We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions of finite orbi-CW-pairs by vector bundles and whose morphisms are stable homotopy classes of (representable) relative maps. The stable representable homotopy category of finite orbispectra admits a contravariant involution extending Spanier--Whitehead duality. This duality relates homotopical cobordism theories (cohomology theories on finite orbispectra) represented by global Thom spectra to geometric (derived) orbifold bordism groups (homology theories on finite orbispectra). This isomorphism extends the classical Pontryagin--Thom isomorphism and its known equivariant generalizations.
构造了有限轨道双谱的稳定(可表示)同伦范畴，其对象是有限轨道- w -对通过向量束的形式悬空，其态射是(可表示)相对映射的稳定同伦类。有限轨道谱的稳定可表示同伦范畴承认一个逆变对合扩展的Spanier—Whitehead对偶。这种对偶性将全局性Thom谱表示的同位共调理论(有限轨道谱上的上同调理论)与几何(衍生)轨道共调群(有限轨道谱上的同调理论)联系起来。这种同构扩展了经典的庞特里亚金-托姆同构及其已知的等变推广。
{"title":"Orbifold bordism and duality for finite orbispectra","authors":"J. Pardon","doi":"10.2140/gt.2023.27.1747","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1747","url":null,"abstract":"We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions of finite orbi-CW-pairs by vector bundles and whose morphisms are stable homotopy classes of (representable) relative maps. The stable representable homotopy category of finite orbispectra admits a contravariant involution extending Spanier--Whitehead duality. This duality relates homotopical cobordism theories (cohomology theories on finite orbispectra) represented by global Thom spectra to geometric (derived) orbifold bordism groups (homology theories on finite orbispectra). This isomorphism extends the classical Pontryagin--Thom isomorphism and its known equivariant generalizations.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134227362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
T. Barthel, Daniel Berwick-Evans, Nathaniel J. Stapleton
The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.
{"title":"Power operations in the Stolz–Teichner\u0000program","authors":"T. Barthel, Daniel Berwick-Evans, Nathaniel J. Stapleton","doi":"10.2140/gt.2022.26.1773","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1773","url":null,"abstract":"The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"68 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120908100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory for this class of singularities was studied by de Jong-van Straten in [dJvS98]; they associated a germ of a singular plane curve to each singularity and described Milnor fibers via deformations of this singular curve. We consider links of surface singularities, equipped with their canonical contact structures, and develop a symplectic analog of de Jong-van Straten's construction. Using planar open books and Lefschetz fibrations, we describe all Stein fillings of the links via certain arrangements of symplectic disks, related by a homotopy to the plane curve germ of the singularity. As a consequence, we show that many rational singularities in this class admit Stein fillings that are not strongly diffeomorphic to any Milnor fibers. This contrasts with previously known cases, such as simple and quotient surface singularities, where Milnor fibers are known to give rise to all Stein fillings. On the other hand, we show that if for a singularity with reduced fundamental cycle, the self-intersection of each exceptional curve is at most -5 in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber).
{"title":"Unexpected Stein fillings, rational surface singularities and plane curve arrangements","authors":"O. Plamenevskaya, Laura Starkston","doi":"10.2140/gt.2023.27.1083","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1083","url":null,"abstract":"We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory for this class of singularities was studied by de Jong-van Straten in [dJvS98]; they associated a germ of a singular plane curve to each singularity and described Milnor fibers via deformations of this singular curve. We consider links of surface singularities, equipped with their canonical contact structures, and develop a symplectic analog of de Jong-van Straten's construction. Using planar open books and Lefschetz fibrations, we describe all Stein fillings of the links via certain arrangements of symplectic disks, related by a homotopy to the plane curve germ of the singularity. As a consequence, we show that many rational singularities in this class admit Stein fillings that are not strongly diffeomorphic to any Milnor fibers. This contrasts with previously known cases, such as simple and quotient surface singularities, where Milnor fibers are known to give rise to all Stein fillings. On the other hand, we show that if for a singularity with reduced fundamental cycle, the self-intersection of each exceptional curve is at most -5 in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber).","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124071164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that L-space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer theoretic invariant for tangles.
我们利用奇异模技术证明了l空间结不存在本质的Conway球，这是缠结的一个花理论不变量。
{"title":"L–space knots have no essential Conway\u0000spheres","authors":"Tye Lidman, Allison H. Moore, Claudius Zibrowius","doi":"10.2140/gt.2022.26.2065","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2065","url":null,"abstract":"We prove that L-space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer theoretic invariant for tangles.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131424903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lukas Braun, Stefano Filipazzi, Joaqu'in Moraga, R. Svaldi
We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index one covers for $n$-dimensional klt singularities. We give an application to the study of local class groups of klt singularities.
{"title":"The Jordan property for local fundamental groups","authors":"Lukas Braun, Stefano Filipazzi, Joaqu'in Moraga, R. Svaldi","doi":"10.2140/gt.2022.26.283","DOIUrl":"https://doi.org/10.2140/gt.2022.26.283","url":null,"abstract":"We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index one covers for $n$-dimensional klt singularities. We give an application to the study of local class groups of klt singularities.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115876047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the trivial band.
