Introduction to Manifolds
August- November 2017
Click here for the home-works.
||Monday 10:30 am, Thursday 11:50am
||Monday: 804, Thursday: 803
|email: pdeshpande AT cmi DOT ac DOT in
||An Introduction to Manifolds by Loring Tu (2nd ed.)
Useful reading material:
and Geometry by Glen Bredon, Springer-Verlag, GTM
- Differential Topology by V. Guillemin and A. Pollack.
- Differential Forms and Applications by M. do Carmo
||First course in topology
and/or permission from the faculty advisor.
- Homework 30%
- Mid semester exam 35%
- Final exam 35%
Manifolds, the higher-dimensional analogs of smooth curves and
surfaces, are fundamental objects in modern mathematics. Combining
aspects of algebra, topology, and analysis, manifolds have also been
applied to classical mechanics, general relativity, and quantum
field theory. The aim of this course is to get aquainted with the
basic theory and lots of examples of manifolds. Towards the end of
the course we will learn to compute, at least for simple spaces, one
of the most basic topological invariants of a manifold, its de Rham
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.
- 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
- Assignment 1 pdf. Due date:
- Assignment 2 pdf. Due date:
- Assignment 3 pdf. Due date:
- Assignment 4 pdf. Due date:
- Assignment 5 pdf. Due date:
- Assignment 6 pdf. Due date:
- Assignment 7 pdf. Due date:
- Assignment 8 pdf. Due date
The mid semester exam and its solutions.
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