Introduction to (smooth) Manifolds

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 cohomology.


Line arrangements and some classical problems, posets and Mobius inversion, hyperplane arrangements, deletion-restriction, Zaslavsky's theorem, graphical arrangements, matroids, the finite field method, ESA, interval order, Shi and Catalan arrangements, free arrangements.


  • Days: Mondays and Thursdays
  • Time: 10:30 am - 11:45 am
  • Place: LH 802
  • Office hours: by appointment
  • Prerequisites: First course in topology, multi-variable calculus and/or permission from the faculty advisor.


  1. An Introduction to Manifolds by Loring Tu (2nd ed.)
  2. Topology and Geometry by Glen Bredon, Springer-Verlag, GTM 139, 1997.
  3. Differential Topology by V. Guillemin and A. Pollack.
  4. Differential Forms and Applications by M. do Carmo

Teaching Assistant



  • Assignments: 20%
  • Mid semester exam: 40%
  • End semester exam: 40%



The assignments will be posted on this page. 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.


  1. Assignment 1 pdf. Due date: 16/08/2018.
  2. Assignment 2 pdf. Due date: 30/08/2018.
  3. Assignment 3 pdf. Due date: 06/09/2018.
  4. Assignment 4 pdf. Due date: 20/09/2018.
  5. Assignment 5 pdf. Due date: 25/10/2018.