Stl: The Turonian Is True For X. 22

By Zorro's (Kannon,Twerk IS A Maximum OPEN U E X Fer wite cat THE Tithonian Is True

.For Any

✓ Containers IN A Coors ware NBin,THE Tithonian Iss True For V

. THE Titan Is Totkw True ForUu U t By NNAX morality

,X= U n g,

Cory : Ix X Is Connie cries Are R -orikurar.us,HI (Xl E R -

"

Cory : If X Is Compact Ano Omkara BEE, Gg = fn -q For Aug

TEI Tn-g - i ,Tq = Torsion Su Boros or Hg .

↳ : It X IS ODD - Dimensions,Compact

,Orientate,q Thieu X (X) =D .

(SA> dinX=n -

-2kt' )

as: xcxt. Efim -- Eiieniei + Eiichi= IE

.

sit !E! " ilium - i

= II. di + E.fi"si= O .

=

Geometry QK Poincare Dutoit- -

Suppose X IS A Loca a > Finite Simplices Complex Ann (Kt Sd X BE THE First Bnn> centric

Subdivision.TheSempach, Of Sd X Ark Or THE Foran fi

,Fie - - - fi

,w 'tKuk ri

,

> ri,> -- - 70in Are

Fi;IS THE BANCENTER OF rig . Partway 0hpm THE Vertices OR Sd X By Dk cries ,nib Dinners 'on 0k

The Simplices Ok Xl Of Uttech THEY ANE THE BAR- centers. This Induces A linen Ones knives On THE UKraces

OR fetch Simplex or Sd X .

Die : Ginsu r In X,Sir DG) To Be The Union Of Au Oren Sonnies Of Sd X or what F

IS THE FINAL Ukntizx.

DG) IS CALLEN THE Bijou Den Thor.Dual or A 2 -Simplex Is Its

#•.

•• a X BAM CENTten .•

E¥¥¥¥¥f¥# six

EE.rs••b

✓,DH D cab)

TIM: l. The Dun Brock, Ana Destour A-o Titian Union Is 1×1

.23

2 . DCI,For A K - Simple . r

,Is A Polytope or Dimension n - k .

3 . D G)°

Is The Union or Au Blocks DC t ) Fon ter .

4.Ix Hi (X

,X- E) I Z Fon i = n t 0 Otherwise

,Tina (DTR)

,

DKP) Has TheHomo Looy or ( D

" -k

,JD"

-k).

THE COLLECTION Of Dun Brocks IS CALLED THEDUI Buick DR oar .THE Union or Blocks 0k

Dimensions Ar Most p ⇒ DE-ones Xp ,THE Den p=Sxu or X

.The Dear CH Coex

DIX ) Is Dee , ran By Dp (X) -- Hp ( Xp, Xp -i ) , with 0 THE Connect. -6 Mm Ok ( Xp

, Xp-i, Xp -z ) .

The : H*( D¥x ) ) E H * ( x) .

Dej : (Er X BE A Compact,TRIANGULATE, n - Manitou. X Is OMkNtE I' It Is Possible To

ORIENT Au THE n - Simplices ri Ss THA- T-- Eri ISA CYCLE.

Titan (Poincare Beauty ) : Ik X IS Omkara Bek, Tina Fort up ,

HP (Xi G) I Hn- p ( Xi C ) .IT

X IS Honoris- Tabuk, THE- THE Titan Hours for G = Ez.

Pie : Thema Is A l - I ConmsspouisEnce Cp (X)- Du - p (X ) Since CP (x) -. How ( Cp CX) , Z),o - Dlr) .

we See Tita (XII Du - p (x) Via o't is Dcr)

,A Generator OF Hn- p ( DII , DGP ) .

THE Truck Is To choose THE Sion d k Tite Generator So That f : C"- Dn - p Marcks THE Dhamar

Commune : CP-'

(x ) Is Dn - pt. ( x)is toCP (x ) Is Dn- plxl

To Do So,Online Tite n- Simplices ri So THE J -

- Eri IS A CYCLE. Omkar The Other Simplices

Arbitrarily,DEE, we f In Dimension n : Ik T IS A- n - Sampley

,

Dlr ) -- I,A vertex . DEE. we

(F) = on,A Generator Or Ho (F) .

