# Floer homology and surface decompositions

@article{Juhsz2008FloerHA, title={Floer homology and surface decompositions}, author={Andr{\'a}s Juh{\'a}sz}, journal={Geometry \& Topology}, year={2008}, volume={12}, pages={299-350} }

Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, \gamma)--> (M', \gamma') is a sutured manifold decomposition then SFH(M',\gamma') is a direct summand of SFH(M, \gamma). To prove the decomposition formula we give an algorithm that computes SFH(M,\gamma) from a balanced diagram defining (M,\gamma) that… Expand

#### Figures from this paper

#### 70 Citations

Sutured Floer homology, fibrations, and taut depth one foliations

- Mathematics
- 2015

For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology (SFH) can be used to determine all fibred classes in H-1(M). Furthermore, we show… Expand

The sutured Floer homology polytope

- Mathematics
- 2010

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is… Expand

Sutured Floer homology and hypergraphs

- Mathematics
- 2012

By applying Seifert's algorithm to a special alternating diagram of a link L, one obtains a Seifert surface F of L. We show that the support of the sutured Floer homology of the sutured manifold… Expand

Partial open book decompositions and the contact class in sutured floer homology

- Mathematics
- 2009

We demonstrate how to combinatorially calculate the EH-class of a compatible contact structure in the sutured Floer homology group of a balanced sutured three manifold which is associated to an… Expand

Graph cobordisms and Heegaard Floer homology

- Mathematics
- 2015

In this paper we construct graph cobordism maps for all flavors of Heegaard Floer homology and prove invariance of such maps. In the process we also finish the proof of invariance of the 4-manifold… Expand

Dehn surgery, rational open books and knot Floer homology

- Mathematics
- 2013

By recent results of Baker, Etnyre and Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact… Expand

A graph TQFT for hat Heegaard Floer homology

- Mathematics
- Quantum Topology
- 2021

In this paper we introduce an extension of the hat Heegaard Floer TQFT which allows cobordisms with disconnected ends. Our construction goes by way of sutured Floer homology, and uses some elementary… Expand

THE SUTURED FLOER POLYTOPE AND TAUT DEPTH ONE FOLIATIONS

- Mathematics
- 2014

For closed 3‐manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsvath and Szabo, Ni, and Hedden. For example, given a closed 3‐manifold Y , there is a… Expand

Bordered Floer homology for sutured manifolds

- Mathematics
- 2009

We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the… Expand

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

- Mathematics
- 2012

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study… Expand

#### References

SHOWING 1-10 OF 18 REFERENCES

Holomorphic disks and genus bounds

- Mathematics
- 2004

We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the… Expand

Knot Floer homology detects genus-one fibred knots

- Mathematics
- 2006

Ozsv\'ath and Szab\'o conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on contact topology and Gabai's theory of sutured manifold… Expand

Foliations and the topology of 3-manifolds

- Mathematics
- 1987

In this announcement we discuss the close relationship between the topology of 3-manifolds and the foliations that is possesses. We will introduce and state the main result, then use it and the ideas… Expand

A cylindrical reformulation of Heegaard Floer homology

- Mathematics
- 2006

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold U a0;1c R, where U is the Heegaard surface, instead of Sym g .U/. We then show that the entire… Expand

Holomorphic disks and three-manifold invariants: Properties and applications

- Mathematics
- 2004

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

Sutured Heegaard diagrams for knots

- Mathematics
- 2006

We define sutured Heegaard diagrams for null-homologous knots in 3‐manifolds. These diagrams are useful for computing the knot Floer homology at the top filtration level. As an application, we give a… Expand

Holomorphic discs and sutured manifolds

- Mathematics
- 2006

In this paper we construct a Floer-homology invariant for a natural and wide class of sutured manifolds that we call balanced. This generalizes the Heegaard Floer hat theory of closed three-manifolds… Expand

Link Floer homology and the Thurston norm

- Mathematics
- 2006

We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its… Expand

An algorithm for computing some Heegaard Floer homologies

- Mathematics
- 2006

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

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… Expand