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.

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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. The standard reference for contact geometry is the book of Geiges. More results on topology of contact manifolds can be found in the book of Ozbagci and Stipsicz.

Morse Homology

This is a beautiful subject that constructs the singular homology of a manifold by studying the gradient flow of a smooth function. I recommend Hutchings' notes (as he warns, there are some mistakes). For more on Morse–Bott theory and the modern approach of cascades, see here. I would still like to better understand Witten's approach and read Bott's expositional work. There is also a book by Schwarz which explains "algebraic topology done Morse."

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.

Analysis

To learn about functional analysis, Fredholm theory, etc., Pedersen's Analysis Now is a comprehensive but quite dense reference. The standard place to learn about Sobolev spaces and elliptic PDEs is Evan's book. For much deeper theory, one can consult the well-known texts of Adams and Gilbarg–Trudinger. A very readable source surveying all the above topics is Brezis' book.

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 but thorough account of (symplectic) Gromov–Witten theory.

Symplectic Floer Homology

For an accessible and broad overview to this subject, I highly recommend the survey of Abbondandolo and Schlenk, which discusses many 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.

Hamiltonian Floer theory has the structure of a 2d TQFT, in particular it has a product which agrees with the quantum cup product (a deformation of the usual cup product by counting holomorphic spheres). This structure and a natural identification of Floer and Morse homology is explained in the original paper on the PSS isomorphism. I gave 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 Lagrangians. 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. More details in the case of cotangent bundles are surveyed by Fukaya–Seidel–Smith. Lagrangian Floer theory has also been generalized to Lagrangian correspondences, which enhances its categorical properties. This is the theory of quilted Floer homology due to Wehrheim and Woodward. A survey of these ideas and the construction of a symplectic 2-category has recently been written by Abouzaid and Bottman.

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. Lagrangian Floer theory also has a version in the non-compact case called wrapped Floer homology and there is a corresponding wrapped Fukaya category.

There are three great surveys of the subject by Oancea, Seidel, and Wendl. There is also a research monograph of Abouzaid focusing on some algebraic operations and the computation for cotangent bundles mentioned below. 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. This paper of Cieliebak, Frauenfelder, and Oancea relates the theory back to symplectic homology and this paper of Frauenfelder and Schlenk defines an S^1-equivariant version.
S^1 equivariant symplectic homology can be related to a form of SFT called linearized contact homology as was proven by Bourgeois and Oancea.
Extending the above result, Ekholm and Oancea show here that symplectic homology has a DGA structure closed related to similar structures in SFT.
Cieliebak and Oancea have a hefty 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.
In favourable settings, one has that the Hochschild homology of the wrapped Fukaya category agrees with symplectic homology. The original proof of Ganatra can be found here.
This page contains a list of many of the above and other papers in the area.
Two important related works which I have not read are the paper of Bourgeois, Eliashberg, and Ekholm on Legendrian surgery and the papers of Ganatra, Pardon, and Shende on Liouville sectors.

Symplectic Field Theory and Contact Homology

SFT is a package of Floer-like theories for contact manifolds. They can be seen as the Morse homology of loops on a contact manifold with the action functional given by integrating the contact form; one defines a chain complex generated by Reeb orbits with differential counting holomorphic curves in the symplectization. Alternatively, SFT can be viewed as an attempt to TQFT-ify the Gromov–Witten invariants. Depending what curves one counts, one can obtain several different theories with differing rich algebraic structures including linearized contact homology, cylindrical contact homology, the contact homology algebra, rational symplectic field theory, and symplectic field theory. Unfortunately, multiply covered curves cause issues with obtaining the transversality and compactness results needed to classically define a Floer theory. Various versions of contact homology are now rigorously defined in certain scenarios, e.g. for hypertight and some dynamically convex contact forms, due to a series of works by Hutchings–Nelson, Bao–Honda, Pardon, et al.. I believe RSFT and SFT are basically still conjectural. Nevertheless, the SFT philosophy has led to a number of significant results, including some classification of contact structures and some new cases of the Weinstein conjecture.

The original paper of Eliashberg, Givental, and Hofer is quite challenging and only schematic; the analytic foundations of SFT can be found here. There is a recent survey by Hind and Siegel which is very good. To seriously learn the details of the subject, I'd like to read Wendl's book in progress. There are some short lecture notes on contact homology by Cristofaro-Gardiner (here) and Bourgeois (here) and notes by Bourgeois relating contact to symplectic homology (here). There's yet another nice survey by Bourgeois on contact homology and its applications in this volume.

Embedded Contact Homology

ECH is a spin on contact homology in three dimensions, discovered by Hutchings, which counts some special collection of embedded J-curves (or rather currents) governed by the mysterious ECH index. Miraculously, this gives a topological invariant, which agrees with certain versions of monopole and Heegaard Floer theories! ECH is totally well defined, although various features of it require passing through Taubes' very difficult proof that ECH agrees with monopole Floer theory. The canonical reference is Hutchings' lecture notes on ECH. The relationship between ECH and Seiberg–Witten theory is central to Taubes' proof of the 3d Weinstein conjecture; Hutchings has a readable survey of the proof.

