Laura has a solution to exercise [15.05] up on her website here:

http://camoo.freeshell.org/roadtoreality.html

Unfortunately it's very sketchy (no working shown), the bits about the antipodean points on S2 are somewhat confusing, and she states that the fibres in S3 map to two full rotations of the tangent vector on S2, but doesn't in any way demonstrate this.

I thought that a solution to the exercise ought to be posted up here on this board, and I thought that I could do a better one. (Sorry Laura!)

...So here it is (attached):

Oops! In my conclusion I say equation (3) and (6) map to S2 and unit tangents on S2 respectively. Actually as written, (6) maps to tangent vectors of length 2. You need to divide the RHS by 2 to get the correct equation to use.