# Periodic Floer homology and Seiberg-Witten Floer cohomology

@article{Lee2009PeriodicFH, title={Periodic Floer homology and Seiberg-Witten Floer cohomology}, author={Yi-Jen Lee and Clifford H. Taubes}, journal={arXiv: Geometric Topology}, year={2009} }

Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer cohomology and the corresponding periodic Floer homology, and describe some immediate consequences.

## 37 Citations

### HF=HM, I : Heegaard Floer homology and Seiberg–Witten Floer homology

- MathematicsGeometry & Topology
- 2020

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the…

### LECTURES ON THE EQUIVALENCE OF HEEGAARD FLOER AND SEIBERG–WITTEN FLOER HOMOLOGIES

- Mathematics
- 2013

This article gives a detailed account of the lectures delivered by the author on the construction, in joint work with Yi-Jen Lee and Clifford H. Taubes, of isomorphisms between Heegaard Floer and…

### HF=HM, V : Seiberg–Witten Floer homology and handle additions

- MathematicsGeometry & Topology
- 2020

This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.

### Floer homology and right-veering monodromy

- Mathematics
- 2022

. We prove that the knot Floer complex of a ﬁbered knot detects whether the monodromy of its ﬁbration is right-veering. In particular, this leads to a purely knot Floer-theoretic characterization of…

### Closed-open morphisms on periodic Floer homology

- Mathematics
- 2021

In this note, we investigate homomorphisms from the periodic Floer homology (PFH) to the quantitative Heegaard Floer homology. We call the homomorphisms closed-open morphisms. Under certain…

### On cobordism maps on periodic Floer homology

- MathematicsAlgebraic & Geometric Topology
- 2021

In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the…

### Embedded contact homology and its applications

- Mathematics
- 2010

Embedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Taubes has shown that ECH is isomorphic to a version of Seiberg-Witten Floer homology (and both are…

### Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

- Mathematics
- 2012

Let A be a dg algebra over F_2 and let M be a dg A-bimodule. We show that under certain technical hypotheses on A, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the…

### HF=HM, IV : The Seiberg–Witten Floer homology and ech correspondence

- MathematicsGeometry & Topology
- 2020

This is the fourth of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is…

## References

SHOWING 1-10 OF 27 REFERENCES

### The periodic Floer homology of a Dehn twist.

- Mathematics
- 2005

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits,…

### Symplectic Floer homology of area-preserving surface diffeomorphisms

- Mathematics
- 2009

The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We…

### The Seiberg–Witten equations and the Weinstein conjecture

- Mathematics
- 2006

Let M denote a compact, oriented 3–dimensional manifold and let a denote a contact 1–form on M; thus a∧da is nowhere zero. This article proves that the vector field that generates the kernel of da…

### Seiberg-Witten and Gromov invariants for symplectic 4-manifolds

- Mathematics
- 2005

1. SW => Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves 1. The Seiberg-Witten equations 2. Estimates 3. The monotonicity formula 4. The local structure of [alpha]1(0) 5.…

### Monopoles and Three-Manifolds

- Mathematics
- 2008

Preface 1. Outlines 2. The Seiberg-Witten equations and compactness 3. Hilbert manifolds and perturbations 4. Moduli spaces and transversality 5. Compactness and gluing 6. Floer homology 7.…

### Reidemeister Torsion in Floer--Novikov Theory and Counting pseudo-holomorphic tori, II

- Mathematics
- 2001

This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, (I_F). (See math.DG/0505013 for part II).
The Floer homology can be trivial in many…

### An index inequality for embedded pseudoholomorphic curves in symplectizations

- Mathematics
- 2001

Abstract.Let Σ be a surface with a symplectic form, let φ be a symplectomorphism of Σ, and let Y be the mapping torus of φ. We show that the dimensions of moduli spaces of embedded pseudoholomorphic…

### Seiberg-Witten invariants of mapping tori, symplectic ﬁxed points, and Lefschetz numbers

- Mathematics
- 1999

Let Y be a compact oriented smooth 3-manifold with nonzero first Betti number. Two nonzero vector fields on Y are called homologous if they are homotopic over the complement of a ball in Y . An Euler…

### Circle-valued Morse theory and Reidemeister torsion

- Mathematics
- 1999

Let X be a closed manifold with (X )=0 , and let f : X! S 1 be a circlevalued Morse function. We dene an invariant I which counts closed orbits of the gradient of f , together with flow lines between…

### Compactness results in Symplectic Field Theory

- Mathematics
- 2003

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic…