COVER in 6 SPACES 20

--

DEI : A MAPp

: E - X Is A C0Vkr MIR IF EACH XEX Has AN OPEN N Bits U Such THAT

p

-

Yu ) Is A Distort U' ' ' ' w

F

'

( Uk¥

% w " "

comp , came :

Sa

I.

EACH Sa O PKN In E,

AnoI

2.

pls,

: Sa → U Is A Homeomorphism For EACH a

.

U X

Such A- OPEN Set U Is Caceres EVENLY Converses AND Time Sa Ark Caulks SHEETS.

ExAnP I.

id : X - X Is A Cover

, .ua For Any X ( obviously )

2.

E -

- IR,

X-

- S

'

, p(a) = EZ

'T ''d

3.

E-

- S'

,

X-

- S

'

, p :S

'

- S'

plz)

-

- Z

n

Q

a

&

IR N : Any Connection OPEN

JnSHEETS S

'

&Uts

'

IS Evenly Cover .es.

LP( I

i

g s

IINFINITELY MANY SHEETS

gSl

(

F7

4.

E= Sn,

X-

- IRP"

, p: S

"

→ 112pm THE Quotient MAP. This Is A Coven WITH 2 SHEERS

( BT Definition, REALLY )

.

5 .

1122 - Six s

'

Ca

,pl ↳ ( ez

" 'd

,

ehtif )

⑦ ① ⑦

⑦ 1122 →

⑦ € ① €

6.

p :Cl → EX = Cl - fo } THIS Is A Cover

, ,→G : Use Polar Coors, ~ # Es w = ft i O

,

ft R

,

OE IR

w , - eat GEIR'

,

Cite lRtxs '

ft id to ( ee

,

eio )

TH 's I > A Coven, .eu Since 112 → Rt X ↳ e× Is A Homeomorphism Ans IR → I Is A Cover

, AB.

DEI :The

FiB Over ye Y Or A Continuous Mar f-

. X -

Y Is f' '

(y

).

Norte :For A Cover

, # 6 Maep

,

THE Fl Bens Ane Discrete.

UNIQUE ( 1kt , -6 Titan

- 1- -

Suppose p

:( E

, eo) → ( X

, Xo

) Is A Covent.ve MAP Ans

f: Yo ) s ( X

, #

o

)Is Continuous

. If

I. →

'E.

e ,

Y IS Connect .es,

THRU There IS

Aims OICL let I : ( Y

, yo) → ( E.

e)

.

. Lp

X. %)Is ( X. to )

PR :Suppose I

,tf

,

Are Litres or f Aae Sto A

= { ye 'll Fly ) -

-

I f. Since yo EA,

A

to.

AISCLOSED ? WE Assume E Hausdorff ( Not NECESSARY But It Simplifies THE Ancrum Ert ).

Time MAP I,

XI : Y - EXE IS Continuous Ares THE Diagonal D= { Ie.

e) I e c- E } Is Closer.

THEN A

= I.XI)

- '

( O ) Is closer.

A-IsOPEC : L to YEA Ans LEE U BE An Eu knit

Counties N Bits

OR fly ).

Writep

' '

( a) -

- LISA .

THERE IsA UNIQUE

doSuch THAT I

,

Cy ) -

-Ily ) E Sao

.

Tina V-

- Ii'

( Sao )n

i. ( Sao ) Is

An OPEN N Bits

Ok

Y .

Ik 2- EV,

THEN I,

ft ) Ann Izft ) E Sao.

Since pot ,

ft ) = f I z )=poId7 )

AND SINCEP

Isao Is I-nsect.ve,

It Follows THAT I,

I t ) =IzCZ ) Ann HENCE ZEA

.

THUS 21

VE A Am A Is OPEN

.Since Y Is Connect .es

,

WE Have.

A -

- Y Am So I,

=I .

"

Pn into THE

+7 p: ( E

, e) → ( X

,

Xo ) ISA Cover, ,u6

,THEN RACH PAT # 8 : ( I

, o ) - ( X

,xo ) Has A UNIQUE

.

LIFT J : ( I, o

) - ( E,

eo ) .

\I • Jus'

1¥ : UNIQUENESS Follows From The PRE vous Result

.

#To Show Existence

,

Coven THE Ihra ok Ok 8 By Every

#

BBCoverts Open Stars.

Since THIS SET Is Compact,

we

[ ⑧TFAaFine A Partition O ' toc t,

a. -

-

a tu =

IOk I Such

#

THAT 8 ( Es - i

,

t ;) c UI Ano U;

Is Evenly Cove neo.

¥8Is.

BaeSee Vs

-

- Nets,

t ;) .

