# Hatcher Vs. May's Algebraic Topology.

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## Main Question or Discussion Point

I must say thusfar I read through chapter one of May's book and chapter 0 of Hatcher's, May is much more clear than Hatcher, I don't understand how people can recommend Hatcher's text.
May is precise with his definitions, and Hatcher's writes in illustrative manner which is not mathematical rigorous from what I know.

I really don't see why there so much hype over Hatcher.

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mathwonk
Homework Helper
Giving you the benefit of the doubt that this is really a question, I will try to answer it. In the first place, to compare "apples to apples", you should compare section I.1 of hatcher (rather than chapter zero) to chapter 1 of May, since those both discuss the fundamental group, compute it for a circle, and apply it to the fundamental theorem of algebra and brouwer's theorem.

If you do this you may notice the large number of times May uses the phrase "it is easy to see that" or "easy to check". where Hatcher just gives the proof. I suggest this difference explains why some people prefer Hatcher. I.e. Hatcher is just as precise as May but far more detailed. Indeed the first 4 chapters of May's book follow exactly the sequence of topics of hatcher's chapter one, but Hatcher devotes 80 pages to May's 35 to this basic material.

But I am happy you have found a source that works for you. May is certainly an excellent concise reference. I like both, but I can easily see why a student would prefer Hatcher. May even states that his book is not written as a text.

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Well, it wasn't a question, merely stated my opinion, and it seems that in amazon's reviews there are some that agree with me.

So to what do I need to compare the zero chapter in Hatcher's, because the part on CW complexes in this chpater was really bad for my taste, for example he defines the smash product:

Code:
The smash product X \wedge Y is then defined to be the quotient XxY/ XvY.
One can think of X \wedge Y as a reduced version of XxY obtained by collapsing away the parts that aren't genuinely a product, the seperate factors X and Y.
I believe that if he just wrote explicitly what is XxY / XvY, I would understnad what's it about, but as it is stated it's too wordy for me, sorry, and I just don't get it.

As for the fact that May writes a lot of times that it's easy to see in this particular chapter that I read it means you should check it for yourself, and if you have taken Topology through Munkres, you would understand why it's the way it is.

I have other books that I lend from the library, Massey's basic course and Bredon's geometry and topology, reading chapter zero of Hatcher's besides the parts on homotopy and retraction that I already knew from topology in UG, were really hard to understand, I don't understand why CW complexes are already covered in chapter zero in Hatcher's but in May's and others they appear much later.

mathwonk
Homework Helper
well, May does not have a chapter comparable to Hatcher's chapter zero. The introduction to May is the only thing comparable, and it isn't really.

I do not like chapters zero myself, where people try to cram in too much to absorb, by way of introduction, but Hatcher has done this there. Spanier also has an early chapter that is more condensed and hard to read than later chapters. In both cases one should skip those.

I.e. the way to read Hatcher is to skip chapter zero and begin in chapter 1. Then you will be in the same boat as reading chapters 1-5 of May.

But you seem like a quick study, and May's brief presentation is probably more suited for you than Hatcher's somewhat chatty one. There is some advantage to a shorter presentation, provided, as you suggest, one fills in the gaps oneself as exercises.

As for CW complexes, they are discussed throughout, including in the appendix, Hatcher, pp. 519-529. Compare Hatcher's discussion of them for difficulty with the first 3 sections of chapter 10 of May. E.g. compare Hatcher's and May's proofs of Whitehead's theorem, to see which you prefer. Hatcher: pp. 346-7, May pp. 76-77.

But you can't go too far wrong with either book, just take your pick.

It is unfortunate we try to read books in the order they are printed sometimes.

E.g. hartshorne's algebraic geometry book should be read in the order chapters 4,5,1,2,3, or maybe 1,4,5,2,3, but not 1,2,3,4,5.