Andras Juhasz's general exam 5/10/2005, 10-11:30 am Committee: Browder, Gunning, Szabo (chair) Special topics: Algebraic Topology, Differential Topology ALGEBRAIC TOPOLOGY (Browder): What is your favorite homotopy group ? I said pi_3(S^2), but Browder said let's do something more complicated, say pi_4(S^3). I offered several proofs, and I had to work out the one using spectral sequences. Freudenthal suspension theorem. Write down a long exact sequence containing the suspension map (embed S^n into the loop space of S^{n+1}). What about pi_6(S^4) ? I sketched the proof using the Pontrjagin-Thom construction, but I didn't remember every detail. DIFFERENTIAL TOPOLOGY (Szabo, Browder): State the h-cobordism theorem. Outline the proof. Where did you use that the cobordism is simply connected ? Why do you need to use gradient-like vector fields ? How do you derive the Generalized Poincare theorem from h-cobordisms ? Why do we only get a homoeomorphism with S^n, not a diffeo. ? What happens in dimension 4 ? I mentioned Freedman's theorem. Characteristic classes: What are Stiefel-Whitney classes ? What are the S-W classes of RP^n ? Prove it. What is the signature of a manifold ? Does the intersection form have to be nondegenerate ? State Hirzebruch signature theorem in dimensions 4 and 8 (I forgot one of the constants). How would you compute the coefficients in the formula ? I said that the oriented cobordism ring over the rationals is generated by complex proj. spaces of even dimensions. What do you know about exotic S^7's ? Group structure induced by connected sum. I had to outline Milnor's constrution of an exotic 7-sphere. How do you obtain rank 4 vector bundles over S^4 ? (You can add homotopy classes in pi_4(BSO(3)).) REAL ANALYSIS (Gunning): Does a convergent sequence in L^1[0,1] have to converge pointwise a.e. ? Give a counterexample. Prove that there is a subsequence that converges pointwise a.e. Fundamental theorem of calculus. Why doesn't it hold for functions of bounded variation ? I mentioned the Cantor function, then they asked me to construct it. COMPLEX ANALYSIS (Gunning): How can an analytic function fail to be conformal (globally and locally)? State the Riemann mapping theorem. Why cannot the domain be the whole plane ? I used Liouville's theorem, then they asked me how would I prove it. Then I had to give an analytic function that is not elementary. I chose the gamma function. I wrote down the functional equation and said it has poles at the negative integers. Then Gunning asked me to write down an explicit formula, but I didn't remember exactly. I just said that we can construct an integral formula by applying Mittag-Leffler to the logarithmic derivative. ALGEBRA (Gunning): How many groups are there of order 8 ? How do you distinguish between the abelian ones ? How would you prove there are no other groups of order 8 ? COMMENTS: I was asked to choose the first topic, so I chose Algebraic Topology. After this I could pick the next subject, which was Differential Topology. I highly recommend that if you are offered the choice, go with the one you know the most, so that you give a good first impression, and also less time remains for the topics you prefer less. Talk about things that you know, many times you can lead the conversation in a favorable direction. The atmosphere was really relaxed, and everyone was very nice during the exam. If you don't remember something, they will give clues. And you can get away with not knowing some things. It's a good idea to talk to your examiners before the exam about what to learn. And check out their favorite questions on these pages. Good luck !