Dimension n - l : Suero sie s Is An Orientis @ - l ) - S impugn. Neko Tf DEF. -e y ( s't ) So THAT

d ( s't) = Y S ( s't ).

Now,S Is A Face OF Exactly Two n - Simplices Go

,r

. .Since 8 IS A Geek

,

THEY Ame Omkar So Thar doo t do,Has Correia ,Err O ou S

. Assume Dro -- - l And do ,-- l ou s

.

Then SSH 6,7- of t So 418 s't ) = F ,-fo

.

we know Thar DTs) IB The Union or live Siemens

From 5 To fo Axs E , . SEE U ( s't ) = Cfo

,5) t [ 5.E.) .

Now Procures Inactive to Drive if IN

Au Dimensions. "

=

⇐ re GID' te CLCXIR)

,ye C

"CXIR )

,LEG, (Xi RI inner Hance)

-- Cceutka)

.

Pie : Ix r :oktl→X ,Tina floret -- theCola

. ... ..us/okvy...u...eD--eklcvo....ucDtkkuk....uk+es)

= @ t) . "

In other words : The war 6 u : Cl- Cktl Is Hoon - Dum To ace .

So Passive TO 24Homolog

,WE GET A Commutative Dinoran Hl(Xii ) 4- Hour ( He Hit) , R )

① do I .( nee ) 't

Hktfxirlh-ltomplltu-ek.tl),R)

IX IU Is A ( Costco,

R- ORIENTE- n - MAN irons,Considine TAE CUP Promo PAIRING

H'' (Mir) x Hn- '' (min) - R

⑥,t) - lat) Cnn)

DKK : A Bilinear Paria p : A X B - R Is Noriko .Ew knee IT Tite https-

A- Hour (B. R ) B- Hour (Air)AND ANE BOTH Isomorphisms,

a ↳ { bits plats)) b - fats plaits) }

Prot : Ix R Is a Fikes THRU THE Cur Prop-o Parma Is NONDEGENERATE. Ie R -- Z THEN

IT IS NonDKG.euEnate Modulo Torsion.

PI : Cons .am H"- "(min) hrs Hour ( Hu-admin) ,R ) Hon

" ( H''

(Mir),R )

we Hwa LD't ht ( t) = { ers + (Cnn) nee) } =@utllCnnD.UustmToteHyrotitBses.hIssAISonnoihPitrsom.Nown Ebisu Erney IS ONE OF THE Variables IS EQUIVALENT To D't BE, An Iso

.

Nono Ebbs ferry IN THE OTHER FOLLOWS FROM Commutativity Of THR (of Product . y,

↳ : Suppose 1h Is closes,Connieerrs

,Anis OMKNTABUE

.A - Rue.me#dEHkCMi2)GExiERnr.ssAv

IN Finite Cyclic Sumatra ⇐ THEME IS A RE It" - "

( ah ,-211 Such THAT LUG Generates

H"

(Nh; 211=21 . With Fi Eun Coefficients Titis Horns For Any 240 .

PI : L Exists Ice : Hk (Mi 2) → 21 With flat = It . B> MONDEGkn tenner OF The Cup proper,

Cf IS Rtauzio By TAKING Cut Prob-et W ur 't ft H " -" (Mi Z) t EVALUATING ON Cnn) .So THE Existence

Of B W '#It d - f GE- Emm .ae It"

(M ; 21 ) Is Evolvement to The Existence or 6 with Celal = El . "

Cory : H't

( Eph ; 21=265/2 " ", dega -- L

IR : THE Inclusion ①p""- EP

"IH Ducks AA IS Omori Ihsan ON Corto.no Looy it

?"Fou ich t So

H"Is GB- Enman By a

" For ion ( Inaction Starts Ar n =L : Ep 1=5).

B> The Coronary,Thane Exists

ME 21 with a - man-'= ma

" GENERATING Hln ( Eph,

' 21 ).

But Timer m= I + THE RESULT Follows. ,,

Cory : H't( IRP " ; 2h ) I 2Nd,/ anti ; dega -- I

Buk - Vien Tem : Ix n > m> l,Thieme Is No MAN g : S

"- SM Commuters with THE #

ANTIPODAL KNAPS.

PI : Such A g woo is Dupre By Passero , tf Tha Quotient A Mar f : IRP"

-IRPmhhn-au.tn#Din-oramCommute.

