Ok... I thought that we pick the last elemeny which is 2 and then we go to 1 then to n then again to 1 and then to 2. How else do we do this? @Semiclassical
@AkivaWeinberger You're about to ask an interesting question, one that depends on what you think a "knot" is. If knots, to you, are continuous/PL embeddings up to locally flat (or, if you prefer, ambient) isotopy, then knots only exist in codimension 2, and they always exist in codimension 2.
If they're smooth embeddings up to smooth isotopy, the story is more complicated; Thom constructed a smoothly knotted $S^3$ in $S^6$.
@AkivaWeinberger: I guess this ties in with your previous discussion: if someone is so foolish as to think the empty set is disconnected, they shouldn't be talking about the empty set. :)
Four $1$-cells (inner and outer boundary of annular surface, one connecting the $0$-cells, and one connecting the $0$-cells the other way — that's is, through the curved surface)
@Ted: "Smooth manifolds" are not geometry, and the statement "Topology is more general than geometry" is a poor understanding of what the phrase 'more general' means.
I'm mostly convinced the separation of fields is somewhat semantics and personal preference. I have seen people with work which to me clearly labels them "topologists" to me call themselves "geometers" and vice versa.
@PVAL, when I finished my Ph.D. and applied for jobs, U Wisconsin had no "geometry" checkbox, and I had to say I was a topologist (was I going to say logician?). But complex integral geometry, despite being filled with Chern classes, was not topology.
main objection i have to symplectic manifolds as an example is that, to me, that equals 'phase space', and one certainly talks about 'phase space volume' etc.
I remember reading a few years ago, about how the "abelian integrals" (integrals of some sort of rational functions over some contours in $\Bbb C$) were a huge object of study in "algebraic geometry". This was an old source, but I think it makes the point that these subdivisions are at least somewhat malleable.
incomplete elliptic integrals (mod periods) map into the jacobian variety, whereas elliptic functions map from the jacobian variety back to the complex plane?
Classically, elliptic functions are doubly-periodic functions on the plane, @Semiclassic, hence map a torus to the Riemann sphere, yes. Just a one-dimensional Jacobian (which is isomorphic to the original elliptic curve).
one thing i've wanted to understand for a while: I know about Picard-Fuchs equations (how periods change if you smoothly vary the underlying elliptic curve) but i had the sense that that should link up with the Jacobian variety somehow
I remember Joan Birman went on a serious rant the year she gave the Cantrell Lectures at UGA ... basically advocating the abolition of traditional journals.
