Missing explanation of Bloch sphere

[thumannw]

I'm not very happy with the exposition of the Bloch sphere. All the necessary mathematics is developed in Part III of the book but the definition of the Bloch sphere is left out. When reading the table on page 341 (Fig. 12.3) one could even get the impression that the Bloch sphere is the unit sphere S^3 in the Hilbert space C^2 which is wrong because the Bloch sphere is S^2 topologically.

Consider the sequence of maps
S^3 --> H\{0} --> PH --> C+ --> S^2
where H is the two dimensional complex Hilbert space C^2, PH is the projective space over H and C+ is the one-point compactification of the complex numbers. Furthermore, the first map is the inclusion onto the unit sphere, the second map is the projection, the fourth map is the stereographic projection and the third map sends [c:1] to c for any complex number c and [1:0] to the point at infinity. Consider the following unit vector in C^2: (cos(t/2), exp(ip)sin(t/2)) for t in [0,pi] and p in [0,2pi]. Then one can calculate that the image of such a vector under the map above (from S^3 to S^2) is the unit vector in R^3 represented by (t,p) in spherical coordinates. In my opinion, a more detailed explanation of these facts would give a good impression of the relationship between state representations in Hilbert space and the actual states on the Bloch sphere.
By the way, the map S^3 --> S^2 is the beautiful Hopf fibration.