S" Is Smf !-7

(Kimura: Turenne Exists A Lier f': IRP " - Sm wit 't p f

'

- f / ,ptp! 1pmPI: USE THE LIFTING Criterion . Ml m=/

,

Then Since THE Orry Anne-

it,( 112PM ,) - it , ( IR p

' ) I 21 Is Trivia,WE Have f

"

.

Suppose m> l. The Map

f't : H*

( RPM ; 2h) - H't ( IRP "

.

- 2h ) Isa Rino Homomorphism.But O=f*(am") -- ( flat)

" '

Implies THAT f* (a) =D Since n > m .( Er i : IRP

'a 112pm Are j : IRP

'- 112pm Be The

D-' C Kus ions OB Taiwan By SETTING ALL but Tite First Two Homoodorous Coomer rares Eaux Tho O .

We know J't: H

' ( IRP'm ; Zz) → It'

( ELP'

; 2h ) Is Av Iso t Ss j* lol I 0

.This Implies Tota

j 't # ( fit'tt so f- if j .

But i t 's Ane GE- E Raions or Tine Furman Enron Groups .It Follows

THE f- * : it , ( IRP" ) → it

,( 11pm) Is THE Zkno Mae t Jo Tite (IR Tene Criterion APPLIES,

Now,WE Hnk p f- 'p

'= pg .

De XES"

,Either glx) -- f

"

p' Cx ) on GC-x) : f

'

p' (x) = f'

p' C-x) ,

Titus,f 'p

't g Ane

Two lifts Ot f p' Witch Aorta As A Point t By UNIQUE lifetime f

'

p'-- g .

But

g.(x) t gl-x) = -glx) WHILE P 't# =p 't- x) , A Contradiction -v

-

DRI : AN a- MANIFOLD it Bout7 ISA Hausdorff SPACE IN IN WHICH Exert Powe Has A

NB HD Homeomorphic To Kirsten 112"on IRI-- f (x. - - , XIE IRN / Xn 30} .

NEE THX IF X Conners Pons, Tf A Point W tiene Xu -- O,THEN Hu (M

,M - 9×4

,

' 2) I HullRI,1124+-9037=0

But It x Hn A Nato Hoouieonaourixc To IR",Hulin

,

ha -Ext ) Z . ( Rt DM -

- fxc.lu/HulM,M-Exsl--O2y.

Titis Is Caxias The Boundary OI th .

=

Miscellaneous Tbpics- -

Universe C ur Tim tIHooT

we know It" (Xi 6) ± Hour ( Hulk

,G) to Ext ( Hn -ilxl

,G )

.

Is THRU A Similar Formula Fa Hix, ?

Akx: YES,Of Course. Cons -Dien THEShore Exact Skagerrak OF CHAIN Complexes :

O - Zn→ Cn Bn - i → ° Note : Cn I Zn⑦ Bu - i

+ on Ldn Ldn .. Io But (* ¥ 2-* to BetO → Zn- , - Cn-i€'s Bn -z - 0

Now Arrive - ④ G : O - Zn④G → Cnxo G ¥1 Bn. .

④ 6 → o

Idiot fdn④ I Ldn- i④ I

O - 2-n. . ④ G - Cn .. ④ G€8' Ba -a ④ G → 0

THE Rows Are Stice Exact Since ④ Distributes Outen ⑦ . Tine LES In HomoLoon Gives 26

- - - - Bu ④ G Zn④ G - Hn ( Cx ; G ) - Bu.. ④ G Zn . . ④ G - n - -

Tins Brutes Into Shorr Exact SkyviewCES :

c)→ coker ( i. ④ i ) - Hn ( Citic) - Kerlin.. ④ D - O

Llerena : IK Ain B Es (→ 0 Is Exact , So Is A ④6¥! B* GE' C④ G - O

.

PI : Exercise. ..Cory : Coker ( in④ i ) = Hnl Cx) ④ G

PI : Bn → Zu - Hn ( Cx ) → 0 Exact ⇒ Bn④ G - Zn ④ G - Coker ( in ④1) - O 'Exact

⇒ color ( in ④ 1) I Hu ( Cx) ④ G . "

Waar About to (in ④ i) ? O - Bu 2- n - H - CCH → o : Arra - ④ 6 :

O- Kerlin④ l ) - Bn ④ G → En④6 → Hn ( (* I ④ G - O

General Stour- -

(Ko H Dk Au Abierto Gree .Construct An Exact Skowronek

. . . - Fa - F. - Fo - H - O

with Each Fi FREE Abidin.

