Main Question or Discussion Point
Usually the hairy ball theorem is cited for proving that [tex]S^2[/tex] is not parallelizable. However, hairy ball theorem is too strong for this. This theorem states that there isn't a nowhere vanishing continuous vector field on [tex]S^2[/tex]. Unparalellizable property means only that there aren't two linearly independent vector fields on [tex]S^2[/tex]. Could somebody tell an example of a nonparallelizable n-dimensional manifold on which hairy ball theorem is false, i.e. on which there is continuous nowhere vanishing vector field (but because of the nonparallelizability, n linearly independent aren't)?