Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3497
Hakho Choi, Jongil Park
In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous complex surface singularity.
{"title":"On symplectic fillings of small Seifert 3–manifolds","authors":"Hakho Choi, Jongil Park","doi":"10.2140/agt.2023.23.3497","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3497","url":null,"abstract":"In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous complex surface singularity.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"38 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135724066","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 : 2023-11-05DOI: 10.2140/agt.2023.23.3849
Thomas Blom, Ieke Moerdijk
We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, such as its functorial behaviour and its relation to Bousfield localization. We compare our construction to the infinity-categorical approach to ind- and pro-categories in an appendix.
{"title":"Simplicial model structures on pro-categories","authors":"Thomas Blom, Ieke Moerdijk","doi":"10.2140/agt.2023.23.3849","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3849","url":null,"abstract":"We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, such as its functorial behaviour and its relation to Bousfield localization. We compare our construction to the infinity-categorical approach to ind- and pro-categories in an appendix.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"38 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135724068","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 : 2023-11-05DOI: 10.2140/agt.2023.23.3707
Daniel Berwick-Evans, Dmitri Pavlov
We prove that smooth 1-dimensional topological field theories over a manifold are the same as vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth generalization of Rezk's complete Segal spaces. With such a definition in hand, we analyze the category of field theories using a combination of descent, a smooth version of the 1-dimensional cobordism hypothesis, and standard differential geometric arguments.
{"title":"Smooth one-dimensional topological field theories are vector bundles with connection","authors":"Daniel Berwick-Evans, Dmitri Pavlov","doi":"10.2140/agt.2023.23.3707","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3707","url":null,"abstract":"We prove that smooth 1-dimensional topological field theories over a manifold are the same as vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth generalization of Rezk's complete Segal spaces. With such a definition in hand, we analyze the category of field theories using a combination of descent, a smooth version of the 1-dimensional cobordism hypothesis, and standard differential geometric arguments.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"38 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135724067","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 : 2023-09-26DOI: 10.2140/agt.2023.23.3221
L. Guerra
We describe a Hopf ring structure on the direct sum of the cohomology groups $bigoplus_{n geq 0} H^* left( W_{B_n}; mathbb{F}_2 right)$ of the Coxeter groups of type $B_n$, and an almost-Hopf ring structure on the direct sum of the cohomology groups $bigoplus_{n geq 0} H^* left( W_{D_n}; mathbb{F}_2 right)$ of the Coxeter groups of type $D_n$, with coefficient in the field with two elements $mathbb{F}_2$. We give presentations with generators and relations, determine additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by De Concini and Salvetti and in terms of their restriction to elementary abelian 2-subgroups.
{"title":"The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost-Hopf rings","authors":"L. Guerra","doi":"10.2140/agt.2023.23.3221","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3221","url":null,"abstract":"We describe a Hopf ring structure on the direct sum of the cohomology groups $bigoplus_{n geq 0} H^* left( W_{B_n}; mathbb{F}_2 right)$ of the Coxeter groups of type $B_n$, and an almost-Hopf ring structure on the direct sum of the cohomology groups $bigoplus_{n geq 0} H^* left( W_{D_n}; mathbb{F}_2 right)$ of the Coxeter groups of type $D_n$, with coefficient in the field with two elements $mathbb{F}_2$. We give presentations with generators and relations, determine additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by De Concini and Salvetti and in terms of their restriction to elementary abelian 2-subgroups.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903419","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 : 2023-09-26DOI: 10.2140/agt.2023.23.3357
You Qi, Joshua Sussan
In arXiv:2009.06498, a link invariant categorifying the Jones polynomial at a $2p$th root of unity, where $p$ is an odd prime, was constructed. This categorification utilized an $N=2$ specialization of a differential introduced by Cautis. Here we give a family of link homologies where the Cautis differential is specialized to a positive integer of the form $N=kp+2$. When $k$ is even, all these link homologies categorify the Jones polynomial evaluated at a $2p$th root of unity, but they are non-isomorphic invariants.
{"title":"On some p–differential graded link homologies, II","authors":"You Qi, Joshua Sussan","doi":"10.2140/agt.2023.23.3357","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3357","url":null,"abstract":"In arXiv:2009.06498, a link invariant categorifying the Jones polynomial at a $2p$th root of unity, where $p$ is an odd prime, was constructed. This categorification utilized an $N=2$ specialization of a differential introduced by Cautis. Here we give a family of link homologies where the Cautis differential is specialized to a positive integer of the form $N=kp+2$. When $k$ is even, all these link homologies categorify the Jones polynomial evaluated at a $2p$th root of unity, but they are non-isomorphic invariants.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135720958","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 : 2023-09-26DOI: 10.2140/agt.2023.23.3395
Daniel J. Woodhouse
Leighton's graph covering theorem states that two finite graphs with common universal cover have a common finite cover. We generalize this to a large family of non-positively curved special cube complexes that form a natural generalization of regular graphs. This family includes both hyperbolic and non-hyperbolic CAT(0) cube complexes.
