Vector fields on spheres

In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers ${\displaystyle \rho (n)}$ determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of ${\displaystyle n}$ odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem), so the case ${\displaystyle n}$ even is an extension of that. Adams showed that the maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the (${\displaystyle n-1}$)-sphere is exactly ${\displaystyle \rho (n)-1}$.

The construction of the fields is related to the real Clifford algebras, which is a theory with a periodicity modulo 8 that also shows up here. By the Gram–Schmidt process, it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis at each point.

The Radon–Hurwitz numbers ${\displaystyle \rho (n)}$ occur in earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) on the Hurwitz problem on quadratic forms.^[3] When we write

${\displaystyle n=(2a+1)2^{4d+c},}$

where ${\displaystyle a,b,c}$ are all integers and ${\displaystyle 0\leq c<4}$, then^[3]

${\displaystyle \rho (n)=2^{c}+8d}$.

If ${\displaystyle n}$ is odd, then ${\displaystyle c=d=0}$ and ${\displaystyle \rho (n)=1}$. The first few values of ${\displaystyle \rho (2n)}$ are (from (sequence A053381 in the OEIS)):

2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ...

These numbers occur also in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real ${\displaystyle n\times n}$ matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by Beno Eckmann. They are now applied in areas including coding theory and theoretical physics.