Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.
{"title":"Classification of tight contact structures on surgeries on the figure-eight knot","authors":"J. Conway, Hyunki Min","doi":"10.2140/GT.2020.24.1457","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1457","url":null,"abstract":"Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2019-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82237159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperk"ahler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete non-compact Spin(7) manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7) metrics on the same smooth 8-manifold.
{"title":"Complete noncompact Spin(7) manifolds from\u0000self-dual Einstein 4–orbifolds","authors":"Lorenzo Foscolo","doi":"10.2140/GT.2021.25.339","DOIUrl":"https://doi.org/10.2140/GT.2021.25.339","url":null,"abstract":"We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperk\"ahler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete non-compact Spin(7) manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7) metrics on the same smooth 8-manifold.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2019-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91265252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article explain the basic property of the circle. It introduce an algorithm to draw the circle on digital devices using matrices which can be multiplied, added, and substrate very fast on GPU in parallel, algorithm added a precision for the real numbers calculation to be used for optimization and accuracy. The aim of this article is to explain that the use of drawing any circle is relative to the plane that it reside on and the plane might use any arbitrary measuring unit like meter, mile, inches,...etc. In addition, the plane location can be to other objects. TX-8-797-987 The Article can be found by this reference number in The United State Copyright Office.
{"title":"The Circle","authors":"Mohammad Alkahtani","doi":"10.36757/math-1-1-001","DOIUrl":"https://doi.org/10.36757/math-1-1-001","url":null,"abstract":"This article explain the basic property of the circle. It introduce an algorithm to draw the circle on digital devices using matrices which can be multiplied, added, and substrate very fast on GPU in parallel, algorithm added a precision for the real numbers calculation to be used for optimization and accuracy. The aim of this article is to explain that the use of drawing any circle is relative to the plane that it reside on and the plane might use any arbitrary measuring unit like meter, mile, inches,...etc. In addition, the plane location can be to other objects. TX-8-797-987 The Article can be found by this reference number in The United State Copyright Office.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75790100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations. The other variant is an equivariant Seiberg-Witten invariant of smooth group actions. We consider several applications of the gluing formula including: obstructions to smooth isotopy of diffeomorpihsms, computation of the mod $2$ Seiberg-Witten invariants of spin structures, relations between mod $2$ Seiberg-Witten invariants of $4$-manifolds and obstructions to the existence of invariant metrics of positive scalar curvature for smooth group actions on $4$-manifolds.
{"title":"A gluing formula for families Seiberg–Witten\u0000invariants","authors":"David Baraglia, Hokuto Konno","doi":"10.2140/GT.2020.24.1381","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1381","url":null,"abstract":"We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations. The other variant is an equivariant Seiberg-Witten invariant of smooth group actions. We consider several applications of the gluing formula including: obstructions to smooth isotopy of diffeomorpihsms, computation of the mod $2$ Seiberg-Witten invariants of spin structures, relations between mod $2$ Seiberg-Witten invariants of $4$-manifolds and obstructions to the existence of invariant metrics of positive scalar curvature for smooth group actions on $4$-manifolds.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88793276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that if $K$ is a fibered ribbon knot in $S^3=partial B^4$ bounding ribbon disk $D$, then with a transversality condition the fibration on $S^3setminusnu(K)$ extends to a fibration of $B^4setminusnu(D)$. This partially answers a question of Casson and Gordon. In particular, we show the fibration always extends when $D$ has exactly two local minima. More generally, we construct movies of singular fibrations on $4$-manifolds and describe a sufficient property of a movie to imply the underlying $4$-manifold is fibered over $S^1$.
{"title":"Extending fibrations of knot complements to ribbon disk complements","authors":"Maggie Miller","doi":"10.2140/gt.2021.25.1479","DOIUrl":"https://doi.org/10.2140/gt.2021.25.1479","url":null,"abstract":"We show that if $K$ is a fibered ribbon knot in $S^3=partial B^4$ bounding ribbon disk $D$, then with a transversality condition the fibration on $S^3setminusnu(K)$ extends to a fibration of $B^4setminusnu(D)$. This partially answers a question of Casson and Gordon. In particular, we show the fibration always extends when $D$ has exactly two local minima. More generally, we construct movies of singular fibrations on $4$-manifolds and describe a sufficient property of a movie to imply the underlying $4$-manifold is fibered over $S^1$.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2018-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75802654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the interplay of the homotopy coniveau tower, the Rost-Schmid complex of a strictly homotopy invariant sheaf, and homotopy modules. For a strictly homotopy invariant sheaf $M$, smooth $k$-scheme $X$ and $q geqslant 0$ we construct a novel cycle complex $C^*(X, M, q)$ and we prove that in favorable cases, $C^*(X, M, q)$ is equivalent to the homotopy coniveau tower $M^{(q)}(X)$. To do so we establish moving lemmas for the Rost-Schmid complex. As an application we deduce a cycle complex model for Milnor-Witt motivic cohomology. Furthermore we prove that if $M$ is a strictly homotopy invariant sheaf, then $M_{-2}$ is a homotopy module. Finally we conjecture that for $q>0$, $underline{pi}_0(M^{(q)})$ is a homotopy module, explain the significance of this conjecture for studying conservativity properties of the $mathbb{G}_m$-stabilization functor $mathcal{SH}^{S^1}!(k) to mathcal{SH}(k)$, and provide some evidence for the conjecture.
