Research level algebraic topology

I became very interested in Algebraic Topology more or less recently, and learnt a lot of "classical algebraic topology", including :

• Hatchers' Book, and more categorical approaches here and there
• Characteristic classes (what is covered by Milnor)
• Basic facts about Model Categories (Definitions, examples of model structures on Top, Simplicial Sets and Chain Complexes..)

So I would like to move on to subjects that would please me as much as these, with the will to approach research-level in a particular subject of modern algebraic topology. However, as I'm on my own, I don't really know where to head. I've heard about Operads, and tried to start reading "Algebraic Operads" but it was not much contextualized and does not really focus on the "applications" to topology.

Hence, what are some research-level/active topics I could try to start and learn ? And what are their purpose/motives ? Operads ? Higher Algebra (following Lurie ?) ? or turn to Algebraic Geometry ? Is there a survey of modern algebraic topology which is an informal introduction to the various domains it is compounded of ?

Solution 1:

You're in luck! A new Handbook of Homotopy Theory is currently being edited by Haynes Miller. I do not believe there is a definitive list of papers that will be included in the final version, but I've seen many of them pop up on arXiv. Moreover, the nLab has compiled a list (I do not know how complete it is). In alphabetical order:

• Gregory Arone, Michael Ching, "Goodwillie Calculus", (arXiv:1902.00803)
• David Ayala, John Francis, "A factorization homology primer", (arXiv: 1903.10961)
• Paul Balmer, "A guide to tensor-triangulated classification" (pdf)
• Tobias Barthel, Agnès Beaudry, "Chromatic structures in stable homotopy theory" (arXiv:1901.09004)
• Mark Behrens, "Topological modular and automorphic forms" (arXiv:1901.07990)
• Julia Bergner, "A survey of models for $(\infty,n)$-categories" (arXiv:1810.10052)
• Benoit Fresse, "Little discs operads, graph complexes and Grothendieck–Teichmüller groups" (arXiv:1811.12536)
• Daniel Isaksen, Paul Arne Østvær, "Motivic stable homotopy groups" (arXiv:1811.05729)
• Wolfgang Lueck, "Assembly Maps" (arXiv:1805.00226)
• Nathaniel Stapleton, "Lubin–Tate theory, character theory, and power operations" (arXiv:1810.12339)
• Kirsten Wickelgren, Ben Williams, "Unstable Motivic Homotopy Theory", (arXiv: 1902.08857)

Judging by your interests (operads and higher categories), I would recommend the papers of Fresse, Bergner, and Ayala–Francis among these. I haven't read all the papers, but the ones I've read were all very interesting and are definitely are the forefront of modern algebraic topology.

If you are really interested in operads, then the book of Loday–Vallette that you mention is a really good starting point, though I will agree that it mainly focuses on the algebraic part (this comes as no shock given the title). If you want to know more about topological applications, then I can recommend Fresse's recent book Homotopy of Operads and Grothendieck–Teichmüller Groups. This mainly focuses on the little $2$-disks operad and its homotopy automorphisms, but there is a good deal of background, especially in the first chapters. If you prefer listening, there's also Willwacher's talk at the 2018 International Congress of Mathematicians.

As for higher categories, Lurie's books (Higher Topos Theory and Higher Algebra) are pretty much standard references now, if you want quasi-categories. Be prepared for a long ride. There are also other, more introductory references available. Be also aware that there are other models for higher categories, and Bergner's paper (see above) is an overview of the different models and their relationships. She also wrote a very nice book (The Homotopy Theory of $(\infty,1)$-categories) on the subject.

I notice that you are also interested in algebraic geometry. This is a bit farther away from my area of expertise, but there are some very interesting connections between algebraic geometry and homotopy theory, in the form of "derived algebraic geometry". Among the foundational references, you have the two papers of Toën–Vezzosi, Homotopical algebraic geometry I and II; and Lurie's series of DAG (Derived Algebraic Geometry) papers. Lurie has written a few words on the prerequisites for the latter. There is an MO question "Derived algebraic geometry: how to reach research level math?" if you want more info.

A word of warning is in order, though. The papers/books above are not courses like Hatcher's book is, for example. They're at the bleeding edge of research, and the difficulty level is proportionate.

I have not covered everything, obviously. But the papers/books will cite references that can be used as a starting point to delve deeper.