Graduate Topology II
January- April 2017
Click here for the homeworks.
||Tuesdays and Thursdays from 3:30 pm to 4:45 pm
||Lecture Hall 4
|email: pdeshpande AT cmi DOT ac DOT in
Topology by Allen
Hatcher, Cambridge University Press, 2002.
Useful reading material:
and Geometry by Glen Bredon, Springer-Verlag, GTM
- 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
- Homology Theory An Introduction to Algebraic Topology
by James Vick, Springer-Verlag.
- Lectures on Algebraic Topology, Albrecht Dold,
||First course in topology
and/or permission from the faculty advisor.
- Homework 30%
- Test 30%
- Quiz 10%
- Final exam 30%
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.
Tests and Quizzes
There will be two tests and three in-class quizzes. There is no
midterm for this course.
Quiz dates: 24 January, 28 February, 28 March
Test dates: TBA
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
The homework problems are assigned from the
- Feel free to work together, but you should submit your own
- 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
- Please turn in a neat stapled stack of papers. Refrain from
using blank / printing paper. Use ruled paper.
- 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.
- 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.
- Do use commonly accepted notation (e.g., for functions, sets,
etc.) and never invent new notation when there is already some
- 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.
- 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.
- 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
- Please be extra careful about the order in which you use your
- 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
|1, 4, 5, 9, 11
|12, 13, 14, 15
|16, 17a, 18, 19
|1, 2, 3
|9a, 10a, 13a
|8a, 8c, 9
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