Corrigendum: A flexible construction of equivariant Floer homology and applications

Hendricks, Kristen; Lipshitz, Robert; Sarkar, Santanu

doi:10.1112/topo.12124

Cited by 5 publications

(10 citation statements)

References 15 publications

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“…Remark 2. 4 We have not yet given an absolute grading for this definition. In fact, Proposition 2.2 is proved in the relatively graded case.…”

Section: Symplectic Khovanov Cohomology For Bridge Diagramsmentioning

confidence: 99%

Bigrading the symplectic Khovanov cohomology

Cheng

2023

Algebr. Geom. Topol.

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We construct a well-defined relative second grading on symplectic Khovanov cohomology from holomorphic disc counting. We use a version of symplectic Khovanov cohomology defined for bridge diagrams rather than braids. We show that our second grading recovers the Jones grading of Khovanov homology up to an overall grading shift over any characteristic-zero field, through proving that the isomorphism of Abouzaid and Smith can be refined as an isomorphism between bigraded cohomology theories. The central idea of the proof is to construct an exact triangle for symplectic Khovanov cohomology that behaves similarly to the unoriented skein exact triangle for Khovanov homology.

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“…Remark 2. 4 We have not yet given an absolute grading for this definition. In fact, Proposition 2.2 is proved in the relatively graded case.…”

Section: Symplectic Khovanov Cohomology For Bridge Diagramsmentioning

confidence: 99%

Bigrading the symplectic Khovanov cohomology

Cheng

2023

Algebr. Geom. Topol.

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“…A knot concordance invariant. Applying equivariant Floer theory to the double branched covers, several knot (concordance) invariants are defined in Heegaard Floer homology, such as [2,16,24,25,27,28,38], and Seiberg-Witten Floer theory [8]. In a certain orbifold setting, several versions of knot instanton Floer homology are developed ( [13,15,46]).…”

Section: 2mentioning

confidence: 99%

Involutions, knots, and Floer K-theory

Konno¹,

Miyazawa²,

Takeuchi³

2021

Preprint

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We establish a version of Seiberg-Witten Floer K-theory for knots, as well as a version of Seiberg-Witten Floer K-theory for 3-manifolds with involutions. The main theorem is a 10/8-type inequality for spin 4-manifolds with boundary and with involutions, and that yields numerous applications to knots, such as genus bounds and stabilizing numbers. We also give applications to get obstructions to extending involutions on 3-manifolds to spin 4-manifolds.

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“…• In [47], Seidel and Smith defined a version of equivariant Lagrangian Floer homology for symplectic involutions (𝐺 = ℤ 2 ). Generalizations to finite group actions appeared in [9,14,25,26].…”

Section: Introductionmentioning

confidence: 99%

“…In [47], Seidel and Smith defined a version of equivariant Lagrangian Floer homology for symplectic involutions (

G = \mathbb {Z}_{2}

). Generalizations to finite group actions appeared in [9, 14, 25, 26]. In [27], Hendricks, Lipshitz and Sarkar defined an equivariant Lagrangian Floer homology for Lie group actions.…”

Section: Introductionmentioning

confidence: 99%

Equivariant Lagrangian Floer homology via cotangent bundles of EGN$EG_N$

Cazassus

2024

Journal of Topology

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We provide a construction of equivariant Lagrangian Floer homology , for a compact Lie group acting on a symplectic manifold in a Hamiltonian fashion, and a pair of ‐Lagrangian submanifolds . We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of . Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are ‐bimodules. In the case when , we show that their chain complex is homotopy equivalent to the equivariant Morse complex of . Furthermore, if zero is a regular value of the moment map and if acts freely on , we construct two ‘Kirwan morphisms’ from to (respectively, from to ). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat ‐connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah–Floer conjecture.

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Corrigendum: A flexible construction of equivariant Floer homology and applications

Cited by 5 publications

References 15 publications

Bigrading the symplectic Khovanov cohomology

Bigrading the symplectic Khovanov cohomology

Involutions, knots, and Floer K-theory

Equivariant Lagrangian Floer homology via cotangent bundles of EGN$EG_N$

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