# 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.

• 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-Álvarez Nov 30 '11 at 4:27

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.
• 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
• Hatcher does explain his choice of $\Delta$ complexes. plus, i had previously learned a.t. thru category language and had learned zero things! – Behnam Esmayli Aug 9 '16 at 19:32

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.

I don't see why I should not recommend my own book Topology and Groupoids (T&G) 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. Also useful is the notion of fibration 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, except a 2016 Bourbaki volume in French on "Topologie Algebrique": and that gives no example applications.

The book is available from amazon at $31.99 and a pdf version with hyperref and some colour is available from the web page for the book. 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 (NAT) 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 page for the book. Motivation for the methods are given by a thorough presentation of the history and intuitions, and the book should be seen as a sequel to "Topology and Groupoids", to which it refers often. The new book 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. You will find relevant presentations on my preprint page. Will this take on? The next 20 years may tell! October 24, 2016 A new preprint Modelling and Computing Homotopy Types: I is available as an Introduction to the above NAT book. This expands on some material presented at CT2015, the Aveiro meeting on Category Theory. • 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 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. • I have discovered this book recently and find it a great pity that he hasn't written more on Algebraic Topology. I couldn't much study Algebraic Topology with Hatcher's book if my life depended on it. Lee's book really stands out, at least fro. what I've seen. The treatment of CW complexes is really exceptional. – polynomial_donut Oct 6 '16 at 14:08 • +1 for John Lee's book. I feel Hatcher's book is okay, but not as good as some make it out to be. From the amount written in the book, at first glance one might think the book will be relatively easy to read because with that much written there should be a lot of details. Once you get into the book, you realize there are many details left out and many jumps in logic or handwaving that is difficult to accept. – TuoTuo Jul 4 '17 at 21:37 If you are taking a first course on Algebraic Topology. John Lee's book Introduction to Topological Manifolds might be a good reference. It contains sufficient materials that build up the necessary backgrounds in general topology, CW complexes, free groups, free products, etc. Here are some books I would prefer to Hatcher, but I don't think they are any easier to read. • Oops. Apparently, the question was posted 5 years ago, and popped up because of someone's edit. I will still leave it here. – user144221 Dec 2 '16 at 11:21 • The book of Peter May "A Concise Course" is a good starting point. Read Chapter 1 and you'll see the difference from Hatcher's more comprehensive work. – Alexander Kartun-Giles Nov 1 '17 at 16:50 I believe that it is very important to think deeply about whether it is a book, the subject matter, or you that makes a book uneasy to read. we have to confess that algebraic topology is a tough subject. it is nothing like any undergraduate course one takes. secondly, you need to be patient. i personally had some hard time with Hatcher's book. but now I find great joy and pleasure reading it on my own, even after my course is finished. Only, later do you come to see why people say his book is so geometric in flavor. l never liked algebra, but Allan's book helped me appreciate it more. it is so motivating to see how groups give us beautiful knowledge about shapes! sometimes, you need to move ahead, leaving things to be re-read later. that makes is fine. Math is tough. that is the sentence that in fact, ironically, helped me get back to work! i started to get harder on something that i couln't understand right away. • nice...................+1 – Bhaskara-III Dec 2 '16 at 10:59 You will take pleasure in reading Spanier's Algebraic topology. It is basically "algebraic topology done right", and Hatcher's book is basically Spanier light. Hatcher also doesn't treat very essential things such as the acyclic model theorem, the Eilenberg-Zilber theorem, etc., and he is very often imprecise (even in his definition of$\partial\$). There is also no treatment of the very crucial spectral sequences method.

There is a really well-written but lesser known book by William Fulton. That's the book I learnt Algebraic Topology from. The chapters are laid out in an order that justifies the need for algebraic machinery in topology. A guiding principle of the text is that algebraic machinery must be introduced only as needed, and the topology is more important than the algebraic methods. This is exactly how the student mind works. The book does a great job, going from the known to the unknown: in the first chapter, winding number is introduced using path integrals. Then winding number is explored in a lot more detail, and its connection to homotopy is discussed, without even mentioning fundamental groups. Then a number of results like the Fundamental Theorem of Algebra, Borsuk Ulam and Brouwer's Fixed Point Theorem are proved using winding numbers. Only in Ch.5 do we see the first algebraic object. Here again, the order is flipped: the first De Rham Cohomology group is introduced and used to prove the Jordan Curve Theorem. Then homology groups of open sets in the plane are discussed, and the connection between homology and winding number is made clear. A number of applications to complex integration etc are discussed, and the Mayer-Vietoris theorem is proved for n=1. Covering spaces and fundamental groups are introduced after homology, another novelty. Higher dimensions are encountered only towards the end of the book, but by the time we get there, we already know the general idea behind all the concepts. Very few books take this point of view of developing intuition clarity before generalising rapidly. I think this is really helpful because before studying the general theory of anything, we need to know what it is we are trying to generalise.
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