THEN 8--8,

-

re.

- . - Tn

.

NoneP! ←THAT 8

,

Lifts To 8,

:( to ,t,

)→ ( E,

ed Since

p

)

p

- i

( u,)

Is A COVERING. THEN Tz Litts Tb

a8

Fi . [ t,

.to ) - ( E,

I, Ct.) )

. Procera

.usIN Doctoral

X ⑦

WE ( 1kt EACH 8;

Tho Is Are Tmrw J -

- I,

.. -

- .fr

Is A LIFT Of 8.

, ,

C outer, .eu/t0inoTfe-iTHan

-- -

Supposep

:( E

,

e ) - ( X,

Xo ) ISA Cover

miceMar Are THA F : Y XI - X Iss A Homotopy

.

Ik THE re Is A LIFT Fo -

- Ye E Of fo : Y → X,

THE There Is A Lier F : YXI - E

Such THE F- ( s

, D= Its )

.

Prior : I.

Ik ALL Of X Is Kueny Cookies,

THEN The Result Is CLEAN.

2.

For EACH

YE

Y

,

Then, Is An Open NB His Ny

ok

yAnn A Partition O

-

- to Lt

, c .

. -

a tu -

-

l

Ok I S - at THAT F ( Ny x C

to -

,

,t ) ) Is Corra

, # Es Ix AN Every Cover .es OPEN Sfo Us

E X

( By Compactness Ok I ).

LETTy (E) = Fly it )

.

THIS IS A PATH In X.

USING THE SAME Aram,

As IN THE PATH LIKING Titan,

WE Have A Lift Ey : Ny XI → E Ok Fy

= Ny

XI . By

UNIQUENESS Of LIFTING,

Ey,

An 's Fy, A Greek ON ( Ny,

n Ny,

) XI . It Follows THAT Fly

, t ) =

Ey (y

't ) Is WELL - Dasan .esA- s Continuous

.

, ,

⇐ :Suppose to

,

8,

:( I

, o) - ( X. yo ) WITH Lifts To ,F ,

:( I, o ) - ( E

,

e ).

Ix 80=8,

rel f 915,

THE,

Jo = f,

rel fo,

R.

In Particular,

Told -

-

I,

C ' ).

Pryor ;Ix Sos

Its E

f f p

By UNIQUENESS,

Els,

D= 5. Cs ),

ie.

F : Fo=F, rel fo

, B.

, ,

IXI Is

X e.g :p:S

's S'

Pati it

,

( s

'

,

,) →

it

,

G'

,, )

2- is Z

'

Colts 292 )

Coy :THE MAP pit : IT

,

( E,

Eu ) → IT

,

( X. xo ) Is INJECTIVE.

2 # Z

PI : If 8 ISA Loor - Ar eo With pot = n*

,

THE VISA Litt Of pot t

neo Is A lift Orn

×,

⇒ 8=4%6/90 ,

I :WHAT

Is The Into orp*

? 22

More THAT A Loop Ar Xo Lhenna To A Loop Ar lo

Is Ceuta,

# ex IN THE IMAGE.

It A loop

REPRESENTS AN ELEMENT OR THE Image Ofp*

,

THEN It Is Homo Topic To A Loot Hau

, noSuch

A LIFT t So by Hon.no Tony

(IR Tim 6 THE Loop ITSELF Hrs Suck A Lift

.

Prof ;THE Num

Beer Ok SHEETS Ok A Coven p :

( E,

eo ) → ( X,

Xo ),

bit't E A mo X PATH Connects

,

Eu-

Ms Tite DuDEx Orp*

( it

,

( E,

ed ) Ih .

it,

( X, Xo )

.

Pp ,

If 8 Is A Looe A- Xo

,

Aas It CLI EIt -

-

P 't ( it

,

( E,

ed ),

THEN The L net I. I Has

THE Stare Run point As I Since I Is ALoop

.

DEFINE OI From THE Ski Ok Costars OFH

Top

- "

(Xo ) B

, IT ( HER ) = I CD.

Since E Is PATH Connects,

OI Is Sunset 'VE

( eo ( An . Beto , nice To Any Point Du p

- '

( Xo) BT A PATH

gPROJECT

, .ve To A Loop 8 Ar Xo)

.

Be Ix OI ( HCA ) -

- OI ( It Ird ) THEN I,

-85

'

Liners TOA Loop IN E Bases A- e.

⇒

[ r,) fire )

-

'

E H.

"

( IET in 6 Criterion-

-

Supposep ,

( E,

e) → ( X,

Xo ) Is A Coven, ,u6 Spree Ana f-

.

