From: "C. Hillman"
Newsgroups: sci.math
Subject: Re: Hopf map question
Date: Wed, 9 Sep 1998 04:51:21 -0700
On Wed, 9 Sep 1998, MindSpring User wrote:
> If each point on a sphere, (x1, x2, x3), maps to
>
> [(x1/(1-x3)), (x2/(1-x3))]
>
> on a plane
This is stereographic projection from S^2 (embedded in R^3 in the usual
way) to R^2.
> and each point of a hypersphere, (x1, x2, x3, x4), maps to
>
> [2(x1x2 + x3x4), 2(x1x4 - x2x3), ((x1)^2 + (x3)^2) - ((x2)^2 + (x4)^2)]
>
> on a sphere,
This is one form of the Hopf map from S^3 --> S^2.
If you consider S^3 as the space of unit norm quaternions, then the
classical Hopf map is simply
q --> q k q^{-1}
where k is the half turn about the z-axis, and where q is a unit norm
quaternion.
If you identify S^3 with the matrix group SU(2), i.e. identify q with
[z w]
[-w* z*] where |z|^2 + |w|^2 = 1, z,w complex, * = complex conjugate,
then k is identified with the matrix
[i 0]
[0 -i]
Multiplying out q k q^{-1} you find that the "complexified" form of the
Hopf map can be written
(z,w) --> ( i(|z|^2 - |w|^2), 2 zw)
Writing z = x1+ i x2, w = x3 + i x4 now gives the "realified" form of the
Hopf map can be written
(x1,x2,x3,x4) -->
(2 x1 x2 - 2 x3 x4, 2 x1 x4 + 2 x2 x3, x1^2 + x3^2 - x2^2 - x4^2)
which is essentially the form given in the book you read.
The Hopf fibers are circles S^1 in S^3 which are obtained as the preimages
of points on S^2. These fibers are pairwise linked. You have probably
seen an illustration showing this picture using stereographic projection
from S^3 to E^3.
For most people, I think the most useful way to think about the Hopf map
is the one most closely connected to Lie theory. Namely, think of the Lie
group SU(2), which can be identified with S^3, and which is the spinorial
double cover of the rotation group SO(3), acting transitively on S^2 in
the obvious way. The isotropy subgroup of say the North pole is then the
spinorial double cover of SO(2). This is geometrically a circle S^1 and
is doubly covered exactly the same way that the central circle of a
Moebius band is covered by its edge (a topological circle). The point is
that the Hopf map is then an example of the canonical fibration G --> G/H.
The fibers are the cosets of subgroup H, the double cover of SO(2). The
Hopf fibration of S^3 then provides an interesting and vivid example of a
nontrivial circle bundle. Moreover, q -> q k q^{-1} = (Ad q)(k).
Another reason this is interesting is because of the light it sheds on
quaternion multiplication. You know how to visualize multiplication of
complex numbers by adding angles and multiplying radii. Multiplication in
SU(2) or the unit norm quaternions is the quaternionic analogue to the
"add angles" part of this prescription, and then the Hopf fibration shows
the effect analogous to z -> z_0^t z where z_0 is a fixed unit norm
complex number and z is on the unit circle S^1. This is of course just
the rotation flow of SU(1), so it is not surprising that SU(2) generalizes
this, although the quaternion connection is interesting.
Compare the conjugation action by SU(3) on itself, which has quite
different fibers.
In order to understand the geometry of the Hopf map, it is helpful to
observe that S^3 is foliated by a one real parameter of Hopf tori, which
are geometrically flat tori T^2. For fixed 0 < theta < pi, the
corresponding torus is parameterized by real u,v like this:
[ cos theta/2 cos u ]
[ cos theta/2 sin u ]
[ sin theta/2 cos v ]
[ sin theta/2 sin v ]
You can easily check that this can be thought of as a rectangle with sides
cos theta, sin theta, with opposite sides identified in the usual way.
Moreover, you can check that it is the preimage of the latitude circle for
angle theta on S^2, and that the Hopf fibers are parallel to the circle
represented by one diagonal of the rectangle before identifying opposite
edges.
Exercise: show the the preimage of a longitude semicircle is a cylinder
S^1 x I with boundary consisting of two linked circles (the preimages of
the N and S poles).
There is -a lot- more to the Hopf map, including connections to more ideas
in Lie theory, uniparameter subgroups, spinors, physics, "twist algebras",
and the like. Without going into details suffice it to say that there are
detailed SU(1,1) analogues for SU(2), including a Hopf map involving the
spinorial covering of SL_2(R) or SO+(1,2), the group of proper
orthochronous Lorentz transformations in two spatial dimensions. Moreover,
both of these are in turn closely related to the well known group
isomorphism between SO+(1,3) and the Moebius group PSL_2(C), which brings
us back to stereographic projection.
There are many research papers on various aspects of the Hopf map. You
probably want to look in particular for on the geometric Hopf map,
published ten or twenty years ago in the Journal of the London Math
Society--- sorry, but I can't remember more information than that--- which
provides details of a few of the things I've mentioned above.
> then what is the general formula?
Formula for what? The Hopf map? Or did you mistake the above for
stereographic projection from S^3 to R^3? That would be
(x1,x2,x3,x4) -> (x1,x2,x3)/(1-x4)
> That is, a spaceform of dimension, n, projects onto an n-1 dimensional
> spaceform by way of the general formula _?_.
It would be a mistake to assume that the Hopf map has a generalization
from S^3 -> S^2 to S^(n+1) -> S^n, if this is what you have in mind.
However, there -is- an obvious generalization to maps S^(2n-1) --> CP^n,
from a real 2n-1 dimensional manifold to a real 2n-2 dimensional manifold,
whose fibers are again circles. This is defined very simply. Let S^1 act
on C^n by
(z1,z2,..zn) --> {e^{i theta}(z1,z2,..zn): theta in R}
The orbit space of this action is CP^n, by definition. Then restricting
this map to the real submanifold S^(2n-1) gives the generalized Hopf map.
Here CP^1 (two real dimensions) may be identified with S^2 and then we
recover the classical Hopf map.
Geometrically, S^{2n-1) also decomposes into an n real parameter family of
flat n+1 real dimensional tori, in which the circular fibers sit in a
manner analogous to the classical Hopf fibers.
This generalization doesn't have the nice Lie group interpretation of the
classical Hopf map, but does suffice to show something interesting to
algebraic topologists, namely that the higher homotopy group
pi_(2n+1)(CP^n) is Z. (IIRC, Hopf introduced the classical Hopf map in
the context of showing that the homotopy group pi_3(S^2) is nontrivial.
In general, computing higher homotopy groups is ferociously difficult.)
Hope that somewhere in the above I have answered your question :-/
Chris Hillman
Please DO NOT email me at optimist@u.washington.edu. I post from this account
to fool the spambots; human correspondents should write to me at the email
address you can obtain by making the obvious deletions, transpositions,
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Thanks!