THE WHITE items Tinn (Er X r Y BE Cw COMPLEXES WITH Basic Points- -- Xo

,Yo O - CEecs . S-f Poste

f : ( X. Xo) - ( Y,yo) IS A Mtr THAT IN Docks Isomorphisms f-* : Tn ( X , Xo ) → In ( Y

, Yo ) For Ace n > d.

THE- Ix Y Is CoonEater,f IS A Homotopy EQUIVALENCE

.

PR: First Prove THE Follow.no : Suppose din X Coo Ans Tn (X) =D For Au no, O . We Show THA

X IS CONTRACTIBLE ( Titis Is THE Spike in Case Xo→ X).Since Ito (x) IS A ONE - Point Ser

,

THE O- Skeleton Can Be DeformRn Tho Xo E X ( ie.

.For Each O- Cece X

,Choose A PATH From x Tho Xo i

TH is G. via A Homotopy From X"to 9×1)

.Use THE Homotopy Extension Property ,*EXTo OBTAIN A Continuous Family ft : X→ X,

OE te 1 WITH foIidx Aref,(X") = Exo} . Now Cons

. Dien X"! THE IMAGE ONNEN f

,or Any Epo .ee In X

")

IS A LOOP AT Xo . Since it,X -- O,WE CAN Deform f

,(e) Throw # Loops BASE n A- Xo To THE CONSTANT

(oof . DOING THIS For EACH 1- CELL Gives A Homotopy From f, I ×c , , To THE Constant Map

f-z: X' "- {Xo}

.Use Tine Ho .ua otopy Extension Property Tb Extreme Titis TO A HomoTom ft: X-X

IE te 2 With fz (Xk) ) = {xo} . CONTINUE In This Fashion Us in , THE FACT THAT In X = 0 For Au u

To GET A Homotopy From Tdx TO THE Constant RA At Xo .

Notte : I.

THE Same A- room Ent Shows THE IF IT,(XI =D For ne N

,THEME IS A Homotopy

ft : X- X,Costel

,with fo -

- Tdx Ann f,(x"" ") -- 9×03 .

It Foucoas Thar Ik (x) -0,ka N

.

2. It dnn X' ⑨

,Maxie THE First Homotopy (Ast Fon OE te t

,THE Second Fon 'KE te 314

,etc .

Ann Taku Dative f, (x) = {Xo ) . Since X Has THE WEAK Tororoby,Tins Hon.noTone Is Continuous.

3 . THERE ISA Rhett, vie Viens, on : Ik Iµ (X, A) = 0 For Ann,THEN THEME IS A DEFORMATION

Rkrraction fi.X - A lie . ft : X - K

,fo -- idx

,f. (HCA

,f- + IA -

- ida OE te 1.).

GtEr CASI : Givin f: X- Y Coals . Dim THE Mappin Cylinder Adf.SINCE Adf RETRACTS TO Y

WE Have Ik (Nfl I Tk ( Y) . It f-* ink (X) → Ik (Yl Is An Isomorphism Fon Au k

,THEN So IS

in : Ty ( X ) - Ik (Mf ) . If Fouoas Tita II ( Mf , X 1=0 For Ace k Am So THERE IS A Retraction

r : Alf → X .THE Composition Y ca Mf S X IS THE Reza, nice Homotopy IN verse TT f . I,

e.g; lRp2× S3 Ans St x IRP's

Haut THE Same Homotopy Groups But THEY Are 1¥ Homotopy

Eau lunk.at ( CHECK Colton.no coax RINGS)- S- There Is No Hohen fi IRP § → 52×112 PB Dv .sc ,# sIsomorphisms ON HOMOTOPY

.

eg : Titanic or Sk As I It. THEME IS A Promo.- Mar IYOI' xI4dE→ I'DI"---

((x , ,xu),(Xz , Xu)) t ( x , ,Xz , Xz, Xy )

This Is A Map f ; 52×52→ 54.NOTE THA flagpg Ans fl ×s2 Ana Noel Horn otome Fo

-A-T P

( since it,( 541--01.8 Since TI ( Stx s') I ite (54×5,15) For Any k, we See THE

f-* : Tk ( 52×5) → TI ( S" ) Is Thurn Fon Au k . But f- Is Not Hormone Ic Tb A Constant Mar

Since f-* : Hy ( 52×5) → Hy (S") ISA- Isomorphism ( Exercise ).

