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MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up some background on homotopy theory. Does the analogue of Hartshorne exist? Are there any must-reads for stable homotopy theory and spectra? What would your advice for a beginning graduate student be? Just to set a starting point, I would ask for suggestions for someone who's familiar with a good chunk of most concepts covered in Hatcher and some differential geometry, but not much more.

I guess in the back of my mind is trying to understand some of this "brave new algebraic geometry", ie geometry over an $E_\infty$-ring spectrum.

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  • $\begingroup$ What's your background on homotopy theory? $\endgroup$
    – Fernando Muro
    Nov 15, 2013 at 21:39
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    $\begingroup$ If you know Hatcher, I'd go for Switzer's book. It's old, but It gets pretty far. Much has been done later concerning applications of new tools, such as monoidal structures on model categories of spectra, but you'll probably be able to pick much of the new stuff if you get familiar with Switzer's. $\endgroup$
    – Fernando Muro
    Nov 16, 2013 at 12:05
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    $\begingroup$ Adams blue book is pretty good, except is treatment of spectra is pretty old fashioned. $\endgroup$
    – Sean Tilson
    Nov 16, 2013 at 19:35

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There have been several questions previously in this vein. This one asks for an advanced beginners book. 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).

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.

Since you mention that you're especially interested in $E_\infty$ ring spectra, let me also point out Peter May's survey article What precisely are $E_\infty$ ring spaces and $E_\infty$ ring spectra?

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    $\begingroup$ looks like I haven't done my homework properly... thanks for the huge compilation! $\endgroup$
    – John Salvatierrez
    Nov 16, 2013 at 0:42
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    $\begingroup$ David, thanks very much for the nice compliments, but I'd like to emphasize that ``More Concise'' is joint with Kate Ponto. $\endgroup$
    – Peter May
    Nov 16, 2013 at 4:18
  • $\begingroup$ @PeterMay: Thanks for your comment. I've edited the answer so it includes Kate Ponto. $\endgroup$
    – David White
    Nov 16, 2013 at 5:04
  • $\begingroup$ Hey David, I think you mean mathoverflow.net/questions/18041/… for the second paragraph, right? $\endgroup$
    – Drew Heard
    Nov 16, 2013 at 16:16
  • $\begingroup$ @DrewHeard. Yes, thanks. That was a silly mistake. Too many links! By the way, it appears you're in Minnesota this semester after all. Maybe we'll see each other at the Joint Meetings in January? $\endgroup$
    – David White
    Nov 16, 2013 at 17:34

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