Actually, it's true that combings are possible for all odd-dimensional spheres. (Or in the language used below, for a sphere in any even dimension:

S^(2n-1) := {x in R^(2n) | ||x|| = 1}.

That can be readily seen because the above definition is equivalent to

S^(2n-1) := {z in C^n | ||z|| = 1}

and so there is a free SO(2) action on S^(2n-1) via

e^it z := (e^it * z_1, e^it * z_2,...,e^it * z_n),

which means that S^(2n-1) is fibred by consistently oriented circles -- hence there is a "combing".

--Dan

--yah, but really, we want d different color hairs, and the green hairs are combed, the red hairs are combed in a direction always orthogonal to the green hairs etc. For that more powerful kind of combing, you can do it for a surface that is 1,3, or 7 dimensional using complex, quaternion, octonion tricks in the d+1 dimensional space...