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Disclaimer: I was really uncertain about posting this question, because it is quite similar to this Algebraic Topology Beyond the Basics: Any Texts Bridging The Gap?. I don't know if it would be best to put my question as a comment on Algebraic Topology Beyond the Basics: Any Texts Bridging The Gap? or rather create this new question. However, I thought that pheraps it brings to new interesting answers.

Question: I am a PhD student in algebraic topology, and I learned the basics of the subject from some of the many valuable texts suitable for a first course. Now, I am searching for more advanced books which I can keep as a reference for more advanced topics that I might encounter in the future.

To be more precise: suppose we divide the literature on algebraic topology in the following categories:

  1. Standard books: they contain the basics of algebraic topology (homology, cohomology, homotopy theory) and are usually used in a first class (e.g. the books by Hatcher, Tammo tom Dieck, Massey)
  2. Non standard books: those with a different approach to the subject and that contain some more advanced topics, for example Algebraic topology from an homotopical point of view by Aguilar, Gitler, Prieto (https://www.amazon.com/dp/1441930051).
  3. Advanced books: those books that they contain more advanced topics but they are not monographs, for example: Differential forms in algebraic topology by Bott and Tu (https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf),
    Homotopical topology by Fomenko and Fuchs (https://link.springer.com/book/10.1007/978-3-319-23488-5), A concise course in algebraic topology by May (https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf), From categories to homotopy theory by Richter (https://www.cambridge.org/core/books/from-categories-to-homotopy-theory/A109E2C4B720337DE19A15EB4FA8C9A6). or Generalized cohomology by Kono and Tamaki https://books.google.it/books/about/Generalized_Cohomology.html?id=3HY4ruJ6BigC&redir_esc=y)
  4. Monographs: those big books where you can find the details of a specific theory (e.g. K-theory by Atiyah, Operads in Algebra, Topology and Physics by Markl, Shnider, and Stasheff, Rational Homotopy Theory by Felix, Halperin, Thomas)

The books I am looking for are those in the second and third category. So, books that approach algebraic topology from an unusual point of view and/or contains a bunch of more advanced topics.

Edit: I know that the concept of "advanced book" is quite subjective. So, let me try to be more precise as I can: suppose you have learned all the stuff contained in Hatcher's book and now you want to learn something about K-theory. You can either study the book of Atiyah (which is a monograph) or you can take a look for example at the book of May which gives you an overview of the theory that you can pick as a starting point.

So as an advanced book I mean a book that contains some topics that you usually do not learn in a basic algebraic topology class (e.g. spectral sequences, generalized cohomology theories, operads) but at the same time it is not a monography. For example the book of Aguilar, Gitler, Prieto contains some chapters about K-theory and other generalized cohomologies, the book of Fomenko-Fuchs contains something on spectral sequences.

So maybe a more fair question could be: what books would you recommend to a graduate/PhD student in order to get the flavour of some more advanced topic (but still classical) without going in all the details of the theory? Are there some big books that contain a lot of stuff and that one can consult everytime he do not know a specific topic?

Other titles would be very welcome, as well as comments about the books I listed above.

Tommaso Rossi
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    May's Concise Course covers a similar selection of material to Hatcher's Algebraic Topology, why is it listed as “advanced”? Also, the material in Richter's book is perfectly standard homotopy theory, there is nothing “nonstandard” about it. – Dmitri Pavlov Dec 29 '21 at 19:45
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    Does Differential cohomology, edited by Araminta Amabel, Arun Debray, and Peter Haine, fit in what you are asking for? See https://math.berkeley.edu/~phaine/files/diffcoh.pdf – D.-C. Cisinski Dec 29 '21 at 19:51
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    Kreck's Differential Algebraic Topology is a nonstandard approach to homology and cohomology of manifolds. – Greg Friedman Dec 29 '21 at 21:39
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    I would also list Bott and Tu as standard. It's one of the most read introductory books, especially by people who come at topology from an analytic perspective. I am concerned this question comes across more as an opinion piece about what words like "advanced" mean to the author. – Ryan Budney Dec 29 '21 at 23:08
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    Related: https://mathoverflow.net/q/18041 – pinaki Dec 30 '21 at 02:45
  • @D.-C.Cisinski, thank you professor for this book I didn't know! Yes, it absolutely fit well with my question. – Tommaso Rossi Dec 30 '21 at 08:21
  • @DmitriPavlov, I put May's book as advanced because it covers also topics that one usually do not learn in a first topology class, I believe it is a great reference for an advanced student but maybe it is not the easiest book to read if you are a beginner (but of course this is an opinion). – Tommaso Rossi Dec 30 '21 at 08:32
  • @RyanBudney, I aree with the fact that this question is partially "dangerous" because what advanced means is subjective. So let me try to reformulate the question. – Tommaso Rossi Dec 30 '21 at 08:39
  • I guess Spanier is a standard textbook, and how about G.W.Whitehead or Switzer? (For me they are standard books like Bott-Tum by the way) Or something less known, Rudiyak? – user43326 Dec 30 '21 at 08:41
  • @user43326 Whitehead and Switzer books are perfectly fine for me as they cover classical topics that one usually do not learn as an undergraduate. I don't Rudiyak's book, thank you for mentioning it, I'll take a look at it! – Tommaso Rossi Dec 30 '21 at 09:58

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