( Y,

- b) → ( X. *

.) Is Continuous

WITH

Y PATH Convectors Anne(

octicy PATH Connkctko.

THRU A LIFT I '

- ( Y

, yo) → ( E,

to )

Exists ⇒ f

*( it

,CY

, % I ) Ep * (

it ,( E

,

ed ).

Prooi i ( ⇒ ) It I Exists,

THE , pot-

- t ⇒f

*I it

,

(4%1) E pale ,

CE,

e )).

⇐ ) Suppose f

* ( it,

CY, yo

) ) E p

* I it

,

( X. xD ).

( Er

yE

Y Ams Lk - 8 Be A Pair

Frum YoTo

y.

THEN

ft Has A Unions ( yetFIT Starts

.io AtEo.

Define I ly ) =It ( I )

.

It 8'

IsAnother S- oh

PATH THE ~ Is'll f 85

' ISA Loop ho Ar x

.

With [ hole f * ( it

,

CY, ) f P * ( It

,

l Kieu) ).

This Three Is A Homotopy ht From he To A door h,

WHICH Lifts Tb A Loop I,

In E Bases

AT Eo. THE Homotopy Lifts To It .

Since I,

Is A Loop A- Eo,

So Is To Any By UNIQUENESS

OK LIFTS,

To Is It'

. ED- '

VIA THE Garmon Mars

Po, no

FT Ci ) = FT'

Cl ).

S.

I Is Wku - Deena?

Continuity Of I IS Not D)

RE , ever

,

.USE Lock PATH Connectivity

.

, ,

Not i IR Y IS Simply Connects,

lifts Aunts Exist.

CLASSIFYING COVERING SPIES-

-

WE Know pet

: IT,

( E

,

e) → IT

,

( X. Xo ) Is INJECTIVE.

QI : DOES Even, Subgroup Ok IT

,

( Xix.

) Arise Aspit

( it,

C Eeo ) ) For Some Cohen ? Existence

QI : Can Two Different Coven was I

, ,

XT Give THE Sanne SUBerror ? UNIQUENESS

Ix Particular,

CAN THE Trivia So B Group Be REALIZE Titis way ? THAT Is

,

Does

X Have A Simply Connectors Cover ?-

-

e.g: XE St

.

IT,

( s

'

) -

-Z Tite SOB Groom Are Ln > For Some n

>o

.

23

For n > 0

,

( Er Xn -

- S

'

WITHPn

: Xn - St Gwen B,

Putz )= Zh.

THE ,

Pit : it,

( Xn ) - it

,

( x ) Is Tite Mme x to nx AnySo

p*( it

,

( xn ) )-

- Ln >.

IT F- O

,Take Xo

= IR.

Ik n # m THEN Xu t Xm Ane Distinct (

oven , nib Spaces.

WHEN Can . X Have A Sammy Connect.es

Cover ?

NECESSARY Condition: Each x

ex Has A N Bits U Such THAT IT

,

( U,

X ) → IT,

( X. x ) Is

TRIVIAL.

THIS Iss CALLIES Skin , LOCAL Sinn PIE CONNECTIVITY Ok X

.

WHT? Suppose

p: I → X Is A Coven

rub With I,

(5) = O.

It Xf X Fins An Runny

Cookies N Bim Uone X Are L Et I BE A SHEET

.

It 8 Is A Loop IN U,

Lift It To

J cUT i I Is NULL Homo Topic Du I ANDTHEN

p o f Nuri Homotopy ) Is A Noel Homotopy OF

8 ex X.

ej: X LOCALLY Simply CONNECTS

X LOCALLY CONTRACTIBLE (eg :

CEKL Complexes )

Prof :IK X IS PATH CONNECTED

,

LOCALLY PATH Correcter,

AND LOCALLY Simply CONNECTS,

THEN X HAS A Simply Correcter Cover I.

PR : Dean,

I-

- { (8) It Is A PATH In X Start,

- is Ar Xo }.

Here [ 8) Denotes THE

Homo Tom Class rel So

, B or t

.THIS Is Just A Set

. Define p : I → X B

, plus ) KD.

Since X I 'S PATH Connected,

P IS SURJECTIVE

.

WHAT'S THE Tbrocoot On I ? Suppose E ME I.

( E- U Be A N Bits IN X ok VCD . LET

↳,

U ) = { [ Vici) /u

Is A PATH IN U BEGINNING A- Tcl ) }

.

WE MAT As WELL Assume HIS

PATH CONNECT Es Anos Simply CONNECTED

.

×

;#U

Claim: THE SE 's

Lt,

US Form A Basis For A Tb Poway,

On I.

