### Second Exam: Algebraic Topology

Today's exam went fine. It was a lot easier than past year's exams so it wasn't that bad. You needed to do 4 of 5 problems.

1. Compute the class of the idagonal embedding of CP^2 in CP^2 x CP^2.

I didn't do this one; I knew what the answer SHOULD be, but I couldn't figure out a good way of writing it up, so I just skipped it.

2. Describe a cell decomposition of RP^2 x RP^2. Compute the cellular chain complex of RP^2 x RP^2. Use this chain compelx to compute the integral homology of RP^2 x RP^2. I did this one, but I screwed up so many times in the computations that it took about 4 times as long as it should have.

3. Prove that Kunneth extends to fiber bundles with compact base. (It didn't actually say this, but that's what it meant. =) ) Pretty easy, once I figured out what the heck it was asking. (The question statement was about half a page, as we never discussed fiber bundles.)

4. Is the twisted circle bundle over S^2 orientable? Compute its integral cohomology and it's cohomology ring with Z/2 coefficients. I did this problem but COMPLETELY screwed up the computation. And I wrote that I was using the universal coefficient theorem when I meant Poincare duality, and I screwed up a computation of products and... pretty much I just screwed up. Hopefully I'll get partial credit since I computated at least part of things correctly, and most of my mistakes were linear algebra errors anyway. But it's kinda annoying, cause it wasn't that hard of a problem.

5. (i) Show that the symmetric group S_3 can act freely on the closed orientable surface of genus 7. This was pretty easy if you drew the right picture. =)

(ii) Define the cap product. (Don't show it's well defined.) For a continuous map f:X->Y show that f_*(x cap f^*(a)) = f_*(x) cap a. REALLY REALLY easy, and the only thing I solved in the first hour (cause I was so busy screwing up the computations in problems 2 and 4).

Maybe I should have written something for question 1... just to see if it worked out. And so I could avoid doing 4. Oh, well. I should get full credit (or almost) for 2,3,5, so I don't think I did too badly. Kinda annoyed aobut problem 4, though.

1. Compute the class of the idagonal embedding of CP^2 in CP^2 x CP^2.

I didn't do this one; I knew what the answer SHOULD be, but I couldn't figure out a good way of writing it up, so I just skipped it.

2. Describe a cell decomposition of RP^2 x RP^2. Compute the cellular chain complex of RP^2 x RP^2. Use this chain compelx to compute the integral homology of RP^2 x RP^2. I did this one, but I screwed up so many times in the computations that it took about 4 times as long as it should have.

3. Prove that Kunneth extends to fiber bundles with compact base. (It didn't actually say this, but that's what it meant. =) ) Pretty easy, once I figured out what the heck it was asking. (The question statement was about half a page, as we never discussed fiber bundles.)

4. Is the twisted circle bundle over S^2 orientable? Compute its integral cohomology and it's cohomology ring with Z/2 coefficients. I did this problem but COMPLETELY screwed up the computation. And I wrote that I was using the universal coefficient theorem when I meant Poincare duality, and I screwed up a computation of products and... pretty much I just screwed up. Hopefully I'll get partial credit since I computated at least part of things correctly, and most of my mistakes were linear algebra errors anyway. But it's kinda annoying, cause it wasn't that hard of a problem.

5. (i) Show that the symmetric group S_3 can act freely on the closed orientable surface of genus 7. This was pretty easy if you drew the right picture. =)

(ii) Define the cap product. (Don't show it's well defined.) For a continuous map f:X->Y show that f_*(x cap f^*(a)) = f_*(x) cap a. REALLY REALLY easy, and the only thing I solved in the first hour (cause I was so busy screwing up the computations in problems 2 and 4).

Maybe I should have written something for question 1... just to see if it worked out. And so I could avoid doing 4. Oh, well. I should get full credit (or almost) for 2,3,5, so I don't think I did too badly. Kinda annoyed aobut problem 4, though.

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