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 abstract partial open book decomposition. As an application we show that every contact three manifold (closed or with convex boundary) can be obtained by gluing tight contact handlebodies whose EH-classes are nontrivial. | Turk J Math 33 (2009) , 295 – 312. ¨ ITAK ˙ c TUB doi: Partial open book decompositions and the contact class in sutured floer homology ¨ gcı Tolga Etg¨ u and Burak Ozba˘ Abstract 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 abstract partial open book decomposition. As an application we show that every contact three manifold (closed or with convex boundary) can be obtained by gluing tight contact handlebodies whose EH-classes are nontrivial. Key word and phrases: Partial open book decomposition, contact three-manifold with convex boundary, sutured manifold, sutured Floer homology, EH-contact class. 1. Introduction A sutured manifold (M, Γ) is a compact oriented 3 -manifold with nonempty boundary, together with a compact subsurface Γ = A(Γ) ∪ T (Γ) ⊂ ∂M , where A(Γ) is a union of pairwise disjoint annuli and T (Γ) is a union of tori. Moreover each component of ∂M \ Γ is oriented, subject to the condition that the orientation changes every time we nontrivially cross A(Γ). Let R+ (Γ) (resp. R− (Γ)) be the open subsurface of ∂M \ Γ on which the orientation agrees with (resp. is the opposite of ) the boundary orientation on ∂M . A sutured manifold (M, Γ) is balanced if M has no closed components, π0 (A(Γ)) → π0 (∂M ) is surjective, and χ(R+ (Γ)) = χ(R− (Γ)) on every component of M . It follows that if (M, Γ) is balanced, then Γ = A(Γ) and every component of ∂M nontrivially intersects Γ. Since all our sutured manifolds will be balanced in this paper, we can think of Γ as a set of oriented curves on ∂M by identifying each annulus in Γ with its core circle. Here Γ is oriented as the boundary of R+ (Γ). Let ξ be a contact structure on a compact oriented 3 -manifold M whose dividing set on the convex boundary ∂M is denoted by Γ. Then it is not too hard to see that (M, Γ) is a balanced sutured .