-

-

Ttt Huz THI : LET X BE A Cw -complex.

If it,dXl=o For Ken THEN

I. Ttklxl -- O Fox Ken

2. It : Tn (x) → Hu (x) Is An Isomorphism

,PROVIDED us l

.

It : WE A-CREASY Know # I ( WHITEHRM Tltml . thx's Show # 2 .

Steel : It Is SURJECTIVE.

Since Ty (x) -- O For Ken THEME IS A Mnr f :X- X Swett

Tano ca) f- side t ( b ) f- ( X"" ) -- Exo} . Leo de Hulk) Ame Lto Ernie! Be A cycle

RikersEntire d .We know That flea, TAKE, Tite Pan Leni

,demi) To (X

, Xo); It Follows

Tutt f- ten , Represents An Element Orc In ( Xl . Since f=td×,we See THA D= f-

*Cal =

{ Emi flee:X).

Since Exit f- ( e?) Is thanksBurks By A SPHENE,This latter Class IS IN THE

IMAGE OR It.

5512 .

.It Is Insect.ve. Lto f: S

"

- X Be A Marse# Tito f-* [5) = 0 In Hu (x ) . WE CAN

Deform f UNTIL UNTIL f-(5) E X") Ars f IS TRANSVERSE TO A Point pi Ia THE Interior Of

Isola CEU e?.

( Er di-xffypd.gg flpi . Tine Chair Edie : Is A Gue IH Cn (x),Theakston

ft f5) c- Hn (Xlt S. Titis IS A Boundary,

- ie.Tinzme Kasem; So Thar Edie! --

d ( Em,- Est' ) .

ADDING A f : S"- X'"Tine CHAN EM,

- de!" MAKES Raut di = O t DOES Not CHANGE THE Homotopy

class or f.WE know There Is A t :X- X with t=id× Am + ( X"") -- fxo's . Tin> liners

Tree t Factors Titnouoit A Wkpbe Of SPHERES ; x'M¥5' ×Since f : S" - X"

'Has Exit di -- O

,Tine Composite sfb

pof: S "- U S" Iss Hoowmouooous To Oi, But THEN

Pu f Is Nuuitonnotorhc Since Tn (US" ) = Hn (Us").Titus

, gop of-- to f Iss Nuu Homotorte Ans

Sinus t-Tdx,f Is Houlton.notoric

. ,,

Nik : THEME IS A Routine UBus.eu : (X ,A) WITH IT,(Ako, Hk (X. A ) -- O, Ken , n >, 2 , Implies THAT

Hi (X. A) =o For Ken Avs H : Tn (X.A) → Hu (X. At ISA- Isomorphism.

↳ : Ik X t -1 Are Simply Connects Cw -Complexes Ams f: X- Y Ik Ducks Aro Isomorphism ON

Homology,THEN f IS A Homo.vn EQUIVALENCE

.

If : ( Er Mf BE THE Amman 6 CYUND.hn.

THEN Y - Adf IS A DKionmnno-tha-rn-c.tt WE Have

it; (4) E ti (Anf ) t Hit 't ) E Hi (MH Fon Avi.

THE Inclusion XcsMf Gives Exact Sawbucks

- - - → Iii (X) fits * icy ) - Iti ( Adf , X ) - Ie, (x) # - - .

- . . - it! Cx) # Htin )- Hiltner ,x) - it ! Htt . - .SINCE f-* Is An Isomorphism ON Homology

,WE SEE THAR Iti (Mf

,X) -- O Fon is, O .

