I am looking for a good book on homology theory. I have taken topology classes up to fundamental groups and covering spaces (Munkers book). I have a very good background in algebra up to categories, homology of groups.

I have considered using Hatcher's book, but I am looking for other books that make more use of categories.

Any suggestions?

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    I imagine that Peter May's book (available freely on his website) is like this, but I haven't read it. – Dylan Moreland Feb 25 '12 at 3:26
  • @DylanMoreland Thanks. I will have a look. – yaa09d Feb 26 '12 at 3:54
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    Peter May's book is lovely, but if you're interested in homology first and foremost, it will provide you with a rather inefficient pathway to your destination. May covers the basics of homotopy theory before homology, so when he covers homology he assumes you're familiar with the basics of homotopy first. – Ryan Budney Feb 26 '12 at 8:08
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    @MarianoSuárez-Alvarez «to make more use of categories?» means use of categorial language. – yaa09d Feb 26 '12 at 21:23
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    In that case, I'd say that —assuming you know the categorical language— it would be most productive to pick a good book, study it and, as an additional exercise, translate it into «categorical language» as you go. – Mariano Suárez-Álvarez Feb 26 '12 at 21:27
up vote 5 down vote accepted

As mentioned in the comments Peter May's Concise Guide is an excellent book.

Some others to consider:

  • Algebraic Topology - Tammo Tom Dieck. Apart from being an excellent text book (see my answer here for example) this has a heavy use of categorical language (already on page 20 he is talking about functors and left adjoints) so if that is your thing, this is the book for you!.
  • An Introduction to Algebraic Topology - Rotman. This has a gentle use of categories and functors.
  • Thanks for the suggestion. How do you compare Dieck's book to the classic book of Spanier? – yaa09d Feb 27 '12 at 2:01
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    Different approach! Rotman's book is more like Spanier (he was his student I believe). I've heard it said that Rotman is 'Spanier-lite' – Juan S Feb 29 '12 at 23:26

There is an algebraic topology book that specializes particularly in homology theory-namely, James Vick's Homology Theory:An Introduction To Algebraic Topology. It does a pretty good job of presenting singular homology theory from an abstract,modern point of view, but with plenty of pictures. It's an underrated book for this purpose that I wish would get more attention-you may find it very useful.

Another book I think you'll find very useful is the aforementioned Joseph Rotman's An Introduction To Homological Algebra. Not only is it the most accessible and clearly written book on the subject, it emphasizes the topological origin of many of the central concepts.

I think both of those,in addition to the other good suggestions here,will help you out a lot.

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    -1 : The OP said that he was interested in the homology of spaces, and Rotman's book has nothing to do with this. – Adam Smith Feb 27 '12 at 4:25
  • @Adam His original post said nothing about that. Of course,a little thing like the facts never kept you from an opportunity to downvote me. – Mathemagician1234 Feb 27 '12 at 5:32
  • His original post only implied it. But his reply to SL2 (written well before you wrote your post) spelled it out. – Adam Smith Feb 27 '12 at 6:45
  • @Adam I was only looking at his question. Besides,if you're serious about homology theory,the topological aspect is only half the story. I would have thought you would have advocated for such a perspective. If anyone else was posting the opinion-you probably would have. – Mathemagician1234 Feb 28 '12 at 5:23
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    They share some history and you need a small amount of homological algebra to set up the topological story, but they are distinct subjects. It is really an accident of history that they share a name. – Adam Smith Feb 28 '12 at 18:16

If you're after something purely homological in nature (i.e. without reference to the applications of (co)homology to topology), then the standard reference would be Weibel's book An Introduction to Homological Algebra.. This is very categorical, but it isn't specifically about homology and cohomology in topology.

If you're looking for something more directly related to (co)homology of spaces, then I'd like to recommend Switzer's book Algebraic Topology - Homology and Homotopy. It has a nice treatment of homology and cohomology from the categorical perspective. He takes the approach of first discussing general (co)homology theories on the category of spaces, and then even goes through Brown representability before turning to singular (co)homology. In fact, I'd recommend this book as a wonderful alternative to Hatcher if you find his geometric arguments and lack of category theory unsatisfying. Switzer seems to me to be more rigorous (less pretty pictures), but he uses more category theory and I prefer the rigour.

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    Thank you. What I am intrested in is homology of topological spaces. – yaa09d Feb 25 '12 at 17:17
  • @yaa09d, I'm curious if you're learning about this stuff so that you can tackle the HoTT book? – Steven Shaw Jun 4 '15 at 9:59

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