Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a2
Yichen Tong
The self-closeness number $Nmathcal{E}(X)$ of a space $X$ is the least integer $k$ such that any self-map is a homotopy equivalence whenever it is an isomorphism in the $n$-th homotopy group for each $nle k$. We discuss the self-closeness numbers of certain non-simply-connected $X$ in this paper. As a result, we give conditions for $X$ such that $Nmathcal{E}(X)=Nmathcal{E}(widetilde{X})$, where $widetilde{X}$ is the universal covering space of $X$.
{"title":"Self-closeness numbers of non-simply-connected spaces","authors":"Yichen Tong","doi":"10.4310/hha.2023.v25.n2.a2","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a2","url":null,"abstract":"The self-closeness number $Nmathcal{E}(X)$ of a space $X$ is the least integer $k$ such that any self-map is a homotopy equivalence whenever it is an isomorphism in the $n$-th homotopy group for each $nle k$. We discuss the self-closeness numbers of certain non-simply-connected $X$ in this paper. As a result, we give conditions for $X$ such that $Nmathcal{E}(X)=Nmathcal{E}(widetilde{X})$, where $widetilde{X}$ is the universal covering space of $X$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784574","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 : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a1
Ambrus Pál, Tomer M. Schlank
{"title":"Two theorems on cohomological pairings","authors":"Ambrus Pál, Tomer M. Schlank","doi":"10.4310/hha.2023.v25.n2.a1","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a1","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135701261","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 : 2022-09-14DOI: 10.4310/hha.2022.v24.n2.a19
Nina Otter
The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that counts the “effective number of points” of the space and has been shown to encode many invariants of metric spaces from integral geometry and geometric measure theory. In 2015, Hepworth and Willerton introduced a homology theory for metric graphs, called magnitude homology, which categorifies the magnitude of a finite metric graph. This work was subsequently generalised to enriched categories by Leinster and Shulman, and the homology theory that they introduced categorifies magnitude for arbitrary finite metric spaces. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded by magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.
{"title":"Magnitude meets persistence: homology theories for filtered simplicial sets","authors":"Nina Otter","doi":"10.4310/hha.2022.v24.n2.a19","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a19","url":null,"abstract":"The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that counts the “effective number of points” of the space and has been shown to encode many invariants of metric spaces from integral geometry and geometric measure theory. In 2015, Hepworth and Willerton introduced a homology theory for metric graphs, called magnitude homology, which categorifies the magnitude of a finite metric graph. This work was subsequently generalised to enriched categories by Leinster and Shulman, and the homology theory that they introduced categorifies magnitude for arbitrary finite metric spaces. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded by magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520648","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 : 2022-08-10DOI: 10.4310/hha.2022.v24.n2.a3
Claudio Quadrelli
Let $p$ be a prime. A pro‑$p$ group $G$ is said to be $1$-smooth if it can be endowed with a homomorphism of pro‑$p$ groups of the form $G to 1 + p mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro‑$p$ Galois groups of fields containing a root of $1$ of order $p$, together with the cyclotomic character, are $1$-smooth. We prove that a finitely generated padic analytic pro‑$p$ group is $1$-smooth if, and only if, it occurs as the maximal pro‑$p$ Galois group of a field containing a root of $1$ of order $p$. This gives a positive answer to De Clercq–Florence’s “Smoothness Conjecture” — which states that the surjectivity of the norm residue homomorphism (i.e., the “surjective half” of the Bloch–Kato Conjecture) follows from $1$-smoothness — for the class of finitely generated $p$-adic analytic pro‑$p$ groups.
{"title":"$1$-smooth pro-$p$ groups and Bloch–Kato pro-$p$ groups","authors":"Claudio Quadrelli","doi":"10.4310/hha.2022.v24.n2.a3","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a3","url":null,"abstract":"Let $p$ be a prime. A pro‑$p$ group $G$ is said to be $1$-smooth if it can be endowed with a homomorphism of pro‑$p$ groups of the form $G to 1 + p mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro‑$p$ Galois groups of fields containing a root of $1$ of order $p$, together with the cyclotomic character, are $1$-smooth. We prove that a finitely generated padic analytic pro‑$p$ group is $1$-smooth if, and only if, it occurs as the maximal pro‑$p$ Galois group of a field containing a root of $1$ of order $p$. This gives a positive answer to De Clercq–Florence’s “Smoothness Conjecture” — which states that the surjectivity of the norm residue homomorphism (i.e., the “surjective half” of the Bloch–Kato Conjecture) follows from $1$-smoothness — for the class of finitely generated $p$-adic analytic pro‑$p$ groups.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520598","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 : 2022-08-09DOI: 10.4310/hha.2023.v25.n1.a13
Pengcheng Li
The self-closeness number of a CW-complex is a homotopy invariant defined by the minimal number $n$ such that every self-maps of $X$ which induces automorphisms on the first $n$ homotopy groups of $X$ is a homotopy equivalence. In this article we study the self-closeness numbers of finite Cartesian products, and prove that under certain conditions (called reducibility), the self-closeness number of product spaces equals to the maximum of self-closeness numbers of the factors. A series of criteria for the reducibility are investigated, and the results are used to determine self-closeness numbers of product spaces of some special spaces, such as Moore spaces, Eilenberg-MacLane spaces or atomic spaces.
