M 392C: Lagrangian Floer Homology (Spring 2014)

Instructor: James Pascaleff

Email: jpascaleff@math.utexas.edu
Office: RLM 11.166
Office hours: By appointment (send me an email).

Lecture

TTh 2:00-3:30 pm in RLM 12.166.

The purpose of this course is to introduce Lagrangian Floer homology, one of the main tools of current research in symplectic geometry and topology. Lagrangian Floer homology is an intersection theory for Lagrangian (= maximal isotropic) submanifolds in a symplectic manifold. Whereas ordinary intersection theory measures properties of the intersection that are unchanged by continuous deformation, Lagrangian Floer homology measures properties that are "symplectically essential," in the sense that they are unchanged by exact or Hamiltonian deformation. For example, this leads to a proof of the Arnol'd conjecture on periodic orbits of time-dependent Hamiltonian flows.

The first part of the course will provide the groundwork for the invariant. This means understanding pseudo-holomorphic curves with Lagrangian boundary conditions. Although this requires some technical analytic material, I will do my best to make the course accesible to students with a background in basic differential topology.

Besides the direct applications to symplectic topology, Lagrangian Floer homology appears in other contexts. I hope to explore some subset of these once the groundwork is done. Which ones we do will be influenced by the students' interests.

• Lagrangian Floer homology can be used as a source of invariants in low-dimensional topology. The most popular seems to be Heegaard Floer Homology and its variants, based on a symplectic interpretation of Seiberg-Witten theory. There are also symplectic versions of Khovanov homology (Seidel-Smith) and instanton homology (Atiyah-Floer). In every case we cook up some symplectic manifold and some Lagrangian submanifolds and take their Lagrangian Floer homology to produce the invariant.
• There is a larger algebraic structure called the Fukaya category (introduced in a simpler form by Donaldson), which uses Lagrangian Floer homology for its morphism spaces. This category is of particular interest because of its role in the Homological Mirror Symmetry conjecture of Kontsevich. In that context, the Fukaya category is the counterpart of (a certain category of) sheaves on the mirror dual space, and Lagrangian Floer homology is the counterpart of sheaf cohomology.

I will be posting some form of lecture notes here as the course progresses. The notes are arranged by topics rather than individual lectures, and may be revised from time to time.

1. Outline of the course.
2. Symplectic linear algebra.
3. Symplectic manifolds.
4. Homework assignment 1.
5. Almost complex structures and pseudo-holomorphic curves.
6. Morse theory.
7. Compactness in Morse theory.
8. Action functional and calculus of variations.
9. Outline of the construction of Floer homology.
10. Linearization and basic elliptic theory.
11. Fredholm differential topology.
12. A transversality theorem.
13. Index theorems.
14. Gromov compactness.
15. Invariance properties, case of cotangent bundle.
16. Arnold conjecture.
17. Oh spectral sequence (incomplete notes).
18. Grading.
19. Dehn twists.
20. Products and the Fukaya category.

I will be updating this list of references as the course progresses. Here are some things to get us started. If you find another paper or book that is useful to you, let me know and I can add it.

Symplectic geometry and topology

Two canonical textbooks are

• A. Cannas da Silva, Lectures on Symplectic Geometry. Lecture Notes in Mathematics, 1764. Springer-Verlag, Berlin, 2001.
• D. McDuff and D. Salamon, Introduction to Symplectic Topology. Second edition. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998.

Morse theory

• J. Milnor, Morse Theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963.
• J. Milnor, Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow Princeton University Press, Princeton, N.J. 1965.
• M. Schwarz, Morse Homology. Progress in Mathematics, 111. Birkhäuser Verlag, Basel, 1993.

Pseudo-holomorphic curves

• D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology.

Lagrangian Floer homology

Here we will work from Floer's original papers. The following four papers contain the essential ideas of Lagrangian Floer homology with characteristic two coefficients. The papers are all interrelated, but I recommend starting with "Witten's complex…" Sections 1–3. If you have trouble obtaining these papers, let me know.

• A. Floer, A relative Morse index for the symplectic action. Comm. Pure Appl. Math. 41 (1988), no. 4, 393–407.
• A. Floer, The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813.
• A. Floer, Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.
• A. Floer, Witten's complex and infinite-dimensional Morse theory. J. Differential Geom. 30 (1989), no. 1, 207–221.

Fukaya categories

Fukaya–Oh–Ohta–Ono provides the foundations in a very general situation, but unfortunately we probably won't be able to delve into that formalism in this course. Seidel's book gives a lucid treatment of the exact case, and is highly recommended.

• K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. AMS/IP Studies in Advanced Mathematics, 46. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.
• P. Seidel, Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.
• D. Auroux, A beginner's introduction to Fukaya categories. arXiv:1301.7056
• I. Smith, A symplectic prolegomenon. arXiv:1401.0269