For Tins,

It Suffices TO SHOW

THA - Ex Lto,

US n LK,

Ui ) €0,

THE - Titters Exists G) c. I Ams A N Bits V ok all ) S- CH

THAT La,

V ) E ( to,

Uo ) n Lt, ,

US.

Suppose G) c- Lto,

MD n Lt,

,U ,)

.

r r

tin:

in:"

.

n: a.

.

.

It Foceows Kasia,

THAT < r

,

Hon USE Cro,

do > n Lt,

,

U,

).

to

Now, p

( Ct,

U ) ) = THE PATH Connects Component Of U Contai

, - tix 6 Hi ) AND SINCE

PATH Components Are OPEN, p

Is Aw OPEN MAP .Since

p( 58

, u

) ) EU, p

Is Continuous.

PIS I'

.

LEE XE X.

Since X IS Looney Simply Connects,

X HasA PATH Connected,

Sunny Connection N Blinn U. THEN

p

- '

( a) = Ugg

,

Lt

,US wait 8107

-

-

Xo,

Kil-

- X.

Giono

Two Such Lk,

u >,

Ct, ,h7

,

we See Easier,

Thar Gro ),

US n Lead,

a) = ¢ ⇐Cro ) -

- EH.

⇒p

- '

( u) Is A Dishonor UNION O " OPE - Stars . lldorhkoukn

,p 1cg

,

a >

ISA

Homeomorphism.

It PH so : TAKE As Basement Eo-

- (2×0).

Time PAH th I rt ) Is 24

APATH

From eo To (8)

,

WHEN ,e rt : si -

Hts ).

×,

K "

I IS Simply Connie cries.

.

Lies a BE A Loop

In I#

--

-

-

Ar e. .

Liar 8 -

-

pox.

By Viviane LIFTING F -

- a bus all )-

-

fn×o) tFci)

-

- (8) ⇒8=7×0

.

This Impress THATp #

( ft ) ) =

(7×0) For ALL a Ann Slack

pit IS INJECT

've

, T

,

( I,

e.) = O.

,

Note : THIS Is A Grint Constructor;

Be IT

Is USELESS the Practice.

e.g: X

-

-

S

'

us

'

Is

.

.g

.

.

Ab

I

5 •5

'

a

WHA Dox,

I Look LIKE?

.

:;IC

Wku ,It Is THE Homo Tora

Classes OF MAPS

!> as Ab

is@

. . .

rb rb→ •STARTING As WRAGE Point

.

n

ya ya § ya

-

I;•

•• •

a. . .

nb nb

. '

TbTHE Gnash Shows THE First

FEW ITERATES, ÷,as:Ab÷ , .

But It IS INFINITE.

' I.ya g

-II•a°

onAb

ro -

!7 @ 7 !

Existence

PRI :Suppose X Is PATH Connects

,

LOCALLY PATH CONNECT Es

,

And Skin, LOCALLY

Sima CONNECTED.

THE- For Even Subareas It

E I,

( X,

Xo)

,

Tiner ISA Couture Senatep

: XH - X

With

pit( I

,( X

,xD ) = It For A Suitably Chosen To C- X

It.

Pilot : LET NX BE THE Snarly Conniecries C oven Construct Es Above AND DEFINE A Rhino

~ f-

,[ 8) n [ 8

'

) Ix ND-

- 8 'll ) Amo [ 8.8

' - '

I E- It.

THIS Is AN Eau valence

RELATIONPRECISELY BECAUSE H IS A Subgroup .

( Er XH BE THE Quotient Of I BY This

RELATION.

None : Ik 8117--8 'll ),

THEN [ r ) no { 8

'

I ⇒ [ t -

n )-

- ( 8

'

.

n ) Foe A Printn .

IN

Particular,

Ile Two Points IN Basic N

Bling28

,

U ) t C 8

'

,

ha'

> Are IDE~tifi.se,

THEN THE WHORE

NBites Ark IDENTIFIES

. It Follows THAT THE Protection X #→ X [ Vb tis

Tl ' ) Is A

Coutee, no

.

Lfo To EXH BE THE Equivalence Class Of (4×0).

Tina THE Into Ok

Px-

- IT,( XH

,

E) → it,

( X, xD IS It : It 8 Is A Loop In X A- Xo

,

Its LIFT Tb I

Start Wb At (4×0) Ears A- [ 8) So THE IMAGE OK THIS Lifters PATH In XH Is A loop

⇐ Cr ) n End ⇐ CHEH. ,

Ominous 25

Dk : Am Isoxnonritissn Ok Coven

, .usSpaces

p ,

:X,

- X,

Pi .

