Nothing makes blogging stop dead like the start of the quarter. Well, that combined with the fact that I sort of reached the logical end to the topic I was posting on. There are lots of neat things we could do from here, but they are all rather technical (like sliding handles around and cancelling them to make our manifold simpler) but have nice simple geometrical interpretation. So I don’t really feel like going into that, since the bulk of the powerhouse tools of Morse theory that the everyday person needs were covered (I guess that isn’t totally true since we’re missing a few quick things for Lefschetz).
There are a few brief corollaries and applications left over that I’m contemplating. The most interesting to me is the one that says if a manifolds admits a smooth function to with exactly two critical points which are both non-degenerate then it is homeomorphic to a sphere. Yes I said homeomorphic! That is what is so interesting. We only have a diffeo if it is dimension less than or equal to 6. There are “exotic ‘s”.
I feel ready to move on in general. The next logical step is to develop some homology, but this is a hard question. I certainly do not have the motivation or patience to build this from the ground up starting with definitions. So I’m not sure how to proceed. I also may give up on Lefschetz for a few months and do things more related to things I’m doing for classes.
In other news, I went to a neat talk on harmonic measure theory. Normally I’m not very fond of measure theory, but this was pretty cool. There was even a result that I don’t remember now that had to do with the set on which the measure was full was an algebraic variety or something.
Anyway, figured I’d update at least once this week.