x topological space - home - mathbromberg/spring2020/5520/day1.pdf · gcs e f cs 1 e. get for f f...
TRANSCRIPT
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