Question

I studied basic algebraic topology elements:

fundamental group, higher homotopy groups, fibre bundles, homology groups, cohomology groups, obstruction theory, etc.

I want to study Rational Homotopy Theory.

Specifically, I want to study Sullivan's model.

What is the short way and what is the complete way to study Sullivan's model?

Answer 1

Griffiths and Sullivan wrote a fine book on the subject. Apart from the obvious attractiveness of learning a theory from its creator, it is written in an amazingly user-friendly style. For example, Chapter XIII is devoted to examples and computations: it starts with the computation of a minimal model for the forms on a sphere and ends with Massey triple products on compact Kähler manifolds, a section inspired by the 1974 Inventiones article of Deligne,Griffiths, Morgan, Sullivan. The first hundred pages (Chapters I to VII) are an introduction to the necessary algebraic topology and you can probably essentially skip it, judging from your description of what you already know.

Reference Griffiths, P.; Morgan, J. (1981), Rational homotopy theory and differential forms, Progress in Mathematics, 16, Birkhäuser

Answer 2

This depends very much on what you want to see (Griffiths-Morgan has been mentioned, and I recommend it as well):

1. A quick introduction: Morita, "Geometry of characteristic classes", chapter 1. He also treats the non-simply connected case.

2. A broad and comprehensive treatise, with tons of examples: Felix, Halperin and Thomas, "Rational homotopy theory". If you fear spectral sequences, this is the book to use for the "complete way".

3. An inspring paper that you'll read 20 times: Sullivan, "Infinitesimal computations in topology".

4. A collection of geometric applications, starting from the historical origin (the de rham models for Lie groups and homogeneous spaces): "Algebraic models in geometry", by Felix, Oprea, Tanre.

5. Model categories in action: Gelfand-Manin, Homological algebra (the last chapter) or Kathryn Hess: "Rational homotopy theory, a brief introduction".

Answer 3

Start with Griffiths-Morgan's green book on Rational homotopy. A quick introduction is also provided in the Springer GTM by Bott and Tu. Also useful is the case of compact Kahler manifolds treated in the paper by Deligne, Griffiths, Morgan and Sullivan in Inventiones Math 1976 (available free at Digizeitschriften)

Answer 4

Here are some video lectures that John Morgan gave at Stoney Brook

http://www.math.sunysb.edu/Videos/dfest/ This also has many other nice mathematical videos.

Answer 5

After reading Griffiths-Morgan, Bott-Tu (not just the chapter on Rational Homotopy Theory, I would say) and Felix-Halperin-Thomas, maybe it wouldn't be a bad idea to be acquainted with:

1. Halperin, Lectures on minimal models, Mémoires SMF 230 -aka "the bible": all technical details you won't find elsewhere.
2. Bousfield, Gugenheim, On PL De Rham theory and rational homotopy type, Memoirs AMS 179 -the model category point of view; Sullivan's results can be stated as an equivalence of categories: find which.
3. Lehman, Théorie homotopique des formes différentielles, Asterisque 45 -if you know French, this is a very nice introduction to the subject.

Answer 6

While not comprehensive, the following book has a nice introduction to the subject in its first chapter or so: J. Oprea, A. Tralle, Symplectic manifolds with no Kähler structure, Lecture Notes in Math. 1661,. Springer–Verlag, 1997.