

rn e -- 414cm

..... Have

....

,vis

r :O"- X

,YE cllxir)

NOI : d ( once ) -- fill ( Orne - rn Se) ( ⇒ Car Provo OKA Gue t Cookie IS A Glock t I.no#o3IjfI ]Pie : done -

- FEI- 'idols. . . ..si . ..> vets ) Mcvea... ..us t IIe 'linoleum

...Wolven

.-fi

..-Nii

on Sy = ¥Ellie blew .. . Ti . -Nett) Haven

. -u'D

compare.W 't ' ' 't

dunce)= it-lather. .

.. .ve)) rlsve....fi

,-Hd

⇒ Insures car Produce n : Hi.Kir)④ Hllxir) - Hu-e (Xie)

WE Also Have RELATIVE Versionsn : HulkAir) ④ He (Xi R) → Hic -e IX.AIR)

n : Hu (X.Air) ④ Hllx,Air ) → Hu-e IX. AIR )

Functionality : f * (a) ne = f-* (L n f'the))

17Ttt ( POINCARE Duality) : IK Hh IS A CLOSED R -ORIENTABLE h - AMAH Ito WITH FUNDAMENTAL

( Cass Cnn) E Hn (Ani R) , THEN THE Mae

D : Hk can ;D) → Hu- i. Carnie)

x 1- Cnn)naIS An Isomorphism For Au K

.

ExaminesT-

- Toros Hom- fa) T Cf ( e) = I ⇐ UkraineDorries line Hase T

CTI -- set -- lait's -a-M as ⇒ t (e) = I ⇐ H * Donahue

"

do,-- 9 - lute ,y

se

drz -- b - ez, ten eij -- Jin of

doz = a - ecstasy H'(Tiz) GEnknn-r.ES By (4) t ft)

Joy = b - easy t e. 4G)n fee) I 6lb) e. y tella) easy - 4lb) ex - Ula) e. y

= O t ez y- O - e.y

= easy - e. y

= b - dry [TIN - : H'

(Ti 21 ) → It,(Tiz )

scans en'

I I :c: l: :L{a)

Similarly, ETI n ft) = - a + do

,⇒ Cnn C,¥,

= - la)

women.... is , sathi Tite Dun Coonce ±'s feta, eco -- i a

ed U -- b -ate (x) -- [ Ut L)O L -- a - btc

gxjn [y) c- Hi ( Xi 2K)⇒ fanged -

- K) -- Eat b)IS U ( b) - C t Uca.) - e = C

=

ALGEBRAIC Doctor

DRI : A DtEs Sit ISA PartiAux Overman See I Such THAT For Am i, i ' E. I , THE me IsAN i'' e I wit 't ie i " - i 's i "

.

Eg : I.I -- IN with THE 05012 Order

2. I-- IN

"wirra ( a

. ..- , ak) s ( b , , .. . bio) ⇒ ai E bi 5=1, -, k .

3. LET X BE A SET A-s LET K Be A Subset

. Leo I-- {As XI KEA } . Define A EA'

⇒ AZA !THEN THIS IS A Dietetics Ster ; Given A.A '

,TAKE A

"= An A !

4.LET ( X

, Xo) BE A Pointer Space t LET I BE THE Saro .. Au Covenant Spaces p : (Eeo) - (X, xd.

Deane ( E,eo;p) E ( E ! eo!- p ' )⇒ F f : (E ! e)→ ( E,e) wit 't pf-- p ! WHAT IS E

" For A Pan E,E' ?

E' '

= { ( e. e ' ) e Ex E' l pce) -- P'Ce ' )} . Co

''s Ceo

,eo' ), p

" 6. e ' )-- p (e) =p'le' !

DER : Suppose I Is A Dickens SET AND {Rdi }ieI IS A Fanny or R - NAD- LIES Such Turnt 18For Any IE i '

,There Is An R- adobe.ve/t0imoimonritism di '

,

i i Nti → Mi ' Such THAT For

is i 's I " Hi ", i' Olli '

,i = di "

,i t Qi

, i= idan ; .

( This Iis Caceres A Direct System 0k Modules. )

A Direct Limit OR TH . > SYSTEM IS A 14100- ee Hh wit 't A Family Of Homomorphisms Cli : MissinSuck THA Vito is

,i= Yi For is i ' ,

which SATISFIES THE Foc cow , no UMP : For Any lidos-re N

Ann Fannie, or Mars ti : Mi →N SATISFYING t.io Cli ',i = ti ,

is it,THERE IS A UNIQUE Homomorphism

+ : the N Suat THA ti-- toy; For Au i . WE WRITE KAI Huy Mi .

