Symplectic homology and the Eilenberg–Steenrod
axioms

Cieliebak, Kai; Oancea, Alexandru

doi:10.2140/agt.2018.18.1953

Cited by 77 publications

(213 citation statements)

References 56 publications

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“…The following definition is due to Lazarev [13, Definition 3.6], and it generalizes the dynamically convex contact structures studied in [1,6,11]. A contact form α for the contact structure ξ is called regular, if all Reeb orbits of α are non-degenerate.…”

Section: Preliminaries On Fillings and Symplectic Homologymentioning

confidence: 99%

“…The differential arises from counting the solutions to the Floer equations; see [4,19,13] for details of the construction. The complex generated by the critical points in W is a subcomplex and the positive symplectic homology SH + * (W ; k) is defined to be the homology of the quotient complex; see [6]. Moreover, we have the following exact triangle.…”

Section: Preliminaries On Fillings and Symplectic Homologymentioning

confidence: 99%

“…Remark 2.11. Our grading convention of SH * (W ; k) follows the convention in [3,6,13]. [18,Theorem 6.6] was stated for the symplectic cohomology SH * (W ; k) and it is related to the symplectic homology above by SH * (W ; k) = SH n− * (W ; k).…”

Section: Proposition 28 ([19]mentioning

confidence: 99%

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Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

Zhou

2018

International Mathematics Research Notices

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For any asymptotically dynamically convex contact manifold Y , we show that SH * (W ) = 0 is a property independent of the choice of topologically simple (i.e. c 1 (W ) = 0 and π 1 (Y ) → π 1 (W ) is injective) Liouville filling W . In particular, if Y is the boundary of a flexible Weinstein domain, then any topologically simple Liouville filling W has vanishing symplectic homology. As a consequence, we answer a question of Lazarev partially: a contact manifold Y admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of Y . The vanishing result provides an obstruction to flexible fillability. As an application, we show that all Brieskorn manifolds of dimension ≥ 5 cannot be filled by flexible Weinstein manifolds.

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Section: Preliminaries On Fillings and Symplectic Homologymentioning

confidence: 99%

Section: Preliminaries On Fillings and Symplectic Homologymentioning

confidence: 99%

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Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

Zhou

2018

International Mathematics Research Notices

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“…It contradicts the no-escape lemma whose particular case is summarised below. The no-escape lemma, whose particular case appears below, is due to Abouzaid and Seidel [4,Lemmas 7.2 and 7.4]; see also [49,Lemma 19.5] and [23,Lemma 2.2]. It generalises the maximum principle for Floer solutions: while the maximum principle only applies inside a Liouville collar (or the symplectisation) of a contact manifold, the no-escape lemma allows an arbitrary Liouville cobordism instead of a collar.…”

Section: 7mentioning

confidence: 99%

From symplectic cohomology to Lagrangian enumerative geometry

Tonkonog

2019

Advances in Mathematics

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We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials.We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains. arXiv:1711.03292v3 [math.SG] 27 Jun 2019As a general prediction, rings of regular functions are mirror to degree zero symplectic cohomology. (This was proven by Ganatra and Pomerleano [35] for the complement to the anticanonical divisor itself, see Section 2.) So one expects:These rings can be complicated (much bigger than C[r]), but they carry a distinguished element, the restriction of W :It is natural to ask what is the symplectic counterpart, or the mirror, of this element. This is answered by Theorem 1.1, which can be summarised in the language of mirror symmetry as follows:The Borman-Sheridan class BS ∈ SH 0 (M ) is mirror to W |M .

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“…Our main goal in this section is to recall the definition of several kinds of symplectic homology groups associated with a compact symplectic manifold W with contact type boundary M . Our treatment of the subject is intentionally brief, for the most part the material is standard or nearly standard, and we refer the reader to numerous other sources for a more detailed discussion; see, e.g., [BO09a,BO09b,BO13,BO17,CO,GG16,Se,Vi99] and references therein. Throughout the paper, all homology groups are taken with rational coefficients unless specifically stated otherwise.…”

Section: Preliminariesmentioning

confidence: 99%

On the filtered symplectic homology of prequantization bundles

Ginzburg

Shon

2018

Int. J. Math.

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We study Reeb dynamics on prequantization circle bundles and the filtered (equivariant) symplectic homology of prequantization line bundles, aka negative line bundles, with symplectically aspherical base. We define (equivariant) symplectic capacities, obtain an upper bound on their growth, prove uniform instability of the filtered symplectic homology and touch upon the question of stable displacement. We also introduce a new algebraic structure on the positive (equivariant) symplectic homology capturing the free homotopy class of a closed Reeb orbit -the linking number filtration -and use it to give a new proof of the non-degenerate case of the contact Conley conjecture (i.e., the existence of infinitely many simple closed Reeb orbits), not relying on contact homology.

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Symplectic homology and the Eilenberg–Steenrod axioms

Cited by 77 publications

References 56 publications

Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

Vanishing of Symplectic Homology and Obstruction to Flexible Fillability

From symplectic cohomology to Lagrangian enumerative geometry

On the filtered symplectic homology of prequantization bundles

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