Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2) This is a bit of an ill-defined question, and I feel I should have been able to resolve it by combining Google with a few library trips, but I'm having difficulty narrowing down the search results to a list I can actually go through practically. Apologies if the question seems too vague or not sufficiently thought through.
What I'm after is a section of a book or published article which could be used by a 3rd-year undergraduate as a source for the fact that the action of SU(2) on the Riemann sphere by Möbius transformations gives rise to a double cover of SO(3). It doesn't need to be too precise about what exactly is meant by a double cover; but I would like something which makes it clear that we are somehow slicing a 3-sphere into 1-spheres (a.k.a. circles) in an unusual way, without saying "let $E$ be a fibre bundle..." or "consider the exact sequence..." In particular, anything that assumes the student has a proper background in algebraic topology or differential geometry is probably at too advanced/sophisticated a level.
Of course, one is tempted to just write down the map and look at some of its properties: but for the present purposes it's important that I can direct the student to a citable source that is reasonably self-contained (at least when it comes to this particular result). Thus although the wikipedia entry, for, say, "Hopf fibration" is along the desired lines, I really need something more "official-looking". For similar reasons, I don't think I can just explain things to the student in person; that wouldn't be correct, whereas "pointing the student to a book" would be.
Anyway: I thought that on MO there might well be people who've had similar ideas/experiences either as teachers or students, and who had therefore come across a handy section of book which could be used. Any suggestions?
 A: Although in the body of your question you mention the action of $SU(2)$ on the Riemann sphere, the simplest (to my mind) answer to the question itself is to understand $SU(2)$ as the unit-norm quaternions acting by conjugation on the imaginary quaternions.  It is easy to see that this defines a homomorphism $SU(2) \to SO(3)$, which is surjective (since both $SU(2)$ and $SO(3)$ are connected) and has kernel consisting of the quaternions $\pm 1$.
You might find this in one of Coxeter's books or in Elmer Rees's Notes on Geometry.
A: cf. Algebra - Michael Artin 
There is a lot on this subject in that book.  It's one of the "special topics" that he included. 
Edit: Page 277
A: One book you might look at is Conway and Smith's "On quaternions and octonions". It discusses the map from the unit quaternions to $SO(3)$, and give applications of quaternions to "geometry in three dimensions" in a reasonable amount of detail. It also discusses the octonions and the Cayley-Dickson construction very nicely, so it's a great book to look at for interesting examples. One thing I liked is that the compact form of $G_2$ is the automorphisms of the octonions and a normed algebra, and thinking in terms of standard bases for octonions calculates the dimension of $G_2$ essentially by showing it's an iterated sphere bundle, in the spirit of how you wanted to talk about the Hopf fibration I think. I was using the book as a resource in a class on classical groups for 3rd year undergrads, and it at least let me find a nonclassical group. (Some of the students knew what a manifold was, some didn't -- it seemed fair to say to the one's who did that they should think through the examples we had and see that they were manifolds/fibrations etc. useful if you have some strong students in the class!)
A: "Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics" by Sattinger and Weaver has Section 4, pp. 10-15 entitled "The covering group of SO(3)". The discussion is self-contained and accessible for kinder-garden kids :)
A: I agree that Artin has this, but also "Group Theory and Physics" by Sternberg has nice discussions early on (circa page 8-15).
A: Undergraduate text about matrix groups: 
http://www.springer.com/mathematics//book/978-0-387-78214-0
