Math GU4053: Algebraic Topology
Columbia University Spring 2020

Instructor: Oleg Lazarev (
Time and Place: Tuesday and Thursday: 2:40 pm - 3:55 pm in Math 307
Office hours: Tuesday 4:30 pm-6:30 pm, Math 307A (next door to lecture room).
Teaching Assistant: Quang Dao (
TA Office Hours: Monday 12:00 pm - 1:00 pm, Wednesday 12:00 pm - 1:00 pm in Math 406 Help Room .

About the course:

In this course, we'll explore certain algebraic invariants of topological spaces, combining computation with theory. In the process we'll get to draw some pretty pictures and learn how to think about high-dimensional spaces. We'll also learn about the associated algebraic structures, which are quite rich and have deep connection to many other areas of math.


The main reference will be Algebraic Topology by Allen Hatcher. It is available for download here. We will cover the four chapters of Hatcher (the main sections, not the additional topics). I will occasionally draw on other references and I will inform you in class if I do so.


It would be helpful to have background in point-set topology (e.g., Math GU4051) and basic topological operations. There is some background in Chapter 0 of Hatcher; also see Topology by Munkres. It is also important to be comfortable with some abstract algebra (e.g., Math GU4041), like group theory and linear algebra.


There will be one midterm exam in class and a take-home final exam. There will be no make-up exams.


There will be weekly homework due at the beginning of class each Thursday. If you can't make it to the class, put it in the assigned box outside of 417 Math. The lowest homework score will be dropped.

Grade distribution:

Homework: 30%, midterm exam: 30%, final exam: 40%

Other info:

Getting help: If you're having trouble in course, get help immediately. I will be available to answer questions during my office hours. Additionally, there is the Columbia help room in 406 Math.

Student with disabilities: Students must register with the Disability Services and present an accomodation letter before the exam or other accomodations that can be provided. More information is available on the Disability Services webpage.

Lecture notes:

Tentative schedule:

Date Topics References
01/21/20 Basic definitions (homotopy equivalence, fundamental group), CW complexes, Hatcher Chapter 0, 1.1
01/23/20 Induced maps, fundamental group of circle, introduction to covering spaces Hatcher Chapter 1.1
01/28/20 Free product of groups, Van Kampen's theorem Hatcher Chapter 1.2
01/30/20 van Kampen's theorem continued, proof and computations Hatcher Chapter 1.2
02/04/20 Introduction to covering spaces Hatcher Chapter 1.3
02/06/20 Classification of covering spaces Hatcher Chapter 1.3, 1.1A
02/11/20 Classification of covering spaces, continued; Cayley complexes Hatcher Chapter 1.1A
02/13/20 Deck transformations on covering spaces Hatcher Chapter 1.3
02/18/20 Group actions Hatcher Chapter 1.3
02/20/20 Delta-complexes and simplicial homology Hatcher Chapter 2.1
02/25/20 Singular homology Hatcher Chapter 2.1
02/27/20 Homotopy invariance, relative homology, exact sequences Hatcher Chapter 2.1
03/03/20 Long exact sequence Hatcher Chapter 2.1
03/05/20 Long exact sequence continued, excision Hatcher Chapter 2.1
03/10/20 Cellular homology. Introduction to degree theory Hatcher Chapter 2.2
03/12/20 Midterm in class
03/24/20 Homology with coefficients, axioms for homology Hatcher Chapter 2.3
03/26/20 Cohomology: definition, examples Hatcher Chapter 3.1
03/31/20 Cohomology: basic properties Hatcher Chapter 3.1
04/02/20 Cup product: definition, examples. Hatcher Chapter 3.2
04/07/20 More on Cup product, cohomology ring Hatcher Chapter 3.2
04/9/20 Orientability and the fundamental class Hatcher Chapter 3.3
04/14/20 Poincare Duality Hatcher Chapter 3.3
04/16/20 Introduction to homotopy groups, Whitehead's theorem Hatcher Chapter 4.1
04/21/20 Fiber bundles Hatcher Chapter 4.2
04/23/20 Long exact sequence and computations Hatcher Chapter 4.2
04/28/20 Connection with cohomology Hatcher Chapter 4.3
04/30/20 Connection with cohomology, continued Hatcher Chapter 4.3