To cite examples, I find chapter 0 unreadable, especially the bits about CW complexes (I feel that the proofs in chapter 0 are at best incomplete but I may be grossly mistaken) and also example 0. In most major universities one of the three or four basic first year graduate mathematics courses is algebraic topology. With this criticism the book presents tons really cool example and intuition behind less-straightforward constructions. We give necessary and sufficient criteria for elementary operations in a two-dimensional terrain to preserve the persistent homology induced by the height function.

So, as a follow-up to first-year algebraic topology - still far from the cutting edge, but very relevant to reaching it - may I recommend reading some of the classic papers of the mid-twentieth century? You can think of that discussion of Hatcher as a "prerequisite quiz" which tests whether you have learned what you need to learn about homotopies. As noted on the page itself, however, the proof contains methodologies borrowed from other sources which, at face value, seem overly complicated. It's nearly impossible to find anywhere now and if a copy is available,I'm sure it'll cost a king's ransom. Most purchases from business sellers are protected by the Consumer Contract Regulations 2013 which give you the right to cancel the purchase within 14 days after the day you receive the item.

If your background in general topology is sufficiently strong, you can go straight to Chapter 5 on cell complexes (in the second edition, the focus is on CW complexes) and work on from there. This is a reminder for me to go back and try simplifying the proof given the insight gained from the original. He covers much more information than any of the other introductory textbooks I have perused, and with tons of explicitly worked out examples.

It covers three topics (Professor May's comment above has details) + an appendix on spectral sequences, which is short but very much to the point. I learned the contents of the first two chapters through a lecture course so I never really read them but rather used them for referencing or looking something up. As a result he sometimes introduces the necessary algebraic structures in a more concrete, less general fashion than an algebraist would. Hatcher's acolytes may counter that this is a book for mature students, so any gaps should be filled in by the reader along the way with pen and paper.I learned -still and will be learning - the fundamentals of Algebraic topology from a professor at my University, Dr. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology.

Homotopy theory in particular has undergone a complete transformation and explosive expansion since Whitehead wrote his book. It’s then redone using a laborious, perhaps-inaccurate-but-also-very-unwieldy method that doesn’t adapt well to the general case. Whether this is a good book or a bad book depends on your background, what you hope to gain from it, how much time you have, and (if your available time is not measured in years) how willing you are to take many things on faith as you press forward through homology, cohomology and homotopy theory. The electronic version has narrower margins than the print version for a better reading experience on portable electronic devices.For example, CW complexes have proved over time to be the most natural class of spaces for algebraic topology, so they are emphasized here much more than in the books of an earlier generation. The more and more algebraic topology that I learn the more I continue to come back to Hatcher for motivation and examples. More Hamburger icon An icon used to represent a menu that can be toggled by interacting with this icon. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics. It deals pretty early with spectral sequences and proves some standard results in classical homotopy theory via them (like Blakers-Massey, Freudenthal.

Serre's thesis is nice, Hatcher's notes are OK, but this seems to be a topic best learned in a good class).But be warned that if you're having difficulty understanding Hatcher, you have zero chance with Spanier.