Topology/Geometry Qualifying Exam

Study Group, Summer, 2005

Here you'll find general information about the Topology/Geometry Qualifying exam study group for the August 2005 exam. To contact me, you can email me at cbtruman (at math etc. etc.) or stop by my office, 4410 where I can be found approximately whenever I happen to come in (which tends to be every weekday between about 9:00 a.m. and 5:00 p.m.). I'm pretty easy to find, though your mileage may vary.

We are meeting Mondays in room 0104 from 3:00-4:00ish and Thursdays in room 1308 from 11:00-12:00ish.

Generally, during the sessions we will work on old qual problems. This tends to work as follows: I pick out a problem, and then pick some poor sucker to work it out on the board, while everyone else in the room nit-picks (or provides hints, depending on any given person's personality and mood). If necessary, I will work some problems on the board, or try to clarify things which I think might be fuzzy. However, when I work problems, I would generally like to give a general outline and leave the details to everyone else. If you would prefer to be left out of the sucker-picking process (for working problems on the board), just let me know, though I will warn you that doing the problems is the best way to learn, especially doing them for other people to critique. On the other hand, if you'd like to work on the board more frequently than you find you are being chosen, I can try to do that as well.

The old quals can be found here. Since some of us have done these quals before, I've also made a couple of practice quals, and I've tried to make them about on par with the old quals. You can get those here (in pdf format; if you want another format you'll have to let me know):

The qual syllabus is posted here. I have found that it is useful to know a couple of other things too (that aren't on the qual syllabus). Generally, it is my understanding that you can use things not on the syllabus as long as it doesn't completely trivialize the problem (i.e. as long as the problem isn't just a restatement or special case of the theorem), but if you use something outside the qual syllabus, you'd better provide a very clear statement of the theorem and why it applies. If I think of more I will add them.

While I find many of the problems are poorly phrased, I think mostly everyone can figure out what was meant. However, problem 6 on the August, 2002 qual had some very important things left out of it, so it isn't as clear what is intended. If I find any more like this, or if anyone would like any particular problem clarified like this, I will post them here:

I also have other general gripes about the qual; for example, connected sum problems come up often, and every time the writer feels it necessary to provide a definition, and I have never seen two that were quite the same. They generally amount to the same thing (although I can think of one in particular, August 2001, that doesn't quite work; try using the given definition with the complements of single points in Sn). My recommendation is for you to figure out for yourself what a connected sum is before you take the qual. Then when you see their definition, simply skim it to make sure that they really are talking about the same thing, and proceed to use your own definition. Another thing which can cause headaches is that sometimes important adjectives (like compact or orientable or sometimes smooth) can be left out in the description of a manifold. Sometimes the person who wrote the problem is not aware that, using Bredon's definitions, a manifold-with-boundary is not, strictly speaking, a manifold. Most of the time one can figure out what is meant. If you can't, it is important to ask the proctor, who can go ask a professor. These mistakes are amusing to people who look at the quals later to try to work over them, but I can personally recall being on the fourth page of a problem before I was told that the manifolds in question were actually compact (August 2001, problem 6). It was somewhat less than amusing then (although it was a relief, since that was the easiest of the three cases I had been considering). One of my favorite mistakes was a problem (August 2002, problem 3) that asked you to assume that X was path-connected, then asked in part (a) to show that X was path-connected. The funny thing is, if you omitted the given condition (that X was path-connected) you no longer had enough information to prove it. Another problem (January 2002, number 1) describes a manifold that the astute reader can prove does not exist (the Euler characteristic of a 4n+2-dimensional manifold must be even - see problem 5, January 2003; I have been told that the problem was do-able otherwise, and that full marks were given to anyone who wrote that the manifold in question did not exist). Some of these have been corrected in the versions that they hand out in the grad office, some haven't. Your best bet when practicing is to ask around.