# 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?

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I could swear there was another question on this exact subject a few days ago. Was it deleted? – Qiaochu Yuan Jan 16 '10 at 23:02
Didn't see it if there was. Unfortunately I still haven't found a way to filter the MO question lists that I find satisfactory, so I may just have missed a prior question through lack of attention – Yemon Choi Jan 16 '10 at 23:08
Probably Qiaochu is thinking of this question: mathoverflow.net/questions/11821/su2-and-the-three-sphere ? – Kevin H. Lin Jan 16 '10 at 23:43
@quid done. FWIW: I don't mind people retagging if it's for sensible reasons and done in drips rather than a deluge. My objection to people tinkering with my posts, or others, is that I feel it sometimes verges on a fetish/totem of false consistency, or spurious accuracy. – Yemon Choi Oct 15 '14 at 23:19
For instance, in the answer below mathoverflow.net/a/12036/763 if I were to put my sub-ed's hat on I would say: cf should be qv (or just "see") and the hyphen should be an em dash. I personally don't think that is a productive way to be an MO user, especially as edits bump questions. But I fear I am repeating myself so I'll stop here. – Yemon Choi Oct 15 '14 at 23:22

http://www.springer.com/mathematics//book/978-0-387-78214-0

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I've not read Stillwell's book. Does it explicitly discuss the Hopf fibration? – Yemon Choi Oct 16 '14 at 10:56
Yes, but without discussing the topological aspect. Uses quaternions. – Deane Yang Oct 16 '14 at 16:02

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

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Well, I used to see so many questions where other people would say "make it CW!" that I vaguely thought I'd pre-empt that. But if no one has objections, and if it were possible, I'd be happy to un-CW it. – Yemon Choi Jan 16 '10 at 23:06
BTW, thanks for the suggestion Harry - I'll check the library tomorrow. – Yemon Choi Jan 16 '10 at 23:07
Agreed. Artin satisfies both requirements of not being too precise about what is meant by a double cover and explicitly describing the fibers as circles. – Qiaochu Yuan Jan 16 '10 at 23:07
I'm happy to help when I can, so it doesn't matter to me. I was just kidding. ;) – Harry Gindi Jan 16 '10 at 23:08
I'm a bit confused here -- if the students know any group theory, then they know that if G is a group and H a subgroup, Lagrange's theorem say all cosets of H are in bijection with H, so the map G -> G/H is clearly some sort of "fibration" -- this would be what I'd use for the double cover and the Hopf map. Then you just need to know that SO(2) is a circle right? All of that can be done with the basic group theory and 2x2 or 3x3 matrices (or complex numbers/quaternions) so surely that's elementary enough? – Kevin McGerty Jan 17 '10 at 0:10

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!)

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"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 :)

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I agree that Artin has this, but also "Group Theory and Physics" by Sternberg has nice discussions early on (circa page 8-15).

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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.

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Of course you know this already José, but I thought I'd say it anyway in case it helps someone with their teaching. One way to say exactly the same thing but without mentioning quaternions is to talk of the adjoint action of SU(2) on its Lie algebra. This preserves the Killing form, so gives a map to SO(3). I sometimes find students are scared of quaternions, but then they are sometimes scared of the adjoint action too so I guess it just depends on their background. – Joel Fine Jan 17 '10 at 0:17
I'm glad you pointed this out. I was tempted to add it to my answer. I guess in hindsight quaternions are not necessarily more elementary than Lie algebras; although they ought to be :) – José Figueroa-O'Farrill Jan 17 '10 at 1:04
@JoelFine and Jose'F.-O'F, it seems to me very funny that at this moment in history quaternions could conceivably be "less elementary" than Lie algebras (although, I hasten to add, I am a very great fan of Lie algebras, etc, etc, etc, ...!!!) Pity that this example of-so-many-things has become obscure... – paul garrett Oct 15 '14 at 23:20
I personally found this quite confusing for some time before I realized it was "simply" the adjoint group of SU(2). This is not "elementary" per se, but it is satisfying. The choice of people to say things like: there is a rank two complex vector bundle associated to a principal SO(3) bundle (if the first Stiefel-Whitney class vanishes) has always bothered me. Writing SO(3) encourages one to think of its (obvious) representation on 3-space. Furthermore, the adjoint group description quickly leads one to write down the isomorphism. I apologize in advance for any hostility... – Andy Sanders Oct 15 '14 at 23:26