Post #1604054

I've been revisiting my hairy ball theorem argument over here: https://blog.sigfpe.com/2012/11/a-pictorial-proof-of-hairy-ball-theorem.html I may try to make a video of the whole argument but here is the key step. If you make the sphere a union of two extended hemispherical caps so they overlap along the band containing the equator, then this band can be looked at in two ways depending on whether you view it as part of the northern hemispherical cap flattened out or the southern cap. If we look at just the band alone, we can see how a vector field on it varies as we flatten it out one way or the other. In this case we have a vector field that starts with a winding number of 1 as we follow around the equator. But when we flatten the other way we get a winding number of -1. This band flipping always introduces a change of winding number of 2. So our vector field can never have a zero winding number around the equator in the northern and southern hemispheres at the same time. See my original blog post for why this makes it impossible for a non-zero vector field everywhere. Does this video help? (In the context of my original article.) We can do the same with a tangential rather than radial vector field so it's clear than the same change happens for any basis and hence for any vector field,