Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session


18.901 Introduction to Topology


This course is a seminar in topology. The main mathematical goal is to learn about the fundamental group, homology and cohomology. The main non-mathematical goal is to obtain experience giving math talks. Lectures will be delivered by the students, with two students speaking at each class. There are no exams. There will be some homework assignments and a final paper.


Buy at Amazon Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2001. ISBN: 9780521795401. [Preview with Google Books]

This book is also available for free online at  Allen Hatcher's webpage.

Buy at Amazon Massey, William S. A Basic Course in Algebraic Topology. Springer-Verlag, 1991. ISBN: 9783540974307.


The final grade is determined as follows:

Lectures and participation 60%
Final paper 30%
Problem sets 10%

Attendance is mandatory. Every three missed classes will result in the drop of a letter grade; thus one can miss up to two classes with no effect on the grade.

Lectures and Participation

Each class two students will give lectures. Each lecture should be about 25 minutes long. Individual lectures will not be graded, but lectures make up a good portion of the final grade. In evaluating your lectures, I will look at their clarity, organization and preparedness. I will also consider how your lectures improve over the course of the semester.

You will give a practice lecture to a small audience before your first lecture. This group will consisting of the course instructor, the mathematics writing instructor working with this class, and the other student lecturing in the same class as you.

Each lecturer will give one or two exercises relevant to the material being presented. These exercises, and their solutions, should be e-mailed to the course instructor as a Latex file. The exercises can be stated during lecture, though this is not necessary. It's ok if the exercises come from a book (although it'd be preferable if they did not, or at least if they were slightly modified), but be sure to give proper attribution.

As a member of the audience, I'd like you to write a few comments on each lecture you observe. I'm not asking for any kind of lengthy analysis; it would be enough to point out that the lecturer is writing too small. However, make sure the comments are useful—don't just say "that proof was good," say why. I will collect these comments at the end of class and e-mail them to the lecturer so that they can have some feedback. (The lecturer will not know who made which comments.)


There will be approximately four problem sets. These will count towards the final grade. Solutions are to be written in Latex. You may work together on the problem sets, but everyone must write up their own solutions.

There will also be exercise sets, mainly composed of exercises given by lecturers. These are optional and do not have to be turned in. If you are interested in learning the material, it is probably a good idea to do at least some of the exercises.

Final Paper

The final paper is an exposition of a topic in algebraic topology that we will not cover in the seminar. It must be at least 10 pages long and written in Latex. Topics will be selected for the papers by session 15. A first draft is due in session 27, and a final draft two weeks later. In the final five meetings of class, students will give talks on their final papers.


1–14 Fundamental Group, Covering Spaces Problem set 1 due in Session 9
15–25 Homology Topic for final paper due in Session 15; Problem set 2 due in Session 20
26–34 Cohomology, Poincare Duality First draft of paper due in Session 27; Final draft of paper due is Session 32
35–39 Student Presentations of their final papers Problem set 3 due in Session 38