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Homology Homotopy and Applications最新文献

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Hyperoctahedral homology for involutive algebras 对合代数的高八面体同调
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-11-06 DOI: 10.4310/HHA.2022.v24.n1.a1
Daniel F. Graves
{"title":"Hyperoctahedral homology for involutive algebras","authors":"Daniel F. Graves","doi":"10.4310/HHA.2022.v24.n1.a1","DOIUrl":"https://doi.org/10.4310/HHA.2022.v24.n1.a1","url":null,"abstract":"Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47920172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Hyperplane restrictions of indecomposable $n$-dimensional persistence modules 不可分解的n维持久性模块的超平面限制
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-10-31 DOI: 10.4310/hha.2022.v24.n2.a14
Samantha Moore
{"title":"Hyperplane restrictions of indecomposable $n$-dimensional persistence modules","authors":"Samantha Moore","doi":"10.4310/hha.2022.v24.n2.a14","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a14","url":null,"abstract":"Understanding the structure of indecomposable $n$-dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed that any finitely presented rectangular $(n-1)$-dimensional persistence module with finite support is a hyperplane restriction of an $n$-dimensional persistence module. We extend this result to the following: If $M$ is any finitely presented $(n-1)$-dimensional persistence module with finite support, then there exists an indecomposable $n$-dimensional persistence module $M'$ such that $M$ is the restriction of $M'$ to a hyperplane. We also show that any finite zigzag persistence module is the restriction of some indecomposable $3$-dimensional persistence module to a path.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43567922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On generalized projective product spaces and Dold manifolds 广义投影积空间与Dold流形
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-10-22 DOI: 10.4310/hha.2022.v24.n2.a13
Soumen Sarkar, Peter Zvengrowsk
{"title":"On generalized projective product spaces and Dold manifolds","authors":"Soumen Sarkar, Peter Zvengrowsk","doi":"10.4310/hha.2022.v24.n2.a13","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a13","url":null,"abstract":"Don Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and studied several topological properties of these spaces. On the other hand, Dold manifolds were introduced by A. Dold long back in 1956 to study the generators of the non-oriented cobordism ring. From then on several interesting properties of Dold manifolds are studies. Recently, in 2019, Nath and Sankaran make a slight generalization of Dold manifolds. In this paper, we generalize the notion of projective product spaces and Dold manifolds which gives infinitely many different class of new manifolds. Our main goal here is to discuss integral homology groups, cohomology ring structures, stable tangent bundles and vector field problems on certain generalized projective product spaces and Dold manifolds.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48750322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
An upper bound on the topological complexity of discriminantal varieties 判别变体拓扑复杂度的上界
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-10-19 DOI: 10.4310/hha.2022.v24.n1.a9
Andrea Bianchi
{"title":"An upper bound on the topological complexity of discriminantal varieties","authors":"Andrea Bianchi","doi":"10.4310/hha.2022.v24.n1.a9","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a9","url":null,"abstract":"We give an upper bound on the topological complexity of varieties $mathcal{V}$ obtained as complements in $mathbb{C}^m$ of the zero locus of a polynomial. As an application, we determine the topological complexity of unordered configuration spaces of the plane.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48886899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 stable hull of an exact $infty$-category 精确$infty$类别的稳定外壳
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-10-10 DOI: 10.4310/HHA.2022.v24.n2.a9
Jona Klemenc
{"title":"The stable hull of an exact $infty$-category","authors":"Jona Klemenc","doi":"10.4310/HHA.2022.v24.n2.a9","DOIUrl":"https://doi.org/10.4310/HHA.2022.v24.n2.a9","url":null,"abstract":"We construct a left adjoint $mathcal{H}^text{st}colon mathbf{Ex}_{infty} rightarrow mathbf{St}_{infty}$ to the inclusion $mathbf{St}_{infty} hookrightarrow mathbf{Ex}_{infty}$ of the $infty$-category of stable $infty$-categories into the $infty$-category of exact $infty$-categories, which we call the stable hull. For every exact $infty$-category $mathcal{E}$, the unit functor $mathcal{E} rightarrow mathcal{H}^text{st}(mathcal{E})$ is fully faithful and preserves and reflects exact sequences. This provides an $infty$-categorical variant of the Gabriel-Quillen embedding for ordinary exact categories. If $mathcal{E}$ is an ordinary exact category, the stable hull $mathcal{H}^text{st}(mathcal{E})$ is equivalent to the bounded derived $infty$-category of $mathcal{E}$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48634340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 cohomology in symmetric tensor categories in prime characteristic 素数特征下对称张量范畴的上同调
