Topics in Topology January - April 2015

 Contact: Lectures: Mondays and Wednesdays from 2 to 3:15 pm Classroom: Lecture Hall 2 Instructor: Priyavrat Deshpande. Office: 403 phone: 962 email: pdeshpande AT cmi DOT ac DOT in Office Hours: Wednesdays from 11:30 to 12:30 Teaching Assistant: Ronno Das Texts: Algebraic Topology by Allen Hatcher, Cambridge University Press, 2002. Lecture Notes in Algebraic Topology by James F. Davis and Paul Kirk, AMS, GSM 35. Useful reading material: Topology and Geometry by Glen Bredon, Springer-Verlag, GTM 139, 1997. A User's Guide to Algebraic Topology, C. T. J. Dodson and P. E. Parker, Kluwer Academic Publishers. Algebraic Topology Homology and Homotopy by Robert Switzer, Springer-Verlag. Homology Theory An Introduction to Algebraic Topology by James Vick, Springer-Verlag. Lectures on Algebraic Topology, Albrecht Dold, Springer-Verlag. Prerequisites: Knowledge of (co)homology theory (graduate course in algebraic topology) and/or permission from the faculty advisor. Grading: Homework 40% Project report 30% Presentation 30% Web: http://www.cmi.ac.in/~pdeshpande/ttop15.html

Course syllabus
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

The homework will be assigned roughly every 2 weeks. It is your duty to submit the solutions on time. Copying and/or plagiarism will not be tolerated. Here are a few writing guidelines you might want to follow.
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.