I also remember taking a class on vector calculus from the same author... which detoured through rudimentary manifold theory and differential forms, and ended with a final week on de Rham cohomology and the Mayer-Vietoris theorem (on vector spaces, to be fair, and not modules in general.)
(And is a very fine K-theorist, too, if I say so myself.)
My bad – that was a misleading thing to say! Thanks for pointing that out. I figured out what I said wrong. (Caveat emptor, I do biostatistics now.)
The context IIRC was this: one of the key results of the class was generalized Stokes' theorem, but this case (since was a 200-level class) we mostly just looked at differential forms on open spaces in R^n, and then said a few quick things about differentiable manifolds.
At this more concrete level, then, I remember that we constructed de Rham cohomology (fixing an open subset of R^n) beginning with the cochain complex given by vector spaces of k-differential forms and exterior derivatives, instead of working more generally with a cochain complex on modules.
But think I said something wrong here, which I why you were (rightly) confused. I'm not sure that the above distinction matters anyway since IIRC, you can get Mayer-Vietoris by showing that de Rham cohomology satisfies the Eilenberg-Steenrod axioms (stated for cohomology), and the Eilenberg-Steenrod axioms only need abelian groups anyway.
But I'm also 90% sure that TFA did something more direct to get to Mayer-Vietoris that I've forgotten, since we didn't use that much homological algebra.
> I also remember taking a class on vector calculus from the same author... which detoured through rudimentary manifold theory and differential forms, and ended with a final week on de Rham cohomology and the Mayer-Vietoris theorem (on vector spaces, to be fair, and not modules in general.)
Any available references for that that you know of?
He once taught an open to all freshman knot theory elective:
https://people.reed.edu/~ormsbyk/138/
I also remember taking a class on vector calculus from the same author... which detoured through rudimentary manifold theory and differential forms, and ended with a final week on de Rham cohomology and the Mayer-Vietoris theorem (on vector spaces, to be fair, and not modules in general.)
(And is a very fine K-theorist, too, if I say so myself.)
The context IIRC was this: one of the key results of the class was generalized Stokes' theorem, but this case (since was a 200-level class) we mostly just looked at differential forms on open spaces in R^n, and then said a few quick things about differentiable manifolds.
At this more concrete level, then, I remember that we constructed de Rham cohomology (fixing an open subset of R^n) beginning with the cochain complex given by vector spaces of k-differential forms and exterior derivatives, instead of working more generally with a cochain complex on modules.
But think I said something wrong here, which I why you were (rightly) confused. I'm not sure that the above distinction matters anyway since IIRC, you can get Mayer-Vietoris by showing that de Rham cohomology satisfies the Eilenberg-Steenrod axioms (stated for cohomology), and the Eilenberg-Steenrod axioms only need abelian groups anyway.
But I'm also 90% sure that TFA did something more direct to get to Mayer-Vietoris that I've forgotten, since we didn't use that much homological algebra.
Any available references for that that you know of?