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.
Best Answer
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.