Re: I'm so confused...

On Sun, 15 Jan 2006 18:26:48 +0000, José Carlos Santos
<jcsantos@xxxxxxxx> wrote:

>David C. Ullrich wrote:
>
>> Say R(t) is the element of SO(3,R) defined by
>> a rotation through an angle t about, say,
>> the axis (0,0,1). Define c : [0,2pi] -> SO(3,R)
>> by c(t) = R(t), and define d : [0,4pi] -> SO(3,R)
>> by d(t) = R(t). Then
>>
>> (i) c is not null-homotopic (as a closed curve in
>> SO(3,R)
>>
>> (ii) d _is_ null-homotopic in SO(3,R).
>>
>> At least I _think_ that that's what Penrose
>> is claiming.
>>
>> At first this seemed clearly impossible since
>> d is just c + c, after all. But this morning
>> I think I see why (ii) holds: We only need to
>> show that c _is_ homotopic to -c (the same as
>> c except traversed in the opposite direction).
>> Say a(s) is a continuous path in the unit
>> sphere in R^3 joining (0,0,1) to (0,0,-1), and
>> let c_s(t) be a rotation about a(s) through an
>> angle t (for t in [0,2pi].) Then (c_s) is a
>> continuous family of closed curves, c_0 = c
>> and c_1 = -c.
>>
>> Right? If that is in fact a correct proof of
>> (ii) then how does one prove (i)? (I mean (i)
>> is obvious, but otoh (ii) is obviously false...)
>
>Consider the multiplicative group G of the quaternions with norm
>1 (which is isomorphic with SU(2)). Topologically, this is the
>3-sphere and therefore it's simply-connected. Now consider the path
>
> g:[0,pi] ----> G
> t |-> cos(t) + i sin(t),
>
>which is clearly not a loop.
>
>The group G acts naturally on the space H of purely imaginary
>quaternions, because if g is in G and q is a purely imaginary
>quaternion, then g*q*g^{-1} is also purely imaginary. It turns out
>that the map from H into itself defined by q |-> g*q*g^{-1}
>preserves the quaternionic norm; since H, as a real vector space, is
>3-dimensional, this defines a homomorphism _f_ from G into SO(3,R)
>and it turns out that it is a double covering.
>
>Now, what's f o g? For each _t_, f(g(t)) is a rotation around
>_i_ whose angle is 2*t. In particular, f o g is a loop. However,
>this loop cannot be null-homotopic, since, if you lift it to G, you
>get _g_, which is not even a loop.
>
>I hope that this helps.

Yes, that seems like exactly what I was looking for.
Had a dim recollection later that there was a double
cover of SU(2) onto SO(3) but I wasn't sure, hadn't
worked out how it worked.

Quaternions. Huh.

>Best regards,
>
>Jose Carlos Santos


************************

David C. Ullrich
.

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