How much algebra and how much topology is there in "algebraic topology?"

Question

I would like to study Hatcher's book, Algebraic Topology - in particular the fundamental group and introductory homotopy theory. I haven't had formal instruction in algebra or topology (my background is primarily in analysis). I've read through the first five chapters of Munkres' Topology and have a fairly good grasp on everything except the proof of Tychonoff's theorem - is that sufficient, or should I continue reading Munkres?

As for algebra, my knowledge is considerably less; it is mostly what I have taught myself, but I've never seriously studied it. I'm familiar with basic notions of group theory but not so much with the major theorems. I am assuming this is where I should focus my efforts on in preparing to study Hatcher's book. What are some topics that I should be familiar with, and some texts to study those from? Would Dummit and Foote be a suitable choice, or should I seek something not quite so heavy? I would like to say I am "mathematically mature," just not specifically familiar with algebra.

I should stress that I'm not looking to become an expert in algebraic topology, just enough to study the fundamental concepts and theorems. My question is mostly whether I should focus more on algebra or topology in my preparation.

Answer 1

Honestly, you don't need a huge algebra background. Also, In algebraic/geometric topology one does not need a huge point set topology. I think you've enough point set topology background.

Basic notions of groups such as groups, subgroups, and homomorphism/isomorphism are needed pretty much all the time. You should be really comfortable with free abelian groups those are the main objects(Homology and homotopy groups) in algebraic topology. When you'll compute fundamental groups, you will find that there are spaces where fundamental groups cannot be easily written explicitly, for example, Kleine bottle. So, you should be comfortable with generators and relations. To compute some homology/cohomology groups sometimes, you will use the tensor product, Free product (many many times), $Hom(A, B)$, $Tor(A, B)$ and $Ext(A, B)$. You can use them as a black box, but understanding them clearly will be fun for sure. If you're familiar with exact sequences, and basic notions of modules that will be extremely helpful.

Fun fact: you'll use the first isomorphism theorem many times. I hope this helps.

Answer 2

In modern (or more advanced) algebraic topology, you will also start to need some more advanced notions and nowadays algebraic topology heavily relies on category theory. This will of course be introduced along the way but you should be prepared to deal with more abstract objects and heavy machinery.

I would say that in modern algebraic toology (e.g. Homotopy theory) you don't see any (or very little) algebra nor topology anymore. It's a field of its own and a very exciting and interesting one !

Answer 3

The first question to address is, how much algebra is there? Universal algebra allows for arbitrary numbers of arbitrary operations of arbitrary arities, but the “interesting algebras”, in the sense of what algebraists actually spend their time working with, and make real progress understanding, are much more limited. Primarily there’s group theory and ring theory, although a few people think about semigroups, monoids arise in computer science, and quivers (directed multi graphs) are technically algebras as well, though I suspect few graph theorists think of themselves as algebraists.

While in principle algebraic topology could study any functor from continuous maps to algebraic homomorphisms, the ones that actually arise in practice are those of groupoids and groups, and of modules and rings. So ring theory and group theory is what’s used. But that’s not far from being all of algebra.

Having said that, homotopy groups are hard to compute, so while all of group theory is in principle relevant to the subject, the spaces for which even some of their homotopy groups can be computed tend to have fairly uncomplicated groups, or were specifically constructed to have a specific such group. Similarly to a lesser degree for the rings and modules arising in cohomology.

Munkres's Topology should be sufficient to start Hatcher. I would also recommend that you read John Baez's notes on Category Theory, as it will be easier to make conncetions.

Algebraic Topology by Tammo Dieck is another fantastic book that you can look. Its my personal favourite, as it more rigourous and offers more explaination than Hatcher