Pub Date : 2020-12-01DOI: 10.2140/agt.2022.22.1417
Pietro Capovilla, C. Loeh, M. Moraschini
Amenable category is a variant of the Lusternik-Schnirelman category, based on covers by amenable open subsets. We study the monotonicity problem for degree-one maps and amenable category and the relation between amenable category and topological complexity.
{"title":"Amenable category and complexity","authors":"Pietro Capovilla, C. Loeh, M. Moraschini","doi":"10.2140/agt.2022.22.1417","DOIUrl":"https://doi.org/10.2140/agt.2022.22.1417","url":null,"abstract":"Amenable category is a variant of the Lusternik-Schnirelman category, based on covers by amenable open subsets. We study the monotonicity problem for degree-one maps and amenable category and the relation between amenable category and topological complexity.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82171803","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 : 2020-11-24DOI: 10.2140/agt.2022.22.3939
David A. Will
For $D$ a reduced alternating surface link diagram, we bound the twist number of $D$ in terms of the coefficients of a polynomial invariant. To this end, we introduce a generalization of the homological Kauffman bracket defined by Krushkal. Combined with work of Futer, Kalfagianni, and Purcell, this yields a bound for the hyperbolic volume of a class of alternating surface links in terms of these coefficients.
{"title":"Homological polynomial coefficients and the twist number of alternating surface links","authors":"David A. Will","doi":"10.2140/agt.2022.22.3939","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3939","url":null,"abstract":"For $D$ a reduced alternating surface link diagram, we bound the twist number of $D$ in terms of the coefficients of a polynomial invariant. To this end, we introduce a generalization of the homological Kauffman bracket defined by Krushkal. Combined with work of Futer, Kalfagianni, and Purcell, this yields a bound for the hyperbolic volume of a class of alternating surface links in terms of these coefficients.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"43 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73289518","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 : 2020-11-20DOI: 10.2140/agt.2022.22.2805
Scott Balchin, J. Greenlees, Luca Pol, J. Williamson
Given a suitable stable monoidal model category $mathscr{C}$ and a specialization closed subset $V$ of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over $V$ and the part supported over $V^c$ spliced with the Tate object. Using this one can show that $mathscr{C}$ is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra from [16] to a Quillen equivalence. In addition, a close analysis of the one step case highlights important features needed for general torsion models which we will return to in future work.
给定一个合适的稳定单轴模型范畴$mathscr{C}$和它的Balmer谱的一个专门化闭子集$V$，可以得到一个Tate平方，将对象分解为$V$上支持的部分和$V^ C $上支持的与Tate对象拼接的部分。使用它可以表明$mathscr{C}$是Quillen等价于由局部扭转对象数据构建的模型，并且拼接数据属于相当丰富的类别。作为应用，我们将有理圆等变谱同伦范畴的扭转模型从[16]提升到Quillen等价。此外，对单步情况的仔细分析突出了一般扭转模型所需的重要特征，我们将在未来的工作中回到这些特征。
{"title":"Torsion models for tensor-triangulated categories: the one-step case","authors":"Scott Balchin, J. Greenlees, Luca Pol, J. Williamson","doi":"10.2140/agt.2022.22.2805","DOIUrl":"https://doi.org/10.2140/agt.2022.22.2805","url":null,"abstract":"Given a suitable stable monoidal model category $mathscr{C}$ and a specialization closed subset $V$ of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over $V$ and the part supported over $V^c$ spliced with the Tate object. Using this one can show that $mathscr{C}$ is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra from [16] to a Quillen equivalence. In addition, a close analysis of the one step case highlights important features needed for general torsion models which we will return to in future work.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"118 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76004864","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 : 2020-11-06DOI: 10.2140/agt.2022.22.2915
Daniel Kasprowski, John Nicholson, Benjamin Matthias Ruppik
We show that the homotopy type of an oriented Poincare 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. This applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3), examples of which are elliptic surfaces with finite fundamental group.
{"title":"Homotopy classification of 4–manifolds whose\u0000fundamental group is dihedral","authors":"Daniel Kasprowski, John Nicholson, Benjamin Matthias Ruppik","doi":"10.2140/agt.2022.22.2915","DOIUrl":"https://doi.org/10.2140/agt.2022.22.2915","url":null,"abstract":"We show that the homotopy type of an oriented Poincare 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. This applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3), examples of which are elliptic surfaces with finite fundamental group.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"33 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87826211","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 : 2020-11-02DOI: 10.2140/agt.2022.22.1969
Pritam Ghosh, Mahan Mj
Answering a question due to Min, we prove that a finite graph of roses admits a regluing such that the resulting graph of roses has hyperbolic fundamental group.
{"title":"Regluing graphs of free groups","authors":"Pritam Ghosh, Mahan Mj","doi":"10.2140/agt.2022.22.1969","DOIUrl":"https://doi.org/10.2140/agt.2022.22.1969","url":null,"abstract":"Answering a question due to Min, we prove that a finite graph of roses admits a regluing such that the resulting graph of roses has hyperbolic fundamental group.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"70 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72930205","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}
Given an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration: $mathcal{A} cap J_2$. We introduce two trace-like operators, inspired by Morita's trace, and show that their kernels coincide with the images by the second Johnson homomorphism $tau_2$ of $J_2$ and $mathcal{A} cap J_2$, respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of $tau_2(mathcal{A} cap J_2)$. By the same techniques, and for a Heegaard surface in $S^3$, we also compute the image by $tau_2$ of the intersection of the Goeritz group $mathcal{G}$ with $J_2$.
