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  Book recommendations for topology

+ 2 like - 0 dislike

I'm interested to learn topology. In particular, my interest lies in algebraic topology. I was wondering if someone could recommend proper maths books/notes (i.e. not books like Nakahara's "Geometry, Topology and Physics" or Naber's "Topology, Geometry and Gauge Fields: Foundations") which are "friendly" enough  for physicists?

I think this topic has the highest chance of succeeding if you could write a very short review. This doesn't have to be elaborate, but maybe you could mention what it is about the book you like and dislike, and what prerequisites are needed to understand the book/notes. If you have any other suggestions to increase the quality of this post, then please let me know.

Although I'm interested algebraic topology and "friendly" maths books, I think it may be worth while to make this topic a general book recommendation related to topology. So if you have any other recommendations than please posts those as well.

asked Apr 18, 2014 in Resources and References by Hunter (520 points) [ no revision ]
recategorized Apr 23, 2014 by dimension10

3 Answers

+ 4 like - 0 dislike

I think you will find that to learn algebraic topology well, you will need a good grounding in point-set topology.  It's probably worth the investment of time because point-set topology is a language which is pervasive throughout mathematics.  I personally like Munkres's Topology: a first course which covers point-set topology and a basic introduction to homotopy; although no homology.  That's covered in a companion book by Munkres called "Algebraic Topology".  That book is perhaps a little old-fashioned, though: algebraic topology has moved on and the old language of (co)homology theories being defined by complexes is being eschewed in terms of the more modern language of spectra, derived functors,...

answered Apr 19, 2014 by José Figueroa-O'Farrill (2,315 points) [ no revision ]

Although this means I will have to study a lot more, this is really useful information; thanks! Are there any other topics that a student, who pursues a theoretical physics degree, might not have come across in order to understand topology? For instance, is Michael Spivak's "Calculus" or a mathematics book on functional analysis needed to understand topology?

@Hunter: Although not technically a necessity, it would be much better if you get exposed to some mathematical analysis before starting point set topology, one is deprived from many concrete examples if he goes into topology without analysis background, and concrete examples are a really useful mental crutch. Functional analysis is necessary for theoretical physics, but not a prerequisite for learning topology. 

@JiaYiyang Thanks for your respons! Good to know.

Point-set topology serves as indoctrination in set theory, in disguise, because it looks like geometry and analysis at first. What it is doing is reformulating obvious elementary concepts from real analysis in set language, and then imposing extra non-obvious set-theoretic conceptions in a language that makes them look natural and seductive (Tychonov's theorem for real-products of spaces example, in lieu of real-number choice). The point-set aspects are totally sterile unless you understand the nuances of set theory well enough to know what they really mean, and that requires learning countable models of set theory, logic, forcing, so that you can answer the point-set questions three ways "yes, no, it depends on the set-theory model", because the most natural questions in point-set topology are set-theoretic questions about uncountable sets, and so have "it depends on the set-theory model" as the best answer.

But still, learning point-set topology is unfortunately necessary. But I personally found it impossible to read point-set topology books after a certain point. It only possible again after learning set theory and formal logic well enough to properly make sense of the set-theoretic nonsense.

By contrast, the ideas of homology are straightforward, and extremely deep, and it all comes from Poincare thinking about simplicies. There is no simpler introduction pedagogically than simplicial complexes, there is no better introduction because these are combinatorial objects, and the combinatorial aspects of simplices are studied today (for example, google shellability, or something like this). These are extremely important extremely active mathematics, and you can't learn homology without learning the classic simplicial homology.

The spectra and derived functors are relatively easy to internalize if you already understand Poincare's idea, but it's not true the other way around. So I found that learning the ideas of actual topology, meaning homology, is best pedagogically using books from the 1960s at the latest, anything later makes it obfuscated for a beginner.

Once you learn homology, then you can easily internalize what categories are for, but it's hard the other way, so I can't imagine learning the way you suggest, it's basically humanly impossible. Humans have to retrace history, more or less, to learn properly.

