Pub Date : 2023-09-26DOI: 10.2140/agt.2023.23.3129
Rostislav Akhmechet, Mikhail Khovanov
We describe equivariant SL(2) and SL(3) homology for links in the solid torus via foam evaluation. The solid torus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams with boundary that may intersect the distinguished line; intersection points, called anchor points, contribute additional terms, reminiscent of square roots of the Hessian, to the foam evaluation. Both oriented and unoriented SL(3) foams are treated in the paper.
{"title":"Anchored foams and annular homology","authors":"Rostislav Akhmechet, Mikhail Khovanov","doi":"10.2140/agt.2023.23.3129","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3129","url":null,"abstract":"We describe equivariant SL(2) and SL(3) homology for links in the solid torus via foam evaluation. The solid torus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams with boundary that may intersect the distinguished line; intersection points, called anchor points, contribute additional terms, reminiscent of square roots of the Hessian, to the foam evaluation. Both oriented and unoriented SL(3) foams are treated in the paper.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"43 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":"134903427","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.2023.23.3205
Christoforos Neofytidis
We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is the trivial bundle. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: Every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is trivial.
{"title":"On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups","authors":"Christoforos Neofytidis","doi":"10.2140/agt.2023.23.3205","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3205","url":null,"abstract":"We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is the trivial bundle. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: Every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is trivial. ","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"1 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":"134904231","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.2023.23.3015
Rok Gregoric
In this note, we show how a continuous action of the Morava stabilizer group $mathbb G_n$ on the Lubin-Tate spectrum $E_n$, satisfying the conclusion $E_n^{hmathbb G_n}simeq L_{K(n)} S$ of the Devinatz-Hopkins Theorem, may be obtained by monodromy on the stack of oriented deformations of formal groups in the context of formal spectral algebraic geometry.
{"title":"The Devinatz–Hopkins theorem via algebraic geometry","authors":"Rok Gregoric","doi":"10.2140/agt.2023.23.3015","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3015","url":null,"abstract":"In this note, we show how a continuous action of the Morava stabilizer group $mathbb G_n$ on the Lubin-Tate spectrum $E_n$, satisfying the conclusion $E_n^{hmathbb G_n}simeq L_{K(n)} S$ of the Devinatz-Hopkins Theorem, may be obtained by monodromy on the stack of oriented deformations of formal groups in the context of formal spectral algebraic geometry.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"77 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":"134904234","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.2023.23.3043
Andrei V. Malyutin, Oleg R. Musin
We introduce and study a new family of extensions for the Borsuk-Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form `a subset that is large in some sense goes to a singleton' with requirements of the milder form `a subset that is large in some sense goes to a subset that is small in some sense'. This approach covers the case of mappings m-sphere to n-space with m
{"title":"Neighboring mapping points theorem","authors":"Andrei V. Malyutin, Oleg R. Musin","doi":"10.2140/agt.2023.23.3043","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3043","url":null,"abstract":"We introduce and study a new family of extensions for the Borsuk-Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form `a subset that is large in some sense goes to a singleton' with requirements of the milder form `a subset that is large in some sense goes to a subset that is small in some sense'. This approach covers the case of mappings m-sphere to n-space with m<n and extends to wider classes of spaces.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"44 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":"134903420","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.2023.23.2925
Giuseppe De Nittis, Kiyonori Gomi
This paper is devoted to the construction of differential geometric invariants for the classification of "Quaternionic" vector bundles. Provided that the base space is a smooth manifold of dimension two or three endowed with an involution that leaves fixed only a finite number of points, it is possible to prove that the Wess-Zumino term and the Chern-Simons invariant yield topological quantities able to distinguish between inequivalent realization of "Quaternionic" structures.
{"title":"Differential geometric invariants for time-reversal symmetric Bloch bundles, II : The low-dimensional “quaternionic” case","authors":"Giuseppe De Nittis, Kiyonori Gomi","doi":"10.2140/agt.2023.23.2925","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2925","url":null,"abstract":"This paper is devoted to the construction of differential geometric invariants for the classification of \"Quaternionic\" vector bundles. Provided that the base space is a smooth manifold of dimension two or three endowed with an involution that leaves fixed only a finite number of points, it is possible to prove that the Wess-Zumino term and the Chern-Simons invariant yield topological quantities able to distinguish between inequivalent realization of \"Quaternionic\" structures.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"23 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":"134904244","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.2023.23.3071
Ishan Banerjee
Let $U_{d,n}^*$ be the universal degree $d$ hypersurface in $mathbb{P}^n$. In this paper we compute the stable (with respect to $d$) cohomology of $U_{d,n}^*$ and give a geometric description of the stable classes. This builds on work of Tommasi and Das .
{"title":"Stable cohomology of the universal degree d hypersurface in ℙn","authors":"Ishan Banerjee","doi":"10.2140/agt.2023.23.3071","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3071","url":null,"abstract":"Let $U_{d,n}^*$ be the universal degree $d$ hypersurface in $mathbb{P}^n$. In this paper we compute the stable (with respect to $d$) cohomology of $U_{d,n}^*$ and give a geometric description of the stable classes. This builds on work of Tommasi and Das .","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"42 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":"135720519","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 : 2023-09-07DOI: 10.2140/agt.2023.23.2593
Hannah Alpert, Ulrich Bauer, Matthew Kahle, Robert MacPherson, Kelly Spendlove
We study ordered configuration spaces $C(n;p,q)$ of $n$ hard squares in a $p times q$ rectangle, a generalization of the well-known"15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which $n$, $j$, $p$, and $q$ the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for"large-scale"nontrivial homology.
