Post on 18-Nov-2021
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