+ 2 like - 0 dislike

Just to present my own experience: I recently sit in an algebraic topology class, the main subjects are fundamental group & covering space, and homology theory. For the fundamental group & covering space part, I feel many textbooks exposit them well(e.g. John Lee); while for the homology theory part, our recommended textbook was Allan Hatcher's renowned textbook, but I find it very hard to follow, and the long winded paragraphs of informal discussions are sometimes just impossible for me to figure out what the core content is. Later I switched to G.Bredon's GTM textbook Topology and Geometry, I feel it is much easier to follow and the content is no less than Hather's.

But as a word of caution, as you are probably already awared, this is just my personal biased experience, and experience differs from person to person. From some informal statistics I suspect Hatcher's book will win higher votes among math guys, and I think it is probably true that the "long winded paragraphs of informal discussions" in Hatcher contain many gems. The problem with me is that, as a physicist without too much appreciation of abstact mathematics(I tried, but failed to appreciate, and not proud of it), I learn subjects like homology because it is going to be useful, constantly doing things like "diagram chasing" just doesn't entertain me, so when I read I tend to make notes about and sometimes even copy down the theorems and proofs, just to prevent my brain from shutting down. This usually doesn't happen when I read a good physics text, where the content just resonates with me and I can think about and remember it even without writing. So probably for me Bredon's book wins simply because he lists the core theorems/lemmas/corollaries in a much more organized manner.

answered Apr 18, 2014 by Jia Yiyang (2,640 points) [ revision history ]
edited Apr 18, 2014 by Jia Yiyang

I am going to disagree with the negative assesment of Hatcher. I would say that Hatcher moves too quickly from easy material to hard material, and could be considered "disorganzed" (each chapter contains a review before a proper treatment). But as a physicist, I rather liked being able to work thorugh the examples, and I feel like that I have a good operational understanding of homology because of his concrete approach. I would focus only on the simple sections in Hatcher if you find the later ones confusing - a very careful study of these will teach you a great deal about algebraic topology.

+ 1 like - 0 dislike

The more advanced topics have been discussed in existing answers, so I'll talk about introductory/general topology instead.

I actually disagree with Ron Maimon's assessment of point-set topology in the comments -- the "analytic" language of point-set topology is natural, it comes from the fact that topology naturally arises as a generalisation the analytic idea of a limit. Specifically, the limit is the "underyling structure" of a topological space -- a map is a homomorphism (note the spelling) of topological spaces, i.e. a continuous map, if it preserves all limits. So for physicists at least, a continuous map is just one that commutes with the limit operation:

$$f\left(\lim_{x\to a}x\right)=\lim_{x\to a}f(x)$$

This is a perfectly sensible "structure" to talk about. It's also perfectly sensible to stop here, but mathematicians like to abstract and generalise things -- the natural way to generalise limits to general spaces is from the notion of a "filter", due to Bourbaki, which leads to the axiomatisation of topology in terms of neighbourhoods. The open set axiomatisation then arises as something to prove to be equivalent to the stuff with neighbourhoods (and the intuition for why this makes sense is that openness and closedness are fundamentally connected to the idea of a limit in a more general setting than sequences -- perhaps a good intermediate axiomatisation to understand this is Kurwatzki's closure, which is precisely the notion of such a generalised limit operator).

As for resources, I wouldn't be too focused on trying to find the best book on the subject, because (1) it's really your introspection about the subject that helps you learn stuff (2) literally everyone in the world has worked with topology and you shouldn't find it too hard to find some content to learn from.

But there are some important insights you should make sure you acquire, and the series of articles I'm writing here are optimal for that: https://thewindingnumber.blogspot.com/p/topology.html. Not everything is done yet (particularly the algebraic topology stuff), but it's at least somewhere to start.

answered Nov 16, 2019 by dimension10 (1,985 points) [ no revision ]

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