Best Algebraic Topology book/Alternative to Allen Hatcher free book?

Allen Hatcher seems impossible and this is set as the course text?

So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online.

Any good intro to Algebraic topology books?

I can find a big lists of Algebraic geometry books on here. On a very old thread on Maths overflow someone recommended that a person should read James Munkres Topology first, then you should read Allen Hatcher book.

It just seems like Rudin's book but crammed with ten times more material.

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Why don't you please indicate what your background is? The alg. topology book by Hatcher is generally so well-regarded that your description of it seeming impossible suggests a mismatch between its intended audience and what you bring to the table as a reader, rather than the book itself really being impossible. –  KCd Nov 22 '11 at 1:00
The free-online-ness is definitely a plus, but/and Hatcher is a very good mathematician (so can be trusted about mathematical fact), and is a better writer than most! There is little reason to object to the choices made in the text, also. One may disagree, as with anything, but Hatcher has put his efforts (and non-collection of royalties, for example) "where his mouth is". And, specifically, I find nothing at all wrong with his choices, presentation, style, etc. It may be more fluid than some styles of 50 years ago, but that's a good thing. –  paul garrett Nov 22 '11 at 1:32
Hatcher's book is very well-written with a good combination of motivation, intuitive explanations, and rigorous details. It would be worth a decent price, so it is very generous of Dr. Hatcher to provide the book for free download. But if you want an alternative, Greenberg and Harper's Algebraic Topology covers the theory in a straightforward and comprehensive manner. It also contains significantly less discussion of motivation and intuition that you seem to dislike, though it does have a nice discussion of the functorial approach to algebraic topology. –  Michael Joyce Nov 22 '11 at 1:38
@simplicity: Like almost every book I have seen on algebraic topology, Hatcher's text is a graduate text. Your other questions posted on this site show that you are still learning undergraduate level algebra and topology. Until you have learned these subjects very well, I suspect you will find any algebraic topology text extremely to prohibitively challenging. –  Pete L. Clark Nov 22 '11 at 2:53
@simplicity: you have asked a few hours ago how to compute the dimension of a matrix algebra: that makes it clear that you are not in the intended audience of Hatcher's book (this is not a judgement on you but simply the statement of a fact) The thing is, your question is written in a tone rather incompatible with this fact; if Hatcher's book is the textbook for a course, that probably means you should wait a bit before taking it, not that the book is «impossible». –  Mariano Suárez-Alvarez Nov 30 '11 at 4:27
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4 Answers

I'm with Jonathan in that Hatcher's book is also one of my least favorite texts. I prefer Bredon's "Topology and Geometry."

For all the people raving about Hatcher, here are some my dislikes:

1. His visual arguments did not resonate with me. I found myself in many cases more willing to accept the theorem's statement as fact than certain steps in his argument.
2. He uses $\Delta$ complexes, which are rarely used.
3. I would have preferred a more formal viewpoint (categories are introduced kind of late and not used very much).
4. There aren't many examples that are as difficult as some of the more difficult problems.
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Another thing to recommend Bredon, if I recall correctly, is that he deals with the differential side of things. Hatcher avoids this. –  Dylan Moreland Nov 22 '11 at 3:44
Eric, your response isn't an answer to the question and should be (at most) a comment. It seems rather counter-productive. –  Ryan Budney Nov 22 '11 at 6:43
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I certainly sympathize with your situation. When I was reading Hatcher as a freshmen for the first time it was very difficult to read for various reasons. But to be honest your post feels quite shallow and awkward, because people usually complain things by making concrete points and your points (cheap, ten times thick, seems impossible,etc) are not really relevant. Would it be better for you to:

0) Ask questions in here or else where (like "ask a topologist") on the problems or sections you found difficult?

1) Register or audit an undergraduate intro level algebraic topology class for next semester? (at a level lower than this course.)

2) Consolidate your mathematical background by working on some relevant classical textbooks first (Kelley's General topology, Dummit&Foote's abstract algebra, Ahlfor's complex analysis, etc). It is not really necessarily for you to learn graduate level algebraic topology at your current mathematics level. It might be condescending for me to suggest this but I believe it is better to read easier stuff than struggle with texts "impossible" for you. The above books are not closely relevant but may be helpful to prepare you to read Hatcher. Also If I remember correctly Hatcher does provide a recommended textbook list in his webpage as well as point set topology notes .

