[Math] How to read Hatcher’s Algebraic Topology

Question

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!

Answer 1

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.