Pub Date : 2021-08-11DOI: 10.4310/HHA.2023.v25.n1.a1
Daniel F. Graves
Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.
{"title":"E-infinity structure in hyperoctahedral homology","authors":"Daniel F. Graves","doi":"10.4310/HHA.2023.v25.n1.a1","DOIUrl":"https://doi.org/10.4310/HHA.2023.v25.n1.a1","url":null,"abstract":"Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45512129","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 : 2021-07-29DOI: 10.4310/HHA.2023.v25.n1.a19
Marianne Lawson, Sven-Ake Wegner
We establish that the category of Silva spaces, aka LS-spaces, formed by countable inductive limits of Banach spaces with compact linking maps as objects and linear and continuous maps as morphisms, is not an integral category. The result carries over to the category of PLS-spaces, i.e., countable projective limits of LS-spaces -- which contains prominent spaces of analysis such as the space of distributions and the space of real analytic functions. As a consequence, we obtain that both categories neither have enough projective nor enough injective objects. All results hold true when 'compact' is replaced by 'weakly compact' or 'nuclear'. This leads to the categories of PLS-, PLS$_{text{w}}$- and PLN-spaces, which are examples of 'inflation exact categories with admissible cokernels' as recently introduced by Henrard, Kvamme, van Roosmalen and the second-named author.
{"title":"The category of Silva spaces is not integral","authors":"Marianne Lawson, Sven-Ake Wegner","doi":"10.4310/HHA.2023.v25.n1.a19","DOIUrl":"https://doi.org/10.4310/HHA.2023.v25.n1.a19","url":null,"abstract":"We establish that the category of Silva spaces, aka LS-spaces, formed by countable inductive limits of Banach spaces with compact linking maps as objects and linear and continuous maps as morphisms, is not an integral category. The result carries over to the category of PLS-spaces, i.e., countable projective limits of LS-spaces -- which contains prominent spaces of analysis such as the space of distributions and the space of real analytic functions. As a consequence, we obtain that both categories neither have enough projective nor enough injective objects. All results hold true when 'compact' is replaced by 'weakly compact' or 'nuclear'. This leads to the categories of PLS-, PLS$_{text{w}}$- and PLN-spaces, which are examples of 'inflation exact categories with admissible cokernels' as recently introduced by Henrard, Kvamme, van Roosmalen and the second-named author.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"42 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435479","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 : 2021-07-24DOI: 10.4310/HHA.2022.v24.n2.a5
A. Libgober, Shoji Yokura
We discuss inequalities between the values of emph{homotopical and cohomological Poincar'e polynomials} of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called emph{stabilization threshold} $frak{pp} (X;varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $varepsilon$, and we show that the Hilali conjecture implies that $frak{pp} (X;1) le 3$.
{"title":"Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties","authors":"A. Libgober, Shoji Yokura","doi":"10.4310/HHA.2022.v24.n2.a5","DOIUrl":"https://doi.org/10.4310/HHA.2022.v24.n2.a5","url":null,"abstract":"We discuss inequalities between the values of emph{homotopical and cohomological Poincar'e polynomials} of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called emph{stabilization threshold} $frak{pp} (X;varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $varepsilon$, and we show that the Hilali conjecture implies that $frak{pp} (X;1) le 3$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48109773","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 : 2021-05-17DOI: 10.4310/hha.2022.v24.n2.a16
K. Mischaikow, C. Weibel
. For a fixed N , we analyze the space of all sequences z = ( z 1 , . . . , z N ), approximating a continuous function on the circle, with a given persistence diagram P , and show that the typical components of this space are homotopy equivalent to S 1 . We also consider the space of functions on Y -shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to S 1 (resp., to a bouquet of circles).
{"title":"Persistent homology with non-contractible preimages","authors":"K. Mischaikow, C. Weibel","doi":"10.4310/hha.2022.v24.n2.a16","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a16","url":null,"abstract":". For a fixed N , we analyze the space of all sequences z = ( z 1 , . . . , z N ), approximating a continuous function on the circle, with a given persistence diagram P , and show that the typical components of this space are homotopy equivalent to S 1 . We also consider the space of functions on Y -shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to S 1 (resp., to a bouquet of circles).","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43617577","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 : 2021-03-25DOI: 10.4310/hha.2022.v24.n1.a5
T. Coconeţ, C. Todea
We show that there is an action of the symmetric group on the Hochschild cochain complex of a twisted group algebra with coefficients in a bimodule. This allows us to define the symmetric Hochschild cohomology of twisted group algebras, similarly to th construction of symmetric group cohomology due to Staic. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.
