Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

22

Chapter

2.

Homotopy

the

topology of spaces.

In

addition

to

loops, which

are

topologically circles,

we

can

consider subs paces

that

are

topologically higher dimensional spheres

sn.

For

example,

JR

3

with

the

unit

ball removed

has

a different

connectivity

property:

all loops

can

be

deformed

into

each

other,

but

2-spheres

in

that

space

may

not

be

shrinkable, if

they

enclose

the

"hole".

So far we have worked

at

an

intuitive

level

and

with

familiar spaces. Now

let us give precise definitions

of

the

objects

in

terms

of

which we will formulate

the

study

of

homotopy.

Definition:

A

path

a(t)

in a topological

space

S,

from

Xo

E

S

to

Xl

E

S,

is a

continuous

map

a:

[0,1]-+

S

such

that

a(O)

= xo,

a(l)

=

Xl

Note

that

the

space

[0,1]

appearing

on

the

left-hand-side

of

the

definition

is

the

familiar closed

unit

interval in

JR,

while

the

space

on

the

right-hand-side

can

be

any

arbitrary

topological space.

When

S.

is a space like

JRn

with

which we

are

familiar,

the

above defi-

nition

of a

path

seems

quite

reasonable.

But

continuous

paths

can

exist

in

arbitrary

topological spaces, even including spaces

with

finitely

many

points!

On

reflection

this

should

be

no

surprise, since

our

definition

of

continuity

in

the

made

sense for

arbitrary

topological spaces. A simple

example

is

provided

by

the

following exercise.

Exercise:

Consider

the

set

S

=

{a,

b,

c}

of

three

elements,

with

the

topology

U

=

((ji,

s,

{a}, {b}, {a, b}).

Find

a continuous

path

from

a

to

b.

It

should

be

kept

in

mind

that

a

path

as

defined above is

not

just

the

image

of a

map,

but

the

map

itself.

One

should

imagine

that

as

the

parameter

t

moves

through

values from 0

to

1, its

image

in

the

topological space moves in

that

space,

and

it

is

the

map

between

these

two

motions

which

we

call

the

path.

It

is convenient

to

think

of

the

parameter

t

as a

time,

then

the

map

gives

the

motion

of

a

point

on

the

topological space as a function

of

time. For example,

given a

path

a(t),

we

can

define a new

path

f3(t)

=

a(t

2

)

which

traces

out

the

same

image

in

the

same

total

time,

but

corresponds

to

a different

map

and

is

therefore

treated

as

a different

path.

Having defined

paths,

we use

them

to

define a new

notion

of

connectedness.

Definition:

S

is

arcwise connected

or

path connected

if

there

always exists a

path

a(t)

-between

any

pair

of

points

Xo

and

Xl.

Theorem:

If

S

is arcwise connected

then

it

is connected. (Recall

that

we

defuied connectedness in

the

without

recourse

to

paths

in

the

space.)

Exercise:

Prove

the

above

theorem.

Assume

S

is arcwise connected,

but

not

2.1.

Loops

and

Homotopies

23

connected,

and

find a

contradiction.

A

path,

as defined above,

has

distinct

end-points

in

general.

But

paths

which close

on

themselves

turn

out

to

be

the

most

useful ones.

Definition:

A

closed

path

or

loop

in

S

at

xois

a

path

aCt)

for which

Xo

=

Xl,

that

is,

a(O)

=

a(l)

=

Xo.

The

loop is

said

to

be

based

at

Xo

(see Fig. 2.2).

Notice

that

a loop is a

map

from a

circle

to

an

arbitrary

topological space

S.

However

it

contains

some

additional

information

in

the

form

of

the

"base

point"

Xo.

In

what

follows, we will always deal

with

based

loops,

in

other

words

loops

with

a fixed base

point.

Figure

2.2: A

based

loop

in

a topological space

S.

We will find

it

useful

to

develop

the

notion

of

multiplication

of

loops.

Each

loop is a

map

from

the

circle

to

the

space,

and

the

product

will

again

be

such

a

map.

We only

admit

multiplication

between

two

based

loops

with

the

same

base

point.

To

multiply

two

loops

based

at

the

same

point

Xo,

define a

map

[0,1]

--->

S

for which,

as

t

goes from 0

to

1,

the

image

first

traces

out

one

loop

and

then

the

other.

This

is formalised as follows.

Definition:

Suppose

aCt), (3(t)

are

two loops

based

at

Xo.

The

product

loop

'Y

=

a

*

(3

is defined by:

'Y(t)

=

a(2t)

=

(3(2t -

1)

1

0<

t

< -

- - 2

~

<

t

<

1

2 - -

(2.1)

It

is easy

to

check

that

this

defines a continuous loop

based

at

Xo.

