9 2 2020

Exe Let X be a convex subsetof IR and x C X Then Leanna

any basedloop in X is lo topic to Cx group IT Xiao istrivialLenny Let X be a top space and p a path from x to x Then themap

IT X x IT X x is an isomorphism ofgroup1In If r PT

In particular theisomorphism class of it CX x of a patched spacedoesnot depend on the choiceof baserent Xo C X

proof o for a notary of loopsbasedat x If r p a homotopy of ashoreat

well defined

t Eris Eris Ep rip If rap If n p ripCxLpig ragtop7 4 Erie Lhd t as in

is an iso 4th averse to r'T Ep y f Ja

Hot ol ftp.xrx.pt ftp.rxpxp In LaCx Go

A spaceX is simplyconnected if it is path connectedand it X 1 0

Leza X is simplycorrected if the y EX thereexists a anyrentuniquehomotopy dais ofpaths x y

Proof existence of a path X path connected Wing Ti o

suppose Tik O and xp toopet between x y Then 2 d f P Pconversely if there is a uniquehomotopy classofperks x 1 X

the I X n D

RentFundamentalgreupofthecircle to E4 sw nd gne berwcrk.TL

theorem T.ES

Explicitly the map lo I IT CS G o is an isomorphismi

n 1 7 w Cs as 25nssi 25ns

PEE Let p IR s wi poet 0161 20.7 poop postTS 1 7 cos245,9245 on I IR anypath on IRIn or I Sts n s from o to n

p t since I Ibylinearhomotopy

lo is a homomorphismth mth

Ilmen p Em TmEon um on 4cm 104translation

Tn IR 1 IR CaB 1 1 Stm Lemma V patha.ISLoop starting at X es ardeato is surjective let a I S representng a givenclass Eep 4 3

acoksen Choi Emetics Fsf fgIfI Rme F lift I I 7112 Then ICD n C Itwith Ico o Since p Ici L G 4,0 b t homotopy

toG post a V Lt S ofpathsstartingat

tis tee surrose tea a o.uaIideII'IIIt I 1112 ofpaths starting

Let Lt be the hotopy Lo um 4 0 at Fo

I it lifts to Ft ab topy ofpaths IRstartingat 0Io In I In byuniqueness inLucaIt L is indep of t I i Ii i V

EdGance a homotopyofpa m n

Lemmata Ca Cbl follow from F Rtopspace

i TI wP

Cc Given a map F x I St and a lifting 4 631

s

F lifting Ip restrictingto thegwenF on

YxI S 4 63F

taking Y point we get LmCataking 4 I we get Lm b

xP f I Use that 7 an open cover UD of S s t th p Oa JayUa's are evenly covered by p Pd

ex

a

EE

co on Y a nbhd CY

can choose CY and apartition 6 6 ate Ltm Ls t F N x Eti ti 17 cue Ug

Ete E is constructedon NxLo t 3 Weknow that FCarita tieD CuithepintF Ui r CIR cala ng Flye ti FC Arlt b CFK p122 replacing N

U with a su Hernbhd

define F on N Eti tied as p Flax ASK.fmtely many repetitionsw

U uwe getEly

Unquenessofthterpt Let F F be two lifts Flo lotof F I Sas before i o t et c Ctm I sit F Etr tie C

Induction assure F F on o tEti tied canceled FCetitie connected Etta tie C Ti r Le some r

p125similarly F Ceti tieD ET r Ui

Ect FCt r r

pe.peF F on Eti tu V

andp jedrive on Vi rF constructed on NxImoreuniquewhenrestrictedholy XI must agree when NxI and N'xI overlapwe have a well defined lift F on YxI Ed

app tFundamental theorem of algebraEvery nonconstant polynomial with coefficients in CC has a not a Q

Proof Wemayassume plz 2 c a Zn t tan Assume p has no rootsin QHr o d g P ret't's pCr defies a loop in theunitcircle S CE

Ip re is par based at 1as r a Ees fr is a homotopy of loopsdoo C 2 3 0 E IT S

cast loop at 2 Hr

take r targe r maxG 2 laitfor lzl r II I r r r fail 171 3 I a zi t tantpolynomial p z 2n t a z t an has no roots on the circle 171_r

her so te lReplacing p with Pt h C andleHag t cEe D

2Tingbe oblan a he topy between Lr and the loop Wncs e ar rep Ptt

ta t oon h L C IT S I

n o do ER Leon11 generator of III Sl

peatMarrununurrunn

inducedLet f X Y cont map 8,8 paths in X Thenif o and 8 have sameendpoints and roof then for fo f IIF't pieif rcn 861 then foCr 8 or f f carpet with

def Let f X Y contmap Themap f Ti Xoxo 1 Ii Yyox Yo ERT for

iscalledthe map of head groups inducedby ffrug

triapft'Xg Y are called 6 topic Tf F a catmap H X't YIt HG o FCK HCox D 67

I'of Y interrooftzapeganib ft X Y

two top spaces X Y are called h py equivalentChronotop c

if F f g XgY st got id go f idy

fEx yo S f 2 EIZ I

g inclusion

we have fog ids go f id yI pt

2 I7 with Hct z iz t I

121

Lenya if f X a CYyo homotopy equivalence of pet tenspaces basepants

then f Ti X x 7 Tilt y is an him

Ex Ti Cellos E it Sl TL