{"title":"Symmetric Hochschild cohomology of twisted group algebras","authors":"T. Coconeţ, C. Todea","doi":"10.4310/hha.2022.v24.n1.a5","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a5","url":null,"abstract":"We show that there is an action of the symmetric group on the Hochschild cochain complex of a twisted group algebra with coefficients in a bimodule. This allows us to define the symmetric Hochschild cohomology of twisted group algebras, similarly to th construction of symmetric group cohomology due to Staic. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48166157","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 : 2021-03-03DOI: 10.4310/HHA.2021.v23.n1.a14
Eoin Mackall
We prove analogs of Whitehead’s theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow groups, or on the Grothendieck group of coherent sheaves, is an isomorphism. As a corollary, we show that there are no nontrivial naive A-homotopy equivalences between smooth projective varieties.
{"title":"$mathbb{A}^1$-homotopy equivalences and a theorem of Whitehead","authors":"Eoin Mackall","doi":"10.4310/HHA.2021.v23.n1.a14","DOIUrl":"https://doi.org/10.4310/HHA.2021.v23.n1.a14","url":null,"abstract":"We prove analogs of Whitehead’s theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow groups, or on the Grothendieck group of coherent sheaves, is an isomorphism. As a corollary, we show that there are no nontrivial naive A-homotopy equivalences between smooth projective varieties.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"23 1","pages":"257-274"},"PeriodicalIF":0.5,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42756526","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 : 2021-02-25DOI: 10.4310/hha.2022.v24.n2.a2
V. Bardakov, Mahender Singh
Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also compare the exact sequence for linear cycle sets with that for their underlying abelian groups via the forgetful functor and discuss generalities on dynamical 2-cocycles.
{"title":"A Wells type exact sequence for non-degenerate unitary solutions of the Yang–Baxter equation","authors":"V. Bardakov, Mahender Singh","doi":"10.4310/hha.2022.v24.n2.a2","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a2","url":null,"abstract":"Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also compare the exact sequence for linear cycle sets with that for their underlying abelian groups via the forgetful functor and discuss generalities on dynamical 2-cocycles.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44729034","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 : 2021-02-12DOI: 10.4310/hha.2022.v24.n1.a2
Eero Hyry, Markus Klemetti
We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product. Furthermore, when the indexing set is a poset, we provide a new characterization for a generalized persistence module being finitely presented.
{"title":"Generalized persistence and graded structures","authors":"Eero Hyry, Markus Klemetti","doi":"10.4310/hha.2022.v24.n1.a2","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a2","url":null,"abstract":"We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product. Furthermore, when the indexing set is a poset, we provide a new characterization for a generalized persistence module being finitely presented.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43939212","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 : 2021-01-01DOI: 10.4310/HHA.2021.V23.N2.A4
R. Frigerio, A. Maffei
Given an open covering of a paracompact topological space X , there are two natural ways to construct a map from the cohomology of the nerve of the covering to the cohomology of X . One of them is based on a partition of unity, and is more topological in nature, while the other one relies on the double complex associated to an open covering, and has a more algebraic flavour. In this paper we prove that these two maps coincide.
{"title":"A remark on the double complex of a covering for singular cohomology","authors":"R. Frigerio, A. Maffei","doi":"10.4310/HHA.2021.V23.N2.A4","DOIUrl":"https://doi.org/10.4310/HHA.2021.V23.N2.A4","url":null,"abstract":"Given an open covering of a paracompact topological space X , there are two natural ways to construct a map from the cohomology of the nerve of the covering to the cohomology of X . One of them is based on a partition of unity, and is more topological in nature, while the other one relies on the double complex associated to an open covering, and has a more algebraic flavour. In this paper we prove that these two maps coincide.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435251","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 : 2021-01-01DOI: 10.4310/hha.2021.v23.n2.a19
Julian V. S. Holstein
{"title":"Erratum to “Properness and simplicial resolutions for the model category $mathbf{dgCat}$”","authors":"Julian V. S. Holstein","doi":"10.4310/hha.2021.v23.n2.a19","DOIUrl":"https://doi.org/10.4310/hha.2021.v23.n2.a19","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70434860","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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