Thinking

of

t

as

time, we

may

say

that

in

the

product

map

defined above,

the

image

point

in

S

moves along

the

loop

a

during

the

first

half

of

the

total

time,

and

along

(3

in

the

second half.

Definition:

The

inverse

loop

a-let)

of

a loop

aCt)

is defined by:

a-let)

=

a(l

-

t),

0

s

t

S

1

24

Chapter

2.

Homotopy

This

is

just

the

same

loop

traced

backwards.

If

we only look

at

their

images

in

the

topological space, a loop

and

its inverse look

the

same,

but

as

maps

they

are

clearly different.

Definition:

The

constant

loop

is

the

map

a(t)

= xo,

0

:::;

t

:::;

1.

The

image

of

this

map

is a single point.

Now we implement

the

intuitive idea

that

some loops

based

at

Xo

can

be

continuously deformed

to

other

loops based

at

the

same

point. Such a defor-

mation,

if

it

exists, should

be

a

map

which specifies how a given loop "evolves"

continuously

to

another

one.

Definition:

Two

loops

a(t)

and

(3(t)

based

at

Xo

are

homotopic

to

each

other

if

there

exists a continuous map:

such

that

H:

[0,1]

x

[0,1]---+

S

H(t,O)

=

a(t),

H(t,

1)

=

(3(t),

H(0,8)

=

H(l,

s)

=

xo,

O:::;t:::;l

0:::;8:::;1

The

meaning

of

this

is obvious

with

a little

thought.

We have

introduced

a new

parameter

s

E

[0,

1]

which we would also like

to

think

of

as a second

.

"time".

The

map

H

(t,

8),

for each fixed

8,

defines a loop

in

the

space

S

based

at

Xo.

Therefore, as

8

evolves,

the

entire loop

can

be

thought

of

as evolving.

At

the

initial value

8

=

0

the

loop was

a(t),

while

at

the

final value

8

=

1

it

has

become a different loop

(3(t).

At

every

intermediate

value

of

8

it

is some

other

loop,

always based at

Xo.

H

is called a

homotopy.

An

example

of

a

homotopy

H

(t,

s) is

illustrated

in

Fig. 2.3.

af.t)

=

H(t,

0)

H(t,

113)

H(t,

213)

~(t)

=

H(t,

1)

[9J~ITZJ~[Ql~[Q

s=O

s=1/3 s=213

s=I

Figure

2.3: A

homotopy

between two

based

loops

a(t)

and

(3(t).

If

two curves

a(t),

(3(t)

are

homotopic

to

each

other,

we

write

a

~

(3

.

Theorem:

Homotopy

is

an

equivalence relation.

This

means

the

following

conditions

are

satisfied:

2.2.

The

Fundamental

Group

25

(i)

a

~

a

(reflexivity)

(ii)

(3

~

a

{:}

a

~

(3

(symmetry)

(iii)

a

~

(3,

(3

~

'"'(

~

a

~

'"'(

(transitivity)

Proof:

Here

is

a sketch of

the

proof, which consists of displaying explicitly a

homotopy

H

in each case.

(i) Pick

H(t,s)

=

a(t)

for all

s.

This

proves reflexivity.

(ii) Given

H(t,

s) :

a

-+

(3,

define a new homotopy

H(t,

1 -

s) :

(3

-+

a.

This

proves symmetry.

(iii) Given

H

1

(t,

s)

:

a

-+

(3,

and

H

2

(t, s) :

(3

-+

,",(,

define

H3(t, s) :

a

-+

'"'(

by

1

O<s<-

- - 2

1

- <

s

<

1

2 - -

This

proves

transitivity

and

completes

the

proof.

Homotopies also satisfy

the

following additional properties:

(iv)

a

~

(3

{:}

a-I

~

(3-1

(

v)

a

~

(3,

a'

~

(3'

~

a

*

a'

~

(3

*

(3'.

However,

it

is

not

in general

true

that

a

~

a-I.

(2.2)

Exercise:

Find

an

example of a topological space

and

a

path

a(t)

in

it

such

that

a

is

not

homotopic

to

a-I.

Find

another

example for which

a

and

a-I

are

nontrivial yet homotopic

to

each other.

2.2

The

Fundamental

Group

A very

important

property

of

an

equivalence relation on a set of objects (in

this

case

the

set of based loops) is

that

it

partitions

the

set

into

disjoint classes,

called "equivalence classes".

Within

each class, all elements are equivalent

to

each

other

under

the

given relation.

