Vector fields on spheres

Remarks

The lower bound is a construction from Radon and Hurwitz: $\rho(n) - 1$ linear maps $A_i$ of $\R^n$ such that the fields $x \mapsto A_i x$ on $S^{n-1}$ are pointwise independent. These maps come from modules over Clifford algebras. The upper bound was proved by J. F. Adams in 1962 using topological K-theory and Adams operations on real projective spaces.

Example: for $n$ odd, $\rho(n) = 1$, so an even-dimensional sphere $S^{n-1}$ has no nowhere-zero vector field; this is the hairy ball theorem. For $n = 4$, $\rho(4) = 4$, and $S^3$ has $3$ independent fields, given by multiplication by the quaternions $i$, $j$ and $k$. For $n = 16 = 2^{4}$, $a = 1$ and $b = 0$, so $\rho(16) = 9$ and $S^{15}$ has $8$ independent fields but not $9$.

As a consequence, the tangent bundle of $S^{n-1}$ is trivial, that is, $S^{n-1}$ is parallelizable, exactly when $\rho(n) = n$, which happens for $n = 1, 2, 4, 8$: the spheres $S^0$, $S^1$, $S^3$ and $S^7$.

Going deeper

For a student

The problem is equivalent to asking for sections of the Stiefel bundle $V_{k+1}(\R^n) \to S^{n-1}$. Adams' proof of the upper bound works with real projective spaces and computes their real K-theory $\widetilde{KO}(\R P^m)$ together with the Adams operations on it. The theorem is closely related to the Hopf invariant one problem and to the classification of normed division algebras over $\R$, which also singles out the dimensions $1, 2, 4, 8$.