In → X Is A Homeomorphism

f: I

,

→ The Wit't

p ,

-

- Prof .

Prof : It X IS PATH Corvettes AND LOCALLY PATH CONNECTED

,

THEN Two PATH Connects

Covens I,

,

ITa

Auk Isomorphic VIA f -

-I

,

- In Taxi .no I,

C- pi

'

( Xo) To The pi

'

(Xo )

⇒p , *

( it,TX

,,

= path ,ID

.

PRo : ⇒ Existence ok f ⇒ p ,

-

- pot,

Pa-

-

p , of

'

' '

⇒p ,

*( it

,

( I, it ,

)) =

Pz #( I

,

( Th,

I. ) )

⇐ ) Suppose THE Subban oops Ane EQUAL.

USING THE Chenab C Ritter , on,

WE ( Ift P , To

A HaeI ,

i ( I, ,

It - ( Iz,

IT With Prof, =p ,

- Similarly, we Ger

Fi . ( In,

→ AT, ,⇒

WITHp ,

of e

-

-

Pa.

By UNIQUENESS OK LIFTS,

F. of .

-

- idxzAndproof,

=

idI

,

Since These

Composites Fix BASE Points .

$0

,f ,

t PT Ark Inverse Isomorphisms.

, ,

-

1117in.

THEN Is A Bisection BTWN BASE Point Pick> KevinISOM one Hisar CLASSES Ok PATH Connects

Couture Spacespi

.

( I,

E) → ( X, %) Ann THE SE

.

- Of Subgroups Ok IT,

( X,xD

.

Ik BASE points

Ark IGNORED,

THIS BIJECTION IS A Combs Poop EnceBtwn Iso CLASSES

Ok Covers Are CONJUGACY

CLASSES Of SUB Groups Of IT,

C X, Xo )

.

PR'

.

It Only REMAINS To Daouk THE Last STATEMENT.

WE Claim Their CHANGING BASE Point

To Withinp

- I

(Xo

) Corresponds Exactly To CHANG , NbPit

( IT,

TX,

To A Couture SUB Grasp.

( is I,

BE Another Boss Point Ihip

- '

( Xo) t lies I BE A PATH From

ToTo I

,

. Tita I

Proves To A Loop 8 IN X,

Reena Entire AN Evan Err GE IT

,

( X, %)

.

SE- Hi

=

pet( it

,I I

,

Ii ) ).

NoneThat IfI Is A Love At To,

F- '

I 8 Is A Loop A- I,

Ams Ssg- '

Hog EH

, .

Similarly,

g.

H, g-

'

s Ho ⇒ Hi

-

-

g

- '

Hog . Conversely,

To CHANGE Ho To Hi -

-

g-

'

Hog,

CHorse A Loop 8

themes quit.net

g.

( let THIS To I Startin A- To Any lkr I,

= JC

,)

.

ThenIt

,

-

- PAH,

TX,

#d), ,

↳ : A Sinn Pl > Connie cites Covin, .us

SPACE Is A Coven OK Kueny OTHEN COVERING SPACE Of

X.

Such A SPACE Is CAUSES THE UNIVERSAL Cover lIt

IS UNIQUE UP To Isomorphism .

).

THE Action ON THE Fiber-

- --

-

( top

:Xd X BE A Covenant Space

. A PATH V IN X HasA UNIQUE Lift I Start in 6 At A

Given Point Inp

- '

( 8101 ). DEE

, reLy :

p

- ' '

( Hd ) →

p

"

( Kill By Lyft ) = Jci ) wit 't Flo ) -

- Z.

THIS It A Bisector :L

ji IS Its Diverse

.

For 8 .

n,

WEHave (

y.

n = (

yLy .

This Rkuensn

Is Bae,

So Rennie Ly By Its Inverse

p

- '

I Kill →

P

- '

( no ) )- THEN Ly .

n

-

- ly . Ln.

Tai> Die Enos

ONLY On Homerton Class t So we GETA Homomorphism

IT,

( X,

Xo ) → Permm(p

- '

( Xo ))

,

(8) to Ly .

Can TH is THE Act' Ii tTH Oe TIE Fiske .

Wk Can Riecovknp

: IT -

X From THIS Action As Follows.

LET XJ →X Be THE 26

UNIVERSAL Covin Constructs Earlier.

LET F-p

- '

(Xo ) Are Dare

, me hi.