给定一个有向曲面包围着一个柄体，我们研究了它的映射类群的子群，这个映射类群被定义为柄体群与Johnson过滤的第二项的交集:$mathcal{A} cap J_2$。受Morita的迹启发，我们引入了两个类迹算子，并分别通过$J_2$和$mathcal{A} cap J_2$的二次Johnson同态$tau_2$证明了它们的核与图像重合。特别地，我们以否定的方式回答Levine提出的关于$tau_2(mathcal{A} cap J_2)$的代数描述的问题。通过相同的技术，对于$S^3$中的Heegaard曲面，我们还通过$tau_2$计算Goeritz群$mathcal{G}$与$J_2$相交的图像。
{"title":"The handlebody group and the images of the second Johnson homomorphism","authors":"Quentin Faes","doi":"10.2140/agt.2023.23.243","DOIUrl":"https://doi.org/10.2140/agt.2023.23.243","url":null,"abstract":"Given an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration: $mathcal{A} cap J_2$. We introduce two trace-like operators, inspired by Morita's trace, and show that their kernels coincide with the images by the second Johnson homomorphism $tau_2$ of $J_2$ and $mathcal{A} cap J_2$, respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of $tau_2(mathcal{A} cap J_2)$. By the same techniques, and for a Heegaard surface in $S^3$, we also compute the image by $tau_2$ of the intersection of the Goeritz group $mathcal{G}$ with $J_2$.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76881715","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}
The homotopy interleaving distance, a distance between persistent spaces, was introduced by Blumberg and Lesnick and shown to be universal, in the sense that it is the largest homotopy-invariant distance for which sublevel-set filtrations of close-by real-valued functions are close-by. There are other ways of constructing homotopy-invariant distances, but not much is known about the relationships between these choices. We show that other natural distances differ from the homotopy interleaving distance in at most a multiplicative constant, and prove versions of the persistent Whitehead theorem, a conjecture of Blumberg and Lesnick that relates morphisms that induce interleavings in persistent homotopy groups to stronger homotopy-invariant notions of interleaving.
{"title":"Rectification of interleavings and a persistent Whitehead theorem","authors":"Edoardo Lanari, Luis Scoccola","doi":"10.2140/agt.2023.23.803","DOIUrl":"https://doi.org/10.2140/agt.2023.23.803","url":null,"abstract":"The homotopy interleaving distance, a distance between persistent spaces, was introduced by Blumberg and Lesnick and shown to be universal, in the sense that it is the largest homotopy-invariant distance for which sublevel-set filtrations of close-by real-valued functions are close-by. There are other ways of constructing homotopy-invariant distances, but not much is known about the relationships between these choices. We show that other natural distances differ from the homotopy interleaving distance in at most a multiplicative constant, and prove versions of the persistent Whitehead theorem, a conjecture of Blumberg and Lesnick that relates morphisms that induce interleavings in persistent homotopy groups to stronger homotopy-invariant notions of interleaving.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"220 2 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90765449","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}
We give a short proof that the $L^p$-diameter of the group of area preserving diffeomorphisms isotopic to the identity of a compact surface is infinite.
给出了紧曲面恒等的保面积微分同胚群的L^p -直径是无限的一个简短证明。
{"title":"A short proof that the Lp–diameter of\u0000Diff0(S,area) is infinite","authors":"Michał Marcinkowski","doi":"10.2140/agt.2023.23.883","DOIUrl":"https://doi.org/10.2140/agt.2023.23.883","url":null,"abstract":"We give a short proof that the $L^p$-diameter of the group of area preserving diffeomorphisms isotopic to the identity of a compact surface is infinite.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75396708","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 : 2020-10-05DOI: 10.2140/agt.2022.22.2395
Daniel Ruberman
Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the classical Levine-Tristram invariant of a knot. In the 3-dimensional situation, a count of singular connections on a knot complement reproduces the Levine-Tristram invariant. We compute the invariant for a number of embedded tori, and compare with what one might expect from Echeverria's invariant. For the simplest example--the product of an ordinary knot with a circle--the answers coincide. But for more general examples, the invariants are different.
{"title":"A Levine–Tristram invariant for knotted\u0000tori","authors":"Daniel Ruberman","doi":"10.2140/agt.2022.22.2395","DOIUrl":"https://doi.org/10.2140/agt.2022.22.2395","url":null,"abstract":"Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the classical Levine-Tristram invariant of a knot. In the 3-dimensional situation, a count of singular connections on a knot complement reproduces the Levine-Tristram invariant. We compute the invariant for a number of embedded tori, and compare with what one might expect from Echeverria's invariant. For the simplest example--the product of an ordinary knot with a circle--the answers coincide. But for more general examples, the invariants are different.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"81 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79297264","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 : 2020-10-01DOI: 10.2140/agt.2022.22.2867
Dimitri Ara, G. Maltsiniotis
Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the classes of $n$-categorical weak equivalences that they define coincide: they are the Thomason equivalences. We give two applications of this result: the first one states that a Dyer-Kan-type equivalence for Thomason equivalences is a Thomason equivalence; the second one, fundamental, is the stability of the class of Thomason equivalences under the dualities of the category of strict $n$-categories.
{"title":"Comparaison des nerfs n–catégoriques","authors":"Dimitri Ara, G. Maltsiniotis","doi":"10.2140/agt.2022.22.2867","DOIUrl":"https://doi.org/10.2140/agt.2022.22.2867","url":null,"abstract":"Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the classes of $n$-categorical weak equivalences that they define coincide: they are the Thomason equivalences. We give two applications of this result: the first one states that a Dyer-Kan-type equivalence for Thomason equivalences is a Thomason equivalence; the second one, fundamental, is the stability of the class of Thomason equivalences under the dualities of the category of strict $n$-categories.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83110303","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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