Pub Date : 2019-01-20DOI: 10.4310/jsg.2021.v19.n6.a2
Zhenkun Li
In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $Ksubset Y$ and a base point $pin K$, we can associate the minus versions, $underline{rm KHM}^-(Y,K,p)$ and $underline{rm KHI}^-(Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $mathbb{Z}$-grading, and there is an $U$-map on the minus versions, which is of degree $-1$. We also prove other basic properties of them. If $Ksubset Y$ is not null-homologous but represents a torsion class, then we can also construct the corresponding minus versions for $(Y,K,p)$. We also proved a surgery-type formula relating the minus versions of a knot $K$ with those of the dual knot, when performing a Dehn surgery of large enough slope along $K$. The techniques developed in this paper can also be applied to compute the sutured monopole and instanton Floer homologies of any sutured solid tori.
{"title":"Knot homologies in monopole and instanton theories via sutures","authors":"Zhenkun Li","doi":"10.4310/jsg.2021.v19.n6.a2","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n6.a2","url":null,"abstract":"In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $Ksubset Y$ and a base point $pin K$, we can associate the minus versions, $underline{rm KHM}^-(Y,K,p)$ and $underline{rm KHI}^-(Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $mathbb{Z}$-grading, and there is an $U$-map on the minus versions, which is of degree $-1$. We also prove other basic properties of them. If $Ksubset Y$ is not null-homologous but represents a torsion class, then we can also construct the corresponding minus versions for $(Y,K,p)$. We also proved a surgery-type formula relating the minus versions of a knot $K$ with those of the dual knot, when performing a Dehn surgery of large enough slope along $K$. The techniques developed in this paper can also be applied to compute the sutured monopole and instanton Floer homologies of any sutured solid tori.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75886002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/jsg.2019.v17.n5.a6
Á. Pelayo, Xiudi Tang
Let M be a noncompact oriented connected manifold and let B be a compact manifold. We give conditions on two smooth families of volume forms { ω p } p ∈ B , { τ p } p ∈ B which guarantee the existence of a smooth family of diffeomorphisms { ϕ p } p ∈ B such that ϕ ∗ p ω p = τ p for all p ∈ B . If B is a point, our result recovers a theorem of Greene and Shiohama from 1979, which itself extended a theorem of Moser for compact manifolds.
{"title":"Moser–Greene–Shiohama stability for families","authors":"Á. Pelayo, Xiudi Tang","doi":"10.4310/jsg.2019.v17.n5.a6","DOIUrl":"https://doi.org/10.4310/jsg.2019.v17.n5.a6","url":null,"abstract":"Let M be a noncompact oriented connected manifold and let B be a compact manifold. We give conditions on two smooth families of volume forms { ω p } p ∈ B , { τ p } p ∈ B which guarantee the existence of a smooth family of diffeomorphisms { ϕ p } p ∈ B such that ϕ ∗ p ω p = τ p for all p ∈ B . If B is a point, our result recovers a theorem of Greene and Shiohama from 1979, which itself extended a theorem of Moser for compact manifolds.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73208579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/jsg.2019.v17.n4.a4
J. Conway
We show that all positive contact surgeries on every Legendrian figure-eight knot in ( S 3 , ξ std ) result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.
{"title":"Contact surgeries on Legendrian figure-eight knots","authors":"J. Conway","doi":"10.4310/jsg.2019.v17.n4.a4","DOIUrl":"https://doi.org/10.4310/jsg.2019.v17.n4.a4","url":null,"abstract":"We show that all positive contact surgeries on every Legendrian figure-eight knot in ( S 3 , ξ std ) result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88922512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/jsg.2019.v17.n5.a2
Nicolina Istrati
We prove that a compact toric locally conformally K¨ahler manifold which is not K¨ahler admits a toric Vaisman structure. This is the final step leading to the classification of compact toric locally conformally K¨ahler manifolds. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally K¨ahler.
{"title":"A characterisation of toric locally conformally Kähler manifolds","authors":"Nicolina Istrati","doi":"10.4310/jsg.2019.v17.n5.a2","DOIUrl":"https://doi.org/10.4310/jsg.2019.v17.n5.a2","url":null,"abstract":"We prove that a compact toric locally conformally K¨ahler manifold which is not K¨ahler admits a toric Vaisman structure. This is the final step leading to the classification of compact toric locally conformally K¨ahler manifolds. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally K¨ahler.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81757668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/JSG.2019.V17.N1.A5
Morimichi Kawasaki
We show that the union of some circles in a closed Riemannian surface with positive genus is superheavy in the sense of Entov-Polterovich. By a result of Entov and Polterovich, this implies that the product of this union and the Clifford torus of C P n with the Fubini-Study symplectic form cannot be displaced by any symplec-tomorphisms.
