# Holomorphic disks and topological invariants for closed three-manifolds

@article{Ozsvath2001HolomorphicDA, title={Holomorphic disks and topological invariants for closed three-manifolds}, author={Peter S. Ozsvath and Zolt{\'a}n Imre Szab{\'o}}, journal={Annals of Mathematics}, year={2001}, volume={159}, pages={1027-1158} }

The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y, equipped with a Spiny structure. Given a Heegaard splitting of Y = U 0o U Σ U 1 , these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U 0 and U 1 .

#### 510 Citations

Holomorphic triangles and invariants for smooth four-manifolds

- Mathematics
- 2001

Abstract In earlier articles, the authors introduced invariants for closed, oriented three-manifolds, defined using a variant of Lagrangian Floer homology in the symmetric products of Riemann… Expand

Holomorphic polygons and smooth 4-manifold invariants

- Mathematics
- 2013

Any smooth, closed oriented 4-manifold has a surface diagram of arbitrarily high genus g>2 that specifies it up to diffeomorphism. The goal of this paper is to prove the following statement: For any… Expand

Holomorphic disks and knot invariants

- Mathematics
- 2002

Abstract We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for three-manifolds defined in an earlier paper. We set up… Expand

Heegaard Floer homology is a collection of invariants for closed oriented three-manifolds , intro

- 2008

In this paper, we give an algorithm to compute the hat version of the Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtrations coming from a… Expand

Lattice cohomology of normal surface singularities

- Mathematics
- 2007

For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of… Expand

Lagrangian matching invariants for fibred four-manifolds: I

- Mathematics
- 2006

In a pair of papers, we construct invariants for smooth four-manifolds equipped with `broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the… Expand

Triple products and cohomological invariants for closed three-manifolds

- Mathematics
- 2006

Motivated by conjectures in Heegaard Floer homology, we introduce an invariant HC(Y) of the cohomology ring of a closed 3-manifold Y whose behavior mimics that of the Heegaard Floer homology… Expand

Perturbed Floer homology of some fibered three-manifolds

- Mathematics
- 2009

Modifying the method of [21], we compute the perturbed $HF^+$ for some special classes of fibered three manifolds in the second highest spin$^c$-structures $S_{g-2}$. The special classes considered… Expand

Link homology theories from symplectic geometry

- Mathematics
- 2006

For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the link polynomial. We use… Expand

Floer theory and its topological applications

- Mathematics, Physics
- 2014

We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and… Expand

#### References

SHOWING 1-10 OF 58 REFERENCES

Holomorphic triangles and invariants for smooth four-manifolds

- Mathematics
- 2001

Abstract In earlier articles, the authors introduced invariants for closed, oriented three-manifolds, defined using a variant of Lagrangian Floer homology in the symmetric products of Riemann… Expand

Geometry of four-manifolds

- Physics, Mathematics
- 1986

1. Four-manifolds 2. Connections 3. The Fourier transform and ADHM construction 4. Yang-Mills moduli spaces 5. Topology and connections 6. Stable holomorphic bundles over Kahler surfaces 7. Excision… Expand

A relative morse index for the symplectic action

- Mathematics
- 1988

The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define… Expand

Holomorphic disks and three-manifold invariants: Properties and applications

- Mathematics
- 2001

In [27], we introduced Floer homology theories HF - (Y,s), HF∞(Y,s), HF + (Y, t), HF(Y,s),and HF red (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spiny structures s ∈ Spin… Expand

Relative Gromov-Witten invariants

- Mathematics
- 1999

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of… Expand

J-Holomorphic Curves and Quantum Cohomology

- Mathematics
- 1994

Introduction Local behaviour Moduli spaces and transversality Compactness Compactification of moduli spaces Evaluation maps and transversality Gromov-Witten invariants Quantum cohomology Novikov… Expand

Lefschetz pencils and the canonical class for symplectic four-manifolds

- Mathematics
- 2000

We present a new proof of a result due to Taubes: if (X,ω) is a closed symplectic four-manifold with b+(X)>1+b1(X) and λ[ω]∈H2(X;Q) for some λ∈R+, then the Poincare dual of KX may be represented by… Expand

Transversality in elliptic Morse theory for the symplectic action

- Mathematics
- 1995

Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder. These results play a central role in the denition of symplectic… Expand

The unregularized gradient flow of the symplectic action

- Mathematics
- 1988

The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this… Expand

Morse-Bott theory and equivariant cohomology

- Mathematics
- 1995

Critical points of functions and gradient lines between them form a cornerstone of physical thinking. In Morse theory the topology of a manifold is investigated in terms of these notions with equally… Expand