Topics in Topology
January - April 2015

1. Poincare duality : This is an important concept that explains the structure of the homology and cohomology groups of manifolds. It says that for an n-dimensional oriented, closed manifolds the k-th cohomology group is isomorphic to the (n − k)-th homology group, for all integers k.
   
2. Cohomology with local coeffecients : It is a cohomology theory which allows coefficients to vary from point to point in a topological space.
   
3. Fiber bundles : An important topological space that is locally a product, but globally may have a different structure.
   
4. Spectral sequences (Leary-Serre and Mayer-Vietoris) : A means of computing (co)homology groups by taking successive approximations. We will learn about the Serre spectral sequence which appears in the context of fiber bundles. We will also learn about the Mayer-Vietoris spectral sequence which is used to compute the cohomology of a space expressed as a union of its subspaces.
   

Homework

1. Feel free to work together, but you should submit your own work.
2. Your questions/comments/suggestions are most welcome. I will also be fairly generous with the hints. However, do not expect any kind of help, including extensions, on the day a homework is due.
3. Please turn in a neat stapled stack of papers. Refrain from using blank / printing paper. Use ruled paper.
4. Your final finished version should be as polished as you can make it. This probably means that you cannot submit sketchy solutions or sloppily written first versions. Please expect to do a fair amount of rewriting. Do not hand in work with parts crossed out; either use a pencil and erase or rewrite.
5. Please write complete sentences that form paragraphs and so forth. It might be a good idea to use short simple sentences; avoid long complicated sentences.
6. Do use commonly accepted notation (e.g., for functions, sets, etc.) and never invent new notation when there is already some available.
7. Make sure you provide a statement clearly indicating precisely what it is that you are about to prove. You can, if you want, label your statement as a Theorem or Claim or whatever. Write the word Proof, and then give your proof.
8. Throughout your writing, constantly tell the reader (i.e. me, :-)) exactly what it is that you are about to demonstrate. Be sure to indicate the end of your proof.
9. Your answers should combine “logic”, your hypotheses, and possibly other mathematical facts (e.g., theorems that we have proved in class) into an argument that establishes the asserted conclusion.
10. Please be extra careful about the order in which you use your quantifiers.
11. In your arguments you can make free use of anything that we have proven in class, and of course all logic rules and basic axioms and definitions. Anything else that you use should be proved.