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 << One idea is to use the famous topology theorem that "it is impossible to comb the hairs on a sphere." (For a 2D surface of a sphere in 3D. Combings are possible, and known, for a sphere in 2, 4, or 8 dimensions.)

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