BY The WHITEHEAD

Tithonian, Mf Deformation Retracts To X. ,,

Cory : Ix X HasThe Homotopy Type 0k An n -Drank Cw - Complex,t It Tilly --O Fon i En

,THEN

X IS ContractBuk.

PI : Ti 1×1--0, ien ⇒ Tt :(X) -- O ,ien . But Aso, Ttilxko For isn. Titus, It* 1×1=0 ⇒ tix

) --O Fahri. ,

Cory : If X Has Tite Homotopy Type or An n - Drank Cw -Cornelia Ann Ii (x) -- O Fon ien - l,

Titta X = Us"

.

In Particular,Ix Han 1×1=-21

,THE X- 5

.( so A- Simply -Connects lton.noLooy SPHENE Is

A Homotopy SPHENE. )

PI : we NAA> Assume X= X"!Am Turns WK Hare X Es US

" Is X.It Follows THE f-* : Hnlvsnts Huh)

⇒ Surname twk IMA> Choose A Basis For Hn ( VI) = ⑦ Hn ( Sh ) Ams Fon Hn (X) so The the

Adrian Or f-* Is Gwen the ('

ar.

.

.

a ,|oµb= rank Hn (x) wit't ai = II fi .

--C -- rank Hn (vs" )

Since In ( V 5) I Hu (US" ),Wexner Fins A Man h Fitters Into S

"

u i -- US"

X×Such Tans h* Has Matrix

c huts#-

b

It Follows TINA j* Is An Isomorphism On Homocoat .+ So ISA Hornstone EQUIVALENCE,,

Cory : Ix itifx) -- O Ans Him-0 Fo- i> n,THEN XI Y

' "" ? If In Amnon Ha CX) Iss Frane

THEN X - y ' ".

Pie : we know THA Hi (x'm) Hi CX) For it n Ams Hn (x"' )- Hn IX)→ O.Osimo Tine LES

Oe Tune Pan ( X,X") : -- → Hi ( x'm ) - Hi Cx)- Hi (X , X

'm ) - Hi -, (xn) - Hi.. #s .. .

WE SEE Tito Hi (X,X'") -- O For it n ti Ans O - Hat, ( X

,X' ") - Hn (x"') It Hn (XH 0 Is Exact

.

Titus,Huh ( X

,X'm) ISA Sub Grue 0k A Free Grow Aws Hence Is three ( Hn (x"') -- Zn (X'" ) c Callin')!

Via THE Reactive Hunan.cz Titan,WE See Thar Int, ( X, X

'm) ⇒ Hat, ( X

,X"') Am So WE Mae

ATTACH ⑥ ti) -Cbus To X'" Tf Kiu THE KENNEL Of i* .Lir Y l

"" BE THE Rissole x. 6 Complex.

THE Inclusion Xl"- X Exams T A Mar yl " " '- X winch Is An Ito on ltomoroo, Ann

Hera On Homotopy. The Other STATEMENT IS AN Exercise

. "

↳ : LET X BE A Simply Connects n - Khan , Four . THEN X Has THE Homotopy TYPK Ok Ann -Dial

Cw Computer Y. "

=

Homotopy THEORY OKA FIBRATION-- -

- -

LET IT : E → B BE A Fibration. LET 8 : I→ B BE A PATH From bo To b. . Co-si Dieu Tine Dianna

IT- ' ( bo) x fo} - E Apply Hormone> LIFTING Tb GERA Anne I

- ' ( bo) x I → E Corkins tops .

f tf THS Inputs THAT IT-" ( bo)x fi} I T

-' ( bi) Ars S Au F. Biters Are✓

IT' ' ( bo) × I Pes IIs B

Homo'The IX B IS PATH Connie comes.