Such A Skewered Is CALLED A"

Free thissuction ' ' of H .Appu

- ④ 6 To Gao A Chaw Complex :

. . . - Fa ④G - F, ④ G - to ④ G - H ④ G - O

Doge : Tori ( H , G) = ith Honore, Orc Tins Complex.

None The Toro ( H ,G) = It ④ G .

Fh: I.It we Choose Another Resolution FI → It

,Uk Ger Isomorphic Tor Groups .

2 . WE CA- Always Fins A Resolution Or THE Form O → F,→ Fo - H → o

+ So Tori ( H ,G) = 0 For i > I

.

Titus,Kerlin ⑦ i ) -- Tor

,( HnCC*)

,

G).

Us Rsn Coecikm- The Ix (* IS A CHA . - Complex Of fake A- Bhanu Groves,THRU

THEne Ane Natura Shorr Exact SEQUENCES

O- Hu (G)④ G - HIC# i G) → Tor, ( Hn- ' ( Cx), G)→ o

THESE Split,But Not Naturally

.

How To Compute Ter- -

P¥ : i.

Tor,( B.A)E Tor . (A ,

2. Tbr.( to Ai , B) I Tor

,

(Ai,B)

3 . Tor , (A , B ) = O Ik A or B IS FREE ( ndone Generally, Torsion- free) .

4.Thor

, ( A ,B) = Tbr. (TCA), B ) , Witten TCA )

-- Torsion Soap OR A

5 . Thor.( Zn

,

A) ± Ker (AIA)

PI : THESE Are All alone or Less CLEAN.(tho 's Do #3 . TAKE A -- Z

.A Resolution or 21 Is 22

O → 2 → I→ O ⇒ O ④ B - I④ B - I④ B - o ⇒ Tbr,(Z ,B)=o . Now Merry # 2.

( O l l

O

#I O - Z Ens Z - In → o

⇒ O- Z ④ A ''2④A

⇒ Tbs.( In

, A)-- Ker (A A )

112 112

A A

eg : Thor , ( Zm ,Zn ) I 21 gcdcm.nl .

Coe : Hn ( Xi QI = Hu Ki 2)④de ⇒ din# n ( Xia) -- Bn .

=

K Na Eat Formula- -

(ki 's Drive Tune Cross Prome .- Ih, Homoway : Hi ( Xi R) x H ; (Yi R)→ Hit

,- (XxYiR).

↳~ Do Tht. s IN GEN Ensz W STA S IN ⑥wenn HomoLoot , But It's Monie ENLIGHTENING TO USE CELLULAR Honewort.

Considine (* ( XxY) .

It ei IS A Cku In X t e' ⇒ A Ceu Ia Y,Then eines Is A Ceu Iv Xxx.

Au CEus DK Xx Y Occur This WAY. We Nicks To Get THE Bounsame NNW In. (* (Xx Y) t - TermsOf THOSE In. X t Y .

Ci (x) Has Basis Seib,But THEme IS A SIGN AMBI Go .-7 For The Basis

Ck u e": THE CHOKE OR Generator 06 Hi (Xi

,Xi - ' ) Cornicheonions To e

".

Such A Choice Is Caves" ( Hoosiers An Omer- moron Or e

" "

.

Q : How Do Orientations or e"

t e' Determine one For eixe' ?

A- : e"← orders Basis Of 112

". So A-no Orientation Of XIE = Ii" Is Obra , was the CHOOSING

AN Ominous Basis Of I" t ONE OF - THEN Concatenated.Reuters , no kitten on It Reverses

THE Pros or So WE ONLY Neko To worry About THE Onsen OR Factors.

PRI : THE Boundary Mar IN Cx ( Xx Y) Is Determines By Those Ih Cx (x) t (* ( Y) VIA

dleixe' ) -- deixeiteilieixde.

If : Do It for In +THEN Carefully CHECK THINGS USING NATURAL it-1. "

S. we Get A Wku -Demuro Bilirubin Mme Hi ( Xix) x H ; (Yi R )→ Hi*j (XxYi R)⇒ A Mar ( Hilxirlxon Hn-ily

.

- RD - Hu ( Xxyir) .

