Lefschetz Connection
There is a closely related argument from algebraic topology, using the Lefschetz fixed point theorem. Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on 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 diffeomorphisms 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.
Read more about this topic: Hairy Ball Theorem
Famous quotes containing the word connection:
“Parents have railed against shelters near schools, but no one has made any connection between the crazed consumerism of our kids and their elders’ cold unconcern toward others. Maybe the homeless are not the only ones who need to spend time in these places to thaw out.”
—Anna Quindlen (b. 1952)