Now, Ek 8 IS A Loop Ar bo,THE RESULTING Homotopy Equivalence it

- ' ( bo) x fi )→ IT- ' Cbo) Is A Homotopy

Automorphism Of it" ( bo)

.Its Homotopy Class DEPENDS ONLY ON THE Class or 8 Ian I

,( B

,Bo) Aro So

WE Have A Homomorphism IT,( B

,bo) → Aut ( H-' ( bo)) = Grove or Homotopy Classes or Homotopy

Raul valencies of it ,- '( bo)

THIS IS CALLES THE Act of IT , ( B ,bo) ON the Fiber .

Titis Iheisuusss Actions ON THE HOMOLOG

Mms Colton Looy Or Tine Fi Bien.

--

Titu: LET IT : E - B Die A Fibration Arno (Er FT IT- '' ( bo) BE THE Fersen

.THEN Tinker Is An Exact

SEQUENCE. . . - Tnt, ( B, b) Is Itn ( F, e.) It In (Eeo) '# In ( B , bo) Is in-, ( Feo) - - - -

Withee E : F→ E IS THE Inclusion.

It: Consider THE LES or The PA - n i : Fcs E :

- - - - Tn # ( E, F)Is In (F) → In (El - In ( E, F)

Is in . . ( R ) - - -.

Cor Also Have A MN Tx : Tnt, ( E, F) → Itn ( B, bo) .

Cn .. IT# IS An Isom onsite San

. E

Pf : Given f: ( TI, din )→ I B. b) Theme IS A LIKING 9) fitIn f- B

Since I"

IS CONTRACTIBLE,ft hbo : I → Ebo} ( The Honorary win Deron

.- DI"Okie both Genera)

By Homotopy LIFTING, Any name WHICH IS Homotopy IN A MAP THAT LIFTS Must Itskik ( Ift. CL Kaney

,

g : ( I',Ott) - (E

,F) Dk team , -Es Au ElkanEnr Ok Tn ( E

,F) £ it* (g)= {f) E tu ( B, bo) . To ProveT*ng*Is I- I

,Wes How Tita If go , g , :( I

"

,

din ) → ( E,Fl Are Such THAT Togo = tog , ,

THEN goes . .

THIS Follows From THE FOLLOWING (Karma By TAKING X-- I" "

.

Lennart : Gwen H : Xx I→ B AND Two lifts II.,Ttc : Xx I -E Witch Aamer on Xx So }

,Tirone

IS A Homeroom of LIFTS or H Connective II,Anis II , Accor Critica Aortas on Xx fo } .

PI : ( E- T -

- Hlxxgog.

We Have A Commutative DIAGmm Tx# /- EComeng J : XxixI - B Is Protector ONTO XXI Followers ¥By H

.Since XXIXI Deforms To f T!* x9oSxI ) u ( xx Ixsol )uCXxa3xI ) u

# #

*

3¥WE Can Texters Tho thxE) UHT Is B

- To A cnet.no Ok J. "

-

Xxix901 Xx IXXI

÷

Applies1. CONS i Dien Tine Fi Bruno, St- S2 htt

*Since Ti ( S" ) = 0 For is I , we Have Ii ( Gen)-5 it :( 5

" ')Ep" For it 2 A.on iz ( Ep " ) I 21 :

. . . - tzl.gs' ) - itzcsgnt

') → teleport -5 it ,,!g' )- it

.( s?!

" )

Z

2 . ( Ee RB De Tine Loor Space on. B.We Have A Fibration RB - PB

j Am Since PB ISCONTRACTIBLE

,WE SEE Tie ,

(RB) I ti ( B).IterationTins D

wie Sais Thar Iti (B) I it. (r :B )

3.An r - FRAME IN Cl

"Is Given By r MUTUALLY Orthogonal Vectors ( e . . .. , er) . T.tk She

--

Oe r -Emma Du E""

IS DENones By Scr,H ) r IS CALLES THE Stiefel Mani Fou .

