Floer Homology
As I try to learn about Floer theory, I am making a list of various sources, surveys, and papers. Mostly to keep track of what I have read and would like to read, but also in case they might be a useful roadmap to someone else trying to learn this vast and beautiful subject.
Basics
History
Hofer has an online talk describing some history of the discovery of Floer homology and the life of Andreas Floer along with a discussion from titans in the field. Apparently he is also working on a biography of Floer with math journalist Siobhan Roberts which I look forward to reading.
Symplectic Geometry
I like da Silva's book for a terse yet wide-ranging perspective. The classical text for symplectic topology is McDuff and Salamon's book.
Morse Homology
I recommend Hutchings' notes (as he warns, there are some mistakes). I would still like to better understand Witten's approach and read Bott's expositional work.
Gauge Theory
I am a big fan of Baez's book Gauge Fields, Knots and Gravity, which develops gauge theory assuming essentially no prerequisites.
The canonical source on Yang–Mills and Donaldson theory is the book of Donaldson and Kronheimer, The Geometry of Four-Manifolds. This book is excellent, but quite challenging, and I would like to revisit it. In Seiberg-Witten theory, amongst several source there is Morgan's great book The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-Manifolds. There are also notes by Hutchings and Taubes. I recommend everyone read Atiyah's original paper on TQFT.
Symplectic Floer Homology
For an accessible and broad overview to this subject, I highly recommend the survey of Abbondandolo and Schlenk, which discusses most of the sources mentioned below.
Hamiltonian Floer Homology
In this first version of Floer theory, one constructs a chain complex generated by periodic orbits of a time-dependent Hamiltonian vector field with differential counting solutions of Floer's equation, i.e. perturbed pseudoholomorphic cylinders, asymptotic to given orbits. The fact the homology of this complex agrees with Morse homology proves the Arnold conjecture in certain cases.
I have read Audin and Damian's book Morse Theory and Floer Homology, which proves the Arnold conjecture for symplectically aspherical manifolds (I would skip the Morse theory section in favour of the resources above and the section on the Conley–Zehnder index in favour of Salamon's notes). With less details but greater clarity are the lectures of Salamon which treats the monotone case and discusses some J-holomorphic curve theory. The extension to the weakly monotone case by use of Novikov rings was worked out by Hofer and Salamon and Ono; these are both contained in the Floer Memorial Volume, which has lots of interesting papers.
The original paper on the PSS isomorphism is worth reading, although lacking details. I have given a talk on this subject with notes you can find here.
Lagrangian Floer Homology and Fukaya Categories
Here one generalizes the theory to study a chain complex generated by intersections of Lagrangian submanifolds with differential counting pseudoholomorphic disks asymptotic to the submanifolds. This proves the Arnold conjecture for Lagrangian intersections. This homology theory also has a richer algebraic structure, defining the morphisms of an A-infinity category called the Fukaya category, which shows up as a central object of study in mirror symmetry (at this point I refer you to one of my friends advised by Nadler).
I would like to read Auroux's and Smith's introductions to the subject. At some point, I would like to read Seidel's book Fukaya Categories and Picard-Lefschetz Theory.
Symplectic Homology
This theory is Floer homology for symplectic manifolds with (contact-type) boundary. It was first developed in two versions by Cieliebak, Floer, Hofer, and Wysocki. The first version for open subsets of complex space was used to answer quantitative questions about symplectic embeddings, in particular to make computations for ellipsoids. The second version was for more general domains and gave qualitative results showing that interiors of domains can detect the dynamics of their boundary. A subsequent theory developed by Viterbo for domains with contact-type boundary has found many applications in answering qualitative questions like special cases of the Weinstein conjecture and providing quantitative capacity invariants. The theory also contains lots of interesting algebraic structure, such as the Viterbo transfer morphisms, invariance under Liouville automorphisms, and an S^1-equivariant version.
There are three great surveys of the subject by Oancea, Seidel, and Wendl. I have been very interested in this theory lately; here is a longer list of papers I have read or would like to read.
- The original work of Viterbo is well worth reading.
- The symplectic homology of the cotangent disk bundle of M (under certain assumptions) is the homology of the free loop space of M. This has been computed several times; one conceptually simple approach is due to Abbondandolo and Schwarz (note this and the other original papers have a mistake coming from orientation issues, addressed in a corrigendum). In this case, the same authors proved that the ring structure of symplectic homology reproduces the Chaas–Sullivan loop product. This establishes a relationship between Floer homology and string topology, expanded upon here.
- Jean Gutt has a great survey of S^1-equivariant symplectic homology. The S^1 theory is related to usual symplectic homology by an exact sequence as shown by Bourgeois and Oancea.
- A related theory is Rabinowitz Floer homology; you can read a survey here. I'd like to read this paper of Cieliebak, Frauenfelder, and Oancea relating the theory back to symplectic homology and this paper of Frauenfelder and Schlenk defining an S^1-equivariant version.
- S^1 equivariant symplectic homology can be related to a form of SFT called linearized contact homology as was proved by Bourgeois and Oancea.
- Cieliebak and Oancea have a quite substantial paper on some more algebro-topological features of symplectic homology and on extending the definition to filled Liouville cobordisms.
- Oancea has a paper describing a Serre-type spectral sequence for symplectic fibrations.
- I'd like to read this paper on symplectic Tate homology.
- The PSS/TQFT structure on symplectic and wrapped Floer homologies is constructed by Ritter. This paper also serves as an excellent introduction to many of the more general important ideas in the theory.
- This page contains a list of many of the above and other papers in the area.
J-holomorphic Curves
The analytical properties of pseudoholomorphic curves are crucial to the details of the homology theories below. The standard reference is McDuff and Salamon's tome J-holomorphic Curves and Symplectic Topology. Chris Wendl has a couple books (here and here) that may be worth reading too. I would like to find a source with a friendly account of (symplectic) Gromov–Witten theory.
Symplectic Field Theory
The original paper by Eliashberg, Givental, and Hofer is supposedly quite challenging and only schematic, yet I would like to read some of it at some point.
(Embedded) Contact Homology
I am reading Hutchings' lecture notes on ECH.
Miscellaneous
I would like to learn more about the subject of Floer homotopy theory. I have read the original article of Cohen, Jones, and Segal. I would like to read Kragh's paper applying the theory to transfer maps in symplectic homology and understand the work of Abouzaid and Blumberg.
I have been recommended Ginzburg's paper on the Conley conjecture and Shelukhin's paper on the Hofer–Zehnder conjecture for some applications of Hamiltonian Floer theory to symplectic questions. The second of these papers uses persistent homology, which I may read about here.
Topological Floer Homology
For an overview of the subject, one should read Manolescu's survey. Everyone should also read Atiyah's classic paper.
Instanton Floer Homology
I gave a talk introducing this subject with notes you can read here. I like the gentle introduction in Saveliev's book Invariants of Homology 3-Spheres and the original paper of Floer. At some point I may like to read Donaldson's book Floer Homology Groups in Yang-Mills Theory. Learning the technical details of this theory is probably not so important, as it has been largely superseded by the Seiberg–Witten theory. Although, I would still like to learn something about progress on the Atiyah–Floer conjecture.
Seiberg–Witten Floer Homology
The standard reference is Kronheimer and Mrowka's book Monopoles and Three-Manifolds, which I would like to read at some point in grad school.
Heegaard Floer Homology
I know nothing about this topic. Eventually, I will read Ozsváth and Szabo's two notes (here and here) introducing the subject.