在Entov-Polterovich意义上证明了具有正格的闭黎曼曲面上一些圆的并是超重的。通过Entov和Polterovich的结果，这意味着该并与具有Fubini-Study辛形式的C P n的Clifford环的乘积不能被任何辛形态所取代。
{"title":"Superheavy Lagrangian immersions in surfaces","authors":"Morimichi Kawasaki","doi":"10.4310/JSG.2019.V17.N1.A5","DOIUrl":"https://doi.org/10.4310/JSG.2019.V17.N1.A5","url":null,"abstract":"We show that the union of some circles in a closed Riemannian surface with positive genus is superheavy in the sense of Entov-Polterovich. By a result of Entov and Polterovich, this implies that the product of this union and the Clifford torus of C P n with the Fubini-Study symplectic form cannot be displaced by any symplec-tomorphisms.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88230654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/jsg.2019.v17.n5.a3
Laurent La Fuente-Gravy
We study obstructions to the existence of closed Fedosov star products on a given Kähler manifold (M,ω, J). In our previous paper [14], we proved that the Levi-Civita connection of a Kähler manifold will produce a closed Fedosov star product (closed in the sense of Connes–Flato–Sternheimer [4]) only if it is a zero of a moment map μ on the space of symplectic connections. By analogy with the Futaki invariant obstructing the existence of constant scalar curvature Kähler metric, we build an obstruction for the existence of zero of μ and hence for the existence of closed Fedosov star product on a Kähler manifold.
{"title":"Futaki invariant for Fedosov star products","authors":"Laurent La Fuente-Gravy","doi":"10.4310/jsg.2019.v17.n5.a3","DOIUrl":"https://doi.org/10.4310/jsg.2019.v17.n5.a3","url":null,"abstract":"We study obstructions to the existence of closed Fedosov star products on a given Kähler manifold (M,ω, J). In our previous paper [14], we proved that the Levi-Civita connection of a Kähler manifold will produce a closed Fedosov star product (closed in the sense of Connes–Flato–Sternheimer [4]) only if it is a zero of a moment map μ on the space of symplectic connections. By analogy with the Futaki invariant obstructing the existence of constant scalar curvature Kähler metric, we build an obstruction for the existence of zero of μ and hence for the existence of closed Fedosov star product on a Kähler manifold.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81660291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/jsg.2019.v17.n6.a8
Jiefeng Liu, Y. Sheng
{"title":"$mathrm{QP}$-structures of degree $3$ and $mathsf{CLWX} : 2$-algebroids","authors":"Jiefeng Liu, Y. Sheng","doi":"10.4310/jsg.2019.v17.n6.a8","DOIUrl":"https://doi.org/10.4310/jsg.2019.v17.n6.a8","url":null,"abstract":"","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89586791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-05DOI: 10.4310/jsg.2021.v19.n3.a2
Kei Irie
We prove that, for a $C^infty$-generic contact form $lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $dlambda$. This is a quantitative refinement of the $C^infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro-Gardiner, Hutchings, Ramos, and inspired by the argument of Marques-Neves-Song, who proved a similar equidistribution result for minimal hypersurfaces. We also discuss a question about generic behavior of periodic Reeb orbits "representing" ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.
{"title":"Equidistributed periodic orbits of $C^infty$-generic three-dimensional Reeb flows","authors":"Kei Irie","doi":"10.4310/jsg.2021.v19.n3.a2","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n3.a2","url":null,"abstract":"We prove that, for a $C^infty$-generic contact form $lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $dlambda$. This is a quantitative refinement of the $C^infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro-Gardiner, Hutchings, Ramos, and inspired by the argument of Marques-Neves-Song, who proved a similar equidistribution result for minimal hypersurfaces. We also discuss a question about generic behavior of periodic Reeb orbits \"representing\" ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72835663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-31DOI: 10.4310/jsg.2021.v19.n6.a4
D. Vela-Vick, C.-M. Michael Wong
We show that the bordered-sutured Floer invariant of the complement of a tangle in an arbitrary 3-manifold $Y$, with minimal conditions on the bordered-sutured structure, satisfies an unoriented skein exact triangle. This generalizes a theorem by Manolescu for links in $S^3$. We give a theoretical proof of this result by adapting holomorphic polygon counts to the bordered-sutured setting, and also give a combinatorial description of all maps involved and explicitly compute them. We then show that, for $Y = S^3$, our exact triangle coincides with Manolescu's. Finally, we provide a graded version of our result, explaining in detail the grading reduction process involved.
{"title":"An unoriented skein relation via bordered–sutured Floer homology","authors":"D. Vela-Vick, C.-M. Michael Wong","doi":"10.4310/jsg.2021.v19.n6.a4","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n6.a4","url":null,"abstract":"We show that the bordered-sutured Floer invariant of the complement of a tangle in an arbitrary 3-manifold $Y$, with minimal conditions on the bordered-sutured structure, satisfies an unoriented skein exact triangle. This generalizes a theorem by Manolescu for links in $S^3$. We give a theoretical proof of this result by adapting holomorphic polygon counts to the bordered-sutured setting, and also give a combinatorial description of all maps involved and explicitly compute them. We then show that, for $Y = S^3$, our exact triangle coincides with Manolescu's. Finally, we provide a graded version of our result, explaining in detail the grading reduction process involved.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79600160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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