X Efim Eiieniei sit !E...' C-x), Titus, f' p ' t g Ane Two lifts Ot f p ' Witch Aorta As A Point t...
Transcript of X Efim Eiieniei sit !E...' C-x), Titus, f' p ' t g Ane Two lifts Ot f p ' Witch Aorta As A Point t...
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) } [email protected] 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
. "