Pub Date : 2021-08-11DOI: 10.2140/agt.2021.21.1445
L. Plachta
The configuration space Fk(Q,r) of k squares of size r in a rectangle Q is studied with the help of the tautological function 𝜃 defined on the affine polytope complex Qk. The critical points of the function 𝜃 are described in geometric and combinatorial terms. We also show that under certain conditions, the space Fk(Q,r) is connected.
{"title":"Configuration spaces of squares in a rectangle","authors":"L. Plachta","doi":"10.2140/agt.2021.21.1445","DOIUrl":"https://doi.org/10.2140/agt.2021.21.1445","url":null,"abstract":"The configuration space Fk(Q,r) of k squares of size r in a rectangle Q is studied with the help of the tautological function 𝜃 defined on the affine polytope complex Qk. The critical points of the function 𝜃 are described in geometric and combinatorial terms. We also show that under certain conditions, the space Fk(Q,r) is connected.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"29 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81802156","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 : 2021-07-31DOI: 10.2140/agt.2023.23.2309
Masaki Kameko, D. Kishimoto, Masahiro Takeda
Let $G$ be a compact connected Lie group with $pi_1(G)congmathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of the moduli spaces of stable vector bundles over Riemann surfaces.
{"title":"Homotopy types of gauge groups over Riemann surfaces","authors":"Masaki Kameko, D. Kishimoto, Masahiro Takeda","doi":"10.2140/agt.2023.23.2309","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2309","url":null,"abstract":"Let $G$ be a compact connected Lie group with $pi_1(G)congmathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of the moduli spaces of stable vector bundles over Riemann surfaces.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"249 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77624239","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 : 2021-07-02DOI: 10.2140/agt.2023.23.2329
Johannes Ebert
For a compact $(2n+1)$-dimensional smooth manifold, let $mu_M : B Diff_partial (D^{2n+1}) to B Diff (M)$ be the map that is defined by extending diffeomorphisms on an embedded disc by the identity. By a classical result of Farrell and Hsiang, the rational homotopy groups and the rational homology of $ B Diff_partial (D^{2n+1})$ are known in the concordance stable range. We prove two results on the behaviour of the map $mu_M$ in the concordance stable range. Firstly, it is emph{injective} on rational homotopy groups, and secondly, it is emph{trivial} on rational homology, if $M$ contains sufficiently many embedded copies of $S^ntimes S^{n+1} setminus int(D^{2n+1})$. The homotopical statement is probably not new and follows from the theory of smooth torsion invariants. The homological statement relies on work by Botvinnik and Perlmutter on diffeomorphism of odd-dimensional manifolds.
对于一个紧的$(2n+1)$维光滑流形，设$mu_M : B Diff_partial (D^{2n+1}) to B Diff (M)$为映射，该映射是通过恒等在嵌入盘上扩展微分同态来定义的。通过Farrell和Hsiang的经典结果，已知$ B Diff_partial (D^{2n+1})$的有理同伦群和有理同伦在调和稳定范围内。我们证明了映射$mu_M$在一致性稳定范围内的两个结果。首先，它在有理同伦群上是emph{内射}的;其次，如果$M$包含足够多的嵌入副本$S^ntimes S^{n+1} setminus int(D^{2n+1})$，它在有理同伦上是emph{平凡}的。同调命题可能不是一个新的命题，它是由光滑扭转不变量理论衍生而来的。该同调陈述依赖于Botvinnik和Perlmutter关于奇维流形的微分同态的工作。
{"title":"Diffeomorphisms of odd-dimensional discs, glued into a manifold","authors":"Johannes Ebert","doi":"10.2140/agt.2023.23.2329","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2329","url":null,"abstract":"For a compact $(2n+1)$-dimensional smooth manifold, let $mu_M : B Diff_partial (D^{2n+1}) to B Diff (M)$ be the map that is defined by extending diffeomorphisms on an embedded disc by the identity. By a classical result of Farrell and Hsiang, the rational homotopy groups and the rational homology of $ B Diff_partial (D^{2n+1})$ are known in the concordance stable range. We prove two results on the behaviour of the map $mu_M$ in the concordance stable range. Firstly, it is emph{injective} on rational homotopy groups, and secondly, it is emph{trivial} on rational homology, if $M$ contains sufficiently many embedded copies of $S^ntimes S^{n+1} setminus int(D^{2n+1})$. The homotopical statement is probably not new and follows from the theory of smooth torsion invariants. The homological statement relies on work by Botvinnik and Perlmutter on diffeomorphism of odd-dimensional manifolds.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"32 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73344606","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 : 2021-05-19DOI: 10.2140/agt.2023.23.2221
Stephan Mescher, Maximilian Stegemeyer
We study the geodesic motion planning problem for complete Riemannian manifolds and investigate their geodesic complexity, an integer-valued isometry invariant introduced by D. Recio-Mitter. Using methods from Riemannian geometry, we establish new lower and upper bounds on geodesic complexity and compute its value for certain classes of examples with a focus on homogeneous Riemannian manifolds. Methodically, we study properties of stratifications of cut loci and use results on their structures for certain homogeneous manifolds obtained by T. Sakai and others.
