Perturbed Floer homology of some fibered three-manifolds

Abstract

In this paper, we write down a special Heegaard diagram for a given product threemanifold † g S 1 . We use the diagram to compute the perturbed c HF for the 3-torus and the perturbed HF C for nontorsion spin c -structures for † g S 1 when g 2. 53D401 IntroductionHeegaard Floer homology was introduced by Ozsváth and Szabó [4;5] and has proved to be a powerful 3-manifold invariant. The construction of the invariant requires an admissibility condition though, which in general is not met by those "simplest" Heegaard diagrams for a given 3-manifold Y with b 1 .Y / 1. A variant of the construction using the Novikov ring overcomes this shortcoming, and in some sense embraces the ordinary homology as a special case. The invariants, usually called perturbed Heegaard Floer homology, have proved to be useful in some situations. For example, Jabuka and Mark made use of them in calculating Ozsváth-Szabó invariants for certain closed 4-manifolds [1].This paper aims to compute the perturbed Heegaard Floer homologies for product three-manifolds † g S 1 . The result is a little surprising as we find that the homology groups are independent of the exact direction of perturbations.We would also like to point out that although the computation is made solely for the product three-manifolds in this paper, the method can in fact be applied to the more general setting of certain fibered three-manifolds; see the author's paper [8].This paper is organized as follows: In Section 2, we review the backgrounds of the Novikov ring A and the perturbed Heegaard Floer homology. Treating homology groups as A-vector spaces, we prove a rank inequality and an Euler characteristic identity. In Section 3, we write down a special Heegaard diagram for T 3 and compute its perturbed Heegaard Floer homology. In Section 4, we compute the homology for nontorsion Spin c structures for † g S 1 .

$U$-action on perturbed Heegaard–Floer homology

“…We also review the construction of a special Heegaard diagram, which will be used throughout the paper. In Section 3, we extract and reformulate a standard argument from [27], and use it as a principal tool in determining the rank of the perturbed Heegaard Floer homology of various mapping tori. In Section 4, we establish the U -action adjunction inequality as a formal consequence of Heegaard-Floer cobordism invariants.…”

“…Hence, for simplicity, we always assume the curves to lie in the standard position in the forthcoming discussions. We are going to compute the rank of HF + (M (φ), g − 2; ω) for various mapping tori by a method based on ideas from [27]. A few simplification is made in the argument although, and it is reformulated in a form most suitable for its subsequent applications.…”

“…(See Definition 2.1 below for the definition of the Novikov ring.) Following [27], where the perturbed Heegaard Floer homology is calculated for the product three-manifolds Σ g × S 1 , we aim to apply a similar method to compute the perturbed HF + for some special classes of fibered three-manifolds. More precisely, viewing each fibered three-manifold Y as a mapping torus Σ g × [0, 1]/(x, 1) ∼ (φ(x), 0), denoted by M (φ), for some orientation-preserving diffeomorphism φ of Σ g , we study the cases where φ can be decomposed as products of Dehn twists along a single non-separating curve, or along a transverse pair of curves.…”

The twisted Floer homology of torus bundles

“…Since we're working over ƒ and d ¤ 0, the middle map is an isomorphism and we see that Tor It is worth noting that alternate proofs of this theorem and Proposition 2.2 are possible through the use of inadmissible diagrams, which have been explored by Wu in [23] as well as by Lekili in [10].…”

Duality and mapping tori in Heegaard Floer homology

“…It is well known that if ω is a non-zero 2-form on T 3 , when HF + (T 3 ; Λ ω ) ∼ = Λ, and furthermore HF + is supported only in the torsion Spin c structure. See the work of Ai-Peters [1, Theorem 1.3], Jabuka-Mark [11,Theorem 10.1], Lekili [21,Theorem 14] and Wu [40]. In particular, our theorem computes that Φ T 4 ,t0 = 1, where t 0 ∈ Spin c (T 4 ) is the torsion Spin c structure, and Φ T 4 ,t = 0 for all other Spin c structures.…”