Foliations seminar 2016

Organizers: Sander Kupers, Kathryn Mann, Bena Tshishiku.

Reference for the first five weeks topics: Candel and Conlon, "Foliations I"

Lecture notes: Notes taken by Bena Tshishiku can be found on his webpage here

Part I:

• Jan. 29: introduction (Bena) and organizational meeting, 1pm in 384I
  


• Feb. 5: Holonomy and Reeb stability (Weston)
  


• Feb. 12: Haefliger's theorem on compact leaves for foliations of codimension 1 (Ben)
  


• Feb. 19: No seminar
  


• Feb. 26: Thurston stability (Katie)
  


• March 4: General position and Poincare-Bendixson theory (Laura)
  


• March 11: No seminar this week
  

Part II

We'll read a recent paper of H. Eynard-Bontemps on the connectedness of the space of (smooth, transversely oriented) codimension 1 foliations on a closed 3-manifolds.

I encourage everyone to read the short and extremely friendly expository version available here
The full paper is here: arxiv version, or -- better published (invent. math.) version.

Other references are listed below week-by-week.
Proposed weekly schedule

• Lecture 0. (March 18)   notes available
  Introduction to the main result and flavor of the proof.
  Examples of constructions and deformations of foliations, and an existence proof (theorem of Novikov, Zieshcang, Lickorish): Every 3-manifold admits a codimension 1 foliation.
  


• Lecture 1 (April 1) Proof of Thurston's theorem: every plane field on a 3-manifold M is homotopic to the tangent plane field of a foliation. (Weston)
  References: Cantwell-Conlon Foliations II (the sequel to our book) Chapter 8 section 5.
  There is a sketch/outline in Eynard-Bontemps' paper (similar in the research paper and the expository version) Another sketch (maybe not so helpful?) is given in Calegari's Foliations book (Example 4.17, book is available here.)
  


• Lecture 2: (April 8) Deformations of plane fields. (Cedric)
  Sketch of some techniques for Eliashberg's deformations of overtwisted contact structures, as applicable to the foliations case (only feasible if someone is familiar with some aspects of this result).
  The technical work for foliations is done in the appendix of Eynard-Bontemps' paper, see references there to results from contact geometry.


• Lecture 3: (April 15) Foliating tori in families. (Bena)
  Given a foliation of the boundary of D^2 x S^1 transverse to the S^1 direction, can one extend it to the interior? Can one do so in a canonical or continuous way?
  The answer is yes, by a construction of Schweitzer and Laranche.
  References: Seciton 2 in the expository version. Section 2 in the paper also. If time, one could say a little extra about foliated bundles.
  


• Lecture 4: (April 22) Proof of main theorem for taut foliations (Sander)
  Reference: section 4 of expository paper.
  This would be a nice time to recap the steps in the proof.
  


• Lecture 5: Diffeomorphisms of the interval. (Katie)
  Kopell's lemma and Szekeres' theorem on the space of commuting diffeomorphisms of the interval (and how a stronger version of these finishes the main proof)
  References: Section 4.1 of "Groups of circle diffeomorphisms" by A. Navas, for the proof of Kopell and Szekeres. (arxiv link)
  Section 5 of the expository paper for an outline of the end of Eynard-Bontemps' proof.
  

Part III?

Future topics, depending on interest: Sander is interested in discussing h-principles for foliations and relationships to diffeomorphism groups. Here is a detailed list of further references. E-mail Sander if you are interested in a particular one.