Let $n$ be a positive integer.

Let $\langle M,\text{charts} \rangle$ be an $n$-dimensional $C^1$ manifold such that the

induced topological space is homeomorphic to the $n$-dimensional sphere.

Does it follow that ((all continuous vector fields on $\langle M,\text{charts} \rangle$ have a zero) if and only if ($n$ is even))?

differentiablemanifold (which is necessary for the problem to even make sense). – Ricky Demer Sep 26 '11 at 6:14