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-08-30 DOI: 10.4310/hha.2022.v24.n2.a8
D. Benson, P. Etingof
{"title":"On cohomology in symmetric tensor categories in prime characteristic","authors":"D. Benson, P. Etingof","doi":"10.4310/hha.2022.v24.n2.a8","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a8","url":null,"abstract":"We describe graded commutative Gorenstein algebras ${mathcal E}_n(p)$ over a field of characteristic $p$, and we conjecture that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)cong{mathcal E}_{n}(p)$, where $mathsf{Ver}_{p^{n+1}}$ are the new symmetric tensor categories recently constructed in cite{Benson/Etingof:2019a,Benson/Etingof/Ostrik,Coulembier}. We investigate the combinatorics of these algebras, and the relationship with Minc's partition function, as well as possible actions of the Steenrod operations on them. Evidence for the conjecture includes a large number of computations for small values of $n$. We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in ${mathcal E}_n(p)$ with a homogeneous system of parameters in $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)$. These parameters have degrees $2^i-1$ if $p=2$ and $2(p^i-1)$ if $p$ is odd, for $1le i le n$. This at least shows that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)$ is a finitely generated graded commutative algebra with the same Krull dimension as ${mathcal E}_n(p)$. For $p=2$ we also show that $mathrm{Ext}^bullet_{mathsf{Ver}_{2^{n+1}}}(1,1)$ has the expected rank $2^{n(n-1)/2}$ as a module over the subalgebra of parameters.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42254505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Structure of semi-continuous $q$-tame persistence modules 半连续$q$-tame持久化模块的结构
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-08-21 DOI: 10.4310/HHA.2022.v24.n1.a6
Maximilian Schmahl
{"title":"Structure of semi-continuous $q$-tame persistence modules","authors":"Maximilian Schmahl","doi":"10.4310/HHA.2022.v24.n1.a6","DOIUrl":"https://doi.org/10.4310/HHA.2022.v24.n1.a6","url":null,"abstract":"Using a result by Chazal, Crawley-Boevey and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper semi-continuous q-tame persistence module can be decomposed as a product of interval modules.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41915393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 the dimension of the mapping class groups of a non-orientable surface 关于不可定向曲面的映射类群的维数
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-07-24 DOI: 10.4310/hha.2022.v24.n1.a17
Cristhian E. Hidber, Luis Jorge S'anchez Saldana, A. Trujillo-Negrete
{"title":"On the dimension of the mapping class groups of a non-orientable surface","authors":"Cristhian E. Hidber, Luis Jorge S'anchez Saldana, A. Trujillo-Negrete","doi":"10.4310/hha.2022.v24.n1.a17","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a17","url":null,"abstract":"Let $mathcal{N}_g$ be the mapping class group of a non-orientable closed surface. We prove that the proper cohomological dimension, the proper geometric dimension, and the virtual cohomological dimension of $mathcal{N}_g$ are equal whenever $gneq 4,5$. In particular, there exists a model for the classifying space of $mathcal{N}_g$ for proper actions of dimension $mathrm{vcd}(mathcal{N}_g)=2g-5$. Similar results are obtained for the mapping class group of a non-orientable surface with boundaries and possibly punctures, and for the pure mapping class group of a non-orientable surface with punctures and without boundaries.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46775496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Homotopy type of the space of finite propagation unitary operators on $mathbb{Z}$ $mathbb{Z}$上有限传播酉算子空间的同伦类型
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-07-14 DOI: 10.4310/HHA.2023.v25.n1.a20
Tsuyoshi Kato, D. Kishimoto, Mitsunobu Tsutaya
{"title":"Homotopy type of the space of finite propagation unitary operators on $mathbb{Z}$","authors":"Tsuyoshi Kato, D. Kishimoto, Mitsunobu Tsutaya","doi":"10.4310/HHA.2023.v25.n1.a20","DOIUrl":"https://doi.org/10.4310/HHA.2023.v25.n1.a20","url":null,"abstract":"The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $pi_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $mathbb{C}$-valued $mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48660857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Cellular sheaves of lattices and the Tarski Laplacian 格的胞槽与Tarski拉普拉斯算子
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2020-07-08 DOI: 10.4310/HHA.2022.v24.n1.a16
R. Ghrist, Hans Riess
{"title":"Cellular sheaves of lattices and the Tarski Laplacian","authors":"R. Ghrist, Hans Riess","doi":"10.4310/HHA.2022.v24.n1.a16","DOIUrl":"https://doi.org/10.4310/HHA.2022.v24.n1.a16","url":null,"abstract":"This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47469950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 18
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