Pub Date : 2020-11-06DOI: 10.4310/HHA.2022.v24.n1.a1
Daniel F. Graves
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.
{"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}
Pub Date : 2020-10-31DOI: 10.4310/hha.2022.v24.n2.a14
Samantha Moore
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.
{"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}
Pub Date : 2020-10-22DOI: 10.4310/hha.2022.v24.n2.a13
Soumen Sarkar, Peter Zvengrowsk
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.
{"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}
Pub Date : 2020-10-19DOI: 10.4310/hha.2022.v24.n1.a9
Andrea Bianchi
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.
{"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}
Pub Date : 2020-10-10DOI: 10.4310/HHA.2022.v24.n2.a9
Jona Klemenc
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 $mathcal{E}$.
{"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}
Pub Date : 2020-08-30DOI: 10.4310/hha.2022.v24.n2.a8
D. Benson, P. Etingof
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.
我们描述了特征$p$域上的分次交换Gorenstein代数${mathcal E}_n（p）$，并推测$mathrm｛Ext｝^bullet_｛mathsf{Ver}_｛p^{n+1}｝｝（1,1）cong｛mathcal E｝_｛n｝（p）$，其中$mathsf{Ver}_｛p^｛n+1｝｝$是最近在｛Benson/Etingof:2019a，Benson/Edingof/Osterik，Coulenbier｝中构造的新的对称张量范畴。我们研究了这些代数的组合数学，以及与Minc配分函数的关系，以及Steenrod运算对它们的可能作用。该猜想的证据包括对小数值$n$的大量计算。我们还提供了一些理论证据。也就是说，我们使用Koszul构造来识别$｛mathcal E｝_n（p）$中的齐次参数系统和$mathrm｛Ext｝^bullet_{Ver}_{p^{n+1}}}（1,1）$。如果$p=2$，则这些参数的阶数为$2^i-1$；如果$p$为奇数，则对于$1le ile n$，这些参数具有$2（p^i-1）$。这至少表明$mathrm｛Ext｝^bullet_｛mathsf{Ver}_{p^{n+1}}}（1,1）$是一个有限生成的分次交换代数，其Krull维数与$｛mathcal E｝_n（p）$相同。对于$p=2$，我们还表明$mathrm｛Ext｝^bullet_｛mathsf{Ver}_｛2^｛n+1｝｝（1，1）$作为参数子代数上的模具有期望秩$2^｛n（n-1）/2｝$。
{"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}
Pub Date : 2020-08-21DOI: 10.4310/HHA.2022.v24.n1.a6
Maximilian Schmahl
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.
{"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}
Pub Date : 2020-07-24DOI: 10.4310/hha.2022.v24.n1.a17
Cristhian E. Hidber, Luis Jorge S'anchez Saldana, A. Trujillo-Negrete
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.
{"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}
Pub Date : 2020-07-14DOI: 10.4310/HHA.2023.v25.n1.a20
Tsuyoshi Kato, D. Kishimoto, Mitsunobu Tsutaya
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.
{"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}
Pub Date : 2020-07-08DOI: 10.4310/HHA.2022.v24.n1.a16
R. Ghrist, Hans Riess
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.
{"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}
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