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?
No comments:
Post a Comment