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Simplicial model structures on pro-categories 亲范畴上的简单模型结构
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-11-05 DOI: 10.2140/agt.2023.23.3849
Thomas Blom, Ieke Moerdijk
{"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":null,"pages":null},"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}
引用次数: 6
The group of quasi-isometries of the real line cannot act effectively on the line 实线的拟等距组不能有效地作用于实线上
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-11-05 DOI: 10.2140/agt.2023.23.3835
Shengkui Ye, Yanxin Zhao
{"title":"The group of quasi-isometries of the real line cannot act effectively on the line","authors":"Shengkui Ye, Yanxin Zhao","doi":"10.2140/agt.2023.23.3835","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3835","url":null,"abstract":"We prove that the group $mathrm{QI}^{+}(mathbb{R})$ of orientation-preserving quasi-isometries of the real line is a left-orderable, non-simple group, which cannot act effectively on the real line $mathbb{R}.$","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726056","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}
引用次数: 1
Smooth one-dimensional topological field theories are vector bundles with connection 光滑一维拓扑场理论是具有连接的向量束
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-11-05 DOI: 10.2140/agt.2023.23.3707
Daniel Berwick-Evans, Dmitri Pavlov
{"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":null,"pages":null},"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}
引用次数: 4
On some p–differential graded link homologies, II 关于一些p微分分级连杆同调，ⅱ
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-09-26 DOI: 10.2140/agt.2023.23.3357
You Qi, Joshua Sussan
{"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":null,"pages":null},"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}
引用次数: 0
The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost-Hopf rings B型和D型Coxeter群无穷族的模2上同调
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-09-26 DOI: 10.2140/agt.2023.23.3221
L. Guerra
{"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":null,"pages":null},"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}
引用次数: 2
Leighton’s theorem and regular cube complexes 雷顿定理和正立方复合体
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-09-26 DOI: 10.2140/agt.2023.23.3395
Daniel J. Woodhouse
{"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":null,"pages":null},"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}
引用次数: 1
Detecting isomorphisms in the homotopy category 检测同伦范畴中的同构
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-09-26 DOI: 10.2140/agt.2023.23.2975
Kevin Arlin, J. Daniel Christensen
{"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":null,"pages":null},"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}
引用次数: 2
Mod 2 power operations revisited Mod 2电源操作重访
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-09-26 DOI: 10.2140/agt.2023.23.2993
Dylan Wilson
{"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":null,"pages":null},"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}
引用次数: 1
On the wheeled PROP of stable cohomology of Aut(Fn) with bivariant coefficients 二元系数Aut(Fn)稳定上同调的轮式PROP
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-09-26 DOI: 10.2140/agt.2023.23.3089
Nariya Kawazumi, Christine Vespa
{"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":null,"pages":null},"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}
引用次数: 4
Operads in unstable global homotopy theory 不稳定全局同伦理论中的算子
3区 数学
Algebraic and Geometric Topology Pub Date : 2023-09-26 DOI: 10.2140/agt.2023.23.3293
Miguel Barrero
{"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":null,"pages":null},"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}
引用次数: 3
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