How to Learn Algebraic Topology

Once in a while, a person finds that boredom has taken over life and then algebraic topology is an excellent remedy. But also anyone who is solving an abstract problem will sometime or another find a connection with a question in algebraic topology, a field that is not so well known outside the world of mathematics.

Algebraic topology is that special part of math in which everything is—at least in the beginning—very intuitive. Only through its study can one really understand what the subjects of study are, so let's begin with that.

(The following is not intended as a guide for professional mathematicians, but rather for the layman.)


  1. Image titled Get a Last Minute Passing Grade on a Mathematics Exam Step 2
    Learn some algebra. Not much is needed to begin, but even that little is not part of your high-school curriculum. A good book to begin with (which looks a bit long, but is very well written, and will take you step-by-step through the theory) is J.B. Fraleigh's A First Course in Abstract Algebra.
  2. Image titled Do Well in Calculus Step 2
    Learn some topology, perhaps with the book by J. Munkres, Topology. You'll do fine with chapters 1 through 6.
  3. Image titled Be Open Minded Step 3
    From here, there are two ways to follow.
    • Get into algebraic topology. Get a good book; you could use the one by J. Munkres, or consult A. Hatcher's highly-illustrated Algebraic Topology. And there you are. Or...
    • Try to understand the theory on your own; prove your own theorems. The first thing you will want to understand is the so-called fundamental group of a topological space X. This is the following: Consider the space of all closed curves (ie, loops) [0,1]-->X beginning and ending at some fixed point x, and consider the equivalence relation that identifies two such curves if they are homotopic (i.e., can be deformed into each other continuously). This space has an (in general non-commutative) binary operation that associates to each (ordered) pair of curves the loop that traverses first the first loop and then the second loop. Now you will want to get theorems of the following sorts: if X can be continuously deformed into another space Y, then their fundamental groups are isomorphic; the fundamental group of a product is the product of the fundamental groups; etc. What happens to the fundamental group when you join two spaces by overlapping open sets? How much freedom do you have to move the point x?


  • Be sure to ask questions whenever you have them. Ask mathematicians; many, many of them know this material. And they are, most of the time, very friendly people. Mathematicians are also available through e-mail. Google a topologist and write your question concisely, trying to use the language used in the books. You are most likely to get an answer.
  • Learn some analysis. It is often believed that some knowledge in analysis is mandatory in order to start understanding topology in the first place; if you feel the need, consult Marsden's Real Analysis.
  • Try all the exercises in the books. They will definitely help you understand better.
  • You can get Allen Hatcher's book online from his webpage.


  • Always be sure to ask very concrete questions when you need to. This means that you can ask about how some phrase should be interpreted, or why some fact mentioned in a book is so. But you cannot say that you simply understand nothing at all and expect anyone to answer your plea. You have to make an effort to understand as much as possible on your own.

Things You'll Need

  • Book on algebra, like J.B. Fraleigh's A First Course in Abstract Algebra.
  • Book on set topology, like J. Munkres's Topology.
  • Book in algebraic topology, like A. Hatcher's Algebraic Topology.

Article Info

Categories: Algebra