0708-1300/Democracy At Last

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The Question

We may have a little time left after we are done with our discussion of ${\mathbb {R} }^{n}$ (i.e., the generalized Jordan curve theorem) and ${\mathbb {R} }P^{n}$ (i.e., the Borsuk-Ulam theorem). If that shall be the case, what should we do?

The Options

Option 1. We could cover the generalized Schoenflies theorem, which in itself is a stronger version of the generalized Jordan curve theorem, saying:

Suppose $f:S^{n-1}\to S^{n}$ is a bi-collared embedding (that is, it has an extension to an embedding ${\bar {f}}:S^{n-1}\times [-1,1]\to S^{n}$, where $S^{n-1}$ is identified with $S^{n-1}\times \{0\}$). Then the closure of each of the two components of $S^{n}-f(S^{n-1}$) is homeomorphic to the closed $n$-disk $D^{n}$.

Dror does not know the proof of this (deep and useful!) theorem, so it will be quite a challenge for him to study it in time. The proof relies on the generalized Jordan curve theorem which relies on our class material, but otherwise the proof seems to involve mostly geometry and general topology, and no further algebraic or differential topology. See pages 236-239 of Bredon's book.

Option 2. We could discuss the de Rham theorem, saying roughly that for manifolds, homology as defined over the last few weeks is dual to de Rham cohomology, as defined briefly last semester using differential forms and the exterior derivative $d$. The theorem is foundational - if you stay near topology, you'll end up using it all over the place, and the proof is a combination of many of the main ideas we have discussed in this class, including integration and Stokes' theorem, Poincaré's lemma and the Mayer Vietoris sequence. Dror would have to brush the details, but that shouldn't be hard. See pages 286-291 of Bredon's book.

Option 3. We could go in a completely different direction and talk a bit about Khovanov homology, one of the hottest topics of current research in Knot Theory. We'd start by talking about the Jones polynomial, which provides one of the simplest proofs that the trefoil knot is really non-trivial, and certainly the simplest proof that it is not equivalent to its mirror image. We will then construct a homology theory whose Euler characteristic is the Jones polynomial. Dror knows this in and out; in fact, we will be following his paper On Khovanov's Categorification of the Jones Polynomial; see also Khovanov's Homology for Tangles and Cobordisms.

The Vote

Please circle your answer to the following questions.

I prefer option number:	1	2	3
I have looked at the relevant pages of Bredon's book, followed the links on this page and considered the options carefully, and therefore my opinion counts double:		Yes	No