THEY Brian : Titis Map Is An Ison.no mention . Arts, 1¥ .

But : Ca ( Xxx ) = (Cil x)④ Cn - i ( Y) )& dleixe dei ④ en - i + ← lie den

-i

eixe"-i ← ei④ en

-i ( TK- son Peover Or Bouman Mars)

So,(Er 's Generalize : Ix (* t (*

'

Ame CHAN Complexes OR R - Morones,THE TRIMPIE

Is ( (* ④pci)n= IO (Ci Hii ) wort d ( c ④c ' ) -- de ④ c ' + C-Dkk ④do' .

I

d4c④c' ) -- d ( etc ④ c ' t " cxodc ' ) 28

= dlc ④ c't

"" '

dcxodc ' + C- 1)"do ④do' + C- it ④ dlc '

= O teal' " - 'ok ④ do 't fill" do ④ do ' to

= O .

⇒ we Get Annas ( HnCC*l④rHn.ilC*'ll- Hal C*④rC*'

)

Titan: IF R IS A PID AND THE R - Moores ( i Ane Freie , THEN THEME Ane Natura Starkers Sees :

O- (Hila)④rHn- i (CI)) → HnCC*④nC*' ) → Tori ( Hila), Hn - in ( CED - O

P : S ,mum TO UCT ( INDEES.Case (*

'= G I- Decorous 0 IS THE UCT) .

,,

KNNE THI : IF X t Y Are CW-Complex.es Arlo R Is A PID,Thame Ane SES :

0→i⑦( HitXie) ④n Hn- ily;D))- Hnlxxyir)- IO Tor? ( Hilxir) , Hn - in Hirt) → o.

Cory : the F Is A FELD : ( HilliF) ④, Hai Cy, ) - Hn ( XxYi F) ISA- Isomorphism . "

eg : Hn ( Risks?- al -. 1,26 ,Hnl "2545721) -- f §, ,

O n =L

21+0212 n =3Z n =3

O n -- 4Be n=4

l Z n -- 5( o n-45 I O n > 6

So 11282×53 4- Rp3×52 Ewer THOUGH In ( IRPZX S3) E Tn ( 5) x in 3) I itncslx HRP 's ) for us ,

& it, ( 1,2 Phx S3) I 212 I it,( S-x IRP

's).

Titus, Tith# Is Neo Mar f- i 71282×53 - R p' xs ' Iupui ,.us/Tpesg

,

t

=

HOMOTOPY THEORY- -

GOAL: ST-by THE CAIRGorey Of C W -Complex,hS UP To Honn OTOPY EQUIVALENCE.

RECALL :

Homotopy Extension Titan : Gcuxeu A Cw -Pain ( Y,X)

,Anne f : X → Z Aus A Homotopy

f : Xx I - Z From f -- F (x. o) To f '= FIX, ' ) , Tithes Is Aw KxtEwsw G : Yx I - Z or

F ( Glx,D= fix.Do For xe X).

Ito Homotopy Tineo .ve WE OFTEN NEED To HAAKE A Construction Regent, ok To Amar f:X-51.

WE'D LIKE the clusters,If Possible

.

Itm : Give- f : X- Y THERE IS A Space Adf,Tite Nhrr £DEn OI £ , Which

Fits INTO THE DiAbram X↳ Mf

µ iyny,W "Kuk Ti 's Ark Inclusions ,t IT t i Ame Homotopy

INVERSES .

It : Define Adf -- (XXI)qY Xx's

j :X - Mf Is x - (x. Bxx Y

Ti : Mf → Y : Ti ( x , t) = fix) t Tty)⇒ . " / ( x. D nfcxleyf. (x )

Note : Ik f ISA Chuan Mae, THEN Hlf (A- BE Given The Structure OR A Cw-Complex.29

RECALL Homotopy LIFTING : A MAP IT : E - B Has THE Homotopy lift PRo-y ItGives A Space Y

,A Map f -

-

Y-E t A Homotopy Gt Ose g-- Tof

,THEME IS A

Homotopy ft Ot f sunt Thar To ft --SI EF bit

Y - Bg-- So

DRI , IR T: E- B HAS the Homotopy lierne Property,Tried IT IS Causes A FiBRm .

THE Fiber Is Fb = it -' ( b).

FAIT : Any Two fibresAme Homotopy Eau,utter- Pennies B Is Para Connects.

