Tom Banchoff and Nelson Max wrote a lovely proof that every sphere-eversion has a quadruple point,
but it was quite long, and was a proof-by-pictures. I realized, just after finishing my Ph.D., that I
could apply the results of my dissertation to prove it in just a few lines, and wrote this paper.
The main ideas were that (i) the regular homotopy class of a 3-sphere-immersion (which you get if you add "caps" to
the track of a 2-sphere eversion) is related to (a) the normal degree of the immersion (just as in the
Whitney-Graustein theorem two dimensions lower), and (b) the normal pontriagin class, and (ii) the map from regular homotopy classes
to the integers mod 2, consisting of "count generic quadruple points" is a homomorphism. One then observes the
values of the homomorphism on a couple of generators for the group of regular homotopy classes, and whammo! the result falls out.
Arguably, this is why I ended up getting a job at Brown, so I'm awfully glad I did write something down.
