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From: GeometricGroup on 14 Jun 2010 02:26
This is Artin's Algebra page 273.

"The fact that the 3-sphere has a group structure is remarkable, because there is no way to make the 2-sphere into a group with a continuous law of composition.
In fact, a famous theorem of topology asserts that the only spheres with continuous group laws are the 1-sphere, which is realized as the rotation group SO_2, and the 3-sphere SU_2."

Why there is no way to make the 2-sphere into a group with a continuous law of composition? What is the famous theorem of topology that the above paragraph is referring to? Any explanation of the above paragraph?

Thanks.
From: Timothy Murphy on 14 Jun 2010 06:47
GeometricGroup wrote:

> This is Artin's Algebra page 273.
>
> "The fact that the 3-sphere has a group structure is remarkable, because
> there is no way to make the 2-sphere into a group with a continuous law of
> composition. In fact, a famous theorem of topology asserts that the only
> spheres with continuous group laws are the 1-sphere, which is realized as
> the rotation group SO_2, and the 3-sphere SU_2."

Doesn't the group of unit quaternions give a group-structure on S_7 ?


--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
From: José Carlos Santos on 14 Jun 2010 06:59
On 14-06-2010 11:47, Timothy Murphy wrote:

>> This is Artin's Algebra page 273.
>>
>> "The fact that the 3-sphere has a group structure is remarkable, because
>> there is no way to make the 2-sphere into a group with a continuous law of
>> composition. In fact, a famous theorem of topology asserts that the only
>> spheres with continuous group laws are the 1-sphere, which is realized as
>> the rotation group SO_2, and the 3-sphere SU_2."
>
> Doesn't the group of unit quaternions give a group-structure on S_7 ?

No. It gives a group-structure on S_3.

Best regards,

Jose Carlos Santos
From: José Carlos Santos on 14 Jun 2010 07:04
On 14-06-2010 11:26, GeometricGroup wrote:

> This is Artin's Algebra page 273.

You should have written "Michael Artin", since Emil Artin also wrote a
book called Algebra.

> "The fact that the 3-sphere has a group structure is remarkable, because there is no way to make the 2-sphere into a group with a continuous law of composition.
> In fact, a famous theorem of topology asserts that the only spheres with continuous group laws are the 1-sphere, which is realized as the rotation group SO_2, and the 3-sphere SU_2."
>
> Why there is no way to make the 2-sphere into a group with a continuous law of composition? What is the famous theorem of topology that the above paragraph is referring to? Any explanation of the above paragraph?

The famous theorem was proved by Hans Samelson in 1941. It says that
S^0, S^1, and S^3 are the only spheres admitting a topological group
structure.

Best regards,

Jose Carlos Santos
From: Timothy Murphy on 14 Jun 2010 07:45
José Carlos Santos wrote:

>> Doesn't the group of unit quaternions give a group-structure on S_7 ?
>
> No. It gives a group-structure on S_3.

Of course, I was being stupid.
I was thinking of octonions,
but then the multiplication would not be associative.

Maybe S_1, S_3 and S_7 are the only spheres
that support a continuous multiplicative norm with unit element?

--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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