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I am fishing for a textbook on basic Algebraic Topology. Almost every where I looked, I saw praises for Hatcher's textbook. Now, I already know a little bit of Homology (at the level of Munkres' Elements of Algebraic Topology), but looking at Hatcher's chapter on Homology I realized that I wouldn't have been able to learn much from it. For a lack of a better phrase, it would have appeared too hand-wavy for someone like me (due to my lack of mathematical maturity).

So, my question is where (apart from Munkres, preferably online) do I turn to learn the basics of Cohomology (say, at the level of Hatcher's chapter 3)?

PS: And, since I am a poor student, online lecture notes would be great.


Thanks everyone. But, I should have stressed that I am looking for lecture notes or textbooks that are freely available online. So I'll wait a bit more before accepting.

@Theo Buehler: Massey looks great. Thanks a lot. Unfortunately, I do not have access to my univ library over the summer. But, if I don't find any notes etc. then I'll accept your answer if you post it.

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Have you looked at Massey's Basic course in algebraic topology? I found it great and it seems a bit more basic than Hatcher. Unfortunately, being a Springer GTM, it's rather expensive. However, you should be able to find it in any good library. – t.b. Jun 6 '11 at 9:10
Glen E. Bredon's "Topology and Geometry" is an excellent book (and it covers cohomology). – Amitesh Datta Jun 6 '11 at 10:27
However, if you already have your hands on Munkres, then (at least in my opinion) it is hard to find a better book. – Amitesh Datta Jun 6 '11 at 12:07
@Dactyl: Thanks, but I think you should accept mixedmath's answer, as I couldn't add much to it (the other answers contain good recommendations, too). Mixedmath's answer is pretty much the same as mine would turn out to be. – t.b. Jun 7 '11 at 16:26
up vote 5 down vote accepted

Firstly, I think that Munkres is excellent. But I also agree that Hatcher's is less accessible than Munkres. In addition to the excellent comments above (Theo's recommendation of Massey, even though expensive, is great - and if you are a student with access to a math library, I bet it will be there).

But I point you to the Autodidact's Guide, something published in Notices of the AMS a while back. It recommends different books for almost every subject, and their topology section is right on.

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I haven't seen it as my library doesn't have it (yet), but there are very good reviews for Tammo tom Dieck's book

It presents the proof of a number of results that were previously unknown to be possible without spectral sequences.

Google link

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You might also want to take a look at William Fulton's A First Course in Algebraic Topology. It covers both homology and cohomology. The style is somewhat similar to Hatcher in that much of it is conceptual in nature. I, like you, don't really benefit from Hatcher's style of exposition. For similar reasons, you might find Fulton's text somewhat frustrating as well. But if you can find it in a library it might benefit you to survey it and see if it helps fill in the gaps and give you additional insight. On the upside, a fair number of exercises in Fulton have hints/answers that can help you if you're stuck; this feature is indeed a rarity in algebraic topology texts. Another thing that is nice about Fulton's text is that he uses problems in calculus/analysis to motivate discussions of co/homology.

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