{"title":"The cosmetic crossing conjecture for split links","authors":"Joshua Wang","doi":"10.2140/gt.2022.26.2941","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2941","url":null,"abstract":"Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the trivial band.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126844915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Lambda^{pm} = Lambda^{+} cup Lambda^{-} subset (mathbb{R}^{3}, xi_{std})$ be a contact surgery diagram determining a closed, connected contact $3$-manifold $(S^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$ and an open contact manifold $(mathbb{R}^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$. Following arXiv:0911.0026 and arXiv:1906.07228 we demonstrate how $Lambda^{pm}$ determines a family $alpha_{epsilon}$ of standard-at-infinity contact forms on $(mathbb{R}^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$ whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on $Lambda^{pm}$. We compute the homology classes and integral Conley-Zehnder indices of these orbits diagrammatically using a simultaneous framing of all orbits naturally determined by the surgery diagram, providing a (typically non-canonical) $mathbb{Z}$-grading on the chain complexes underlying the "hat" version of contact homology as defined in arXiv:1004.2942. Using holomorphic foliations, algebraic tools for studying holomorphic curves in symplectizations of and surgery cobordisms between the $(mathbb{R}^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$ are developed. We use these computational tools to provide the first examples of closed, tight, contact manifolds with vanishing contact homology -- contact $frac{1}{k}$ surgeries along the right-handed, $tb=1$ trefoil for $k > 0$, which are known to have non-zero Heegaard-Floer contact classes by arXiv:math/0404135.
{"title":"Combinatorial Reeb dynamics on punctured\u0000contact 3–manifolds","authors":"Russell Avdek","doi":"10.2140/gt.2023.27.953","DOIUrl":"https://doi.org/10.2140/gt.2023.27.953","url":null,"abstract":"Let $Lambda^{pm} = Lambda^{+} cup Lambda^{-} subset (mathbb{R}^{3}, xi_{std})$ be a contact surgery diagram determining a closed, connected contact $3$-manifold $(S^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$ and an open contact manifold $(mathbb{R}^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$. Following arXiv:0911.0026 and arXiv:1906.07228 we demonstrate how $Lambda^{pm}$ determines a family $alpha_{epsilon}$ of standard-at-infinity contact forms on $(mathbb{R}^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$ whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on $Lambda^{pm}$. \u0000We compute the homology classes and integral Conley-Zehnder indices of these orbits diagrammatically using a simultaneous framing of all orbits naturally determined by the surgery diagram, providing a (typically non-canonical) $mathbb{Z}$-grading on the chain complexes underlying the \"hat\" version of contact homology as defined in arXiv:1004.2942. Using holomorphic foliations, algebraic tools for studying holomorphic curves in symplectizations of and surgery cobordisms between the $(mathbb{R}^{3}_{Lambda^{pm}}, xi_{Lambda^{pm}})$ are developed. \u0000We use these computational tools to provide the first examples of closed, tight, contact manifolds with vanishing contact homology -- contact $frac{1}{k}$ surgeries along the right-handed, $tb=1$ trefoil for $k > 0$, which are known to have non-zero Heegaard-Floer contact classes by arXiv:math/0404135.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"137 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122770285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.
给出了任意域上任意有限图的无序位形空间的渐近Betti数的显式公式。
{"title":"Asymptotic homology of graph braid groups","authors":"B. An, Gabriel C. Drummond-Cole, Ben Knudsen","doi":"10.2140/gt.2022.26.1745","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1745","url":null,"abstract":"We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130276907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the second part of cite{Bre18} on the uniqueness of ancient $kappa$-solutions to higher dimensions. We show that for dimensions $n geq 4$ every noncompact, nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is $kappa$-noncollapsed is isometric to a family of shrinking round cylinders (or a quotient thereof) or the Bryant soliton.
{"title":"Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions","authors":"S. Brendle, Keaton Naff","doi":"10.2140/gt.2023.27.153","DOIUrl":"https://doi.org/10.2140/gt.2023.27.153","url":null,"abstract":"We extend the second part of cite{Bre18} on the uniqueness of ancient $kappa$-solutions to higher dimensions. We show that for dimensions $n geq 4$ every noncompact, nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is $kappa$-noncollapsed is isometric to a family of shrinking round cylinders (or a quotient thereof) or the Bryant soliton.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132491362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M to mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b text{Coh}(Y)$, and the wrapped Fukaya category $mathcal{W} (M)$ is isomorphic to $D^b text{Coh}(Y backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.
给定一个具有极大边界$D$的log Calabi-Yau曲面$Y$和不同的复杂结构，我们解释了如何构造一个镜像Lefschetz纤维$w: M 到$ mathbb{C}$，其中$M$是一个Weinstein四流形，使得$w$的有向Fukaya范畴同构于$D^b text{Coh}(Y)$，而包装的Fukaya范畴$mathcal{w}(M)$同构于$D^b text{Coh}(Y 反斜线D)$。我们构造了$M$与Gross-Hacking-Keel工作中产生的近环振动的总空间之间的显同构;当$D$为负定时，预期这是$D$的双尖平滑的米尔诺纤维。我们还将我们的镜像势$w$与一系列特殊情况$(Y,D)$的现有结构相匹配，特别是在Auroux-Katzarkov-Orlov和Abouzaid的工作中。
{"title":"Homological mirror symmetry for log\u0000Calabi–Yau surfaces","authors":"P. Hacking, Ailsa Keating","doi":"10.2140/gt.2022.26.3747","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3747","url":null,"abstract":"Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M to mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b text{Coh}(Y)$, and the wrapped Fukaya category $mathcal{W} (M)$ is isomorphic to $D^b text{Coh}(Y backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132828806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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