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If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, what is the maximum number $k(n)$ of linearly independent vector fields? Rewriting $n=2^{4a+b}(2s+1)$, Adams computes $k(n)=2^b+8a-1$.

In particular, on the $(4n+1)$-spheres, there is only one nowhere-vanishing vector field up to linear-independence, whereas in every other (odd) dimension there are more.
Example: on the circle $S^1$ there are the vector fields generated by (counter)clockwise rotation, but these are the same up to a scalar. This makes sense: I start flowing along this single dimension and then I have to continue flowing in that direction until I come back to my starting point.
I tried considering the difference between $S^3$ and $S^5$, which fiber over $\mathbb{C}P^1$ and $\mathbb{C}P^2$ respectively. A nowhere-vanishing vector field in both cases is given by taking the standard nowhere-vanishing vector field on the $S^1$-fiber. But for $S^3$ there are three linearly-independent fields (the $i,j,k$-directions when representing $S^3$ as the unit quaternions -- is there a way to see this using the fibration picture?), whereas for some reason $S^5$ can only admit the one.

What can be the differential/topological reasoning behind this? I.e. is there a down-to-earth way to deduce this result on $S^{4n+1}$, or for starters, $S^5$?

Could there possibly be an analogous index theorem going on here, in the same way that the Poincare-Hopf theorem provides us the Hairy Ball result?

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The result for spheres of dimension $4n+1$ was first proven in 1942 independently by Eckmann and Whitehead. Eckmann's paper is in German (and I will not comment on it), but Whitehead's 1942 Annals paper "Homotopy Properties of Real Orthogonal Groups" is readable and is low-tech by comparison to Adams' paper. It is still much harder than Poincare-Hopf, and the proof is homotopy-theoretic. If you are really motivated, you may want to run Whitehead's arguments for $n=1$ and see if you find them sufficiently geometric for your taste. – Misha Apr 30 '13 at 5:26
You might want to check out Lawson & Michelson's ``Spin Geometry''. It contains a discussion about how Clifford algebras can be used to study the number of linearly independent vector fields on spheres. I can't recall off the top of my head if this involves the Hopf fibration in an essential way, but you figure at least in these low dimensions the algebras are simple enough -- in the non-technical sense of the expression -- to understand how things play out concretely. At worst, the book contains many references to classic papers on the problem. – Russell Apr 30 '13 at 7:30
Chris: You should also specify what "down-to-earth" means: I am nearly sure that every proof (of linear dependence) will involve homotopy groups at some point and in some form. – Misha Apr 30 '13 at 13:00

The Radon-Hurwitz number $k(n)$ is the largest $k$ such that there exists an orthogonal multiplication $\mathbb R^k\times \mathbb R^n\to \mathbb R^n$; so for an ONB $x_1,\dots, x_k$ of $\mathbb R^k$ and a unit vector $y\in \mathbb R^n$ the vectors $y, x_1.y, x_2.y,\dots x_k.y$ are orthogonal in $\mathbb R^n$. This describes vector fields on $S^{n-1}$. The orthogonal multiplications were constructed by Radon [Lineare Scharen orthogonaler Matrizen, Abh. Math. Sem. Univ. Hamburg 1, 1-14, 1921] who extended the construction of Hurwitz for $k=n=1,2,4,8$. They extend to representations of Clifford algebras $C(\mathbb R^k, -\langle\quad,\quad\rangle)$ which explains the "periodicity" in $k(n)$ with respect to 8 and 2.

Radon writes: "For real matrices the solution offered itself to me by a particular reduction method using matrices whose elements are complex numbers or quaternions." (My translation) Maybe, there is an inkling of the fiber method you are looking at.

Adams showed that there are not more linearly independent vector fields on the sphere $S^{n-1}$.

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Correct, but this is what sparked my questions. – Chris Gerig Apr 30 '13 at 5:36
up vote 2 down vote accepted

Thanks Misha for the reference! (Just rewriting it here to complete this thread).
It seems that an off-the-cuff calculation (what I refer to as "down-to-earth") probably isn't going to suffice; there is some intricate stuff going on in the proofs involving the homotopy groups of the rotation groups of the spheres and orthogonal frames.

Homotopy Properties of the Real Orthogonal Groups (Whitehead), 1941, which focuses precisely on our case in question. The proof comes down to the obstruction of certain maps induced from rotation groups and their fixed subspaces on spheres.

Vector Fields on the n-Sphere (Steenrod, Whitehead) 1950, which studies particular fibrations of Stiefel manifolds over the spheres. It proves that for $n=2^k(2m+1)-1$ we have $k(n)<2^k$, which includes our case in question.

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