Pub Date : 2023-09-07DOI: 10.2140/agt.2023.23.2715
Jens Harlander, Stephan Rosebrock
Wirtinger presentations of deficiency 1 appear in the context of knots, long virtual knots, and ribbon 2-knots. They are encoded by (word) labeled oriented trees and, for that reason, are also called LOT presentations. These presentations are a well known and important testing ground for the validity (or failure) of Whitehead's asphericity conjecture. In this paper we define LOTs of Coxeter type and show that for every given $n$ there exists a (prime) LOT of Coxeter type with group of rank $n$. We also show that label separated Coxeter LOTs are aspherical.
{"title":"Ribbon 2–knot groups of Coxeter type","authors":"Jens Harlander, Stephan Rosebrock","doi":"10.2140/agt.2023.23.2715","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2715","url":null,"abstract":"Wirtinger presentations of deficiency 1 appear in the context of knots, long virtual knots, and ribbon 2-knots. They are encoded by (word) labeled oriented trees and, for that reason, are also called LOT presentations. These presentations are a well known and important testing ground for the validity (or failure) of Whitehead's asphericity conjecture. In this paper we define LOTs of Coxeter type and show that for every given $n$ there exists a (prime) LOT of Coxeter type with group of rank $n$. We also show that label separated Coxeter LOTs are aspherical.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"18 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":"136365009","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}
M. Cárdenas, Francisco Fernández Lasheras, A. Quintero, R. Roy
{"title":"Proper 2–equivalences between infinite ended\u0000finitely presented groups","authors":"M. Cárdenas, Francisco Fernández Lasheras, A. Quintero, R. Roy","doi":"10.2140/agt.2023.23.1","DOIUrl":"https://doi.org/10.2140/agt.2023.23.1","url":null,"abstract":"","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75758387","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}
In this article a recognition principle for $infty$-loop pairs of spaces of connective commutative algebra spectra over connective commutative ring spectra is proved. This is done by generalizing the classical recognition principle for connective commutative ring spectra using relative operads. The machinery of idempotent quasiadjunctions is used to handle the model theoretical aspects of the proof.
{"title":"Recognition of connective commutative algebra spectra through an idempotent quasiadjunction","authors":"Renato Vasconcellos Vieira","doi":"10.2140/agt.2023.23.295","DOIUrl":"https://doi.org/10.2140/agt.2023.23.295","url":null,"abstract":"In this article a recognition principle for $infty$-loop pairs of spaces of connective commutative algebra spectra over connective commutative ring spectra is proved. This is done by generalizing the classical recognition principle for connective commutative ring spectra using relative operads. The machinery of idempotent quasiadjunctions is used to handle the model theoretical aspects of the proof.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"71 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91228865","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}
Let $A_1$ be any spectrum in the class of finite spectra whose mod-2 cohomology is isomorphic to $mathcal{A}(1)$ as a module over the subalgebra $mathcal{A}(1)$ of the Steenrod algebra; let $tmf$ be the connective spectrum of topological modular forms. In this paper, we prove that the $tmf$-Hurewicz image of $A_1$ is surjective.
{"title":"On the surjectivity of the tmf–Hurewicz image\u0000of A1","authors":"Viet-Cuong Pham","doi":"10.2140/agt.2023.23.217","DOIUrl":"https://doi.org/10.2140/agt.2023.23.217","url":null,"abstract":"Let $A_1$ be any spectrum in the class of finite spectra whose mod-2 cohomology is isomorphic to $mathcal{A}(1)$ as a module over the subalgebra $mathcal{A}(1)$ of the Steenrod algebra; let $tmf$ be the connective spectrum of topological modular forms. In this paper, we prove that the $tmf$-Hurewicz image of $A_1$ is surjective.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78186712","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 : 2022-12-31DOI: 10.2140/agt.2022.22.3855
Viet-Cuong Pham
Let $A(1)$ be any of the four finite spectra whose cohomology is isomorphic to the subalgebra $A(1)$ of the Steenrod algebra. Let $E_{C}$ be the second Morava-$E$ theory associated to a universal deformation of the formal completion of the supersingular elliptic curve $(C) : y^{2}+y = x^{3}$ defined over $mathbb{F}_{4}$ and $G_{24}$ a maximal finite subgroup of automorphism groups $mathbb{S}_{C}$ of the formal completion $F_{C}$. In this paper, we will compute the homotopy groups of $E_{C}^{hG_{24}}wedge A(1)$ by means of the homotopy fixed point spectral sequence.
