Welcome to the wonderful world of undergraduate algebraic topology!

Topology, more or less, is the study of spaces, or more specifically spaces where you can reasonably talk about continuity and convergence. Topological spaces can be quite crazy, but we will usually consider fairly nice spaces (like "manifolds" and "CW-complexes"). Some of the most basic questions we can ask are

  • What do these things look like?
  • How many are there?
  • How can we tell two of them apart?
  • What interesting properties do they have?

Algebraic topology is a set of tools we can use to try to answer some of these questions. In particular, we will associate algebraic things --- like numbers, vector spaces, polynomials, groups, modules, rings --- to topological spaces. With these algebraic invariants we will be able to do some amazing things. Initially we will consider some questions that seem rather basic, like "how can you tell the difference between the surface of a ball and a doughnut?" and "how can you tell if a closed loop is knotted?", but we will see that in answering these questions we learn surprising things, like "if you want a non-zero vector field tangent to the boundary of a region in three space, say a magnetic field to contain some reaction, then that region better, more or less, look like a doughnut" and "at any given moment there are two antipodal points on the earth that have the same temperature and humidity" and "can certain knots bound a disk like membrane in space-time", and “how can you decompose a group into basic pieces".


Lecture Notes:

Please find the course lecture notes here.


Announcements:

  • Update, the second midterm exam has been moved to Monday, April 12 and there will be a review session on Friday, April 9
  • The second midterm exam will cover the material in Sections 1V through VI, that is the material on homework assignments 4 and 5. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems (even the ones that were not turned in for grading), and (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
    • There will be 4 questions of which 3 will be similar to the homework problems and one will consist of several True/False or short answer questions.
    • As a "practice test" work problems 3, 7, 9, and 13 from Homework 4, problems 2, 5, and 8 from Homework 5, and compute the fundamental group of the projective plane RP^2 using the Seifert-Van Kampen theorem.
  • Update, the midterm exam has been moved to Wednesday, March 3 and there will be a review session on Monday, March 1
  • The first midterm exam will cover the material covered through the end of Section III, that is the material on homework assignments 1, 2, and 3. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems (even the ones that were not turned in for grading), and (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
    • There will be 4 questions of which 3 will be similar to the homework problems and one will consist of several True/False or short answer questions.
    • As a "practice test" work problems 5, 7, and 9 from Homework 1 and problems 1, 3, and 5 from Homework 2, and 1, 6, and 9 from Homework 3.
    • I guarantee that at least 2 of the problems on the test will either be among these problems or very similar to them.

Course Information: