The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best possible: there are never more linearly independent vector fields.

More precisely, this gives the following number: if $n+1 = 16^a 2^b c$ with $c$ odd, $0 \leqslant b \leqslant 3$, we get $\rho(n) = 2^b + 8a$ and there are exactly $\rho(n) - 1$ linearly independent vector fields on $S^n$. This lower bound comes by construction of vector fields from Clifford module structures on $\mathbb{R}^{n+1}$, and figuring these out isn't too hard, it follows from the classification of real Clifford algebras with negative definite quadratic form. This is detailed for example in *Fibre Bundles* by Husemöller; the material comes from the paper *Clifford Modules* by Atiyah, Bott, Shapiro. This classification hinges on a particular mod 8 periodicity for real Clifford algebras.

**Question:** How does this description of vector fields on spheres relate to Bott Periodicity in the real case (either for real $K$-Theory, in the form $KO^{n+8} \cong KO^{n}$, or for the homotopy groups of the infinite orthogonal group, $\pi_{n+8}(O) \cong \pi_n(O)$)?

In particular, I'm inclined to think there should be a rather direct relationship: after all, $K$-theory is talking about vector bundles, sections of which are vector fields! Surely the formula for the number of vector fields on spheres should have a concrete interpretation in terms of $K$-theory? The (underlying) mod $8$ periodicities must be linked!

In addition, the result of periodicity mod 8 for Clifford algebras is also often called Bott Periodicity; what is the deeper relationship here? This other post mentions that the periodicity for Clifford algebras relates to the periodicity for complex K-Theory and so it mentions BU and not BO.

Fiber Bundleshas details, if I recall correctly. $\endgroup$