{"title":"On homotopy groups of EChG24∧A1","authors":"Viet-Cuong Pham","doi":"10.2140/agt.2022.22.3855","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3855","url":null,"abstract":"Let $A(1)$ be any of the four finite spectra whose cohomology is isomorphic to the subalgebra $A(1)$ of the Steenrod algebra. Let $E_{C}$ be the second Morava-$E$ theory associated to a universal deformation of the formal completion of the supersingular elliptic curve $(C) : y^{2}+y = x^{3}$ defined over $mathbb{F}_{4}$ and $G_{24}$ a maximal finite subgroup of automorphism groups $mathbb{S}_{C}$ of the formal completion $F_{C}$. In this paper, we will compute the homotopy groups of $E_{C}^{hG_{24}}wedge A(1)$ by means of the homotopy fixed point spectral sequence.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"210 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73973810","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 : 2022-12-31DOI: 10.2140/agt.2022.22.3533
Ensil Kang, J. Hyam Rubinstein
{"title":"Spun normal surfaces in 3–manifolds, I :\u00001–efficient triangulations","authors":"Ensil Kang, J. Hyam Rubinstein","doi":"10.2140/agt.2022.22.3533","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3533","url":null,"abstract":"","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"33 17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82552295","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 : 2022-12-31DOI: 10.2140/agt.2022.22.3485
Jone Lopez de Gamiz Zearra
A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG and that the membership problem is decidable for their finitely presented subgroups.
{"title":"Subgroups of direct products of limit groups over Droms RAAGs","authors":"Jone Lopez de Gamiz Zearra","doi":"10.2140/agt.2022.22.3485","DOIUrl":"https://doi.org/10.2140/agt.2022.22.3485","url":null,"abstract":"A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG and that the membership problem is decidable for their finitely presented subgroups.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"92 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72616655","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 : 2022-10-25DOI: 10.2140/agt.2022.22.2419
Daniel López Neumann
We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with an homomorphism of the fundamental group into $text{Aut}(H)$ and possibly with a $text{Spin}^c$ structure and an homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.
{"title":"Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion","authors":"Daniel López Neumann","doi":"10.2140/agt.2022.22.2419","DOIUrl":"https://doi.org/10.2140/agt.2022.22.2419","url":null,"abstract":"We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with an homomorphism of the fundamental group into $text{Aut}(H)$ and possibly with a $text{Spin}^c$ structure and an homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"7 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83190117","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 : 2022-02-01DOI: 10.1142/9789811245039_0013
J. Warner
Let A be an abelian category. Definition 1.1. A complex in A, A•, is a collection of objects A, i ∈ Z and boundary morphisms d : A → A such that d ◦ d = 0 for all i ∈ Z. If A• and B• are complexes, a map f : A• → B• is a collection morphisms f i : A → B commuting with the boundary morphisms. Two maps f, g : A• → B• are said to be homotopic if there are morphisms k : A → Bi−1 such that f i − g = di−1 B ◦ k + kdA. Two complexes are homotopy equivalent if there exist maps f : A• → B• and g : B• → A• such that the compositions are homotopic to the appropriate identity map.
{"title":"Cohomology of Sheaves","authors":"J. Warner","doi":"10.1142/9789811245039_0013","DOIUrl":"https://doi.org/10.1142/9789811245039_0013","url":null,"abstract":"Let A be an abelian category. Definition 1.1. A complex in A, A•, is a collection of objects A, i ∈ Z and boundary morphisms d : A → A such that d ◦ d = 0 for all i ∈ Z. If A• and B• are complexes, a map f : A• → B• is a collection morphisms f i : A → B commuting with the boundary morphisms. Two maps f, g : A• → B• are said to be homotopic if there are morphisms k : A → Bi−1 such that f i − g = di−1 B ◦ k + kdA. Two complexes are homotopy equivalent if there exist maps f : A• → B• and g : B• → A• such that the compositions are homotopic to the appropriate identity map.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75914883","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 : 2022-02-01DOI: 10.1142/9789811245039_0011
{"title":"Derived Functors, δ-Functors, and ∂-Functors","authors":"","doi":"10.1142/9789811245039_0011","DOIUrl":"https://doi.org/10.1142/9789811245039_0011","url":null,"abstract":"","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81233324","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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