Fundamental Group

http://cis.k.hosei.ac.jp/~yukita/

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.