In

the

present case, denote

by

[aJ

the

class of all loops homotopic

to

a(t)

(always

with

reference

to

a fixed base point). One

can

check

that

multiplication

of classes

is well-defined.

This

amounts

to

showing

that

the

above

product

defines

the

same

class irrespective of which loops we choose as representatives for

the

classes

on

the

left

hand

side.

Definition:

The

collection of

all distinct

homotopy classes of loops in

X

based

at

Xo

is:

7rl

(S,

xo)

==

{

[aJ

I

a(t)

is a loop in S based

at

Xo

}

26

Chapter

2.

Homotopy

71"1

(S,

xo)

is a group

under

multiplication.

It

is called

the

fundamental

group

or

first

homotopy

group

of

S

at

Xo.

The

group

property

of

71"1

needs

to

be

proved. We do

this

by checking

the

group

axioms:

(i)

Closure:

(ii)

Identity:

[i]

is

the

equivalence class

of

loops homotopic

to

the

constant

loop

at

Xo.

Clearly

[aJ

*

[i]

=

[i]

*

[a]

=

[a *iJ

=

[a]

for all

raj.

(iii)

Inverse:

[a]-l

=

[a-I],

because

(iv)

Associativity:

To show

that

([aJ

*

[,6])

*

b]

=

[aJ

*

([,6J

*

b])

is

straightforward

but

requires some

thought,

and

this

part

of

the

proof

is left

as

an

exercise

to

the

reader.

To proceed further, we will find

it

useful

to

define

the

product

operation

even

on

paths

which

are

not

closed, namely,

paths

a(t)

such

that

a(O)

and

a(l)

are

distinct.

Definition:

The

product

of

two

open paths

a(t)

and

,6(t)

is defined only if

a(l)

=

,6(0),

in

other

words, if

the

final

point

of

a

is

the

same

as

the

initial

point

of

,6,

and

is given by

"(

=

a *,6

where:

"((t)

=

a(2t)

,

=

,6(2t

-1)

,

0<

t

<

~

- - 2

1

- <

t

<

1

2 - -

(2.3)

A~

one might expect,

the

initial

point

of

the

product

path

a

*

,6

is

the

initial

point

of

a,

and

the

final

point

of

a *,6

is

the

final

point

of

,6.

Now

returning

to

the

fundamental

group, we

note

that

in

principle

this

group

depends

on

the

base

point

with

respect

to

which

it

is defined.

Thus,

71"1

(S,

xo)

is a different

group

from

71"1

(S,

x

h

for

Xo

=f.

Xl

.

But

in

fact

under

very

general conditions

the

two

are

the

same.

To specify

what

we

mean

by two groups being

the

"same", let us first

define

the

concept

of

"homomorphism" between groups

(not

be

confused

with

"homeomorphism" between topological spaces!).

This

is a

mapping

from one

2.2.

The

Fundamental

Group

27

group

to

another

which preserves

the

group

operations.

Definition:

If

G

and

H

are

two groups, a

homomorphism

¢ :

G

.......

H

is a

map

g

E

G

.......

¢(g)

E

H

such

that:

¢ :

g-l

E

G

~

¢(g-l)

=

(¢(g))-l

E

H

¢ : gl .

g2

E

G

~

¢(gl .

g2)

=

¢(gd

. ¢(g2)

E

H

¢ :

i

E

G

~

¢(

i)

=

i'

E

H

(here

"¢

:"

should

be

read

as

"¢

maps"

(a

given element

of

G

to

an

element

of

H)).

Definition:

A

homomorphism

¢

is

an

isomorphism

if

it

is also bijective (one-

to-one

ami onto).

If

an

isomorphism exists between two groups,

the

groups

are

said

to

be

isomorphic,

and

can

be

thought

of

as completely equivalent

to

each

other.

We now show

that

the

fundamental

group

of

a topological space is inde-

pendent

of

the

base

point

under

some conditions.

Figure

2.4: Isomorphism

of

¢dS,

xo)

and

¢l

(S,

xd

for a

path-connected

space.

Theorem:

If

a topological space

S

is

path-connected,

7rl

(S, xo)

and

7rl

(S,

Xl)

are

isomorphic as groups.

Proof

This

is

illustrated

in Fig. 2.4. Consider

o:(t)

based

at

Xo

along

with

its

equivalence class

[0:]

E

7rl

(S, xo).

Let

,(t)

be

an

open

path

from

Xo

to

Xl.

Now

define a

map

(2.4)

by

¢([o:])

=

[,-1

*

0:

* ,]

(2.5)

Clearly

[0:]

E

7rl(S,XO)

~

¢([o:])

E

7rl(S,xd.

