# 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 R3) 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:

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:

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.

1. Aug 23. Intro and course overview.
2. Aug 25. Definitions of foliations and contact structures. Got halfway though page 9, also mentioned co-orientability on page 10.
3. Aug 27. Examples of foliations and contact structures. Did through page 14, with a very brief preview of 15-16.
4. Aug 30. Holonomy, gluing, and foliating the 3-sphere. Did through page 20, and also 21 very quickly.
5. Sept 1. Reeb stability. Covered everything.
6. Sept 3. Limits of leaves and the proof of Reeb stability. Covered everything.
7. Sept 8. Foliating all 3-manifolds I. Ended halfway through page 35. See Etnyre's lecture notes for more on the contact story.
8. Sept 10. Foliating all 3-manifolds II. Did through the top half of 39; added more detailed account of handle decompositions.
9. 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.
10. Sept 15. Incompressible surfaces in 3-manifolds. Here is an old lecture on the Virtual Haken Conjecture. Covered everything.
11. Sept 17. Taut foliations. Through page 54.
12. Sept 20. Properties of taut foliations. Covered everything.
13. Sept 22. More on taut foliations. Everything but the theorem at the bottom of the last page.
14. Sept 24. Universal covers of taut foliations. Covered everything.
15. Sept 27. Thurston's Universal Circle. Did through 73.
16. Sept 29. More on Thurston's Universal Circle. Covered everything.
17. Oct 1. The L-space conjecture I. Did through 85.
18. Oct 4. The L-space conjecture II. Covered all but the theorem on the last page.
19. Oct 6. The Thurston norm. Through page 95.
20. Oct 8. The Thurston norm and foliations.r Covered everything.
21. Oct 11. Essential laminations. Covered everything.
22. Oct 13. Branched surfaces. Covered everything.
23. Oct 15. Triangulations to foliations. The end.