LAST TIME IT S Th

I I g fpl 9yd IR

in der R nontriv ett it S1

S 1 1 as zits or 2TS

Lemmy a

Foreach path 2 I S starting at andeach To E P Xo

there exists a unique lift I R startingat YoI Rr L Ptie poI 2 or

I s2

Lemma b

For each homotopy at I St of paths starting at Xo andeachFootpadhereexists a unique liftedh.noopy It I 7112 of pathsstartingatYo

Ti e p It Lt

homotopy liftng property Fo RLemma e in IPr IlGiven a map F Y XI St and a lifting 4 63 Ff s s

R

there exists a unique lifting Yx I t restricting togiven on Yx a

Y pt Ca Y I b

rq U3 ru

og T

ptuf Il s

U

s

U It

can choose a cover of S Ud set V 2

P U2 I 02 r Us are evenly coveredr

Usconstruct F Locall

f

S Y F apt yo y

f ft.it riii i

t t tsc

for some L

ydxI cpt fu Many Cat Lt Ove II can choose N c Y and a partition6

Yoo to tactic ctm s.t.EC VxEti ti iD

cUa.uFIt TEsui Iu

We kno that FCNxetixgt.it CUIT r CIR catering tu pint FcyostiP IRu FCNxtt.is cu r

mightneedtoreplaceNarth a smallern4hd

Narth asmallern4hd

define F p FN Eti twig

INxLtigt.itTUY r U

after fu many repetitions we get FNxt

Uniqueness

Y pt Let E E of F I S Eco F o

as before o t t c ctm 1 at FEET t D CU

Induction assume F E on Eat 3Eti ti connected FCEfyt.is also connected

ECEti.tieiDcUT.rpffzors

nesinilarlyEtCEti.tD CUT r2 Uir r because F ti F'Cfi r r

since p pT F F on Eti t

on Eti Ait D and p I r Uproves uniqueness

concede on overlap by uniquenessor lift exists on Y and S unique

mmmm

Induced map on IT2

fi X Y r 8 path in X thenIf 867 867 rc fa and nf then

for fof

IIIT ftp.ftsxiss

ceorxcfthe map pushforward

a cont n p themapf T1 X xD IT Y yo is called of fundamental2 1 fo 2 groups inducedby f

f it X sco IT CY yo doesn't have to be surjectiveor injective

e if f is a homeo then fix is a isomorphism

f f 1

Homotopy invariance of it

X Y t y are henotopic frug k f a coat napH Xx I Y s t HCx o fcx

II y fi ys

Taps interpolating Lehre fuel g

spaces X Y are utopia if

X Y sit 409 rid 904 yidyne Tif X Xo Y y horotopies preserve

Xo ythe

then Kat Xpw IT CY yoEx IT E to e IT S I

IT Xx Y Ti X x it Y

Ex it T I TL