Two applications of twisted Floer homology

Ai, Yinghua; Ni, Yi

doi:10.48550/arxiv.0809.0622

Cited by 3 publications

(10 citation statements)

References 11 publications

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“…is an isomorphism. Since HF + (Y, ω, [F ], 1 2 x(F ); Λ) = 0 and U n a = 0 for any a ∈ HF + (Y, ω, [F ], 1 2 x(F ); Λ) and sufficiently large n, we get a contradiction. The following lemma will be used in the proof of Theorem 3.8.…”

Section: The Topmost Nontrivial Termmentioning

confidence: 96%

“…It is easy to construct knots that violate Property G. However, if we make some assumption on Y or K, then we can get Property G. For example, one can show that non-prime knots have Property G. In [6], Gabai proved that if K is a null-homologous knot in a reducible manifold Y , such that H 1 (Y ) is torsion-free and Y − K is irreducible, then K has Property G. This result has overlap with our Theorem 1.4. Moreover, using Heegaard Floer homology, we can show that if HF red (Y ) = 0 then K has Property G. (For Property G2, the proof can be found in [10,1]. The proof for Property G1 is similar.…”

Section: Introductionmentioning

confidence: 91%

“…Let U ⊂ H 1 (Y ; R) be the dual of U * . Now for any ω ∈ U , the argument in [15, Section 4] shows that 1 2 x(F ); Λ) = 0, then the map…”

Section: The Topmost Nontrivial Termmentioning

confidence: 99%

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Nonseparating spheres and twisted Heegaard Floer homology

2013

Algebr. Geom. Topol.

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Section: The Topmost Nontrivial Termmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 91%

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Nonseparating spheres and twisted Heegaard Floer homology

2013

Algebr. Geom. Topol.

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“…Observe that a Dehn twist along γ does not introduce any new intersections between α i and β i . Consequently, there is no additional generator, other than the (2g − 1) pairs we initially had for Σ g × S 1 .…”

Section: Multiple Dehn Twists Along a Non-separating Curvementioning

confidence: 99%

“…This paper is aimed to compute the perturbed Heegaard Floer homologies for product three manifolds Σ g × S 1 . The result is a little bit surprising as we find that the homology groups are independent of the exact direction of perturbations.…”

Section: Introductionmentioning

confidence: 99%

Perturbed Floer homology of some fibered three-manifolds

2009

Algebr. Geom. Topol.

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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 .

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Manifolds with small Heegaard Floer ranks

Hedden

2010

Geom. Topol.

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We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink. 57M27; 57M25

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Two applications of twisted Floer homology

Cited by 3 publications

References 11 publications

Nonseparating spheres and twisted Heegaard Floer homology

Nonseparating spheres and twisted Heegaard Floer homology

Perturbed Floer homology of some fibered three-manifolds

Manifolds with small Heegaard Floer ranks

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