(Rs F BE

Any SPACE Havin THE Hormone> TYPE OF Fb.Noir, ow : F → E

ditB

EXAMPLES-

I. Co uterine Spaces I:( I

,

ID → (X , Xo) .THE Fibrin T

- ' (xo) IS A Discrete Skt wit't

[AND wait-1 [ IT,(x) : it

, TX)) .

2.locally Trivia Fiber BUNDLES

Term , no loot . Ik G IS A Tbeoroo .cn Grue t F A spark, AN Actor Of G on F IS A MAP n : Gxfsf-

Soo#Tith n Is Continuous, Ne, y )--Y,719,92

, y) = 9 ( 9, , 'll ga , y ) ) ht g. g, EG, YE F. THE AereoIs Effective Ik gy -- y Vy c- E ⇒ g -- e

A (Ottery Trivia Febkn BUNDLE Consists 0k

I.A SPACK E

2. A Senate B

3 . A ANH IT -- E- B Caceres THE PRoe-

4.A Spree F Chucks The Fiber .

5. An Reflective Group Action OR ⑨ o- Fi G Is Cavies THE GRIOT THE BUNDLE

G. Are Often Coven { Vj}jet Of B .THE Us Amie Causes Corrine Nims .

7. For EACH JET, A Homeomorphism Uj : V; XF → IT

- ' ( Vj ) ( Cacus THE Coordinate Fun).8.THE Q; SATISFY IT Uj (x ,y) -- X , Xt Vj , y EF

9.Define Qs ,× : F- T- ' (x ) BT Q; ,×(y)

-

- es Cx,) . Tina For Au i. g- EJ , XE Vin Vs

THE Homeomorphism Q×ofi. × : F - F Coincides WITH THE Action OR SOME g EG .

10. For Au is- E J, gji : Vin Vj → G Deceives By 9ji (x) -- I × - Qi,× IS Continuous

.

A Skc OF T -

-E - B IS A AAN s : B- E with ios -- id

.

'

: l. Promo Bowser : IT : Bx F → B

,G -- id

30

I -t-

z. Mops.us Bars .

B -- s ',E- B

f.IS#.+Note Tttnr Am, Two Stations Must A- Grier Ar At LEAST ONE Point

.WE Have 6--2/2

,REFLECTION

Across Midpoint Of F.

3 . Kcrw Bottle it: K - s'

,G -- Ze ( Hanno To Draw)

4 . If F = 112"t G-- Glu ( IR)

,THE Grier Of nxn Inventive AMATRicks

,Titter Such A Fiber

Bonnie IS CALLEN A VE Brink .

eg : Lhs M BE A cases n - MANIFOLD.At EACH XEN WE Have THE TANGENT SPACE

TIM.

As A space This Is Just 112". THE TanoErr Burak ¥Hf⇒ETxm- -

Iss TM = { ( x. u) ( x earn, VEIN IT : TM- M,think 18,814=-1

A Skc' - IS A Victor Fie on M ITH' ' " Brea Gutt) )5.If I : E - B Is A VElton Burak

, THE Assoc narco

SPIE Bungee Is obtainer. By Bheem, no 112" -- T- ' lb ) Tx'M

B, S" - '

c R? -

G.Les X Ba A Spree

. The PnSPA PCX,xD Is Tine Soo Oro Au Pants w :I→x

w/o) = Xo,WITH THE Compact OPR- TOPOLOGY.[ S- BBAS.is : KC I CoverAer, U C X OPEN

,

LK,us = { w : I-XI w (K) cu}) . Define it :P(x. xo ) → X Be it ( w) -- wt . ) .

Pie : it :P(X.xo) → X IS A Fibration.

PI : Gives A PATH g : I- X Aro Foo EP CX) wit't it = glo) ( ie .

, Gives A PATH g In. X

AND A PATH BEGINNING A- Xo Ano Evans Ar glo) ) , DEFINE g-t EP (X. Xo) By

Jets) -- { 5o(scltt') Ossett

g. ( s ( Itt)- 1) It ESE 1

Titan it (St)-- g ft) Ars t↳ It IS A Continuous Mor OE I Duno PCX,xD .Titis Proves

Homotopy ( IFF.ae Four Points . IT'S EASY TO SEE THAR THIS VARIES Continuously IN THE

ONG intr Dara t Soo Gives Homotopy LIFTING For ALL SPACES. ,,

DIR : I- ' ( xd ⇐ P (X.xD Is Denotes R(X. xo) t Canio Tine Lee SPICE OR X Basion Xo

.