研究了同伦conveau塔、严格同伦不变束的Rost-Schmid复形和同伦模之间的相互作用。对于严格同伦不变束$M$、光滑$k$ -方案$X$和$q geqslant 0$，构造了一个新的循环复合体$C^*(X, M, q)$，并证明了在有利情况下，$C^*(X, M, q)$等价于同伦conveau塔$M^{(q)}(X)$。为此，我们建立了罗斯特-施密德复合体的移动引理。作为应用，我们推导了Milnor-Witt动力上同的循环复模型。进一步证明了如果$M$是严格同伦不变轴，则$M_{-2}$是一个同伦模。最后，我们推测对于$q>0$, $underline{pi}_0(M^{(q)})$是一个同伦模，解释了这一猜想对于研究$mathbb{G}_m$ -镇定函子$mathcal{SH}^{S^1}!(k) to mathcal{SH}(k)$的保守性的意义，并为这一猜想提供了一些证据。
{"title":"Towards conservativity of 𝔾m–stabilization","authors":"Tom Bachmann, Maria Yakerson","doi":"10.2140/GT.2020.24.1969","DOIUrl":"https://doi.org/10.2140/GT.2020.24.1969","url":null,"abstract":"We study the interplay of the homotopy coniveau tower, the Rost-Schmid complex of a strictly homotopy invariant sheaf, and homotopy modules. For a strictly homotopy invariant sheaf $M$, smooth $k$-scheme $X$ and $q geqslant 0$ we construct a novel cycle complex $C^*(X, M, q)$ and we prove that in favorable cases, $C^*(X, M, q)$ is equivalent to the homotopy coniveau tower $M^{(q)}(X)$. To do so we establish moving lemmas for the Rost-Schmid complex. As an application we deduce a cycle complex model for Milnor-Witt motivic cohomology. Furthermore we prove that if $M$ is a strictly homotopy invariant sheaf, then $M_{-2}$ is a homotopy module. Finally we conjecture that for $q>0$, $underline{pi}_0(M^{(q)})$ is a homotopy module, explain the significance of this conjecture for studying conservativity properties of the $mathbb{G}_m$-stabilization functor $mathcal{SH}^{S^1}!(k) to mathcal{SH}(k)$, and provide some evidence for the conjecture.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2018-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82365583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then X admits a branch covering over a product of curves inducing an isomorphism on rational cohomology in degree 1. We also study the class of Kodaira fibrations possessing a holomorphic section, and demonstrate that having a section imposes no restriction on possible monodromies.
{"title":"On Kodaira fibrations with invariant cohomology","authors":"Corey Bregman","doi":"10.2140/gt.2021.25.2385","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2385","url":null,"abstract":"A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then X admits a branch covering over a product of curves inducing an isomorphism on rational cohomology in degree 1. We also study the class of Kodaira fibrations possessing a holomorphic section, and demonstrate that having a section imposes no restriction on possible monodromies.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77501163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered continuation elements to prove that the Lagrangian spectral norm controls the barcode of the Hamiltonian perturbation of the Lagrangian submanifold, up to shift, in the bottleneck distance. Moreover, we show that it satisfies Chekanov type low-energy intersection phenomena, and non-degeneracy theorems. Secondly, we introduce a new averaging method for bounding the spectral norm from above, and apply it to produce precise uniform bounds on the Lagrangian spectral norm in certain closed symplectic manifolds. Finally, by using the theory of persistence modules, we prove that our bounds are in fact sharp in some cases. Along the way we produce a new calculation of the Lagrangian quantum homology of certain Lagrangian submanifolds, and answer a question of Usher.
{"title":"Bounds on spectral norms and barcodes","authors":"A. Kislev, E. Shelukhin","doi":"10.2140/gt.2021.25.3257","DOIUrl":"https://doi.org/10.2140/gt.2021.25.3257","url":null,"abstract":"We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered continuation elements to prove that the Lagrangian spectral norm controls the barcode of the Hamiltonian perturbation of the Lagrangian submanifold, up to shift, in the bottleneck distance. Moreover, we show that it satisfies Chekanov type low-energy intersection phenomena, and non-degeneracy theorems. Secondly, we introduce a new averaging method for bounding the spectral norm from above, and apply it to produce precise uniform bounds on the Lagrangian spectral norm in certain closed symplectic manifolds. Finally, by using the theory of persistence modules, we prove that our bounds are in fact sharp in some cases. Along the way we produce a new calculation of the Lagrangian quantum homology of certain Lagrangian submanifolds, and answer a question of Usher.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2018-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75231322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the nu^-(K). Moreover, the difference l(K)-nu^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.
{"title":"Knot Floer homology and the unknotting number","authors":"Akram Alishahi, Eaman Eftekhary","doi":"10.2140/gt.2020.24.2435","DOIUrl":"https://doi.org/10.2140/gt.2020.24.2435","url":null,"abstract":"Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the nu^-(K). Moreover, the difference l(K)-nu^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2018-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88243943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.
{"title":"Barcodes and area-preserving homeomorphisms","authors":"F. Roux, Sobhan Seyfaddini, C. Viterbo","doi":"10.2140/gt.2021.25.2713","DOIUrl":"https://doi.org/10.2140/gt.2021.25.2713","url":null,"abstract":"In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. \u0000Our main dynamical application concerns the notion of {it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. \u0000In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2018-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76072927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：