proof of Scott-Wiegold conjecture

proof of Scott-Wiegold conjecture

Suppose the conjecture were false. Then we have some

w

∈

C

p

*

C

q

*

C

r

with

N

⁢

(

w

)

=

C

p

*

C

q

*

C

r

. Let

a

,

b

,

c

denote
the of

w

onto

C

p

,

C

q

,

C

r

respectively. Then

a

,

b

,

c

are all non-trivial as otherwise

N

⁢

(

w

)

would be

contained

in the kernel of one of the .

For

0

∘

<

θ

<

360

∘

we say that a spin through

θ

consists of a unit vector,

u

→

∈

ℝ

3

together with the
rotation of

ℝ

3

through the angle

θ

anticlockwise about

u

→

.
In we have a single spin through the angle

0

∘

and
a single spin through

360

∘

. Thus the set of spins
(usually denoted Spin(3)) naturally has the

topology

of a
3-sphere.

The spin through

θ

about a unit vector

u

→

has the
same underlying rotation as the spin through

360

∘

-

θ

about

-

u

→

. Hence there are precisely two spins
corresponding to each rotation of

ℝ

3

about the origin.

is well defined on spins as you can compose the
underlying rotations and continuity determines which of the two
spins is the correct result. For example a

350

∘

spin about

u

→

composed with a

20

∘

spin about

u

→

is a

350

∘

spin about

-

u

→

(not a

10

∘

spin about

u

→

which would be at the other end of the 3-sphere).

Let

n

→

denote the unit vector

(

0

,

0

,

1

)

. Fix an arc,

I

,
on the unit sphere connecting

n

→

and

-

n

→

. Let

t

→

be a vector on this arc. Let

u

→

be an arbitrary
unit vector. We may define a

homomorphism

ϕ

t

→

,

u

→

:

F

{

a

,

b

,

c

}

→

Spin

⁢

(

3

)

by:

ϕ

t

→

,

u

→

:

a

↦

the spin through

(

p

-

1

2

)

⁢

360

∘

p

(or

180

∘

if

p

=

2

)
about

n

→

ϕ

t

→

,

u

→

:

b

↦

the spin
through

(

q

-

1

2

)

⁢

360

∘

q

(or

180

∘

if

q

=

2

) about

t

→

ϕ

t

→

,

u

→

:

c

↦

the spin
through

(

r

-

1

2

)

⁢

360

∘

r

(or

180

∘

if

r

=

2

) about

u

→

(Here

F

{

a

,

b

,

c

}

denotes the

free group

on

a

,

b

,

c

).

So

ϕ

t

→

,

u

→

⁢

(

a

)

,

ϕ

t

→

,

u

→

⁢

(

b

)

and

ϕ

t

→

,

u

→

⁢

(

c

)

are spins of between

120

∘

and

180

∘

, all having non-trivial underlying rotations.

Let

w

~

be a word in

F

{

a

,

b

,

c

}

representing

w

, such that

a

,

b

,

c

occur in

it

1

Mod

(

2

⁢

p

)

times,

1

Mod

2

⁢

q

times and

1

Mod

(

2

⁢

r

)

times, respectively.

We have a homomorphism

ϕ

′

:

C

p

*

C

q

*

C

r

→

S

⁢

O

⁢

(

3

)

induced by

ϕ

. If

ϕ

t

→

,

u

→

⁢

(

w

~

)

has a

trivial

underlying rotation for
some

t

→

and

u

→

, then

N

⁢

(

w

)

will only contain

elements

in the kernel of

ϕ

′

. In particular, we would have

a

,

b

,

c

∉

N

⁢

(

w

)

. So
we may assume we have a map:

h

:

I

×

S

2

→

S

2

which maps

(

t

→

,

u

→

)

to the unit vector corresponding to

ϕ

t

→

,

u

→

⁢

(

w

~

)

.

By we have

h

⁢

(

n

→

,

R

⁢

u

→

)

=

R

⁢

h

⁢

(

n

→

,

u

→

)

for any rotation

R

about

n

→

. Thus

h

⁢

(

n

→

,

_

)

:

S

2

→

S

2

maps latitudes to latitudes (possibly rotating them
and / or moving them up or down).

Also

h

⁢

(

n

→

,

n

→

)

=

-

n

→

, as

ϕ

n

→

,

n

→

⁢

(

a

)

,

ϕ

n

→

,

n

→

⁢

(

b

)

and

ϕ

n

→

,

n

→

⁢

(

c

)

are spins of between

120

∘

and

180

∘

anticlockwise about

n

→

, so the sum of the
angles will be greater than

360

∘

. Similarly one may
that

h

⁢

(

n

→

,

-

n

→

)

=

n

→

. Thus, as

h

⁢

(

n

,

_

)

maps latitudes to latitudes, it must be

homotopic

to a

reflection

of

S

2

.

Again by we have

h

⁢

(

-

n

→

,

R

⁢

u

→

)

=

R

⁢

h

⁢

(

-

n

→

,

u

→

)

for all rotations

R

about

n

→

.
Hence

h

⁢

(

-

n

→

,

_

)

:

S

2

→

S

2

also maps latitudes to
latitudes.

Further,

h

⁢

(

-

n

→

,

n

→

)

=

n

→

and

h

⁢

(

-

n

→

,

-

n

→

)

=

-

n

→

. Thus

h

⁢

(

-

n

→

,

_

)

is homotopic to the .

But

h

gives a

homotopy

from

h

⁢

(

n

→

,

_

)

to

h

⁢

(

-

n

→

,

_

)

, yielding the desired

contradiction

.

Title

proof of Scott-Wiegold conjecture

Canonical name

ProofOfScottWiegoldConjecture

Date of creation

2013-03-22 18:30:31

Last modified on

2013-03-22 18:30:31

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

5

Author

whm22 (2009)

Entry type

Proof

Classification

msc 20E06

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