# Why not a Roadmap for Homotopy Theory and Spectra?

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.

• What's your background on homotopy theory? – Fernando Muro Nov 15 '13 at 21:39
• 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. – Fernando Muro Nov 16 '13 at 12:05
• Adams blue book is pretty good, except is treatment of spectra is pretty old fashioned. – Sean Tilson Nov 16 '13 at 19:35

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?
• David, thanks very much for the nice compliments, but I'd like to emphasize that More Concise'' is joint with Kate Ponto. – Peter May Nov 16 '13 at 4:18