In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in
n
{\displaystyle n}
-dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that if
ρ
(
n
)
{\displaystyle \rho (n)}
is the Radon-Hurwitz number of the n-sphere (definition below), then there were at least
ρ
(
n
)
−
1
{\displaystyle \rho (n)-1}
such fields. Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found. Hence
ρ
(
n
)
−
1
{\displaystyle \rho (n)-1}
is the exact number of pointwise linearly independent vector fields that exist on an (
n
−
1
{\displaystyle n-1}
)-dimensional sphere.
Contents
Technical details
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
ρ
(
n
)
{\displaystyle \rho (n)}
determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of
n
{\displaystyle n}
odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem), so the case
n
{\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 (
n
−
1
{\displaystyle n-1}
)-sphere is exactly
ρ
(
n
)
−
1
{\displaystyle \rho (n)-1}
Radon–Hurwitz numbers
The Radon–Hurwitz numbers
ρ
(
n
)
{\displaystyle \rho (n)}
occur in earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) on the Hurwitz problem on quadratic forms. When we write
n
=
(
2
a
+
1
)
2
4
d
+
c
,
{\displaystyle n=(2a+1)2^{4d+c},}
where
a
,
d
,
c
{\displaystyle a,d,c}
are all integers and
0
≤
c