P (X, xD Is Contract ' BRE .

PI : Drew's. F :P(X.xo) x Ins P(X. Xo ) By Flw,t) ( s) -- WITH .

Then Flw, o) -- 4×0 - Aw,

Haw. "

WE know Wh Can Rename Any MAP f : X- Y By An Inclusion Of The Hormone-1 .WE CAN Also Bl

RRPlack It By A FIBRATION UP TO Hhosunotopy . TAKE THE Star

* = { ( x.nl/w1o)=fc*fcXxpcy , ←SACRO" AIK PATHS IN

DEFINE IT : I- Y By it ( x. D= wht s DIE fine i : X - I Be ilx) -- ( x, Mfc,) . Then

Toi (x) -- I(x. Mfm) = Nf ( 1) = flxx).

(n: i : X - NX IS A Homotopy Buuunten CE r T : I- Y ISA Fibration .

PI'. Exercise .( e.g : p : I-X p (x. nd -

- X Satisfies poi (x) -e p (X , Mfc, ) -- X t iop (x.w) - i Cx) -- (x. Af,×, )

Bvi wt Mf,×, Gives = iop - tdx.

)#

#

Hirt GroveIT X t Y Ane Species

,Dino.ae Bu [X

,-1) Tite Sk.- Or Homotopy Cassie, ok Maes X - Y

.

( THERE ARE Rhett . uk Ukrsrons,Too)

eg : IT. (X. xd - (d. it , (X. xd)

DIE : Tn (X, xd = ( ( 5. is) , ( X.xD) -- [ ( In, din) , ( X , ) ( p e5 Ca - Be Any Point )

Nik : to (X.Xo) = [ 15, i) , (X , xd) = SET OF PATH Components Or X

PRI : Tn (X, Xo) Is A- ABEL in- v Gear For nai .PI : Titanic 0k Talk, xo) Abs [En ,

d- In ),( X

,xd)

.

Ik {ft,

C- In (X,Xo) Define Cfl .fg)

th f -g--

fist Coon . .no#.e.i.e.f.glltn---iti--ffgYztf, ,

' In , I },

WH-its Tms Ameen - ?

HI = = = IREce G : Tn (X, A. ad = [ ( In, din , dIn-IB) , ( X , A. so))

Same Grove (Aw,A-Behan For no, 3 , Gwen f : (X,A) → CY

,B ) GER f* : Tul X.A) → In ( Y, B) .

(end Exact SEoeEi :(A

,xD - (X

,xD

,I :(X. xoxo) ↳ (X.A

,xD ⇒

- - -

d-in (A. Xo) Hulk

,xD Tal X.A , xo) Is in - ,

CA, Xo) - -

- .Isa

,(A. xo) IT

,(X.xo)

Is Exact. WHA Is d ? fflc. In (X. A,xo) ⇒ f : (IidIn , dI'x9i3)- ( X. A. Xo)aft. fflo.sn) :(OF ,

dE-Ii3 ) → (A. xD .

⇐*

Eg : im 's* '- herd : Suppose L - THE Tn (X.xD . This f Represents 5*2 . BT DEE . - it, ou f Mars In-'

INTO Xo

⇒ of-- f ten -i ⇒ Of - o I- in-, ( X.xD i ie . 05*-0 ⇒ This⇒ E lard .Coo vk.rs,zuy

,Ik da -- O

,THEN

It Is Noel Homotopic . etc., ,

NIK : I.Homotopy Is Easier Tf DEF ,_ k THA- Homo Looy

,But NdHarston Tb (Account

.

32

2. Claim: tic ( 5^1=0 For Ken t iz (5) to .

PI: (Et f : Sk→ S"

Be A Mar,Ken

. WE May Assume Tito f Is Cklwcan t Since THE

K-SKELETON OK S"

May BE TAKEN To Be A Point,THIS Deformation CANNES f Th A Constant Mnr.

(In Gerakan Tins Shows Thar Tf (X'm" ) tick) For Kem- I t Ik (x"" ) I ,d*) !

Tf See THE Tz (5) to, Comes coin Ep'= Ep

'

Uf e" Witten f : Sss Epp' = 52 IS

THE Hoek Mar. THE Homotopy Ty Pk Darkness Onu ON THE Homotopy Class Off .