{"title":"Leighton’s theorem and regular cube complexes","authors":"Daniel J. Woodhouse","doi":"10.2140/agt.2023.23.3395","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3395","url":null,"abstract":"Leighton's graph covering theorem states that two finite graphs with common universal cover have a common finite cover. We generalize this to a large family of non-positively curved special cube complexes that form a natural generalization of regular graphs. This family includes both hyperbolic and non-hyperbolic CAT(0) cube complexes.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903225","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 : 2023-09-26DOI: 10.2140/agt.2023.23.2975
Kevin Arlin, J. Daniel Christensen
We show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms by giving an explicit counterexample involving large symmetric groups. We also show that, in contrast, the spheres jointly reflect equivalences in the homotopy 2-category of spaces. The non-existence of such a set in the homotopy category was originally claimed by Heller, but his argument relied on the statement that for every set of spaces, long enough transfinite sequential diagrams admit weak colimits which are privileged with respect to the given set. Using the theory of graphs of groups, we show that this statement is false, by proving that for every ordinal with uncountable cofinality, there is a diagram indexed by that ordinal which admits no weak colimit that is privileged with respect to the spheres.
{"title":"Detecting isomorphisms in the homotopy category","authors":"Kevin Arlin, J. Daniel Christensen","doi":"10.2140/agt.2023.23.2975","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2975","url":null,"abstract":"We show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms by giving an explicit counterexample involving large symmetric groups. We also show that, in contrast, the spheres jointly reflect equivalences in the homotopy 2-category of spaces. The non-existence of such a set in the homotopy category was originally claimed by Heller, but his argument relied on the statement that for every set of spaces, long enough transfinite sequential diagrams admit weak colimits which are privileged with respect to the given set. Using the theory of graphs of groups, we show that this statement is false, by proving that for every ordinal with uncountable cofinality, there is a diagram indexed by that ordinal which admits no weak colimit that is privileged with respect to the spheres.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134904246","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 : 2023-09-26DOI: 10.2140/agt.2023.23.2993
Dylan Wilson
In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of $mathbb{E}_{infty}$-ring spectra. The main advance is a quick proof of the Adem relations utilizing the Tate-valued Frobenius as a homotopical incarnation of the total power operation. We also give a streamlined derivation of the action of power operations on the dual Steenrod algebra.
{"title":"Mod 2 power operations revisited","authors":"Dylan Wilson","doi":"10.2140/agt.2023.23.2993","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2993","url":null,"abstract":"In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of $mathbb{E}_{infty}$-ring spectra. The main advance is a quick proof of the Adem relations utilizing the Tate-valued Frobenius as a homotopical incarnation of the total power operation. We also give a streamlined derivation of the action of power operations on the dual Steenrod algebra.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134904245","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 : 2023-09-26DOI: 10.2140/agt.2023.23.3089
Nariya Kawazumi, Christine Vespa
We show that the stable cohomology of automorphism groups of free groups with coefficients obtained by applying Hom(−, −) to tensor powers of the abelianization, is equipped with the structure of a wheeled PROP H. We define another wheeled PROP E by Ext-groups in the category of functors from the category of finitely generated free groups to k-modules. The main result of this paper is the construction of a morphism of wheeled PROPs ϕ : E → H such that ϕ(E) is the wheeled PROP generated by the cohomology class h 1 constructed by the first author.
{"title":"On the wheeled PROP of stable cohomology of Aut(Fn) with bivariant coefficients","authors":"Nariya Kawazumi, Christine Vespa","doi":"10.2140/agt.2023.23.3089","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3089","url":null,"abstract":"We show that the stable cohomology of automorphism groups of free groups with coefficients obtained by applying Hom(−, −) to tensor powers of the abelianization, is equipped with the structure of a wheeled PROP H. We define another wheeled PROP E by Ext-groups in the category of functors from the category of finitely generated free groups to k-modules. The main result of this paper is the construction of a morphism of wheeled PROPs ϕ : E → H such that ϕ(E) is the wheeled PROP generated by the cohomology class h 1 constructed by the first author.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903428","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 : 2023-09-26DOI: 10.2140/agt.2023.23.3293
Miguel Barrero
In this paper we study operads in unstable global homotopy theory, which is the homotopy theory of spaces with compatible actions by all compact Lie groups. We show that the theory of these operads works remarkably well, as for example it is possible to give a model structure for the category of algebras over any such operad. We define global $E_infty$-operads, a good generalization of $E_infty$-operads to the global setting, and we give a rectification result for algebras over them.
{"title":"Operads in unstable global homotopy theory","authors":"Miguel Barrero","doi":"10.2140/agt.2023.23.3293","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3293","url":null,"abstract":"In this paper we study operads in unstable global homotopy theory, which is the homotopy theory of spaces with compatible actions by all compact Lie groups. We show that the theory of these operads works remarkably well, as for example it is possible to give a model structure for the category of algebras over any such operad. We define global $E_infty$-operads, a good generalization of $E_infty$-operads to the global setting, and we give a rectification result for algebras over them.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903227","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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