# Holomorphic triangles and invariants for smooth four-manifolds

@article{Ozsvath2001HolomorphicTA, title={Holomorphic triangles and invariants for smooth four-manifolds}, author={Peter S. Ozsvath and Zolt{\'a}n Imre Szab{\'o}}, journal={Advances in Mathematics}, year={2001}, volume={202}, pages={326-400} }

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 surfaces. The aim of this article is to introduce invariants of oriented, smooth four-manifolds, built using these Floer homology groups. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology…

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## References

SHOWING 1-10 OF 19 REFERENCES

Holomorphic disks and topological invariants for closed three-manifolds

- Mathematics
- 2001

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…

Holomorphic disks and topological invariants for rational homology three-spheres

- Mathematics
- 2001

The aim of this article is to introduce and study certain topological invariants for oriented, rational homology three-spheres Y. These groups are relatively Z-graded Abelian groups associated to…

Holomorphic triangle invariants and the topology of symplectic four-manifolds

- Mathematics
- 2002

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169.…

The symplectic Thom conjecture

- Mathematics
- 1998

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together…

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…

4-manifolds and Kirby calculus

- Mathematics
- 1999

4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic…

THE SEIBERG-WITTEN INVARIANTS AND SYMPLECTIC FORMS

- Mathematics
- 1994

(Note: There are no symplectic forms on X unless b and the first Betti number of X have opposite parity.) In a subsequent article with joint authors, a vanishing theorem will be proved for the…

IMMERSED SPHERES IN 4-MANIFOLDS AND THE IMMERSED THOM CONJECTURE

- Mathematics
- 1995

The introduction of the Seiberg Witten monopole equations ([SW1],[SW2],[W]) has served to make the study of smooth 4-manifolds more accessible. Many of the important earlier theorems regarding…