( NAAN irons ? Note That Scr, N ) c 52" " x -- - X Sl" " wit 't Tite IH Ducker Tbroeoot. )

-

Claim ; it ; ( Scr , xD) -- O Fon i c CN - Zrtl Are t.v-zrncslr.nl ) E 21.

Pryor: Note That SCI,al ) = S2" ' ' Ans This Is Correct For r=l. Assume the D- a-way THE Result Is

True Fon r - I Am Constantine F. bear ,oo S"""" - ( ' r -2)

→ Scr,all

uTle, . .. , er) = le, , .-" er.. )

[ Fl Bien '. (House + Unit. Ukc.vn Ortho born Tf Tmz S2 't -2rt 't 'T

Scr- l,N )

Cr - H - Pennie Determines Bee le, . -"er. . )

.We Have

⑦ N - r) - 2 ( r - l) there Dimensions LEFT.]

Co-sirs.sn THE Homotopy SEQUENCE

. . . - Tin ( Scr- i , N ))- ti ( 5"-" t ') - ti ( Slr , Nl ) - ti ( Slr-it, N) → . - -

we knowIti ( Str- I , m))

-- O Fon ICZN -Rcr-htt = 2N-Zr +3

* 2N -2+3 (Str- I , nm)) I 21

⇒ Tzu -2h33 (Slr- I , N) ) → Tavern (Sl""" " ) → Tzu -z.r+z(Scr, N ))→ Tzu-zr*z ( r - 1. N))

Ill

-> Tzu-art, (S2

"'t' ' ' )in Is Tzu-zr+, r

,

ND) - Tzu-art , !§r-'int )O

2

So, Tzu-zr+, ( Slr, N) ) I 21 An's ti ( Slr.nl/--OFoeic2N-2rt1. "

4.LET U (r ) BE THE Group or Unitary rxr Complex Matrices ( ie

.A FF -- I )

.

Noir Titan THE

Column .us OK A UN 'Tan-7 llhttnx Give Au Ontkoneon ante Basses Of fr Ams So U ( r ) = Slr,r )

.

LEE, Glr , N ) Bk Tite Grassia AAAioun ok r - PLANES the ¢"

.NOTE THA GLI

,N ) = EP

""

.

WHY IS THIS A MAN , Fours ? Timers Are atAny APPROACHES,Bei HERE's A Nick ONK . Dkk, ne

IT : Scr, N ) - G ( r , N) By it ( e , . .. , er ) = l , A-- her = r- PLANE Annie B , e , . .. , er . WHAT IS THE FIBER

Of it? If WE Fix AN r - PLANK,THE SEE OF OrthoNaram BASE, For THE PLANE IS Isomorphic TO

Ukr) . So WE Hmat Fibration Ucr) → Slr, Nl

d. it .None : Ix rel : u , - S2" ' '

Glr,N) L

Gpa-1

USING with WE knowAbout ti (Slr,N)) WE SEE THA Iti ( Gir, nil) I tie ,( Ucr) ) For ie 2N -Zrtl .

5.Consider Tune Inclusion Uln)- Ulu tu) A te ( Aff ) .

Dex, ne T: U Int .) → 52" ' By

it (Al -- last Cow.mn. or A e- 5" '? Note :

+- i ( ¥ ) = (Aff ) ,

Ae Ulnl .So we Have A

Fi Barrow Ulu ) - Until.

§zn* , Titus, Ti KUHNE Iii (Until) , Fon ie

2n.

Tzu, ( S2" ") - Tzu ( Ulu) ) - Tzu ( acute ) ) - Tzu 4,5"

') - Tzu. . (Ulu)) Es Tzu - , ( ulna) ) → O

H2 O

21

So THE Hormone -1 OK THE Unitary Group Is STABLE '

. Ti ( Ulu)) I Ti (U ( utm)) ,ie 2n

,m? O

.

BOTT PeriodicityTiti : LET U = line Ulu) .THEN Tz it , (a) EZ Ano title) = O . "

- -