3) In case you decide you must learn some algebraic topology, and favor "short" books. You may try this book: introduction to algebraic topology by V.A. Vassilev. This is only about 150 pages but is difficult to read (for me when I was in Moscow). It seems to be available in here. Vassilev is a renowned algebraic topologist and you may learn a lot from that book.

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I don't see why I should not recommend my own book "Topology and groupoids" as a text on general topology from a geometric viewpoint and on 1-dimensional homotopy theory from the modern view of groupoids. This allows for a form of the van Kampen theorem with many base points, chosen according to the geometry of the situation, from which one can deduce the fundamental group of the circle, a gap in traditional accounts; also I feel it makes the theory of covering spaces easier to follow since a covering map of spaces is modelled by a covering morphism of groupoids. A further bonus is that there is a theorem on the fundamental groupoid of an orbit space by a discontinuous action of a group, not to be found in any other text.

The book has been adopted for a course at Harvard, and more details are available at http://www.bangor.ac.uk/r.brown/topgpds.html It is available from amazon at \$31.99 and an e-version with hyperref and some colour is available at £5 from http://www.kagi.com.

The book has no homology theory, so it contains only one initial part of algebraic topology.

BUT, another part of algebraic topology is in the new jointly authored book "Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids"published in 2011 by the European Mathematical Society. The print version is not cheap, but seems to me good value for 703 pages, and a pdf is available on my web site. There is a thorough presentation of the history and intuitions, and it should be seen as a sequel to "Topology and groupoids", to which it refers often.

It gives a quite different approach to the border between homotopy and homology, in which there is little singular homology, and no simplicial approximation. Instead, it gives a Higher Homotopy Seifert-van Kampen Theorem, which yields directly results on relative homotopy groups, including nonabelian ones in dimension 2 (!!), and including generalisations of the Relative Hurewicz Theorem. Part I, up to p. 204, is almost entirely on dimension 1 and 2, with lots of figures. You'll find little, if any, of the results on crossed modules in other algebraic topology texts.

Will this take on? The next 20 years may tell!

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This is a nice book; haven't spent a whole lot of time with it but it seems easy to read and has helpful illustrations. It's helpful in that it seems to cover the material that a general topology course typically omits but is more-or-less a prerequisite for algebraic topology –  ItsNotObvious Apr 10 '12 at 14:35
I don't think the omission of homology theory in a book pitched at this level is a problem, Ronnie.It also has a very good introduction to basic category theory.Unfortunately,it is missing a few topics you'd really like to see in an introductory topology book,such as combinatorial topology (i.e.the classification of surfaces,etc.) and the more analytic aspects of point-set theory,such as generalized convergence.The latter,to be honest,would work against the overall theme of the book,so it's omission is understandable.(Thanks for quoting my comments on MO on your website,btw). : ) –  Mathemagician1234 Apr 23 '12 at 16:00
A nice book to be sure, but I do not believe appropriate for the OP! –  user641 Apr 30 '12 at 22:22
@Steve D: An anonymous user posted the following in the form of a suggested edit: $${}$$ In reply to Steve D, below: I am not sure which book he refers to, or what OP stands for (ordinary person, old person?)$${}$$ The new book is a sequel tp T&G, and will surely take a while to digest, as is shown by the fact that even the 2-d van Kampen theorem, published in 1978, and which calculates homotopy 2-types, is not well known. The history and intuitions behind this work should be for all. –  t.b. May 8 '12 at 10:30
Re classification of surfaces: This topic was omitted from all editions of "Topology and Groupoids" because of space considerations and because I did not see how to improve on the account in the book by Massey. Note that Ross Geoghegan in a 1986 Math. Review of an article by Armstrong wrote: "This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." This material on orbit spaces is covered in Chapter 11 of T&G, using groupoids in an essential way. –  Ronnie Brown Jun 30 '13 at 14:50
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If you want a more rigorous book with geometric motivation I reccomend John M. Lee`s topological manifolds where he does a lot of stuff on covering spaces homologies and cohomologies. As a supplement you can next go to his book on Smooth Manifold to get to the differential case. I especially like his very through and rigorous introduction of quotient spaces/topologies and so on which are used very heavily and which hatcher explains mostly in a very pictorial and unsatisfying way.

However let me also note that Hatchers through examination of the covering space of the circle (which also lee does) has been a very helpful example for me to keep in mind whenever I am thinking of covering spaces in general. So I propose that you should read that part.

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