{"title":"Geodesic complexity of homogeneous Riemannian manifolds","authors":"Stephan Mescher, Maximilian Stegemeyer","doi":"10.2140/agt.2023.23.2221","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2221","url":null,"abstract":"We study the geodesic motion planning problem for complete Riemannian manifolds and investigate their geodesic complexity, an integer-valued isometry invariant introduced by D. Recio-Mitter. Using methods from Riemannian geometry, we establish new lower and upper bounds on geodesic complexity and compute its value for certain classes of examples with a focus on homogeneous Riemannian manifolds. Methodically, we study properties of stratifications of cut loci and use results on their structures for certain homogeneous manifolds obtained by T. Sakai and others.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"57 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78921620","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 paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of the coordinate ring of a complement of a hyperplane arrangement. We give a presentation of these rings by generators and defining relations. This presentation was used by Holler and I. Kriz to calculate the $mathbb{Z}$-graded coefficients of localizations of ordinary $RO((mathbb{Z}/p)^n)$-graded equivariant cohomology at a given set of representation spheres, and also more recently by the author in a generalization to the case of an arbitrary finite group. We also give an interpretation of these rings in terms of superschemes, which can be used to further illuminate their structure.
{"title":"Equivariant cohomology and the super reciprocal plane of a hyperplane arrangement","authors":"S. Kriz","doi":"10.2140/agt.2022.22.991","DOIUrl":"https://doi.org/10.2140/agt.2022.22.991","url":null,"abstract":"In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of the coordinate ring of a complement of a hyperplane arrangement. We give a presentation of these rings by generators and defining relations. This presentation was used by Holler and I. Kriz to calculate the $mathbb{Z}$-graded coefficients of localizations of ordinary $RO((mathbb{Z}/p)^n)$-graded equivariant cohomology at a given set of representation spheres, and also more recently by the author in a generalization to the case of an arbitrary finite group. We also give an interpretation of these rings in terms of superschemes, which can be used to further illuminate their structure.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"232 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77583961","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 introduce a bounded version of Bredon cohomology for groups relative to a family of subgroups. Our theory generalizes bounded cohomology and differs from Mineyev--Yaman's relative bounded cohomology for pairs. We obtain cohomological characterizations of relative amenability and relative hyperbolicity, analogous to the results of Johnson and Mineyev for bounded cohomology.
{"title":"Bounded cohomology of classifying spaces for families of subgroups","authors":"Kevin Li","doi":"10.2140/agt.2023.23.933","DOIUrl":"https://doi.org/10.2140/agt.2023.23.933","url":null,"abstract":"We introduce a bounded version of Bredon cohomology for groups relative to a family of subgroups. Our theory generalizes bounded cohomology and differs from Mineyev--Yaman's relative bounded cohomology for pairs. We obtain cohomological characterizations of relative amenability and relative hyperbolicity, analogous to the results of Johnson and Mineyev for bounded cohomology.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"286 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80267999","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 : 2021-05-09DOI: 10.2140/agt.2023.23.2369
R. Huang
Let $M$ be the $6$-manifold $M$ as the total space of the sphere bundle of a rank $3$ vector bundle over a simply connected closed $4$-manifold. We show that after looping $M$ is homotopy equivalent to a product of loops on spheres in general. This particularly implies the cohomology rigidity property of $M$ after looping. Furthermore, passing to the rational homotopy, we show that such $M$ is Koszul in the sense of Berglund.