Pryor : THE Direct Linnet Exists.PI ; LET Kate ¥±Mi with 6¥ : Hi → IN THE Inclusion AsTine ith Factor. LET N De Tine

Submodule or ant Generates By Au 6itc@ii.iLx.D - UE (xD For EE i'

, Xie Mi .SEAN= Ant IN

,T : hate AN THE Quotient Mar

.LET Lei -- to it

. THEN ( kN,Gili's) Is THEDeke .- Linn

Prof : Suppose I Hasa care.es.- Beemer m .Tuan Ting hear @in

: Adm→ light Is An Isomorphism.

PI : For Au i,WE Have @m.i : Miss NAM Conn Parsee with ti : Adi → th .

So WE Have A

UNIQUE Map 4 : Nl -- lies Adi→ Mm.By THE UMP

,Aus ti ⇐ toQi ⇒ tu

, ie t - Yi ⇒

id -- Lu

, m-- to

in .Titus Clu Is Ihl Ther - uk t t Is Surjective . D-i Also

, Clu Is Surtout wet to

Is INJECTIVE. y,

lA : Sunrise Thx For tacit if I,Mi -- Ni⑤ Pi Ave For is it

,Tine Mar di '

,i Dk composes

Accountably : Cli;; -- ti : i to fit, i . (Er N -

- Ing Ni Aris D= Ing Pi , So THAT Wx Ger Insures Mars

µ : Ns M, e : P → an S-ca THA tti -- film, a eli =4i/p

.

..

Then t ⑤ e : No P → IN

IS AN Isomorphism.

If: Construct THE Iv ukase As Fouows. Given X EM,Choose XIE Mi with X -- fi (Xi) . Whine

Xi -- Yi + Zi UN auk-y WITH YEE Ni, 2- i E Pi .

Define @(x ) = ( ti Yi , fi Zi ) E N① P . IT 's Etsy to

Citizen THAR O(x) Is Insane→But OK THR (Hoick OK X i t O Is Ihwirnsis To t ⑤ f. ,,

Dej , A Ster Tc I Is Finn IT T IS A Dink ones Ster brr.EE THE Inducts Orman Ans

IT For Am i c- I,Thieme IS A JET wit't EE j .

Form THE Limit Ovkn J.WE GET A Camonica Anne f : life Mi - they Mi

(¥wA ; t ISAO Isomorphism.

PI : Lfo an'= long Mi , d ;

''

- Mj →an ' Tite Ca-onion ANAP ( so do'j= Q; VJEJ) . Gwyn xexn,

write x -- Qi (Xi), Xie Mi . Choose JET with ies .

Then x =P ; (x; ) , Xi-- Rj

, ; (xi) ⇒

X -- de ! (x; ) ⇒ X Suntiectiuie. It Xx ' -- O

,write x'= es! (Xi )

.x ; EM,

-

.Titan 6,

- (x; ) = O .

SuBLkmmA_ '. It Qi (Xi) -- O , Theme IS A- i ' , if i'

, wit't Cli '

,i (Xi) -- O .

Assuming Tuns,There Is A- i 'EI

, j si'wit't Cli '

, ;(x ; )

-- O

. Since J Is Finn,THEME IS A j

'EJ

wit 't i 's 5'. THEN QI

, ;(x; ) -- Aj

, i, ( x; ) =D⇒ x'= Qs, of ;; (x; ) = O .

So d IS INJECT 've.

Pio's SumA : S >nice Qi Gil- o, CLE (x;) ISA Sum Or Elements 19

GE Chi,klyii

,# ) - 4h44 ii. k) , Yun. Ethic

By Tite Dna - tier or clit,UR Hms H) Xi

-- if , 4inch i. K) - £, 'hi, i

wit we Fon h # i ⇐*) O = ¥+444 (Yuk ) - ,€hYk'in(Hoosier A- Innis x i' Greenan Titanic k

"

which Occur . Arrive ceil, ;To *) Ars fish To (* *)

Ano ADD . THE 12kWh Is Qi '.ilxi)=(§f, di 'ni Rd,

, kid,u) - I ii.k (yw .ie ) ,WHICH IS 0 BY DREW

. now.

,

Suppose WK Hauk Three Inactive Systems on I w -Tut Homan owns.us Ad ! Isi Mi dis Mi"

SCH Tano Koch SEQUENCE Is Exact Ams Such Tite Foa ie i'

, Mi' Ii Mi dis Mi

' '

Press Tf Tite Limit : M' Is in Isn" di :it pi::L Lei:c. Commutes

.

dei' -- cedi,e Yi -- ce.iq. Hi Mel Isi' Mi. Ei mi's

(Emma THE Llano Shauna Is Exact.