Tho XF →

X B,

h ( [ 8 ),

= JLI ),

Winery J IS A LIFT Ok 8 Start, # b A- Io

.

h IS Continuous & Kuku A Loon

Homeomorphism Since A N Bite Ok ( C 8),

XT ) In . Tho XF Consists Ok Pains ( ( 8-

n)

,

XT )w

'T 't

yA PATH In A Small

N BHO Of Jli ).

h IS SURJECTIVE SINCE X IS PATH CONNECTED.

h Is

Almost Ckntainky Not Extractive . Suppose h ( (8),

XT ) =h ( I 8

'

),

Kj ) . Then 8 t 8

'

Auk

PATHS From Xo Tf THE Stink Endpoint AND To

-

- Ly

'

. g- I ( To )

.

( Er D= 8

!8

"

,

A Loop IN.

THEN

h HH,

To ) -

-

h ( [ did,

Lahti ).

Conversely,

For Au Looe d weHave hkrl

,

E) -

. hat -81,41×51 ).

So

,

h Induces A Mae Rox Fln-

X Witten ( 181,

Io ) - ( C d. 8) ↳ I IT ),

CNET.CL/,xol.Cn-uTtnsQuotikm-XpWitEnefit ,

( X. xo ) - Perm ( F) IS THE Action

.

NKi Xp ANAKES Sense For Aux Action f OF it

,

( X. xo ) ONA SE.

- F : The -

X,

( ED,

→ 8C 's

ISAC oven.

Now,

Ie - Tf Is A Bisection t Titus A Homeomorphism Since h IS A Loca Homeomorphism.

SINCE It TAKES FIB Rns To FIBERS

,

It Is An I > oinonrttism.

,

DECK Transformations

--

( Etp

:I - X Be A Cove rave Are Denote By G ( E) Tine Set or Acc Isomorphisms I - X

.

THIS Is A Grier UNDER Composition,

CALLED THE Group Or Deck TRANSFORMATIONS.

eg.

. piIR → St

,

G ( IRIE & Since THE Isomorphisms Auk Those Translations 4th at in

,

ME I.

p: S

'

→ St,

Z ↳ En,

G ( I ) I In ( Rotations Of S

"

Throw it AirlocksIt KH

Nork THAT Bt UNIQUE Cacti . -16

,

A DECK TRANS formation IS Completely DET Enna

, .niz⇒ By WHEN ,e

It Stuns A SINGLE Point

,

Assuming I PATH CONNECTED.

Die.

.

A Cover we p :

I - X Is Caccia Norma Ix For knew

XEX Are RACH I

,

I 't p

-

'

( x

!

Twerk Is A DECK Transformation TAKING I Tho I !

Note - Example p: I → S

'

v S

'

Tacks Acc Norse, In

- -

i

••?

•

*

# yiIT THE Wks Gie Point . None

••• it

,

TX) IZ Amspi

: it

,

(D) → it ! sis

'

)

IX

a

.

Is Ca ) to fab ) c Fe

• a

°

THE wk Is 1¥ DECK

TRANSFORMATIONS~

X Taeko I Too I

'

Since 5*6 ]

Lovers Be A Loop Ar I

'

,

But There

ARE No Nontrivial ONES

.

Pip .

.

( E -

p: ( I

, Io ) - ( X. xoI Be A Nick Cover

was A - s Let It -

- path,( I

,

Eit,

( X. xo )27

I .

THE Cover, no Sister Is Noun

.mn ⇒ HOT,

( X,

Xo)

.

2.

G ( f) IS Isa .name#cTbNlH)/H,

Wineries ht ( H ) Is THE Normalizer Ur It Ie IT,

( Xx ,

).

IN Particular,

G ( I ) I T,

( X. x. DIH IT I Is Norma Are Foe THE Univision Coven The X

Cox ) = it

,

( X. xo)

.

PR :RECALL THAT CHANGING BASE

point To Ep

- '

(Xo ) A I

, Ep- '

(Xo ) Correspond Tb Constantin

Ok H By (8) f it

,

( X, xo ) where 8 Lifts Th A PATH I From EAT

, .

So,

[ 8) t NCH ) #

Pet (Hi

( F. Folk

pit( It

,

( I,

I,

) ),

WHICH Be Tune Lik two Criterion Is Kaurava

.

- To THE

Existence 0k A DECK Transformation TAKING To A IT .

So THE Cover, - co Is

No name

⇐ NCH ) -

- it

,

( X. xo)

.

Now Dee, we Q : NCH ) → Gtx ) By 4 ( Cst )

-

- T,

Withers T TAKES To TTX,

.

THEN yeIs

A Homomorphism : It 8

'

⇒ T

'

TAKING To TF ! Tinea ,8.8

'

Lifts A JT. ( I

. ) )

,

A Pato

From Io Tbt ( I! ) IT t

'

( xo )⇒ It

'

Cunmssponns To [ 8) fr'

).

