Math 607: Floer Homology

Robert Lipshitz, Winter 2017.

Basic course information

Time: MWF 2:00–2:50 p.m.
Place: 205 Deady Hall.
Textbook: None; see below for a list of papers, books, and resources.
Office hours: By appointment.
Final exam: None.

Prerequisites

This course is intended for mathematics Ph.D. students (who have passed their qualifying exams). All other students should consult with the instructor before enrolling.

The course will assume a basic understanding of smooth manifolds (smooth maps, derivatives, differential forms) and algebraic topology (homology, cohomology).

Description and goals

Since its introduction in the late 1980s, Floer homology has become one of the most important tools in symplectic and low-dimensional topology. This course will focus on one version of Floer homology: Lagrangian intersection Floer homology. The first third of the course will give a rapid introduction to symplectic topology, the definition of Lagrangian intersection Floer homology, and a sketch of its technical underpinnings. The middle third is a survey of some of Floer homology's triumphs in symplectic geometry. The final third will focus on Ozsváth-Szabó's Heegaard Floer homology, one way of using Lagrangian intersection Floer homology to prove results in low-dimensional topology, including detecting exotic smooth structures on 4-manifolds.

Policies

Grading

Homework 100%

 

Exercises will be mentioned in class from time to time. Every enrolled student must solve and turn in one exercise by the middle of the course and a second exercise by the last day of class.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Homework

Here are a few suggested problems. All are intended to be solvable, though some are harder than others. The textbooks [MS98], [daS01], and [AD14] also have lots of good problems about the material they cover. Pascaleff's lecture notes [Pas] also have lots of good problems, and are closer to the topics we will cover in the first third of the course.

Homework 1. Homework 2. Homework 3. Homework 4. Homework 5. Homework 6. Homework 7.

References

This list is not intended to be comprehensive. I did try to list original references for results proved since 1985. Beyond that, I have listed expository books and papers I found helpful; they mainly reflect where I learned the material. In the schedule for the first third of the course, I have tried to consistently give references to the most closely related material in McDuff-Salamon's books. Audin-Damian's book also seems like a helpful and nicely written reference for this material. (It's newer, so I don't know it as well.)

Expository books, papers, and notes

  • [AD14] Michèle Audin and Mihai Damian, Morse theory and Floer homology. Translated from the 2010 French original by Reinie Erné. Universitext. Springer, London; EDP Sciences, Les Ulis, 2014. MR 3155456.
  • [daS01] Ana Cannas da Silva, Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764. Springer-Verlag, Berlin, 2001. MR 1853077.
  • [daS06] Ana Cannas da Silva, "Symplectic geometry," Handbook of differential geometry. Vol. II, 79–188, Elsevier/North-Holland, Amsterdam, 2006. MR 2194669.
  • [EM02] Yakov Eliashberg and Nikolai Mishachev, Introduction to the h-principle. Graduate Studies in Mathematics, 48. American Mathematical Society, Providence, RI, 2002. MR 1909245.
  • [GS99] Robert Gompf and András Stipsicz,. 4-manifolds and Kirby calculus. Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI, 1999. MR 1707327.
  • [Hu97] Christoph Hummel, Gromov's compactness theorem for pseudo-holomorphic curves. Progress in Mathematics, 151. Birkhäuser Verlag, Basel, 1997. MR 1451624.
  • [Hut02] Michael Hutchings, "Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves)," available from his website here.
  • [MS98] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology. Second edition. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616.
  • [MS12] Dusa McDuff and Dietmar Salamon, J-holomorphic curves and symplectic topology. Second edition. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2012. MR 2954391.
  • [Pan94] Pierre Pansu, "Compactness." In Holomorphic curves in symplectic geometry, 233–249, Progr. Math., 117, Birkhäuser, Basel, 1994. MR 1274932.
  • [Pas] James Pascaleff's lecture notes from "M 392C: Lagrangian Floer Homology" at UT Austin (Spring 2014).
  • Leonid Polterovich, The geometry of the group of symplectic diffeomorphisms. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2001. MR 1826128.
  • [Smi15] Ivan Smith, "A symplectic prolegomenon." Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 3, 415–464. MR 3348443.

