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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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!

Answer 4

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.