# How to read Hatcher's Algebraic Topology?

I have to read Allen Hatcher's textbook and I am having a really hard time with the book. To cite examples, I find chapter 0 unreadable, especially the bits about CW complexes (I feel that the proofs in chapter 0 are at best incomplete but I may be grossly mistaken) and also example 0.7 where he says that the three graphs are homotopy equivalent because they are deformation retract of a disk with two holes. Similarly, when I tried reading the proof of the Van Kampen's theorem, I felt the proof was not so clear with words like "perturb the vertical sides" making a cameo. In other words, I feel that the treatment is not rigorous.

Am I reading the book wrong? To clarify, this is not a rant against the book for the heck of it: I really want to try and read this book. I know that Hatcher's text is followed all over the world, so I am just trying to understand how to really read the book. Should I be spending a lot more time trying to fill in the gaps or am I supposed to gloss over the details? Thanks in advance for your suggestions!

• It is, in my opinion, an excellent book in algebraic topology, but a non optimal first book in algebraic topology. – Randall Mar 8 at 10:54
• Just try a different book. Although I think Hatcher's book is littered with great, well-thought out examples, I personally didn't like it. I much preferred Spanier to Hatcher and read that as my first course in AT. Not everyone is going to benefit from the same approach, so just try to find your own. – Tyrone Mar 8 at 10:59
• @Randall What book do you suggest for learning algebraic topology for a beginner with no prior exposure to the topic? I also have seen some online lectures on YouTube, recorded in an Italian institution. I think they're worth checking out. (to the OP) – stressed out Mar 8 at 11:11
• Check out math.stackexchange.com/q/689680/118539 on what to read alongside Hatcher's Algebraic Topology. – J W Mar 8 at 14:51

How good is your background in topology? For example, have you mastered Munkres' book?

The point of view in Hatcher's book requires you to have already mastered several important topics in topology including these two key topics:

• Quotient maps and quotient topologies, which are the key to CW complexes;
• Homotopies, which are the key to deformation retractions and homotopy equivalences.

Just as an example, I would expect someone who has mastered Munkres' book to be able to write down an explicit formula for a subset of $$\mathbb R^2$$ that is homeomorphic to one of the graphs in that discussion of Hatcher, to write down an explicit formula for a specific deformation retraction from a disc with two holes to that graph, and to write down the specific formulas for the homotopies needed to prove that map to be a deformation retraction. You can think of that discussion of Hatcher as a "prerequisite quiz" which tests whether you have learned what you need to learn about homotopies.

So, if you find yourself unable to write down such maps and such homotopies, or if you have any other deficiencies in those two topics, or in any other basic topics in topology, you should shore up those topics with another book such as Munkres as you proceed into Hatcher's book.

• Thank you very much for your comments. I shall try to spend more time filling in the details. – user448760 Mar 9 at 13:51

As others have suggested, one solution could be to try another book. There are many possibilities out there, but a good one for beginners is Lee's Introduction to Topological Manifolds. Even though it is about manifolds, it takes plenty of time to introduce key topics from general and algebraic topology. If your background in general topology is sufficiently strong, you can go straight to Chapter 5 on cell complexes (in the second edition, the focus is on CW complexes) and work on from there. Chapter 10 is on the Seifert-Van Kampen theorem. Lee is very careful and thorough in his presentation, which could help you with the gaps you have encountered.

• Thank you for letting me know about Lee's text. It looks fairly comprehensive for the topics it covers. :) – user448760 Mar 9 at 13:52
• @user448760: Glad if it helps. Lee has a webpage for the book at sites.math.washington.edu/~lee/Books/ITM. – J W Mar 10 at 6:24