Research papers and books

  • [Che98] Yuri Chekanov, "Lagrangian intersections, symplectic energy, and areas of holomorphic curves." Duke Math. J. 95 (1998), no. 1, 213-226. MR 1646550.
  • [Che00] Yuri Chekanov, "Invariant Finsler metrics on the space of Lagrangian embeddings." Math. Z. 234 (2000), no. 3, 605–619. MR 1774099.
  • [Flo88] Andreas Floer, "Morse theory for Lagrangian intersections." J. Diff. Geo. 28 (1988), no. 3, 513-547. MR 965228.
  • [Flo89] Andreas Floer, "Symplectic fixed points and holomorphic spheres." Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR 987770.
  • [FOOO09] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. AMS/IP Studies in Advanced Mathematics, 46.1. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465.
  • [FQ90] Michael Freedman and Frank Quinn, Topology of 4-manifolds. Princeton Mathematical Series, 39. Princeton University Press, Princeton, NJ, 1990. MR 1201584.
  • [GH11] Vinicius Gripp Barros Ramos and Yang Huang, "An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields." arXiv:1112.0290.
  • [Gro71] Mikhail Gromov, "A topological technique for the construction of solutions of differential equations and inequalities." Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 221–225. Gauthier-Villars, Paris, 1971. MR 420697.
  • [Gro85] Mikhail Gromov, "Pseudoholomorphic curves in symplectic manifolds." Invent. Math. 82 (1985), no. 2, 307–347. MR 809718.
  • [Hed07] Matthew Hedden, "Knot Floer homology of Whitehead doubles." Geom. Topol. 11 (2007), 2277–2338. MR 2372849.
  • [Hee98] Poul Heegaard, "Forstudier til en topologisk teori for de algebraiske fladers sammenhaeng." Available here. English translation by Hans Munkholm available here.
  • [KM93] Peter Kronheimer and Tomasz Mrowka, "Gauge theory for embedded surfaces. I." Topology 32 (1993), no. 4, 773–826. MR 1241873.
  • [KM07] Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds. New Mathematical Monographs, 10. Cambridge University Press, Cambridge, 2007.
  • [Kru03] Boris Kruglikov, "Non-existence of higher-dimensional pseudoholomorphic submanifolds." Manuscripta Math. 111 (2003), no. 1, 51–69. MR 1981596.
  • [Lee76] Alexander Lees, "On the classification of Lagrange immersions." Duke Math. J. 43 (1976), no. 2, 217–224. MR 410764.
  • [Lip06] Robert Lipshitz, "A cylindrical reformulation of Heegaard Floer homology." Geom. Topol. 10 (2006), 955–1097. MR 2240908.
  • [Mil68] John Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61 Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1968. MR 239612.
  • [MOS09] Ciprian Manolescu, Peter Ozsváth and Sucharit Sarkar, "A combinatorial description of knot Floer homology." Ann. of Math. (2) 169 (2009), no. 2, 633–660. MR 2480614.
  • [MOSzT07] Ciprian Manolescu, Peter Ozsváth, Zoltán Szabó and Dylan Thurston, "On combinatorial link Floer homology." Geom. Topol. 11 (2007), 2339–2412. MR 2372850.
  • [MW95] Mario Micallef and Brian White, "The structure of branch points in minimal surfaces and in pseudoholomorphic curves." Ann. of Math. (2) 141 (1995), no. 1, 35–85. MR 1314031.
  • [Nov81] S. P. Novikov, "Multivalued functions and functionals. An analogue of the Morse theory." (Russian) Dokl. Akad. Nauk SSSR 260 (1981), no. 1, 31–35. MR 630459. English translation available here.
  • [OSz03] Peter Ozsváth and Zoltán Szabó, "Knot Floer homology and the four-ball genus." Geom. Topol. 7 (2003), 615–639. MR 2026543.
  • [OSz04a] Peter Ozsváth and Zoltán Szabó, "Holomorphic disks and topological invariants for closed three-manifolds." Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019.
  • [OSz04b] Peter Ozsváth and Zoltán Szabó, "Holomorphic disks and three-manifold invariants: properties and applications." Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. MR 2113020.
  • [OSz04c] Peter Ozsváth and Zoltán Szabó, "Holomorphic disks and knot invariants." Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507.
  • [OSz06] Peter Ozsváth and Zoltán Szabó, "Holomorphic triangles and invariants for smooth four-manifolds." Adv. Math. 202 (2006), no. 2, 326–400. MR 2222356.
  • [Per08] Timothy Perutz, "Hamiltonian handleslides for Heegaard Floer homology." Proceedings of Gökova Geometry-Topology Conference 2007, 15–35, Gökova Geometry/Topology Conference (GGT), Gökova, 2008. MR 2509747.
  • [Ras02] Jacob Rasmussen, "Floer homology and knot complements." Thesis (Ph.D.)–Harvard University. 2003. arXiv:math/0306378. MR 2704683.
  • [Sar11] Sucharit Sarkar, "Grid diagrams and the Ozsváth-Szabó tau-invariant." Math. Res. Lett. 18 (2011), no. 6, 1239–1257. MR 2915478.
  • [Sei99] Paul Seidel, "Lagrangian two-spheres can be symplectically knotted." J. Differential Geom. 52 (1999), no. 1, 145–171. MR 1743463.
  • [Sei08] Paul Seidel, Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. MR 2441780.
  • [Sma65] Stephen Smale, "An infinite dimensional version of Sard's theorem." Amer. J. Math. 87 (1965), 861–866. MR 185604.
  • [Ush13] Michael Usher, "Hofer's metrics and boundary depth." Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 1, 57–128 (2013). MR 3087390.
  • [UZ16] Michael Usher and Jun Zhang, "Persistent homology and Floer–Novikov theory." Geom. Topol. 20 (2016), no. 6, 3333–3430. MR 3590354.