So If

it,( 54=0

,we won. Have ICP

'

= 5054.

But THEN Tate Cup Pinault Lua,

& E H'

( Ep')

,Woo-e Bk O

,A Contradiction.

Similarly,WE CAN USE IHP

"To Show Tyne, ( S

" ) # O).

3 . If I→ X Is THE www.ensn Coven,Tineo Ii (x) I still)

,i>2.

¥: I.⇒§ .pLift,no ⇒ p* : In (E) → Tn ( X ) Is SunTKCTWE.cn>22 REQUIRES HERE)

S"

f- X suppose I -. S"- IT SATISFIES Po I =f = hxo .THEN USING Homotopy ( IF Tino

WK Get A Homotopy It.

By UNIQUENESS OF LIFTS,I = My

.

⇒ P't INJECT'VE.

Now,Ix f: S

"

- X,we Have An thanes Mar f-

* : Hal 5)→ Hnlxl . But Hunt 5)IZ

So WE Have f-* ( HE Hn (x) .Titis DeterminesA Anne H : talk → Hn ( x)

[ f) 1- f-* ( i)

Titis Iss (Avis Tine Hurkwicz Homomorphism.

IT X= 5,Hkf ) = desgf.

-

Titu : H : in (5)→ HIS" )-- 21 Is An isomorphism.

PI : S , nice Hlidsn) =L,

It Is Surtees .VE . IT 's Trick , ten To SHOW INJECT writ . Suppose

It (A)= 0,ie

. degf -- O . We adust SHOW f-= Constant. WE MADEform f Until It Is

Smooth t (Es p E S"

BE A REGULAR VALUE.WE know ( last Semester) Tito

degf = ¥fyp,degf/×, .

By Assumption THIS IS O . WE Proceed BY INDUCTION ON THE Number 0"

Points the f-' ( p ) . Assume n 7,2 .Suppose First Thar f

-"

( p ) = 9×3045 , degfk, = I , degfly = - l .A Connkctinb X dy Ans MEETING 5%90}'Fi! ÷:! " "I:}.mu?I::I:.s.smauDiscsnno-xmm-sma-

sA Tubular NBHD N or A : N I enx I . Notre Tita f- Is Drinks

ON e' +903 t e" x fi} t Since THE Color DEGREESOK f Ar x - y Haute

OPPOSITE SIGNS,THE DEGREES or f O- S

" -'

* 903 r S"-' xf i } Are The Stone .

By Induction,tf Extends To F : S"

- '

x I → d Dn.

We NAT ketene F Tf A MAP

F : en c- I → D"

t So f Externs to I : 5×903 UN- S" SoTita Ii ON→ ODES? Since

5- int Dh Iss Contract ibuE, WE CAN Externs E To A MAP ON THE REST Of SmxI . THS who

It p As A REGULAR VALUE WITH PhelanAGE A .The Resort , no Mar On Smx { i} WILL Miss p

+ So IS NULL Homo Topic .

It f-"

( p ) Has More THAN 2 Points t n > , 2 , THIS An#mint A-cows06 To Deform f Tf 33

" CANCEL" Two Or THE Points with Opposite Loch DEGREE.

Rice Ear Until f- ' (p) IS EMPTY t

Take deegf -0 ⇒ f Nluu Homotopic . Ik n -- I,Monk Cane Is Rewires

,But WE know it ,

(5) = 21. ,.

Tin: H : in ( Vi sa)- Hn ( Vi 5) I 21 Is An Isonaorwitism For n > l .

PI : Similar ARGUMENT .

. .

Cory : Excision Fails For Homotopy ,

PI ' . Consider ( D2 , S' ) :

Tz (D' )→ IT, (D

'

,5) Is it ( s ' ) → Tz (D

') ⇒ Is (D',S' ) -- O

' ll , 11

O o O

X--S2But it, ( S2, D

') Etz ( S2) -1-0 .

So Consider Tite Excisiue TRIPLE ( X. A. Z)Ecaocx A ft( X - Z

,A -Z ) - ( X IA )

11 11

( D' , s ') - (5,15 )

S Ix Excision workers For Homotopy,we warn Have IT, ( DR, S

' ) Es 53 ( S2 , D' ),But we Don't

. "