And

moreover

this

is

an

isomor-

28

Chapter

2.

Homotopy

phism. For example,

the

product

law holds:

¢([a]

*

[,6])

=

¢([a

*

,6])

=

b-

l

*a*,6*,]

=

b-

l

* a

*,]

*

b-

l

*

,6

*,]

=

¢[(a)]

*

¢([,6])

(2.6)

with

[aJ,

[,6]

E

1fl

(S,

xo).

Similarly one

can

check

the

other

properties

of a

homomorphism, as well as

the

fact

that

this

map

is bijective which finally makes

it

an

isomorphism. •

Because

of

the

above theorem, for

path-connected

spaces we denote

the

fundamental

group

as

1fl

(S)

instead

of

1fl

(S,

xo).

It

must

be

kept

in

mind,

however,

that

in

the

process

of

finding

1fl

we

must

always work

with

loops

based

at

some

(arbitrary)

point

Xo.

The

final result for

1fl

will

then

be

independent

of

Xo.

2.3

Homotopy

Type

and

Contractibility

So far we have discussed

maps

from

the

closed interval

[0,

1]

to

a topological

space

S,

and

defined two such

maps

to

be

homotopic if

there

exists a

suitable

map

from [0,1] x [0,1]

to

S.

This

has

a

straightforward

generalisation involving

two topological spaces.

Definition:

Let

Sand

T

be

two

arbitrary

topological spaces,

and

consider two

different continuous

maps

fo:

S

---+

T

ft:S---+T

(2.7)

The

maps

fo

and

ft

are

said

to

be

homotopic

to

each

other

if

there

exists a

continuous

map

such

that

F:

S

®

[0,1]

---+

T

F(x,O)

=

fo(x)

F(x,

1)

=

ft(x)

(2.8)

(2.9)

In

the

special case where

S

is

the

closed interval

[0

,

1]

on

the

real line,

this

homotopy

of

maps

reduces

to

homotopy

of

paths

or

loops,

but

one should

note

that

there

is no reference

to

a base-point. So

we

will use

th

e

symbol'"

to

denote

the

homotopy

of

maps

defined above (which does

not

involve a base

point),

as

against

~

which always denotes

the

homotopy

of

based

loops.

Homotopy

of

maps

between general topological spaces is useful because

it

helps us identify

properties

that

the

two spaces

may

have in common. For

2.3.

Homotopy Type and Contractibility

29

example, let us find

the

condition

that

two different path-connected spaces

S

and

T

have

the

same fundamental group:

71"1

(S)

=

71"1

(T).

Definition:

Two topological spaces

Sand

T

are

of

the

same homotopy type

if

there

exist continuous maps

f:

S-+T

and

g:

T-+S

such

that

(we use

the

symbol

"0"

for composition of maps):

fog:

T

-+

T

iT

go

f:

S

-+

S

is

where

iT

is

the

identity

map

T

-+

T

and

similarly for

is.

The

property

of "being of

the

same

homotopy type"

guarantees

that

71"1

(S)

and

71"1

(T)

are

the

same group.

This

is

embodied in

the

following result.

Theorem:

If

Sand

T

are

two path-connected topological spaces of

the

same

homotopy type,

then

71"1

(S)

is

isomorphic as a group

to

7I"l(T).

(The

proof

is

somewhat complicated

and

we will skip it.)

Theorem:

If

Sand

T

are

homeomorphic as a topological spaces

then

in par-

ticular

they

are of

the

same

homotopy type,

and

hence have

the

same

71"1.

This

is obvious from

the

above, since a homeomorphism

is

a

pair

of continuous

maps

f:

S

-+

T,

g:

T

-+

S

such

that

g

=

f-

1

,

i.e.

fog

=

iT,

gof=is.

(2.10)

Thus

the

theorem

is

true.

Summarizing, we have found

out

two

important

facts:

(i) Homotopy is a

topological invariant.

Two spaces which are topologically

equivalent (homeomorphic) have

the

same

homotopy properties.

(ii) However,

the

converse

is

not necessarily true. Two topological spaces may

have

the

same

71"1

(if

they

are of

the

same homotopy type),

but

this

does

not

imply

that

they

are homeomorphic.

The

following definitions, theorems

and

examples

tend

to

crop

up

fairly

often in physical applications.

Definition:

A topological space

is

contractible

if

it

is of

the

same

homotopy

type

as a single point.

Definition:

A topological space

S

for which

71"1

(S)

=

{i}

is

called

simply con-

nected.

Otherwise

it

is called

multiply connected.

A simply connected space has no "nontrivial" loops, in

other

words all loops

are deformable, or homotopic,

to

each other.

