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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$\begingroup$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:
- 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.
- He uses $\Delta$ complexes, which are rarely used.
- I would have preferred a more formal viewpoint (categories are introduced kind of late and not used very much).
- There aren't many examples that are as difficult as some of the more difficult problems.
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.
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ 5 $\begingroup$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.
$\endgroup$ 2 $\begingroup$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.
$\endgroup$ $\begingroup$Here are some books I would prefer to Hatcher, but I don't think they are any easier to read.
One of my favorite books is A Concise Course in Algebraic Topology by Peter May. It is really... concise, and it's freely available from May's webpage:
Another modern textbook is Algebraic Topology by Tammo tom Dieck.
Finally, there's Algebraic Topology from a Homotopical Viewpoint by Marcelo Aguilar, Samuel Gitler, Carlos Prieto. As the title suggests, it is based on homotopy theory.
The other answer suggests Spanier's Algebraic topology. It was published in 1966, but it's still a great reference. Another classic is Algebraic Topology — Homotopy and Homology by Robert Switzer, published in 1975. It is rather terse, and probably not suitable as an introductory text.
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.
$\endgroup$ $\begingroup$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.
$\endgroup$ 1 $\begingroup$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.
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