-i

PI: Exercise ,

Cory : If EACH fi Is Initiative,So Is f . If RACH di Is Surjective

,So Is A

PRI : (Ks x Be Any Point IN A HausDomer Space X. THEN

y : the Itu (X , X- U ) → HIX

,X- Sd) Is A- Isomorphism .

X EUOPEN

PI : Since WE Have A Mar Hn ( X,X-UI → Hn ( X

,X- E3) For Any Such U

,we Ger Tine

Mare Q.RhamesEnt A RELATIVE n-Cycle IN (X

,X- Ext ) Be A Finite Sum Of Sonoran Simplices

.

Time Imma 0k This Is Coon pros Are So Lian In. Some ( X,X -u) ; Tms Imputes Q IS

Sortie cave.For INJECTiv.it

,If A CT cue In Some (X

,X-H) IS A Bourbon In (X

,X- 9×9)

,Totten Compactness Forces IT TO BE A Boomsma Is Soan , ( X,X- V )

,U E V ⇒ It Is Zeno Io

Hug Hnlx, Knut .,=

Back To Dutch-1--

Walk Actuary Prove A STRONG.hn Statement.

Suppose X IS A Spack,Nor NECKSsaucy Compact .

THE Compact Subspaces Of X Form A Dirkcites SYSTEM UND En Inclusion..THE MOD-res

H" ( X

,X -K ) THE Form An Inactive System : K E K'⇒ X- K 's X -K ⇒ ( X

,X-K

') - (X,X- K)

⇒ H"

( X,X- K)→ It

"

( X,X- K

' ).

DKK - E THE Cottonmouth with Compact Supports :

HI (x) -- lip H " (X,X- K ) ( Ik X Coomera

, turn It! CX) = H" Cx) )

( Krinke Rickey . A Corto.no Loo> Class Ix H { ( X) Is Rtrrhsssiutiss By A Cochet , no Tetro VANISHES-

Off Some fourth Subset K j Ie. It ANN, itunes Acc Curious with Support In X- K.

20Functor : Gwen f : X- y,THEN For Any Kc X Competes

,

f-(K) CY IS Coomera.

B f holy .ir Nos Mar X- K Inis Y- f( K ) t So WE Mio .to Not Ger Am I.ua#Es Mar ONIt !

.

If WE Assume THE f Is Proper ( CY ⇒ f-' '

( L) Compte .- In. X,Then f-

annus X - f-'

(c) Into Y- L ⇒ Ahly,

Y - its HMX,X- f

-'(4)→ HIX) ⇒ FI : Andy ) → HIM.

NOI .. If u CX Is one we CIN Deane HI (ul - HI ( X) : In Kc U Ies Compte,we

Have Tite Inverse Orc Fine Excision Iso H"

( U,U - K ) → H

"

( X,X- K )

,Compatible worn

THE Inclusion Homomorphisms. Do snob To THE limit GivesA UNIQUE RAAP MAKING THE Dltbnssn

Commute Fox Au K : HI (ul - HI Cx )T T

H"

(U,U-K)- Hncx

,X-K )

eye. Liu U = 42" Cons iron.sn As S

"- 9×4

.Time Sars S

"- K Fon Compact Kc u Fon.ua

SYSTEM OR N Bits Ok x with cut Contains A Finite SYSTEM OR CONTRACTIBLE N Bites.

For THESE,

HEC s",S"- K) H' 45) Is An Isomorphism tso H { ( IR" ) -= It Els" )

. "

Dej : IK X IS AN n - Dimension AAA Nitrous,F.x Ann R -Orkut#od t Leo [X) BE A Funston Evin

CLASS In Hn (X ,

- R). If Kc X Is Compact

,Denote By Ex)z The tanto, Ove Ix) In. Hu ( X

,X- K )

.

WE Now Hwa Tite Recurve Cnn Pro.- ca : G)kn - : HECX,X - K) → Hn

-glx)IR KEK ! The Ditonnm It

'(X,X- KI Mun -

T \ Hn - q (x )Communes

.

H4x.x-iilf.in .Pass into T Tote limit we Get A Homomorphism : D : H{ (x) → Hu

-z(x )

TI : IF X Is An R -Orient.es n - MANIFOLD,Tina D : HI (x) - Ha-glx ) Is An Iso Fou Au q > O

.

IDIt0Pi"

As K ENLARGES To Become Au or X ,THE Moores

-.÷€. . . .

