# Algebraic Topology and Homotopy Theory prerequisites

I would like to know what are the prerequisites for studying algebraic topology (with a homotopical viewpoint). The books I have in mind are

• Selick ("Introduction To Homotopy Theory")
• May ("A Concise Course in Algebraic Topology")
• Strom ("Modern Classical Homotopy Theory")
• Fomenko, Fuchs ("Homotopic Topology")

I know that general topology and abstract algebra are assumed. But how much of those one need to start studying AT? What chapters of Munkres for point-set topology? What topics in algebra/linear algebra?

One of the classic references to studying algebraic topology is Hatcher's Algebraic Topology, which is available online at Hatcher's webpage. He says the following on the topic of prerequisites:

In terms of prerequisites, the present book assumes the reader has some familiarity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identification spaces as they are sometimes called, which are quite important for algebraic topology.

You should probably study the following collection of topics: topological spaces, continuous maps, connectedness, compactness, separation, function spaces, metrization, embedding theorems, and the fundamental group. You should also know what is taught in a "standard undergraduate course in algebra". A nice collection of notes written by Professor Richard Elman is here.

Since you say that you want to study algebraic topology with a "homotopical viewpoint", you should check out this MathOverflow question. Note: you can actually study homotopy theory through combinatorial objects called simplicial sets, but it's probably better to learn homotopy theory the classic way.

Hope this helps.

• I've read mixed opinions about AT prerequisites. Some say it's only group theory, others say one needs all undergrad-grad abstract algebra( except for Representation Theory )I remeber one comment explicitly saying you only need group theory for Hatcher. I heard a guy saying AT is easier via simplicial sets, though he was somewhat extremistic in his views about mathematical education, so I thought it was another rash statement. I wanted to learn simplicial homotopy theory anyway in the future. On general topology side, I understood it, thank you(for the entire answer). Sep 5, 2015 at 19:47
• @AlexPH7 If you plan on going into stable homotopy theory, it's perhaps best that you have a solid foundation in algebra which contains more than group theory. Simplicial homotopy theory is very fun, and it does simplify ideas in algebraic topology (like homotopies), but it is best for one to study algebraic topology the "normal" way and then proceed to simplicial homotopy theory.
– user122283
Sep 5, 2015 at 19:49
• I will learn general/commutative algebra fully anyway, because it's major in my university, plus for algebraic geometry. It's just I wonder wether I can start reading Homotopy Theory/Algebraic Topology( right now I prefer Strom's book) texts just now or should I just continue abstract algebra textbook. Well, in any case, I'm still going to finish Ring theory and modules before picking up anything else. Sep 6, 2015 at 17:19