# introductory reference for Hopf Fibrations

I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to learn about vector bundles. )

If anyone with more experience could point me in the right direction this would be really helpful.

I think there are several sources to learn the definition of the Hopf fibrations (wikipedia in particular), but to really "understand" them is up to you.

The best starting place is the fibration $S^3\rightarrow S^2$. Before getting to this map, I would highly suggest trying to think, learn and understand everything about $S^3$ from every possible direction, i.e., stereographic projection, two 3 balls with their boundaries glued, two solid tori with their boundaries glued, etc. Then think about how $S^3$ is the unit ball in $\mathbb{R}^4$ = $\mathbb{C}^2$ and think about things like rotating one orthogonal place in $\mathbb{R}^4$ while keeping the other fixed and its effect on your pictures of $S^3$. Can you follow the orbits? How about rotating both planes? You can make up and answer all these types of questions about $S^2$, so learn how to do the same for $S^3$.

Once you get an intuition and a level of geometric understanding of $S^3$, go back to the definitions of $S^3\rightarrow S^2$ and you'll realize that this is just another way to view $S^3$ and , in fact, you might have already discovered it. When you do, it will fit in nicely with all of the other pictures of $S^3$ you have built in your mind.

The fibration $S^3\rightarrow S^2$ is just the starting place, but I think understanding it enables you to easily feel comfortable with all the other Hopf fibrations from spheres to projective spaces.

These notes might also add some motivation to the topic coming in from Physics:

http://www.itp.uni-hannover.de/~giulini/papers/DiffGeom/Urbantke_HopfFib_JGP46_2003.pdf

This notes in arXiv are supposed to be accessible for high school students, although I think they would be better for undergraduate studens.

http://arxiv.org/abs/0908.1205

I hope it helps you.

The book Topology and Groupoids (pdf available) has Chapter 5 on "Projective and other spaces" and p. 150 gives the Hopf map $p_V: S(V) \to P(V)$ where $V$ is a finite dimensional normed vector spaces over the reals, complex numbers or quaternions; the latter are fully explained.

Another source is "Topological geometry" by I. Porteous, which also does Clifford algebras.