= 2) - How would you compute the fundamental group of the Klein bottle? (Polygonal complex picture). What does Hurwitz tell you? (abelianization of pi_1 is H_1, so here H_1 is Z + Z/2Z) What is H_*(Klein bottle)? - Give an example of a topological space with Z/3Z first homology group (disk with triangular boundary where boundary edges are identified in order and direction induced by the orientation) - Compute H_* of this last space. - What is the homology with Z/3Z rather than Z coefficients? - Classify complex line bundles over S^2. (I identify with CP^1, talk about first chern class, classifying bundles by degree) - What about higher rank complex bundles? (Chern classes no longer are sufficient to classify) - How would you classify bundles in general, say over some nice CW complex? (Homotopy classes of maps into the classifying space) At this point I comment that one can do a more direct argument in the case of S^2, taking a cover by northern and southern hemispheres, and consider homotopy classes of maps of the intersection (homotopically a copy of S^1) into the structure group. - What is the classifying space for complex rank 1 bundles? (CP^infinity) - What is the classifying space for complex rank 2 bundles? (Infinite grassmanian of 2-planes) ADVICE: The summary above is much smoother than the exam was, or at least much smoother than the exam *felt*. I had many long awkward pauses when trying to answer questions, and sometimes began a problem in entirely the wrong way, but they were very patient and helpful. They were most happy when I could answer, after a struggle, a question I had clearly never thought about before. The advice others have given on these "generals help" pages is quite good, but you shouldn't expect that the questions you get are just repeats of questions from old generals exams. You have to be on your toes! On the other hand, the committee drops many quite explicit hints, and sometimes builds to a question (although at the board it can be hard to see the links). Don't be scared to admit you're totally stuck on a problem, at least once you've made some serious attempts. Also, although it's helpful to say what you know about a problem, be aware the committee picks up on what you say and may steer the questioning in that direction, so be sure to have a good understanding of any auxillary concepts you introduce (e.g., structure equations in my Differential Geometry section helped lead into integrability discussion). Finally, I have two important (although perhaps "obvious") suggestions for preparation: (a) WORK EXAMPLES. It's always good to have an example for every occasion. At the very least you can use it to test conjectures at the board, and having them at your fingertips, ready to use, is occasionally seen as a sign of maturity. Besides, it makes preparation for the generals MUCH more fun, fulfilling, and (in my opinion) of lasting long-term benefit. And best of all, you can impress folks back home with them! You'll be the life of every party! (b) PRACTICE ORAL EXAMS WITH OTHER STUDENTS. I wish I did this more. I've lectured classes before, and so had board experience, but lectures are prepared. This is different. The little bit I did with others in the days before the exam proved very useful. Learn to use the board effectively (my computations during the exam were sprawled all over the board, and so difficult to correct when I made an error). Most important, learn to recover from mistakes in front of a friendly audience, and to clarify points they find confusing without getting tongue-tied.