{"title":"Loop homotopy of 6–manifolds over\u00004–manifolds","authors":"R. Huang","doi":"10.2140/agt.2023.23.2369","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2369","url":null,"abstract":"Let $M$ be the $6$-manifold $M$ as the total space of the sphere bundle of a rank $3$ vector bundle over a simply connected closed $4$-manifold. We show that after looping $M$ is homotopy equivalent to a product of loops on spheres in general. This particularly implies the cohomology rigidity property of $M$ after looping. Furthermore, passing to the rational homotopy, we show that such $M$ is Koszul in the sense of Berglund.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"131 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72805002","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 prove that the tunnel number of a satellite chain link with a number of components higher than or equal to twice the bridge number of the companion is as small as possible among links with the same number of components. We prove this result to be sharp for satellite chain links over a 2-bridge knot. We also observe that the links in the main result satisfy the genus versus rank conjecture.
{"title":"On unknotting tunnel systems of satellite chain links","authors":"D. Girão, J. Nogueira, António Salgueiro","doi":"10.2140/agt.2022.22.307","DOIUrl":"https://doi.org/10.2140/agt.2022.22.307","url":null,"abstract":"We prove that the tunnel number of a satellite chain link with a number of components higher than or equal to twice the bridge number of the companion is as small as possible among links with the same number of components. We prove this result to be sharp for satellite chain links over a 2-bridge knot. We also observe that the links in the main result satisfy the genus versus rank conjecture.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"44 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76698212","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 : 2021-04-16DOI: 10.2140/agt.2023.23.2213
Tetsuya Ito
By estimating the Turaev genus or the dealternation number, which leads to an estimate of knot floer thickness, in terms of the genus and the braid index, we show that a knot $K$ in $S^{3}$ does not admit purely cosmetic surgery whenever $g(K)geq frac{3}{2}b(K)$, where $g(K)$ and $b(K)$ denotes the genus and the braid index, respectively. In particular, this establishes a finiteness of purely cosmetic surgeries; for fixed $b$, all but finitely many knots with braid index $b$ satisfies the cosmetic surgery conjecture.
{"title":"A remark on the finiteness of purely cosmetic surgeries","authors":"Tetsuya Ito","doi":"10.2140/agt.2023.23.2213","DOIUrl":"https://doi.org/10.2140/agt.2023.23.2213","url":null,"abstract":"By estimating the Turaev genus or the dealternation number, which leads to an estimate of knot floer thickness, in terms of the genus and the braid index, we show that a knot $K$ in $S^{3}$ does not admit purely cosmetic surgery whenever $g(K)geq frac{3}{2}b(K)$, where $g(K)$ and $b(K)$ denotes the genus and the braid index, respectively. In particular, this establishes a finiteness of purely cosmetic surgeries; for fixed $b$, all but finitely many knots with braid index $b$ satisfies the cosmetic surgery conjecture.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81489801","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}
For a simply connected closed orientable manifold of dimension $6$, we show its homotopy decomposition after double suspension. This allows us to determine its $K$- and $KO$-groups easily. Moreover, for a special case we refine the decomposition to show the rigidity property of the manifold after double suspension.
{"title":"Suspension homotopy of 6–manifolds","authors":"R. Huang","doi":"10.2140/agt.2023.23.439","DOIUrl":"https://doi.org/10.2140/agt.2023.23.439","url":null,"abstract":"For a simply connected closed orientable manifold of dimension $6$, we show its homotopy decomposition after double suspension. This allows us to determine its $K$- and $KO$-groups easily. Moreover, for a special case we refine the decomposition to show the rigidity property of the manifold after double suspension.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86493067","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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