Algebraic Topology
August - November 2013

Click here for the homeworks.
Lectures: Monday from 10:30 to 11:40  and Tuesday from 9:10 am to 10:25 am
Classroom: Lecture Hall 4
Instructor: Priyavrat Deshpande.
Contact: Office: 403
phone: 962
email: pdeshpande AT cmi DOT ac DOT in
Office Hours:
 Thursdays from 2:00 to 3:00 p.m.
Texts: Algebraic Topology by Allen Hatcher, Cambridge University Press, 2002.

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:   First course in topology and/or permission from the faculty advisor.
Grading:
  • Homework 50%
  • In-class tests 25%
  • Final exam 25%
Web: http://www.cmi.ac.in/~pdeshpande/atop.html
Course description

This course will serve as an introduction to the concepts of Algebraic Topology, emphasis being homological methods, and with several applications. The main idea of algebraic topology is to assign algebraic invariants to topological spaces. These invariants are not only interesting in their own right but also capture geometric information.They are expected to be accessible during actual calculations. First example we will encounter is that of singular homology theory. Here the invariants are abelian groups. Towards the end of the course, we will study the closely related theory of singular cohomology. It turns out that singular cohomology with coefficients in a commutative ring carries more algebraic structure. For example, the product structure makes the singular cohomology of a space into a graded-commutative algebra.

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.
  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.
The homework problems are assigned from the textbook.
HW No. Page No. Problems Due Date
1 131 1, 4, 5, 9, 11 19/08/2013
2
 132
 12, 13, 14, 16, 18
 03/09/2013
3
 132
 19, 20, 21, 26, 27a
 16/09/2013
4
 155
 3, 4, 6
 30/09/2013
5
 156
 9a, 9b, 10, 12a, 13a
 25/10/2013
6
 156-7
 14a, 19, 20, 28a, 28b
 28/10/2013
7
 158
 30a, 30b, 31, 32, 34
 5/11/2013
8
 205
228 		8a, 8c, 9
3a, 6, 7
 19/11/2013

In-class tests

There will be two such tests during the semester. The test will consist of at most 2 problems based on the material covered prior to the test.
Test 1 : 02/09/2013
Test 2 : 07/10/2013
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