Hopefully it's interesting, you enjoy it, and it doesn't contain too many errors.
ERRATA:
1) A sort of minor correction/elucidation.... but it should be noted that while the tangent bundles of Ci^{1} and As^{1} are diffeomorphic to cylinders that are infinitely extending in both positive and negative directions, the tangent bundle of Sq'^{1} is a cylinder that is only extending infinitely in the positive direction. Note that "direction" here is arbitrary. The point is that Sq'^{1} has a tangent bundle that terminates in one direction (except at {CP_{Ci^{1}}}, which do extend infinitely in both directions) while the other two 1-manifold in question do not. So the tangent bundle of Sq'^{1} ends up being infinitely extensible in the positive direction only, but with 4 spikes that extend infinitely in the negative direction.
2) Less a correction than an explanation: I didn't include a proof of the Hairy Ball Theorem because it's one of those proofs that has been given in many ways, all better than I could conjure up. Some famous, and some not as famous but still neat proofs. Here's a couple (click name of proof to go to the proofs):
Milnor's Analytic proof
McGrath's Proof Via Winding Numbers
Eisenberg and Guy's Proof Via Fundamental Group of the Circle (couldn't find a free version)
and so on and so forth. Those 3 are just the "top of the front page of google" proofs. For my work, and as maybe a more natural progression from last summer's paper, Eisenberg and Guy's proof would probably be most appropriate to use with my work. For this paper, as a stand alone paper, I'd be more inclined to use Milnor's proof. Any which way, the point remains that I just chose not to include a self-generated proof because it's already been said in better ways by better mathematicians.
3) Less to do with this paper than with previous papers (especially some of my earliest papers regarding gL), but this paper (link) by Rosca and Plonka seems to confirm (and add necessary rigour to) a lot of my earlier conjectures regarding maps from the Sphere to the Octahedron and the Hyperbolic Octahedron. While their paper doesn't include the Hyperbolic Octahedron, it's not a giant step (intuitively, at least) to extend the result. Their paper does predate even my earliest paper (evidently by more than a decade), so that's sort of comforting, in that it confirms that I wasn't just some crazy undergrad going on some "rouge mathematician" journey into the realm of total quackery. My only disappointment is that I didn't find the paper when I was writing mine, so that I could include it as a reference. Consider it a retroactive reference, then.