I keep naming my posts “handlebodies”, so I think it is officially time to define what one is. A handlebody is a manifold obtained from by attaching various -handles successively. Thus a general handlebody will look like .
If you’re familiar with how to construct a CW-complex, this is pretty similar. You just inductively attach the handles using smooth maps, and then smooth out the manifold so that at each step we have a legitimate smooth manifold. It may be useful to introduce a notation for this. The first attaching with attaching map will be denoted . So inductively denote the i-th attaching by .
After i steps, we will always have i attaching maps even if some are formally meaningless (attaching a 0-handle is a disjoint union, so there is no attaching).
If we express M as a handlebody we call that a handle decomposition of the manifold.
Next time I’ll prove the result that everyone that has been reading the posts will have already guessed. Given a Morse function on a closed manifold, determines a handle decomposition of . Moreover the handles of this handlebody correspond to the critical points of , and the indices of the handles coincide with the indices of the corresponding critical points.
I’m short on time today, so I’m going to put off proving it.
I’m not sure how much to say about my other news, since it is still sort of up in the air. I passed two of my qualifying exams (which was all that was necessary for now), and I may officially lock myself into the path of algebraic geometry as my field in the next couple of days. Before I say too much about this, I’ll just say that I should have more information on Monday about what I’m officially doing.