Math 595 FTM, Foliations and the topology of 3-manifolds
Fall 2021
- Time and Place: MWF from 11:00-11:50am in 441 Altgeld Hall. First
half of semester only, ends Oct 15.
- Section: FTM CRN: 57739
- Instructor: Nathan Dunfield
- E-mail: nmd@illinois.edu
- Office: 378 Altgeld. Office Phone: (217) 244-3892.
- Office Hours: 3-4pm Monday and Wednesday and also other
times by appointment.
- Web page: http://dunfield.info/595A
- Lecture notes
Course Description
As with many areas of topology and geometry, the study of
3-dimensional manifolds begins with its codimension-1 submanifolds,
namely embedded surfaces. From there, it is very natural to consider
foliations of 3-manifolds by surfaces, that is, locally trivial
decompositions into (typically noncompact) surface leaves. While
every closed orientable 3-manifold has a foliation, the seemingly mild
condition of requiring the foliation to be taut restricts both the
topology of the ambient manifold (e.g. the universal cover must be
R^{3}) and of the leaves, and forces the existences of interesting
transverse dynamics.
I will start with an overview of foliations, focusing on examples, and
then discuss the basic properties of taut foliations on 3-manifolds.
Then I will then turn to the foundational work of Gabai, specifically
his tool of sutured manifold hierarchies for constructing
foliations. Next, I will discuss connections to contact geometry,
namely the work of Eliashberg-Thurston which builds from a taut
foliation a pair of tight contact structures. Finally, I will explain
how taut foliations fit into the various Floer homology theories of
3-manifolds (the Heegaard, instanton, and monopole Floer homologies).
Prerequisites: Basic knowledge of smooth manifolds and
algebraic topology, e.g. Math 518 and Math 525. No prior knowledge of
foliations or 3-manifolds will be assumed.
References
I will not follow any particular text, but recommend the
following:
- A. Candel and L. Conlon, Foliations I, Graduate Studies in
Mathematics 23, AMS, 1999.
- A. Candel and L. Conlon, Foliations II, Graduate Studies in
Mathematics 60, AMS, 2003.
- D. Calegari, Foliations
and the geometry of 3-manifolds, Oxford Univ. Press, 2007.
The most relevant parts of these books are Chapters 1-3 of Foliations
I, Chapters 8-11 of Foliations II, and Chapters 4-5 of
Calegari. General background on the topology and geometry 3-manifolds
can be found in:
Grading
Students registered for the course will need to write a short (2-4
page) paper which will be due on Friday, October 15th. This paper is
largely free-form, and can be about any subject related to the content
of this course. For instance, it could be a brief account of a result
not covered in class, a review of the some related results explaining
why they are interesting, a detailed work-out of a proof only sketched
in class, or careful solutions to problems from class or
taken from one of the texts or other readings.
Lecture notes
Notes from each lecture will be posted here.
- Aug 23. Intro and course
overview.
- Aug 25. Definitions of
foliations and contact structures. Got halfway though page 9,
also mentioned co-orientability on page 10.
- Aug 27. Examples of
foliations and contact structures. Did through page 14,
with a very brief preview of 15-16.
- Aug 30. Holonomy, gluing, and
foliating the 3-sphere. Did through page 20, and also 21 very
quickly.
- Sept 1. Reeb
stability. Covered everything.
- Sept 3. Limits of leaves and
the proof of Reeb stability. Covered everything.
- Sept 8. Foliating all
3-manifolds I. Ended halfway through page 35. See Etnyre's lecture notes
for more on the contact story.
- Sept 10. Foliating all
3-manifolds II. Did through the top half of 39; added more
detailed account of handle decompositions.
- Sept 13. Dehn surgery on
links. One reference is Chapter 9 of Rolfsen's classic Knots
and links. For Lickorish's theorem, see e.g. Section 6.5 of
Martelli. Covered everything.
- Sept 15. Incompressible
surfaces in 3-manifolds. Here is an old
lecture on the Virtual Haken Conjecture. Covered everything.
- Sept 17. Taut
foliations. Through page 54.
- Sept 20. Properties of taut
foliations. Covered everything.
- Sept 22. More on taut
foliations. Everything but the theorem at the bottom of the last
page.
- Sept 24. Universal covers of
taut foliations. Covered everything.
- Sept 27. Thurston's Universal
Circle. Did through 73.
- Sept 29. More on Thurston's Universal
Circle. Covered everything.
- Oct 1. The L-space
conjecture I. Did through 85.
- Oct 4. The L-space conjecture
II. Covered all but the theorem on the last page.
- Oct 6. The Thurston
norm. Through page 95.
- Oct 8. The Thurston
norm and foliations.r Covered everything.
- Oct 11. Essential
laminations. Covered everything.
- Oct 13. Branched
surfaces. Covered everything.
- Oct 15. Triangulations to
foliations. The end.