Home page for the course "Seminar topology and geometry"

This talk will be given by the organisers.

Show some examples of foliations, in particular the Reeb foliation of the solid torus and construct a foliation on the 3-sphere. Describe the operation of turbulization (or spinning). Give a sketch of the proof that every closed orientable 3-manifold has a foliation. For the last part some familiarity with 3-manifold theory could be useful. 

References: 

• Examples can be found in Section 1, Chapter 2 in [1] and Section 1, Chapter 1 in [2]. The Reeb foliation is described in Example 3 of Section 1, Chapter 2 in [1].

• The operation of turbulization can be found in Note (4) to Chapter 2 in [1]. The operation of spinning is described in 3.3.B in Section 3, Chapter 3 of [2]. 

• A very quick sketch of the fact that every 3-manifold has a foliation is given in Note (5) to Chapter 2 in [1].  A more detailed version is given in Example 4.11 of [4] and an even more detailed version can be found in Section 1, Chapter 8 in [3].

Define Reebless foliations, state Novikov and Rosenberg theorems and use it to show that the 3-sphere has no Reebless foliations. Define taut foliations and observe that a taut foliation is Reebless. Prove that a foliation is taut if and only if it has no dead end components  if and only if there exist a flow transverse to it that preserves some volume form on the 3-manifold.

References:

• Most of the material is covered in Chapter 4, Section 4 of [4].

• Novikov and Rosenberg theorems are stated in Theorem 4.35 of [4].

Introduce minimal sets with some examples and show some properties and existence results. Describe the construction of a smooth foliation of a 3-manifold with an exceptional set due to Sacksteder and compare it to the 2-dimensional torus case.

References:

• Minimal sets are defined and studied in Section 4, Chapter 3 in [1].

• The work of Denjoy on minimal sets of foliations on the 2-dimensional torus is discussed in Note (2) to Chapter 3 in [1] and Sacksteder's construction is described in Section 2, Chapter 5 in [1].

Define holonomy and prove the Global Stability theorem in the transversely orientable case. Stick to the case of foliations of 3-manifolds with simply connected leaves.  Deduce that an orientable 3-manifold with such a foliation must be diffeomorphic to S^2 x S^1. If there is time, state Sacksteder's theorem.

References:

• Holonomy is defined in Chapter 4 of [1] and the Global Stability theorem is proved in Section 5 of the same chapter.

• The statement of Sacksteder's theorem is given in Note (2) to Chapter 5.

Define coorientable contact structures, contact forms, Reeb vector fields [8, section 1.1]; give examples of contact structures to build some visual intuition (for example on R^3, T^3, S^3), see section 1.2 in [7]. Introduce notions of equivalence of contact structures (contactomorphisms, homotopies of contact structures, contact isotopies). Conclude the class by proving Gray's stability theorem which says that two contact structures are homotopic iff they are isotopic.  For this we suggest you follow the more geometric 3-dimensional proof in [7, section 1.3.1] as opposed to the more general one that works in all dimensions [8, chapter 2.2].  For this you will need Cartan's magic formula for Lie derivatives of time dependent 1-forms along the flow of time dependent vector fields, a proof of which can be found in [8, Appendix B].  A full proof of this fact is not necessary, however you should at least give a plausibility argument. Finally, contrast Gray stability with the lack of such a stability statement for foliations.

Prove the Darboux theorem for contact structures, either following [7, Theorem 2] or [8, Theorem 2.5.1] (the first again being more in line with the 3-dimensional intuition we want to build). Introduce characteristic foliations ([7, section 2.1 and part of section 2.2] as well as [8, section 4.6.1]) and prove the local and global reconstruction Lemma which roughly speaking tell you that contact structures are (locally) determined by the way the intersect surfaces in the manifold. Prove that analogous statements are wrong for foliations.

Define overtwisted contact structures and tightness (use section 4.5 in [8] as a main source, maybe sections 4.5 in [9] and end of chapter 1 in [7] help as well); briefly expand on significance of the notion of overtwistedness and tightness ([8, section 4.7]) compare overtwisted disks with a cross section of a Reeb component; introduce lutz twists (section 4.3 in [8]),  compare with turbulizations, and use it to prove that every three manifold has a contact structure ([8,section 4.1]).

Define symplectic manifolds and describe their relation to contact structures (e.g. strong fillability). Define (weak) semi-fillability for contact structures and foliations and state Theorem 3.2.4 in [5]. Prove that taut foliations have closed dominating two-forms and hence are semi-fillable.

References:

• Symplectic manifolds and strong fillability can be found in [8, chapter 5] note that the various notions of fillability mentioned in [8] and [5] don't exactly agree (but this is not a real problem,  just be aware of the slightly different definitions)

• (Weak) semi-fillability for contact structures and foliations is discussed in Section 2, Chapter 3 of [5].  The proof that taut foliations admit closed dominating two-forms and hence are semi-fillable is also contained in the same Section.