Quantitative Symplectic Geometry

Action filtrations and the associated barcodes on various versions of symplectic Floer give rise to quantitative invariants, like capacities and spectral invariants, which help solve embedding problems and dynamical questions. For a broad introduction to embedding problems, I recommend the survey of Schlenk which contains many open problems and interesting references. There is a long history of finding obstruction to embeddings via Floer theory going back to the work of Floer–Hofer–Wysocki.

One important example of these quantitative invariants are the ECH capacities, explored in Hutchings' ECH notes. Hutchings has several papers explaining additional quantitative invariants that can be obtained from ECH (see here, here, and here). Gutt and Hutchings also have an analogous sequence of capacities obtained from S^1 equivariant symplectic homology defined here. More recently, there are some ideas for building capacities from SFT (see here, here, and here) to understand stabilized embeddings.

Miscellaneous

Here are some other papers I find interesting or are important to various Floer theories.

For gradings in Floer theory, it's helpful to understand Robbin–Salamon's work on spectral flow.
For orientations in Floer theory, the fundamental papers are here and here.
To understand how to compute Floer theory in settings with symmetry, one should look at Bourgeois' work on Morse–Bott techniques.
Explicit examples in Floer theory are pretty rare, so let me shout out some other things we know. Here is how to find the symplectic cohomology of a subcritical Stein domain. Here is how to study the symplectic homology of Lefschetz fibrations. Here is the ECH of a prequantization bundle.
There are many interesting papers I'd like to read that deduce dynamical consequences just from a thorough study of Hamiltonian Floer theory. For example, Ginzburg's paper on the Conley conjecture and Shelukhin's paper on the Hofer–Zehnder conjecture. The second of these papers uses persistent homology, the background for which is explained here.
I would like to understand this paper of Seidel that detects the topology of the space of Hamiltonian diffeomorphisms.
Seidel's paper fits into a more general framework of family Floer theory axiomatized and investigated by Hutchings. There should be lots of interesting new math lurking here.
I'd like to understand the proof of the Weinstein conjecture for overtwisted (or plastikstufe) contact manifolds.
I'd also like to understand why contact homology vanishes for overtwisted contact 3-manifolds.

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

This is a Floer theory generated by flat connections on a 3-manifold Y with differential counting instantons (ASD connections) on IxY; it is a TQFT-ification of the Donaldson invariants and categorifies the Casson invariant. 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.

Monopole Floer Homology

This is a Floer theory generated by Seiberg–Witten solutions on a 3-manifold Y with differential counting 4d solutions on the cylinder IxY; it is a TQFT-ification of the Seiberg–Witten invariant. There are several different flavours of monople Floer theory based on how one treats reducible vs. irreducible solutions (or equivalently how one implements S^1-equivariant homology). The fact this is an abelian gauge theory and there is a priori compactness in all cases makes it a highly appealing successor to instanton Floer theory. The standard reference is Kronheimer and Mrowka's book Monopoles and Three-Manifolds.

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.

Floer Homology of Knots

Heegaard Floer theory is closely related to various versions of Floer homology of knots. Some of this theory is surveyed here and here. There is also such a knot Floer theory coming from instanton Floer theory due to Kronheimer and Mrowka, which is closely related to Khovanov homology.

On the symplectic side, some combination of Lagrangian Floer theory and SFT gives rise to Legendrian contact homology which gives a new invariant of Legendrian knots. This theory is surveyed here. Given a contact manifold with nice boundary, one can construct its sutured contact homology and sutured ECH, as described here. In the case where your contact manifold is the complement of a tubular neighbourhood of a Legendrian knot, this gives rise to knot invariants.

Floer Homotopy Theory

This is a program to upgrade various Floer homologies to stable homotopy types. As a result, one could compute Floer homology with coefficients in some spectrum, e.g. Floer K-theory or Floer cobordism; these novel invariants may contain new information not observed by the basic homology groups. I am a little wary of the analytic foundations of this subject given that it requires one to understand all the corresponding moduli spaces, instead of just the low-dimensional ones. Nevertheless, the topic is rising in popularity and has yielded some interesting results.

To begin, one can read Floer's work and the original article of Cohen, Jones, and Segal demonstrating a proof of concept Morse homotopy theory, although the subject is apparently being built on new foundations by Abouzaid and Blumberg. Some early symplectic applications include Kragh's paper applying the theory to transfer maps in symplectic homology, the work of Abouzaid and Blumberg on the Arnold conjecture mod p, and the work of Lee Tanaka and Traynor on contact homology for Legendrians.

On the topological side, Manolescu constructed a Seiberg–Witten stable homotopy type building on work of Bauer and Furuta, who define a stable homotopy refinement of the Seiberg–Witten invariant. Manolescu has a nice survey describing his construction and its application to the triangulation conjecture through Pin(2)-equivariant monopole Floer theory. I also found this survey quite enlightening to explain why there are so many versions of monopole Floer theory and what they all represent. I have some notes from a talk on this topic. There is also a construction of a Khovanov stable homotopy type by Lipshitz and Sarkar.