Schedule (tentative)

Week Date Topic Relevant resources
1 1/9 Overview and goals  
  1/11 Symplectic geometry 1: symplectic manifolds and their submanifolds. [daS01] §1, 2, 3.
[daS06] Ch. 1, 2.
[MS98] Ch. 1, 2, 3.
[Smi15] §1, 2.
  1/13 Symplectic geometry 2: symplectic linear algebra, almost complex structures. [MS98] §2.1, 2.2, 2.5.
[daS01] Ch. 5.
[Gro85]
2 1/16 Martin Luther King Day (no class)

This is the last day to drop the class without a W.		  
  1/18 Symplectic geometry 3: local neighborhood theorems. [daS01] §6, 7, 8, 9.
[daS06] §1.4, 1.5, 2.2, 2.3.
[MS98] §3.2, 3.3.
  1/20 Pseudo-holomorphic maps. Elliptic regularity. [MS12] Ch. 2, Appendix B.
[Gro85], [Kru03], [MW95]
3 1/23 Lagrangian intersection Floer homology: definition, key theorems. [Hut02]
[Flo88], [Sei08], [FOOO09]
  1/25 Transversality. [MS12], Ch. 3, Appendix A.
[Sei08], [FOOO09]
[Sma65], [Flo88]
  1/27 Expected dimensions and orientations. Transversality problems.  
4 1/30 Compactness. [Pan94], [Hu97]
[MS12] Ch. 4.
[Gro85], [Flo88]
  2/1 More compactness.  
  2/3 Gluing. [MS12], Ch. 10
[Flo88]
5 2/6 Floer homology: definition, invariance. Self Floer homology. Non-displaceable Lagrangian submanifolds. The Arnold Conjecture. [Flo88], [Flo89]
[Gro71], [Lee76], [EM02]
  2/8 Novikov coefficients. More non-displaceability and displacement energy. [Nov81], [Che98], [Che00], [Ush13], [UZ16]
  2/10 A moment to catch up.  
6 2/13 Symplectic capacities and embeddings. First homework problem due.

We're behind schedule, so this week never happened.
  2/15 The nearby Lagrangian problem.  
  2/17 Knotted Lagrangians. [Sei99]
7 2/20 Heegaard diagrams, Morse functions, Heegaard moves. [GS99] [Hee98], [OSz04]
  2/22 Heegaard Floer homology (HF^): definition (for rational homology spheres), first examples. [OSz04], [OSz04b], [Per08], [Lip06], [KM07], [GH11]
  2/24 Invariance of HF^.

The last day to withdraw from the class is 2/26.		 [OSz04a]
8 2/27 Knot Floer homology (CFK-): definition, examples. [OSz04c], [Ras02], [MOS09], [MOSzT07]
  3/1 Tau and the 4-ball genus. [OSz03], [Sar11]
  3/3 The Milnor conjecture. [Mil68], [KM93], [OSz03], [Sar11]
9 3/6 tau of Whitehead doubles. [Hed07]
  3/8 Topologically slice knots; an exotic R4. [FQ90], [GS99]
  3/10 Spinc structures; HF+, HF-, HFred. [OSz04a]
10 3/13 Cobordism maps and the mixed invariant. [OSz06]
  3/15 More cobordism maps and the mixed invariant. [OSz06]
  3/17 The adjunction inequality, Thom conjecture, and exotic closed 4-manifolds.

Second homework problem due.		 [OSz06]