Editing Hairy ball theorem (section)

==Lefschetz connection==
There is a closely related argument from [[algebraic topology]], using the [[Lefschetz fixed-point theorem]]. Since the [[Betti number]]s of a 2-sphere are 1, 0, 1, 0, 0, ... the ''[[Lefschetz number]]'' (total trace on [[homology (mathematics)|homology]]) of the [[identity mapping]] is 2. By integrating a [[vector field]] we get (at least a small part of) a [[one-parameter group]] of [[diffeomorphism]]s on the sphere; and all of the mappings in it are [[homotopic]] to the identity. Therefore, they all have Lefschetz number 2, also. Hence they have fixed points (since the Lefschetz number is nonzero). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general [[Poincaré-Hopf index theorem]].