Handlebodies III

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 D^m by attaching various \lambda-handles successively. Thus a general handlebody will look like D^m\cup D^{\lambda_1}\times D^{m-\lambda_1}\cup \cdots \cup D^{\lambda_n}\times D^{m-\lambda_n}.

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 D^m\cup_{\phi_1} D^{\lambda_1}\times D^{m-\lambda_1} with attaching map \phi_1: \partial D^{\lambda_1}\times D^{m-\lambda_1}\to \partial D^m will be denoted \mathcal{H}(D^m; \phi_1). So inductively denote the i-th attaching by \mathcal{H}(D^m ; \phi_1, \ldots , \phi_{i-1}, \phi_i).

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 f: M\to \mathbb{R} on a closed manifold, f determines a handle decomposition of M. Moreover the handles of this handlebody correspond to the critical points of f, 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.


11 thoughts on “Handlebodies III”

  1. Well, congrats on being close to a decision. From the interest in Morse Theory, I assume CAG rather than abstract scheme theory? Or is that still up in the air?

  2. That’s sort of still up in the air. I actually find scheme theory to be really fascinating and amazing (I’m still really uncomfortable with it, so I haven’t posted on it yet). Although, the classical side is also really interesting, but I’m not sure that I could see myself doing it in the long run.

    So I guess I’m not sure.

  3. Well, funny thing about CAG…though schemes, perhaps, show up less often directly, the more abstract structures are a requirement for many of the interesting problems, like stacks in order to talk reasonably about moduli spaces or quotients by groups, or gerbes (which I know little about, but know that we need them in geometric Langlands.)

  4. Let’s just say in theory that you had to read something (on the scale of a small book or half a large book or so) over 2 quarters (20 weeks) that would culminate in an oral exam on the subject. What would you choose?

    I should maybe be a little more specific. This is not a “general exam”, but a replacement for a qualifying exam that typically covers a year-long first year sequence. In theory, the amount of effort should be about 5 out of 10 registered hours worth of devotion for both of the quarters, so it is a significant amount.

    To be slightly more specific. My school allows people that have passed 2 out of 3 written exams (out of a choice of 5 total) to substitute the last one with an oral exam that is not one of the 5 standard topics to help jump-start them into their chosen field: mine being algebraic geometry.

    Also, Hartshorne is probably not a valid choice, since it is the text for the standard course in AG here (but I haven’t checked with anyone on that yet).

  5. Hmm, some possibilities in AG for something you can learn in 20 weeks…
    Miranda “Algebraic Curves and Riemann Surfaces”
    Harris “Algebraic Geometry: A First Course”
    Schenck “Computational Algebraic Geometry”
    Shafarevich “Basic Algebraic Geometry I”
    Fulton “Algebraic Curves”
    maybe part of Liu “Algebraic Geometry and Arithmetic Curves” for a more abstract thing
    maybe part of Eisenbud and Harris “Geometry of Schemes”, my favorite schemes book.
    At the least, that’s what I come up with looking at my bookshelf. I would probably choose Miranda, considering that it and extensions of it’s stuff in the Griffiths and Harris and the Hatcher directions were my orals, but there’s a LOT to choose from these days other than Hartshorne.

  6. Wow. Thanks. Great list. I was actually glancing through the Geometry of Schemes book as a possibility.

    I was supposed to talk with the person that is going to oversee it yesterday/today, but schedule conflicts have pushed it back to Friday. In any case, he might have something in mind, but I figured if not I should have some things to bring to the table.

  7. Nice thing about bibliophilia: I can recommend books to taste. I really can’t recommend Eisenbud/Harris enough, definitely one of my favorites. If you need any more help narrowing stuff down, picking books, or the like, or have a specific taste you’d like to indulge, I might be able to help.

  8. I glanced through the Miranda today. It seems important and like things I should learn, but it also is scary. It says one semester of graduate complex analysis is the only prereq, but I’ve had three quarters and passed a qualifying exam in it, and it still scares me (and my final project was to prove the uniformization theorem for Riemann surfaces, so I even have some experience there).

    Tomorrow I might actually try reading some of it. It might not be that bad when I actually try it.

    I’m actually sort of interested in moduli spaces, but can’t seem to find anything elementary there. Do you know of anything along those lines?

  9. Miranda is actually a lot LESS intimidating than it sounds, though there are a few sections that need chapter 1 of Hatcher’s algebraic topology book (covering spaces, mostly). If you just read it, it’s pretty ok.

    For moduli spaces…well, I’m actually doing stuff there myself (see my recent series on the Moduli of Vector Bundles) but as for books…there’s Harris and Morrison “Moduli Space of Curves” or maybe it’s “Moduli of Curves” or the like. But it’s at least a second book. Geometry of Schemes has some stuff on Functors of Points, Representability of Functors, and Sheaves in the Zariski Topology, which are all things to get down before REALLY talking moduli spaces, but really, moduli spaces are kind of a second course topic, from what I understand. Geometry of Schemes, though, is good for getting started in that direction.

  10. Yeah. I don’t really know anything about moduli spaces. I just know the idea, and the whole concept is really fascinating.

    I started reading Miranda, and just read right through the first chapter in one sitting. I had seen everything in it before. The second chapter doesn’t look any worse, so I guess I was scared for nothing.

  11. Chapter 3 has some serious stuff in it, thouhg, actually. The Group Actions stuff gets you most of the way to proving that in genus g>=2, there are only fnitely many automorphisms, and the Monodromy stuff is used to show that the moduli space of curves of genus g is irreducible (in characteristic 0). Actually, that was one of my orals prep problems. And once you get into chapter 4, it starts getting more serious.

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