{"title":"Homology of configuration spaces of hard squares in a rectangle","authors":"Hannah Alpert, Ulrich Bauer, Matthew Kahle, Robert MacPherson, Kelly Spendlove","doi":"10.2140/agt.2023.23.2593","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2593","url":null,"abstract":"We study ordered configuration spaces $C(n;p,q)$ of $n$ hard squares in a $p times q$ rectangle, a generalization of the well-known\"15 Puzzle\". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which $n$, $j$, $p$, and $q$ the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for\"large-scale\"nontrivial homology.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135097193","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 : 2023-09-07DOI: 10.2140/agt.2023.23.2627
Sherry Gong, Marco Marengon
We use unoriented versions of instanton and knot Floer homology to prove inequalities involving the Euler characteristic and the number of local maxima appearing in unorientable cobordisms, which mirror results of a recent paper by Juhasz, Miller, and Zemke concerning orientable cobordisms. Most of the subtlety in our argument lies in the fact that maps for non-orientable cobordisms require more complicated decorations than their orientable counterparts. We introduce unoriented versions of the band unknotting number and the refined cobordism distance and apply our results to give bounds on these based on the torsion orders of the Floer homologies. Finally, we show that the difference between the unoriented refined cobordism distance of a knot $K$ from the unknot and the non-orientable slice genus of $K$ can be arbitrarily large.
{"title":"Nonorientable link cobordisms and torsion order in Floer homologies","authors":"Sherry Gong, Marco Marengon","doi":"10.2140/agt.2023.23.2627","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2627","url":null,"abstract":"We use unoriented versions of instanton and knot Floer homology to prove inequalities involving the Euler characteristic and the number of local maxima appearing in unorientable cobordisms, which mirror results of a recent paper by Juhasz, Miller, and Zemke concerning orientable cobordisms. Most of the subtlety in our argument lies in the fact that maps for non-orientable cobordisms require more complicated decorations than their orientable counterparts. We introduce unoriented versions of the band unknotting number and the refined cobordism distance and apply our results to give bounds on these based on the torsion orders of the Floer homologies. Finally, we show that the difference between the unoriented refined cobordism distance of a knot $K$ from the unknot and the non-orientable slice genus of $K$ can be arbitrarily large.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135096603","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 : 2023-09-07DOI: 10.2140/agt.2023.23.2561
Andrew McCullough
We define the notion of a knot type having Legendrian large cables and show that having this property implies that the knot type is not uniformly thick. Moreover, there are solid tori in this knot type that do not thicken to a solid torus with integer sloped boundary torus, and that exhibit new phenomena; specifically, they have virtually overtwisted contact structures. We then show that there exists an infinite family of ribbon knots that have Legendrian large cables. These knots fail to be uniformly thick in several ways not previously seen. We also give a general construction of ribbon knots, and show when they give similar such examples.
{"title":"Legendrian large cables and new phenomenon for nonuniformly thick knots","authors":"Andrew McCullough","doi":"10.2140/agt.2023.23.2561","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2561","url":null,"abstract":"We define the notion of a knot type having Legendrian large cables and show that having this property implies that the knot type is not uniformly thick. Moreover, there are solid tori in this knot type that do not thicken to a solid torus with integer sloped boundary torus, and that exhibit new phenomena; specifically, they have virtually overtwisted contact structures. We then show that there exists an infinite family of ribbon knots that have Legendrian large cables. These knots fail to be uniformly thick in several ways not previously seen. We also give a general construction of ribbon knots, and show when they give similar such examples.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136364065","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 : 2023-09-07DOI: 10.2140/agt.2023.23.2545
Ramin Naimi, Andrei Pavelescu, Elena Pavelescu
We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $nge 10$, and another family on $n$ vertices and $m< frac{25n}{12}-frac{1}{4}$ edges for all $nge 13$. The latter significantly improves the lowest edge-to-vertex ratio for any previously known infinite family. We construct a family of graphs showing that the class of maximal linklessly embeddable graphs differs from the class of graphs that are maximal without a $K_6$ minor studied by L. Jorgensen. We give necessary and sufficient conditions for when the clique sum of two maximal linklessly embeddable graphs over $K_2$, $K_3$, or $K_4$ is a maximal linklessly embeddable graph, and use these results to prove our constructions yield maximal linklessly embeddable graphs.
{"title":"New bounds on maximal linkless graphs","authors":"Ramin Naimi, Andrei Pavelescu, Elena Pavelescu","doi":"10.2140/agt.2023.23.2545","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2545","url":null,"abstract":"We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $nge 10$, and another family on $n$ vertices and $m< frac{25n}{12}-frac{1}{4}$ edges for all $nge 13$. The latter significantly improves the lowest edge-to-vertex ratio for any previously known infinite family. We construct a family of graphs showing that the class of maximal linklessly embeddable graphs differs from the class of graphs that are maximal without a $K_6$ minor studied by L. Jorgensen. We give necessary and sufficient conditions for when the clique sum of two maximal linklessly embeddable graphs over $K_2$, $K_3$, or $K_4$ is a maximal linklessly embeddable graph, and use these results to prove our constructions yield maximal linklessly embeddable graphs.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136364066","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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