"""""""" """ " """"

(Ei K = Compact Portion ABOVE DASHED line . A RinksButtrick Ork THE Dun or A ONE - D)omissions

Homeobox Cates Is Shown ( IHRE ) .Then H'(X

,X-Kleiss.ae/tECw,-w-k)#s-Hu-z(w)

worksSo Cornishmen To G.to .no Loo> OR X SEEN BY HE ( X , X- K ), Titanic Is Homoway or X witness Sorrow↳ IES Ih W . knln-no.rs K By Unions PRODUCES B- LANG Ennius OR W

.

PRook One Poincare Duality 21- - - -

Steel : If THE Titanium Hours For Oreo Stars U ,U,t B -- UN

,THEN It Horns For YI Uu V

.

(Er K c U,L c U Be Commer

. Use Motyka - Viktoris For Triple ( Y,Y - K

, Y - L) '.

. - -→ HE ( B, B - kn )- Held ,u -e) ⑤ H'(V.v - c)→ H'(Y,y - Kui) - HE " ( B , B - line ) - -- -

Nunn- L Mun- ④HI - CHI L Hun

- .. → Huq ( B )-Hn-Elul Ha-eh)- Ha-2K ) - Huq, (B)

The Tbr Row IS that Bu - Virions t Excisions Of Forum (W,W- s) E ( Y

,

Y- S).THE Bottom Row Is

1Mt-ibn -VIE tons Fae ( Y, U ,ul

.Commutativity 0k Two (Kit SQUAME, IS NATURAL#7 Ok Car Product

.

£BLnnna : THE Follow , no Dharam Communes we To f - 1)Et ' H'(Y

,Y- Kul) Is It

"' (Y,Y- Kur )II

/ Hunn - H't' ( B

,B -Ksk)

Assume -G Titis,Notre Tita Kuku

, Compact Ster Iv Y Has THE V t

%' 'm Kul

. Pass, .no To Tine Cima .- Gives A Sion- Commutative Hn-g ( Y )

Nd Hh-E- i ( B )

DAA Canon :

. . .- HI CBI → HE cut ⑦ HEH) → HI cyl Is HITE )- . . .

D L ID D DI L- -- s Haz (B ) - Hn-du) to Hn-eh) → Hash ) Is Hu-g. .

(B) → - -

BY THE Five learner, It THE Tutu Hours Fon U,v, B, It Huns Fon Y

.

Step : Lee Gui} Be Assam OR OPEN Stars Tbtauy Omkar Bt Ihc erosion & Lk. Ul = U Ui .

It THE Tithonian Howes Fon Kaat Ui,It Hours For it .

THIS Amounts Tb UkniE4 , no Isomorphisms to,i Iif Hn-q(LAD → Huzur)

tan. ↳ it ! cut - H :(us

Note The Fae A-y Compare Kc U,we Have Kc U ; For Soan , i ( b sie Tite Then One kn)

.It Fouows

Tans 4,

IS An Isomorphism B, Cons , Diarra Tite Coomera Support OR Any CHAIN. Similar Arcement Forte.

Step : THE Tithonian Is True If U IB Containers 5nA Coordinate NB Ito.

Rk ornis UE 112".IT U IS Convex

,THRU U IS Homeomorphic To THE Interior Of The Closers

n - BALL D".In Computers The Linux 1¥ HEL 10

,D)

°

- K ),IT ⇐Facts TO Cons . Dan Tine Finn

System 0k Cluster Baus OR RADIUS L l CK~tqn.gs Ar O . For Such 14 THESE MuDorks Are Au 0 Except

Fon g -- n s (x)kn - i Hn(DCT,@" T - K )- Ho (①7)⇒ R

,Is Eteria . we > An Isowww.trs.n ( Tite

GE- Error Of It"

TAWES ✓Awa l ou ①"T t By THE Dkkcrit, on UK Car Provo, UK Ger THE Vkntisx Gerner no

Ito (④"t)).

It Follows Tans Tune (limit . if Adar IS An Isomorphism.

If U Its Not CONVEX,known# name A DE-se Soo Of Points In U Havin Ration Coordinates 2

(Hoosick A Convex Open Sto Uj Containers In U About THE j th Dani . (Er U,= V

,t Uli = Ui - , WE

,ist

Tite THEone,m Hours For M, .Assume Ianwww.zu, It Horns For A Union of Kei Convex or .sn Stars . None

Tito Ui -, n Vi IS THE Union Of Ar Alost i - I Cowin OPEN Sk.-s .

By STEP I t In Doctor THE

Titan Hans Fa Ui + THEN Ba Step 2 D- Hours For W.