As I recall, there’s a bit of one proof of the homotopy invariance of Morse-Bott homology where you draw a diagram, and then do Morse theory on the diagram you just drew.

On a somewhat separate note, it’s one of my longterm goals to collect a bunch of proofs by contradiction, and then publish a book where they’re all reduced to the contradiction 1=0. The Hairy Ball Theorem is a pretty good intermediate step, if you can cook up an appropriate vector field. Maybe I’ll put in a Hairy Ball chapter.

It would be nice if someone came along and showed how to “nudge” one one of these proofs into another by a kind of “proof-theoretic homotopy”, to put it impressionistically.

The idea is to triangulate your manifold and put positive and negative “charges” on all the simplices, depending on whether their dimension is odd or even. What little work there is goes into showing that your triangulation can be chosen so that all the lower-dimensional faces are transverse to the non-zero vector field. Now use your vector field to “nudge” the charges into the interiors of the top-dimensional faces. A simple count shows that total charge in each face is zero – and so the Euler characteristic has to be zero too.

I agree, Milnor is a technical genius, and his proof is very slick. But as with many technically elegant proofs, the result ends up seeming somewhat mysterious, to me at least. Thurston gives a really clear idea of “why” it’s true.

I mean, the ingenious proof seems to be very geometrical (as opposed to topological) in nature — it seems to rest upon the particular standard embedding of the unit sphere in R^n. This embedding induces a number of differential-geometric structures (e.g., Riemannian metric, Levi-Civita connection, curvature, etc.) which have homological import, and I have a feeling some of this may be sneaking in through a back door.

Maybe to be just a tiny bit more explicit, there’s this theory of tubes which I really know nothing about, but which goes back to Weyl, where he apparently gives a formula for tube volumes in terms of the curvature tensor. That I suspect is secretly connected to the tube volume computations of Milnor, but for now that’s just a guess.

For example:

It’s very pretty, but is it fundamentally different to the usual proof, or homotopic to it in some sense? Why can this theorem, typically considered a classic application of homology theory, be done using calculus? How many other “classic applications of homology theory” can be done using calculus. When I think “calculus” and “homology theory” together, I think of de Rham theory. Does Milnor’s proof secretly involve de Rham theory?

I’ve never made much progress on any of these questions.

I don’t know if “thet(os)” can also mean angle; I overreached there.

At any rate, the etymology suggests that “homothety” should mean something like “similarly placed”, which is at least in the neighborhood of the mathematical meaning.

[I'm curious about "theta = place" as well; is there something you can cite to that effect? Does this bear on why was originally chosen to denote angles (by Euler, perhaps)?]