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From: José Carlos Santos on 14 Jun 2010 08:25
On 14-06-2010 12:45, Timothy Murphy 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?

If you are talking about norms, that I guess that you are thinking about
vector spaces. Do you know Hurwitz's theorem on normed division
algebras? See:

http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28normed_division_algebras%29

Best regards,

Jose Carlos Santos

From: GeometricGroup on 14 Jun 2010 04:20
> 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

Is there any intuitive explanation about this? or any reference?

Thanks.
From: José Carlos Santos on 14 Jun 2010 08:46
On 14-06-2010 13:20, 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.
>
> Is there any intuitive explanation about this?

None that I know of.

> or any reference?

http://retro.seals.ch/digbib/view?rid=comahe-001:1940-1941:13::175&id=hitlist

I can provide a proof of the fact that S^2, with its usual topology,
cannot become a topological group.

Best regards,

Jose Carlos Santos
From: Timothy Murphy on 15 Jun 2010 20:24
José Carlos Santos wrote:

>> Maybe S_1, S_3 and S_7 are the only spheres
>> that support a continuous multiplicative norm with unit element?
>
> If you are talking about norms, that I guess that you are thinking about
> vector spaces. Do you know Hurwitz's theorem on normed division
> algebras? See:
>
http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28normed_division_algebras%29

No, I meant the theorem that there is no continuous multiplication on S_n
with identity element except for n=1,3,7.


--
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
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