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I am a student of mathematics, and have some background in

  • Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff),
  • Differential Geometry (Lee, Kobayashi-Nomizu),
  • Riemannian Geometry (Do Carmo),
  • Symplectic Geometry (Ana Cannas da Silva) and
  • Differential Topology (Hirsch, Minor (Morse Theory), Milnor (h-cobordism theorem)).

I would like to learn Floer homology and Khovanov homology.

Q1. Is my background sufficient to learn these topics right now? What are the standard first-level and second-level sources for these topics?

Q2. At the research level, will I need to know any other areas to work on these topics? In particular, would I need to know algebraic geometry and to what extent? To give you an idea of my present knowledge, I currently know absolutely no algebraic geometry, and even my commutative algebra background has several gaps, especially in parts about DVRs.

Q3. Once I finish books recommended in Q1, what are some good papers to start reading on these areas?

I would greatly appreciate reasoning behind any comments that aren't strictly factual and are opinions of the writer.

Thank you in advance! :)

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    $\begingroup$ I am not sure if Floer homology or Khovanov homology are really under the "algebraic topology" umbrella. At least I've almost never seen them discussed in algebraic topology seminars. $\endgroup$ – Denis Nardin Jun 19 '19 at 15:30
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    $\begingroup$ @DenisNardin You should come to Paris 7 more often then :) Knot theory is part of topology, and Khovanov homology is an algebraic invariant, in any case. $\endgroup$ – Najib Idrissi Jun 19 '19 at 16:25
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    $\begingroup$ @AsuraPath This is farther away from what I know so I can't really say. I know some people prefer calling the field "symplectic topology" rather than "symplectic geometry". $\endgroup$ – Najib Idrissi Jun 19 '19 at 16:31
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    $\begingroup$ @Najib Oh I know the division is fickle and arbitrary :) I just have the impression they are usually lumped with differential topology or geometric topology instead $\endgroup$ – Denis Nardin Jun 19 '19 at 16:41
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    $\begingroup$ But those aren't topics in algebraic topology. People working on those topics are some combination of geometric topologists, geometric representation theorists, and symplectic geometers. $\endgroup$ – Noah Snyder Jun 19 '19 at 18:54
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I'm not an expert in Floer homology or Khovanov homology, but if that's your goal I don't think you need quite as wide a background as suggested in the other current answer (though admittedly that answer was written when the title of the question was much broader). For example, in the talks I've seen about these things, I don't think much in the way of spectra comes up, or even homotopy theory at the level of May. My more modest suggestion is first to acquaint yourself with some more classical knot theory. Rolfsen is a very good start for the really classical stuff, while there are now many places to learn about the more modern skein-type invariants. My personal recommendation would probably be Lickorish, but there are several good books and surveys now about that material. After that, some symplectic topology might be useful, though you say you've already got some of that. Beyond that background, I think that this is an area that does not yet have many secondary textbook sources, but there are very many survey articles and I think the primary material tends to be written fairly well compared to some other areas of mathematics. Just Googling "Introduction to Khovanov homology" and "Introduction to Floer homology" brings up a lot of surveys and lecture notes. I would suggest diving into that stuff and then referring outward once you hit something specific that you don't understand or want more background about. Based on what you wrote in your question, I think your broad topological background is already in pretty good shape to get started.

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There have been several questions previously in this vein, but yours is more general. My present answer is adapted from an answer to a question asking for a "Road Map" to Homotopy Theory. Your question is a bit different, so I'll write some different things. First, we need to define what we mean by "Algebraic Topology" so I'll take the subfields listed on the wikipedia article. I think the books you have suggested for manifolds (smooth, Riemannian, etc) are already sufficient to get a working grasp.

In general, you have a good list of "first level" sources. The background from the linked answer will give you great second level sources (Q1), plus papers (Q3), for homotopy theory:

  • Here is a question asks for an advanced beginners book (and here is another, that was closed as a duplicate). The consensus seemed to be that it was difficult to find a one-size-fits-all text because people come in with such diverse backgrounds. Peter May's textbook A Concise Course in Algebraic Topology is probably the closest thing we've got. If you like that, then you can also read More concise algebraic topology by May and Ponto. I also recommend Davis and Kirk's Lecture Notes in Algebraic Topology. I think these would be a very reasonable place for a beginning grad student to start (assuming they'd already studied Allen Hatcher's book or something equivalent). I'll add that nice books for simplicial things include Curtis, and Goerss-Jardine.

  • Another question asked for textbooks bridging the gap and got similar answers. Finally, there was a more specific question about a modern source for spectra and this has a host of useful answers. Again, Peter May and coauthors have written quite a bit on the subject, notably EKMM for S-modules, Mandell-May for Orthogonal Spectra, and MMSS for diagram spectra in general. Another great reference is Hovey-Shipley-Smith Symmetric Spectra. On the more modern side, there's Stefan Schwede's Symmetric Spectra Book Project. All these references contain phrasing in terms of model categories, which seem indispensible to modern homotopy theory. Good references are Hovey's book and Hirschhorn's book.

Now for Q1 and Q3 for knot theory. A book that was standard reading for graduate students at Wesleyan learning knot theory is here. If the start is a bit rough, I can highly recommend this book for an exposition aimed at undergraduates. After those books, Here is a question that gave recommendations for papers to learn Khovanov homology.

As for Q2, if you take the knot theory route, you'll need to know about the Alexander Polynomial and the Jones Polynomial. A good book to start with would be Dummit and Foote. For any of the above subfields of algebraic topology, it would be good to know some commutative algebra, e.g. Matsumura or (the classic, but harder) Atiyah-MacDonald. For both fields, you also need homological algebra, and a great book would be Hilton-Stammbach. A second level book, more suited for homotopy theory, would be Weibel.

As for algebraic geometry, I have not seen much used in knot theory. If you go the homotopy theory route, you will need to know about sheaves, and eventually about schemes and stacks. A reasonable book would be Hartshorne (but only after the algebraic background above). In my opinion there's no reason to rush into trying to teach yourself algebraic geometry, but if you can take classes on it, do. Much of the algebraic geometry needed for homotopy theory has been reformulated beautifully by Jacob Lurie over the last 10 years, and his writings are also great if you intend to do algebraic topology (after learning homological algebra and learning about simplicial things from the first block of links above).

Lastly, there is the relatively new field of topological data analysis, and applied algebraic topology. Many excellent sources to learn in that field (from the basics all the way up, including what's needed from homological algebra) are at Peter Bubenik's page. In particular, I highly recommend Ghrist's book, and the surveys Bubenik links to.

Good luck!

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  • $\begingroup$ Incidentally, one of the reasons I included Matsumura is because that's where I learned about DVRs, and the OP mentioned feeling a gap for DVRs. $\endgroup$ – David White Jun 19 '19 at 18:14
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    $\begingroup$ Someone has just edited the title. It used to say "Background needed to understand modern research in algebraic topology". My answer is to the old title of the question, but includes an answer about knot homology theory. Also, I'll add a recent book: cgp.ibs.re.kr/~yongoh/all.pdf $\endgroup$ – David White Jun 19 '19 at 18:33
  • $\begingroup$ Thanks a lot for the recommendations and the roadmap. Just one particular question regarding the immediate next step: will covering May's first book be sufficient to give me the categorical background I need, or will a book (Riehl/ MacLane) on category theory itself be necessary as well? $\endgroup$ – user142103 Jun 19 '19 at 19:25
  • $\begingroup$ I think Riehl's books are a good addition (both the undergrad category theory one and the categorical homotopy theory as a 2nd/3rd level) $\endgroup$ – David White Jun 20 '19 at 21:44

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