pathf.iso.isx

n

X topological space

will usually assure X is

path connected

Homotopy of paths

me

Perturbation of pqth.gr gTXD 2 paths f and g are homotopic

if 3 F 0,13 6,1 X

5 f f fo g f D

with fish Fest

Notation tf f ard g are

homotopic we write frog

LEMMAN Homotopy is an equivalencerelation

a frig geta fry f

frig gel fzhP

f ng 7 Danks Xf Ca Dx 0,13 X w tf fo ga f

LetG do DxLo 1 X be

defined byGCS E F Cs 1 E

get

For f f we define the homotopy

by Fcs fCsl constant homotopy

DE n X

Transitivity fzg g h so

we haredF set Gcsitl with

f s F s 07 gcs FCS 1 GCS o

hist Gcs D ya1 EsDefine J

H Cs t7F 924 o Ete Yz

GCS at 17 42 Lts 1

As It is continuous this definesone teh

a homotopy from f to h

EXERcIHere we're using the following

fact A B C Yf A

And

s B xB

continuous functions with

f g on A n B

Then ha zf CH KA

g CN XEB

is continuous on A OB

PROPD If X is path connectedther t f g Lon X

we have fag

PR IN e first assume that f gare constant napsf Loch X EX g Lois ye X

Since X is path connected we

havex so.is x

with Hoi x OG y

Dil

y

project tothe vertical side

Define IT Cse7 t

F sat Jolt Git 2 t

Need to show that an

arbitraryf 913 X is

homotopic to a point

C IlyE I

f Csce f sci t

sires a homotopy of f to

a constant nap

Given f g To D X

f const g e const2 court raps are homolopic

tis

Exan Let K lo.is

f o.is Laid withKoko I f 17 1 note that

F fixesThen f kid o 21

Definestraight f sat Ct t for Is

line

homotopyF is continuous and focstFG.hnfcsand f Cs2 FCSc s

We need to check thatthe image of F is GD

Note that for any aebelR

a ECI Ela tb Eb if Cc Loisince Ci teeth C Hatta a

l Ha Eb E L Abts b

112

f s a E o

l X

E IX

t Xo Ct El X

t f O D

fM f

More generally if f g Lois Lag

we have the homotopy

F Sitt I tlfcsl

tgcslcotuposITIONot.tl

7P

o.is Is X 4 yung

If f Ig then hof z hog

F Cs.tl is the homotopy between

f and g

Define H hot thenIt define a homotopy betweenhof and hog

We can combine A reparameterizationof a path is homotopic to the

original path

If T Lois Lois is a homeomorphism

f so.is X is a path

ther f for

We have seen that if X is pathconnected ther all paths are komotopic

That is the equivalence relation ofhomotopy has only one equivalenceclass

This is not very interesting

To make things more interesting we

can replace acid with any topologicalspace

We can also look at maps of pairs

In general it is much harder toshow that maps are not homotopic

The circle S

S LYN E 1122 X2ty2 13 E

S z e e 171 13 121 KeyE Xtiy O

S's to.is where 905 i3

S's lR n where t C tu k ne 7L

S's Cx E ITE 1 1 191 13 fs's Cx.de Rt nax 1 1,1913 1 1

S's lR n where t C tu k ne 7L

TO

General definition of homotopy

f g X Y

are homotopic frog if 3

F Xx D Y

set f fo g f

EXERCISED This is still an equivalencerelation

How can we define maps fromS to S

S's z e E 121 13

Deline

fn S S

byf n z Z

Since Iz't IH 1 L this is

a map from S to S

Alternative constructionDefine

In IR 112

byIn H7 nt

Note that Factin Ktvu ht um

so in our equivalence relation on 112

we have Intel In Cam

In takes equivalere classes toequivalence classes

112 Its 112

I las s

fachoose some Es CR s E

TAK XDeline fake IT In ft