30

Chapter

2.

Homotopy

For a

contractible

space

5,

the

fundamental

group

7l'1

(5)

=

{i},

the

identity

element. Therefore

it

is simply connected. However a simply connected space

need

not

be

contractible, as

we

will see

in

examples below.

Theorem:

7l'1(X

®

y)

=

7l'1(X)

EB

7l'1(Y)'

Here

the

direct

sum

of

groups,

EB,

is

their

Cartesian

product

as sets:

(2.11)

The

proof

of this

theorem

is

quite

simple

and

is left as

an

exercise.

Examples:

(i)

lR"

is contractible. To show this, we need

to

find continuous

maps

f :

lR

n

----7

{p}

and

9 :

{p}

----7

lRn.

Clearly

the

only available choices

are

f(x)

=

p,

the

constant

map,

and

f(p)

=

0,

where 0 is some chosen

point

in

lR

n

(which we

may

call

the

origin).

Then,

go

f :

lR

n

----7

lR

n

is

the

map

go

f

(x)

=

0

(2.12)

Define a

homotopy

F:

lR

n

x

[0,

l]----7lRn

by

F(x,

t)

=

tx.

For

t

=

0,

this

is

the

constant

map

go

f

which sends all points

to

O.

For

t

=

1,

it

is

the

identity

map

iRn.

Thus

9

0

f

'"

iRn.

Of

course

fog

is

trivially'"

i.

Thus

we have shown

that

lR

n

is

of

the

same

homotopy

type

as a point,

and

hence contractible.

It

follows

that

7l'1(lR

n

)

=

{i}.

(This is also intuitively obvious, for

any

loop

in

lR

n

is homotopically trivial.)

(ii)

52

is

not

contractible

(this is

illustrated

in

Fig. 2.5). To prove

contractibility

of

a space, we

must

find a way

to

"move" all

its

points

continuously

to

one point.

On

52

we could

try

to

move everything

to

the

south

pole along

great

circles,

but

then

the north

pole

cannot

move in

any

direction

without

breaking continuity!

Nevertheless,

any

based

loop

on

52

can

be

continuously deformed

to

a

point,

therefore

7l'1

(52)

=

{i}

and

52

is simply connected.

We see

that

from

the

point

of

view

of

contractibility

52

is nontrivial,

but

from

the

point

of

view

of

the

fundamental

group

it

is trivial.

(iii)

51,

the

circle, is

not

contractible.

The

proof

is

just

as in

the

But

it

also

has

a nontrivial

fundamental

group, unlike

the

an

(t)

be

a loop which winds

n

times

around

the

circle

in

an

anticlockwise direction for

n

>

0,

and

Inl

times clockwise for

n

<

O.

For

n

=

0,

take

the

constant

loop, namely, a

point

Xo

E

51.

Clearly

[an]

-I

[am]

for

m

-I

n,

and

the

collection

[an],

n

E

7L.

exhausts

all

homotopy

classes.

Under

multiplication,

(2.13)

Thus,

7l'1(5

1

)

= {

[an],

n

E

7L.

},

with

[an]

*

[am]

=

[a

n

+

m

].

Clearly

this

group

is isomorphic

to

the

integers

under

addition.

The

isomorphism is:

(2.14)

2.3.

Homotopy Type and Contractibility

31

N

s

Figure

2.5: 8

2

is

not

contractible:

we

can

move all points

except

the

north

pole

to

the

south

pole.

The

isomorphism

permits

us

to

write

7r1

(8

1

)

= 7L

The

number

n labelling

the

class

to

which a given loop

a(t)

belongs is called

the

winding number

of

the

loop.

N

s

Figure

2.6: A closed loop in 8

2

/8.

(iv) Consider

the

two-sphere 8

2

with

diametrically opposite points identified.

This

is

the

first seriously nontrivial space we

are

studying!

If

the

identification

map

is called

8

then

we

may

think

of

the

space as

the

quotient

8

2

/8.

A rep-

resentative

set

of

points

is

the

upper

hemisphere, which is identified

with

the

lower hemisphere, as well as

half

the

equator,

which is identified

with

the

other

half.

Now all loops which were closed in

8

2

will

remain

closed in 8

2

/8

and

of

course

they

are

still shrinkable. However,

there

are

paths

which

are

open

in 8

2

but

closed in 8

2

/8.

For this, consider

any

path

that

starts

at

the

north

pole

and

ends

at

the

south

pole, as is

illustrated

in Fig. 2.6.

This

path

is closed in

8

2

/8

because

the north

and

south

poles

are

identified,

but

it

is

not

shrinkable

to

a point.

Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

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