Its Sortie ⇐ weAm Its KENNEL

Consists Of CLASSES ( 8 ) ( IK Tini 6 Tf Loops In I,

ie.

Ker U =H

.

, ,

More GENERALLY,

WE Have THE IDEA Ok A Group Action

.

LE-

G Be A Grove Are Y

A SPACE.

AN Action Of G On Y Is A Homomorphisme : G → Home

o( Y )

i urineg

:Y -

y

Fore (g) E Home

.

CY )

.

Note: g

,

(Sz

Cy

,) =

⑨, gal

C ylI

g, ,gzC

.

G,

Y E Y

.

WE USUAL it Assume

fInitiative

.

USE Kuc CONDITION For ACTIONS

--

--

EACH

yE

Y Has A NB Hs U S- CH THA Are IMAGES

g( h ) Foer

gf

G Ark Distant,

'

ie.

g.( lil

n gz( Ulfd⇒ g ,

-

- 92 .

egoG ( K ) A crime on TT

: Suppose Tl CI Protects Honnieonnonritucaccy A X.

Ik

g. ( Ti) n g.( I ) # ¢

,

Titian g.( I ) =

ga( Ir ) For Some I

,

Iaf U - B- r

Since I,

,xT Lie In

Somep

- '

(x ) Are

p

- '

(x ) n

UN Consists OK AS

an Guy Point,

I,

-

- In.

THENg

? gz Fixes THIS

Point Ann So91=92 .

,

Given AN Action,

We Cnn Form Time Quotient Snack 4/8: yn GG )

,g

t G. Time Points

Ok

4/6 Are Tine Orbits Gy-

- 9 gyI

gEG }

.

Eg: For A Norman Coven

. .ie I → X

,

NXIGCX) I X.

eg.

.

Zz Acts On Sh ;X te

- X 5/212 = IRP

"

AN - Titis Action SATI sneaks THE Condition

Since It X IS Ite THE Orfu Orren Highness meeee U, g

( uln

U = 0.

;the G Actinic on Y

Is Nick,

Titta , 28

I. pi

Y →4/6 y

1- Gy Is A Norma Cove rink Space.

2. Ik

Y IS PATH Connections,

THEN 6=614 ).

3.

G I IT,

(4/6) /py( it,

( Y ) ) It Y PATH Connkctkntloctccy PATH Connection

.

PRIOR : I.

LET U CY BE An OPEN Ski SATISFY,

-16 THE Condition . THENp

IDK notifies Au THE

Distant Homeomorphic Stars { scull gEG } To A Smock Open Ster

p(a) C

'

1/6 .

By Definition

OF THE Quotient Topo Loot

,

P Restricts Tb A Homeomorphism Fru. . n

g( U ) TO

p( U ) For Kaew

g fG

.

THOS, P

! Y - Y 16 Is A Cover in 6.

Each't

gtf Acts As A DECK TRANSFORMATION t THE Covering

Is Norm Sinceregzgj

'

Tacksg

,

( ht Tf gzlu ).

G E GET ) with EQUAL# T Ie

Y Is PATH

CONNECTED Since EE f E Gl Y ) THEN For Any ,

y EY,

yAns fly ) Are

In THE Same OrbitAre

Tinker IsA

gE

G Wit 't

gly ) ' fly )⇒ f

=gSince DE ex TRANS Formations Are Unico Bey

Dieskau was

By Action 0nA SING- k Point

. , ,

eg: 2h A crime On S

"

Xt - x

Titis Is A Cover # GS

"

- 5/22 = IRP

"

Ann Since it,

( Sn Ko,

we Have

it,

( IRPME ti ( key it

,

Csn ) I 2h.

( Et G= Symmetry Group Of This Grin

.

G Corra, us A COPY

es :

1122r.ua

Or 2×21 : ( X,

y) to

( Xtnn,

ytln ); Can This Soooooo H

.

•

x But THE arts Mork : ✓ I > THE Course Reelection : TRANSLATE

UP I UNit t REFLECT Across Ukraine L INK .

ONLY THE IDENT Takes A Savanna To Its BeeS

- Titis Action Is Nick.

None THE Follow, .ro :

I.

111246 Is THE Kkk, a

Bottle

2. It Has INDEX 2 In

G tso Ho G.

IRYH -

- T Ans IR t - IR ?16 Is A 2

'

.

I

→ K

Coven

GALOIS Correspondence

Univ . I ← it,

(5) -

- o Fibers

Cover

, F Gell

Flc

)u

,→ a

,

( x

'

) -

-H x

! RIH

×61×1 ) INCHYH

I LIK GA cars

I-

-

G Ht he L ⇒Gal ( He )

u

X

'

N . name0 Coal ( HK )

X ← tix )-

- 6I

Gall LIK )

K

Is:f.

