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Fundamental Group
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Fundamental Group 2
Path homotopy
.),1(),0()()1,()()0,(
such that :function
continuous a is path the topath thefromhomotopy path A
.in paths be :,Let
10 xrHxrHtgtHtftH
XH
gf
XXgf
1
r
0
0 t 1
g
f
hr g
f
hr
x0
x1
XH
Fundamental Group 3
Simply connected space
not. is whileconnected,simply
is ballunit closed or the ballunit open The
.in homotopic-path are endpoints same with thepaths any two
and connected-path isit that provided is spaceA
21
nnnn D
X
Examples.
connectedsimply
Fundamental Group 4
Path homotopy classes
relation. eequivalencan is relation homotopy path The
).(on relation homotopy path theby denote We
.in paths all ofset thedenote )(let , spaceany For
Lemma. 1.1
XP
XXPX
Fundamental Group 5
Proof of Lemma 1.1 Reflexivity
. and allfor )(),(such that :
as taken becan path the topath thefromhomotopy path A
.in path any be :Let
trtfrtHXH
ff
XXf
:yReflexivit
1
r
0
0 t 1
f
fx0
x1
f
f
XH
Fundamental Group 6
Symmetry
. and allfor )1,(),(such that :
where,: Then, .:Let
trrtHrtKXK
fgKgfH
:Symmetry
1
r
0
0 t 1
g
f
h1-r g
f
x0
x1
Kh1-r
X
Fundamental Group 7
Transitivity
.2
10 if)2,(
12
1 if)12,(
),(
such that :
where,: Then, .: and :Let
rrtH
rrtLrtM
XM
hfMhgLgfH
:Symmetry
1
1/2
0
0 t 1
g
f
h
g
f
hx0
x1
L
K
XM
Fundamental Group 8
Fundamental set
).(by denoted
and of thecalled is /)(set quotient The
X
XXP
set lfundamenta
Fundamental Group 9
1.2 Naturality
.: then ,in : If
.in homotopic-path
are :, paths then the,in homotopic-path are
:, paths theIf function. continuous a be :Let
gFfFHFXgfH
Y
YgFfFX
XgfYXF
Proof.
Fundamental Group 10
1.3 Theorem
).()(:1)1( (c)
).()()(:)(
, and : functions continuous any twoFor (b)
).()(:
:functionset a induces :function continuousEach (a)
:properties functorial following esatisfy th sets lfundamenta The
)(#
###
#
XX
ZYXFGFG
ZGYXF
YXF
YXF
XX
Fundamental Group 11
Multiplication in P(X) and in (X)
system. algebraican is )),((
.2
1t0 if),2(
12
1 if),12(
)(
),,,()1,0,(: and ),,()1,0,(:Let 2110
XP
tf
ttgtgf
xxXIgxxXIf
Fundamental Group 12
1.4 Lemma
.210 if),,2(
121 if),,12(
),(
. and defined is product then the
defined, is product theif and and If
trtH
trtKrtL
gfgfgf
gfgfgf
Proof.
1
r
0
0 1/2 1
gf g
f
x0
x1
XL
f’ g’
H K x2f' g'
Fundamental Group 13
1.5 Lemma
. thusand ,1 1.2, lemma naturalityBy
.1 connected,simply is Since .Let
homotopic.-path are )1,0,()1,0,(:parameter
of change ajust by differ that ),,()1,0,(:, paths Two 10
gfpff
ppfg
p
xxXgf
Proof.
Fundamental Group 14
1.6 Theorem
. ][][][ and ][][][
:inverse sided- twoa has )(][element Every (c)
].][[][][][
:unitright a andunit left a has )(][element Every (b)
][])[]([])[]([][
:eassociativ istion multiplica defined sometimes The (a)
follows. as groupoid, a of structure algebraic thehas )(set lfundamenta The
*1
*0
*0
*1
xffxff
Xf
fxfxf
Xf
hgfhgf
X
Fundamental Group 15
1.6 Proof (a)
.parameter of changelinear
piecewise theis where))(()( that Observe
order.given in the lemultipliab be )(][],[],[Let
pphgfhgf
Xhgf
p
0 f ½ g ¾ h 1
0 ¼ ½ 1
f g h
Fundamental Group 16
1.6 Proof (b)
parameter. of changelinear piecewise
are and where and *1
*0 qprfxfqffx
0 x0* ½ f 1
q
f
r
0 ½ 1
f x1*
Fundamental Group 17
1.6 Proof (c1)
.0 gives 0 ,naturalityBy
. space connectedsimply in the )0,1,0,()1,21,0,(:
parameter of changelinear piecewise theis
where),,()1,0,(:path theequals
*0
**
00
xfufffu
u
u
xxXufff
0 f ½ f 1
u
f
v
0 ½ 1
f f
Fundamental Group 18
1.6 Proof (c2)
.1 gives 1 ,naturalityBy
. space connectedsimply in the )1,0,1,()1,21,0,(:
parameter of changelinear piecewise theis
where),,()1,0,(:path theequals
*1
**
11
xfvfffv
v
v
xxXvfff
0 f ½ f 1
u
f
v
0 ½ 1
f f
Fundamental Group 19
1.7 Theorem
]).([])([][][
)]()[()]([])([])[]([
pointwise. checked becan )()()()(
defined. is :)()( defined is :
).(])([])([])[]([ :groupoids lfundamenta theof
smhomomorphi a is ][])([by defined )()(:
function induced the,:function continuouseach For
##
##
###
##
gFfFgFfF
gFfFgfFgfFgfF
YPgfFgFfF
YgFfFXgf
YgFfFgfF
fFfFYXF
YXF
Proof.
Fundamental Group 20
Summary
.1)1( and )( properies
functorial thesatisfying )()(: smhomomorphi
groupoind a ,:function continuouseach To
).( groupoind a
spaceeach toassociates functor. valued-groupoid a is
)(####
#
XXFGFG
YXF
YXf
X
X
Fundamental Group 21
Fundamental Group at a Basepoint
).,(),(),(
operationbinary a torestricts )(on tion multiplica The
).(),(by denoted is classeshomotopy path ofset The
).(),(by denoted is loopssuch ofset The
.point terminaland
initial has that :path a is at based in loopA
111
1
xXxXxX
X
XxX
XPxX
x
XfxX
Fundamental Group 22
3.1 Theorem (X,x) is a group.
).,(][ allfor ][][][][][
:inverse sided- twoa gives loop reverse The (c)
).,(][ allfor ][][][][][
:identityan is ][ (b)
).,(][],[],[ allfor ][])[]([])[]([][
shown.already hasty Assciativi defined. always istion multiplica The (a)
1*
1**
*
1
xXfffxff
xXffxfxf
x
xXhgfhgfhgf
Fundamental Group 23
3.2 Theorem The fundamental group have the functorial properties:
.1)1( and )( properies
functorial thesatisfying ))(,(),(: smhomomorphi
group a have we,:function continuouseach To
functor. valued-group a is
),(####
11#
1
1 xXXFGFG
xFYxXF
YXf
Fundamental Group 24
3.3 Corollary Any homeomorphism F:XY induces an isomorphism F#:1(X,x)1(Y,F(x)).
).,(),(),(:)()1(1
and
),(),(),(:)()1(1
have we3.2, TheoremBy
.),(),(: and ),(),(:
smshomomorphi group inverse induce ),(),(:
and ),(),(: functions continuous Inverse
111####),(
111####),(
11#11#
1
1
yYxXyYGFGF
xXyYxXFGFG
xXyYGyYxXF
xXyYG
yYxXF
YyY
XxX
Proof.