Basically, the Hopf fiber is a fiber structure on the -sphere . Or, it describes a decomposition of into and .
Start from the definition of , it consists of points such that . On defines a projection where stands for the first coordinates of points in . Note that, the image of indeed lies in . We will show that is surjective and for each , the fiber is a circle. And this describes a fiber structure on .
Indeed, for any , we want to find such that
Yet this is almost trivial, since we have that
This system is always solvable. Moreover, if , then . This can also be seen from the above system of equations. Note that sometimes we are likely to take this projection, . One problem with this projection is that, its image is not the entire ( is nover negative), another point is that , yet for the point , its fiber is only itself, only one point, not a circle.
Another interesting way to describe this fiber is to use some Lie groups. Note that, for , the Lie group acts transitively and freely on (because each matrix in can be written as , in general, a matrix in can be written as ).
But, acts also mysteriously on ! This is a rather important fact. Note that, in some sense, this is also reasonable: that is acts also on the tangent space of some point on , and has a group structure(that is exactly ). After giving this tangent space a metric(induced from that of , of course), acts thus on preserving the metric, thus acts also on . Yet, we know that, acts on , we can show that the action of on induces a homomorphism , with kernel . Moreover, acts transitively on , and for each point , the stabilizer of on is a circle. Note also that, for the opposite point , the stabilizer of on is the same as the previous, . This shows that, not rigorously, that acts on with for each point, a stabilizer group the same as one copy of circle.
More concretely, since consists of matrices of the form with , we consider the conjugate action of on itself. This is indeed a group action, and we note this as
Now take derivative with respect to the variable , and transform this action to the Lie algebra
Note that, the Lie algebra of consists of matrices such that . More specifically, we can find a -basis for (note that, is not a complex Lie group, only a real Lie group, thus its Lie algebra is only a real vector space, not a complex one):
We can show that the metric induced from on is preserved by , and forms an orthonormal basis. Now, we want to show that the stabilizer of each point is a circle, since acts transitively on , we can verify this for just one point, say . Note that, if preserves , then . So, after some routine yet simple calculations, we find that, under the basis , (why, you may wonder, there is an increase in the ‘speed’? This can be explained by taking another derivative with respect to in , thus we get that . For , we have that . It is readily verified that , this little is the origin for this speed increase! And this accounts also for the fact that ).
So, in this way, we describe another way of the Hopf fiber on .
The second method enables us to generalize to higher dimension case.
A nice demonstration of the Hopf fiber is here.