.name#ei....Sc-.Esmwo:oiir.n.BLl-H--fgeGalIEIkl/

*

, #

" 'EH

F -

Ci 29

Did ;A Grant Is A I - Dimensions C W - Complex .

A TREE Is A CONTRACT

, Brie Gannett.

PRI : Every CONNECT Es Guns It X Count, .us A MAX

, xmx Three (A Trick CONTAINING ALL

Ukrticesok X ).

In Fact,

Every TREE Is Container IWA Maxims TREE.

Pryor : ACTUALLY Prove THE Following: ( ET Xo CX BE AN Arbitrary SUBGRAPH

.

WE Will Construct

A Subban,

no

Y C X ConIFA,

Nino Are Uk races Or X Such THE Xo IS ADKfoninm.com RETRACT

Ok Y

. TA kink X = { Xo } Y leans THE RESULT.

First CONSTRUE XoC X,

C -

.-

BY LETTING Xi+ ,

BR OBTAINED

From Xi By ATTACHING Closures EI Ok All Robes• •

X :

•

KE X-

Xi Huan .io A- LEAST ONE ENDPOINT In Xi.

••Nork Thx U Xi Is OPEN IN X Since A NB Itn Of A Point Xo BeIH Xi Is Contai

-

i Es In Xi

# .

Also U Xi Is Closer• • • •

Since It Is A Union Ok Closer EDGES Am X HAS THE. .

WEAK Tfpoeooy.

Since X Is Connie cries,

U Xi

= X.

.•

• °

Now,

SET Yo -

- X ,.

Assume Yi C Xi Has BEEN Constructs•

.

To Contain Au VENTI CES In Xi.

( RT

Yin Bk OBRA,

.IE - X,

Xz-

- X

From Yi By Antoine ING ONE EDGE CONNECTING

EACH Ukntfx. .

Ok Xiu'

Xi Tb Yi.

LET 4=0 Yi

.

THEN Yin Retracts TO

•

Yi.

DOING THIS Retraction OVEN ( ¥ '

,

¥ ) Yikes>

@ D

A thorn-

crow Y → Xo.

,

Y,

•

PRI: L E #X Be A Connect Es Grant Arlo ( RE T BE A

AdAxim TREE

.

THE - IT ,( Xl Is A Fekete Group w . - it Basis I fa ) Connes Pons, as

Tf Eats EaIn XT

.

Pilot :Fix Xo ET

.

EACHla Determines A Loop EH X By CHOOSING A PATH 82 From X -

To ONE

END O 'c

la,

THEN Axons la,

THEN BACK To Xo Aonb A PatMa ( to Ars

MaLIE INT )

.

( It ta -

.

8,

-

la .

Ma .

Since T Is Simply Connecting Cfa) Darkness Owe ON

G. THE Quotient Mar X → XIT IS A Homerton Eau .UA Canice Since ÷TE fxo )

.

But Xlt Is A Content with ONE Vkrtkx;

THA Is,

Xl TISA WRs

OYEOf Greeks Am IT

,

( X HT) Is A FREE Group With Basis G. view By Time Into .es Ok ffa )

.

, ,

1¥ : Every Coverline Space ok A Grant Is A GRAPH

.

30

PRIOR :l E '

P: I → X Be A Cover

.

For Tite Vertices Ok I WE Taek Xo ?

p

"

( Xo ).

None

THATX Is A

Quotient XI Xoall

Ia Wit a RACH In Connors Poms , -6 Tb An Kobe I - X

.

APPLYING PATH LIFTING The Etch IL → X WE Get A UNIQUE LIFT IN £ PASSING THROUGH

EACH Point In

p

- '

I x) For XE Is

.THESE Lifts Dee we THE Eso .es In I

.

Since I → X

IS A LOCAL ltosnihonronr Hasan

,

THE RESULTING Tf polo on ON I Is THE SAME AsITS ORIGINAL 'T

Poway,

Ttm ? Every Suborns 0¥ A Knee Group Is FREE.

PRI :Given A Free Group F

,

Choose A Grant X With IT

,

X EF.

Ix G E F Is A

S - A Group,

Tatem Is A Coven. - is Setae pix → X wit 't

p

*( I.

( It ) =G ⇒ I,

(5) I 6.

Since I Is A Grant,

IT,

(F) IS FREE ⇒

G IS FREE.

, ,

No ;

THIS Is A D- rear ALGEBRAIC RESULT Proven Via Topology !