I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them. I have a solid background in Abstract Algebra, and also have knowledge on Homological Algebra(in fact I am now study Tor and Ext functors). But my knowledge in topology is poor, very poor. I could only remember the concepts, but I got no idea with problems. Please feel freely suggesting me some book to work on. Thank for reading.

1For a gentle introduction to the subject I recommend: Algebraic Topology: An Intuitive Approach by Hajime Sato – Dec 09 '11 at 07:41
4 Answers
If you would like to learn algebraic topology very well, then I think that you will need to learn some pointset topology. I would recommend you to read chapters 23 of Topology: A First Course by James Munkres for the elements of pointset topology. If you would like to learn algebraic topology as soon as possible, then you should perhaps read this text selectively. In particular, I would recommend you to focus mainly on the following (fundamental) notions, reading more if time permits:
 Topological space
 Basis for a topology
 The product topology (on a finite cartesian product; if time permits, you can read about the case of an infinite cartesian product but this is not urgently needed as far as algebraic topology is concerned)
 Subspace topology
 Closed set and limit point
 Continuous function
 Metric space
 Quotient topology (this is a very important, but sometimes ignored, prerequisite for algebraic topology)
 Connected space
 Component and path component
 Compact space
 Hausdorff space
 The separation axioms, Urysohn's lemma and the Tietze extension theorem (if time permits; these are very useful and inteteresting concepts but you can take the Urysohn lemma and Tietze extension theorem on faith if you desire)
I think that as far as algebraic topology is concerned, there are two options that I would recommend: Elements of Algebraic Topology by James Munkres or chapter 8 onwards of Topology: A First Course by James Munkres. The latter reference is very good if you wish to learn more about the fundamental group. However, the former reference is nearly 450 pages in length and provides a fairly detailed account of homology and cohomology. I really enjoyed reading Elements of Algebraic Topology by James Munkres and would highly recommend it. In particular, I think a good plan would be:
 Learn the elements of pointset topology as outlined above.
 Read chapter 8 of Munkres' Topology: A First Course to learn the rudiments of the fundamental group.
 Read Elements of Algebraic Topology by James Munkres.
You will not need to know anything about manifolds to read Elements of Algebraic Topology but I believe that it is good to at least concurrently learn about them as you learn algebraic topology; the two subjects complement each other very well. I think a very good textbook for the theory of differentiable manifolds is An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby (but this is a matter of personal taste; there are (obviously) many other excellent textbooks on this subject). The advantage of this textbook from the point of view of this question is that there is a flavor of algebraic topology present in one of the chapters.
I hope this helps!
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4Munkres' ELEMENTS is extremely readable,visual and details constructions at a level of depth few other books do. However,there are 2 problems with the book:Firstly,it's VERY old fashioned; it focuses almost entirely on simplicial complexes and their computationsit barely mentions category theory or homological algebra.While the tremendous detail and visual nature of the calculations makes the presentation incredibly clear,the down side is they are so belabored at times, it's almost ugly. I'd supplement it with either Rotman or Vick. The other problem is the priceit's VERY expensive. – Mathemagician1234 Nov 24 '12 at 08:47

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3@Mathemagician1234 Well, The second problem is not a problem, since you can find that book and download it on Library Genesis for FREE, here's the link for it: http://gen.lib.rus.ec/book/index.php?md5=437A0F15AC08A6F43970FC5AB742AD7D. – Kevin.S Apr 14 '20 at 13:21
I'm going to be controversial here, and suggest that you start with Spanier's Algebraic Topology, supplemented by Switzer's Algebraic Topology: Homology and Homotopy. These are very good and comprehensive books which have stood the test of time; books that present algebraic topology properly and algebraically. Many great algebraic topologists grew up on these books.
That having been said, I am also a fan of Munkres Elements of Algebraic Topology which works out examples very nicely using simplicial decompositions. Not to sound too oldfashioned, but I'm also a huge fan of Schubert's Topology as a book which works out the simplicial story properly. But, given your background, I strongly recommend that you dive straight into Spanier and Switzer, and enjoy their beautiful, classical, algebraic treatment of the material.
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14Egad! I must strenuously object to your suggestion to use Spanier as a first text (supplemented or not) for algebraic topology. In fact, I cannot conceive of a worse choice. If you say it's good as a reference, I will take your word for it but as a pedagogical device I can say, from first hand experience, it is a disaster. – ItsNotObvious Dec 09 '11 at 16:39

8I can't deal with these kind of recommendations. It's like the poster who recommended Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS as a CALCULUS textbookI find it very difficult to believe that's a serious recommendation and isn't simply trying to go for the "awe" factor to make themselves look like Karl Guass to the question author. – Mathemagician1234 Dec 09 '11 at 17:17

1Respectfully, I strongly disagree. My recommendation is serious, although I'm wellaware that it is controversial. – Daniel Moskovich Dec 09 '11 at 19:52

2@Mathemagician1234 : I suspect that you mean "Carl Gauss", not "Karl Guass"... – Adam Smith Dec 09 '11 at 20:42

1@Adam That's how I was taught to spell it and it's stayed with me,despite my knowing now that it's wrong. – Mathemagician1234 Dec 12 '11 at 04:10

7@Mathemagician1234 : The "Karl" thing is understandable since that spelling is common in German. But "Guass"? – Adam Smith Dec 12 '11 at 04:23

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Spanier isn't too bad but Switzer would be an awful textbook for most introductory students. Bredon's book covers the same introductory material yet in a far more pleasant way than these two books. IMO May and Hatcher's texts are a great way to start. – Ryan Budney Sep 15 '12 at 23:42
If your background in point set topology is insufficient, Munkres' Topology is a great book for foundations. Once you understand the basics in Munkres, you can move on to Armstrong's Basic Topology or Massey's Algebraic Topology: An Introduction (the former also contains all of the point set topology necessary to read it in its entirety) . Either of those should prepare you for Hatcher's Algebraic Topology, provided you have the necessary algebra background (which it sounds like you do). There will be some overlap between Hatcher and the other two, but attempting Hatcher without familiarity with the classification of surfaces and basic homotopy theory may be overwhelming.
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Firstly, let me qualify this answer by acknowledging that I don't know algebraic topology very well and have only assimilated the rudiments of homotopy and homology theory. That being said, I find the two texts by Prosolov, Elements of Combinatorial And Differential Topology and Elements of Homology Theory, to be very promising. These texts are meant to be read sequentially and each text introduces problems in a justintime fashion and to which hints and, in many cases, full solutions, are provided. This feature makes these texts very suitable for selfstudy and is a rarity among algebraic topology texts. The author doesn't expect you to have a thorough background in differential geometry and introduces these concepts as needed. Although in the first volume he does cover basic definitions from point set topology, you will probably need to supplement it with a good general topology text as others have suggested, e.g., Munkres.
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2+1 for recommending these 2 outstanding and underappreiciated textbooks,3Sphere.They are extremely visual and yet at the same time,very modern and complete.Supplementing them with J.P. May's A CONCISE COURSE IN ALGEBRAIC TOPOLOGY as collateral reading for learning the abstract point of view concurrently and the reader's all set. – Mathemagician1234 Dec 09 '11 at 17:19

2These are definitely nice books, and not nearly wellknown enough! – Daniel Moskovich Dec 09 '11 at 19:55