{"title":"Self-closeness numbers of product spaces","authors":"Pengcheng Li","doi":"10.4310/hha.2023.v25.n1.a13","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a13","url":null,"abstract":"The self-closeness number of a CW-complex is a homotopy invariant defined by the minimal number $n$ such that every self-maps of $X$ which induces automorphisms on the first $n$ homotopy groups of $X$ is a homotopy equivalence. In this article we study the self-closeness numbers of finite Cartesian products, and prove that under certain conditions (called reducibility), the self-closeness number of product spaces equals to the maximum of self-closeness numbers of the factors. A series of criteria for the reducibility are investigated, and the results are used to determine self-closeness numbers of product spaces of some special spaces, such as Moore spaces, Eilenberg-MacLane spaces or atomic spaces.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44378956","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 : 2022-02-14DOI: 10.4310/hha.2023.v25.n1.a18
Mark Blumstein, J. Duflot
This paper generalizes a result of Lynn on the"degree"of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincar'{e} series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: $$deg(H^*_G(X)) = sum_{[A,c] in mathcal{Q'}_{max}(G,X)}frac{1}{|W_G(A,c)|} deg(H^*_{C_G(A,c)}(c)).$$ We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.
推广了Lynn关于等变上同环“度”的一个结果$H^*_G(X)$。梯度模的度是其庞卡罗级数的一定系数，与多重性密切相关。本文研究了等变上同调环的交换代数不变量。主要定理是阶的可加性公式:$$deg(H^*_G(X)) = sum_{[A,c] in mathcal{Q'}_{max}(G,X)}frac{1}{|W_G(A,c)|} deg(H^*_{C_G(A,c)}(c)).$$我们还展示了这个公式是如何与交换代数中的可加性公式联系起来的，展示了阶不变量的代数和几何特征。
{"title":"A degree formula for equivariant cohomology rings","authors":"Mark Blumstein, J. Duflot","doi":"10.4310/hha.2023.v25.n1.a18","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a18","url":null,"abstract":"This paper generalizes a result of Lynn on the\"degree\"of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincar'{e} series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: $$deg(H^*_G(X)) = sum_{[A,c] in mathcal{Q'}_{max}(G,X)}frac{1}{|W_G(A,c)|} deg(H^*_{C_G(A,c)}(c)).$$ We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47713122","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 : 2022-01-01DOI: 10.4310/hha.2022.v24.n1.a11
M. Benkhalifa
{"title":"Realisability of the group of self-homotopy equivalences and local homotopy theory","authors":"M. Benkhalifa","doi":"10.4310/hha.2022.v24.n1.a11","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a11","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70434940","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 : 2022-01-01DOI: 10.4310/hha.2022.v24.n2.a10
Catarina Mendes de Jesus Sánchez, María del Carmen Romero Fuster
{"title":"Graphs associated to fold maps from closed surfaces to the projective plane","authors":"Catarina Mendes de Jesus Sánchez, María del Carmen Romero Fuster","doi":"10.4310/hha.2022.v24.n2.a10","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a10","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435023","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 : 2022-01-01DOI: 10.4310/hha.2022.v24.n1.a10
Ippei Ichigi, K. Shimomura
. Consider Hopkins’ Picard group of the stable homotopy category of E (2)-local spectra at the prime three, consisting of homotopy classes of invertible spectra [1]. Then, it is isomorphic to the direct sum of an in(cid:12)nite cyclic group and two cyclic groups of order three. We consider the Smith-Toda spectrum V (1) and the co(cid:12)ber V 2 of the square (cid:11) 2 of the Adams map, which is a ring spectrum. In this paper, we introduce imaginary elements which make computation clearer, and determine the module structures of the Picard group graded homotopy groups (cid:25) ⋆ ( V (1)) and (cid:25) ⋆ ( V 2 ).
{"title":"On the Picard group graded homotopy groups of a finite type two $K(2)$-local spectrum at the prime three","authors":"Ippei Ichigi, K. Shimomura","doi":"10.4310/hha.2022.v24.n1.a10","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a10","url":null,"abstract":". Consider Hopkins’ Picard group of the stable homotopy category of E (2)-local spectra at the prime three, consisting of homotopy classes of invertible spectra [1]. Then, it is isomorphic to the direct sum of an in(cid:12)nite cyclic group and two cyclic groups of order three. We consider the Smith-Toda spectrum V (1) and the co(cid:12)ber V 2 of the square (cid:11) 2 of the Adams map, which is a ring spectrum. In this paper, we introduce imaginary elements which make computation clearer, and determine the module structures of the Picard group graded homotopy groups (cid:25) ⋆ ( V (1)) and (cid:25) ⋆ ( V 2 ).","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70434895","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 : 2022-01-01DOI: 10.4310/hha.2022.v24.n1.a15
P. Bhattacharya, B. Guillou, A. Li
{"title":"An $R$-motivic v1-self-map of periodicity $1$","authors":"P. Bhattacharya, B. Guillou, A. Li","doi":"10.4310/hha.2022.v24.n1.a15","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a15","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70434956","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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