Pub Date : 2024-02-21DOI: 10.4310/hha.2024.v26.n1.a4
Sacha Ikonicoff
$defP{mathcal{P}}$We work over the finite field $mathbb{F}_q$. We introduce a notion of unstable $P$-algebra over an operad $P$. We show that the unstable $P$-algebra freely generated by an unstable module is itself a free $P$-algebra under suitable conditions. We introduce a family of ‘$q$-level’ operads which allows us to identify unstable modules studied by Brown–Gitler, Miller and Carlsson in terms of free unstable $q$-level algebras.
{"title":"Unstable algebras over an operad II","authors":"Sacha Ikonicoff","doi":"10.4310/hha.2024.v26.n1.a4","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a4","url":null,"abstract":"$defP{mathcal{P}}$We work over the finite field $mathbb{F}_q$. We introduce a notion of unstable $P$-algebra over an operad $P$. We show that the unstable $P$-algebra freely generated by an unstable module is itself a free $P$-algebra under suitable conditions. We introduce a family of ‘$q$-level’ operads which allows us to identify unstable modules studied by Brown–Gitler, Miller and Carlsson in terms of free unstable $q$-level algebras.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924572","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 : 2024-02-21DOI: 10.4310/hha.2024.v26.n1.a7
Emily Roff
Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the magnitude of graphs is its behaviour with respect to an operation known as the Whitney twist.We give a homological account of magnitude’s invariance under Whitney twists, extending the previously known result to encompass a substantially wider class of gluings. As well as providing a new tool for the computation of magnitudes, this is the first new theorem about magnitude to be proved using magnitude homology.
{"title":"Magnitude, homology, and the Whitney twist","authors":"Emily Roff","doi":"10.4310/hha.2024.v26.n1.a7","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a7","url":null,"abstract":"Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the magnitude of graphs is its behaviour with respect to an operation known as the Whitney twist.We give a homological account of magnitude’s invariance under Whitney twists, extending the previously known result to encompass a substantially wider class of gluings. As well as providing a new tool for the computation of magnitudes, this is the first new theorem about magnitude to be proved using magnitude homology.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924628","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 : 2024-02-21DOI: 10.4310/hha.2024.v26.n1.a5
Muriel Livernet, Sarah Whitehouse
Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.
{"title":"Homotopy theory of spectral sequences","authors":"Muriel Livernet, Sarah Whitehouse","doi":"10.4310/hha.2024.v26.n1.a5","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a5","url":null,"abstract":"Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924630","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 : 2024-02-21DOI: 10.4310/hha.2024.v26.n1.a8
William Balderrama, Nicholas J. Kuhn
A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if $A$ is a finite abelian $p$-group of rank $r$, then any finite $A$-space $X$ which is acyclic in the $n$th Morava $K$-theory with $n geqslant r$ will have its subspace $X^A$ of fixed points acyclic in the $(n-r)$th Morava Ktheory. This is a chromatic homotopy version of P. A. Smith’s classical theorem that if $X$ is acyclic in mod p homology, then so is $X^A$. The main purpose of this paper is to give an elementary proof of this new theorem that uses minimal background, and follows, as much as possible, the reasoning in standard proofs of the classical theorem. We also give a new fixed point theorem for finite dimensional, but possibly infinite, $Atextrm{-CW}$ complexes, which suggests some open problems.
T. Barthel、M. Hausmann、N. Naumann、T. Nikolaus、J. Noel 和 N. Stapleton 最近提出了一个定理。斯特普尔顿说，如果 $A$ 是一个秩为 $r$ 的有限无性 $p$ 群，那么任何在第 $n$th 莫拉瓦 $K$ 理论中具有 $n geqslant r$ 的非循环性的有限 $A$ 空间 $X$ 都会在第 $(n-r)$th 莫拉瓦 K 理论中具有非循环性的定点子空间 $X^A$。这是 P. A. Smith 经典定理的色度同调版本，即如果 $X$ 在 mod p 同调中是非周期性的，那么 $X^A$ 也是非周期性的。本文的主要目的是给出这一新定理的基本证明，它使用了最少的背景知识，并尽可能遵循经典定理标准证明中的推理。我们还给出了有限维，但可能是无限维的 $Atextrm{-CW}$ 复数的新定点定理，并提出了一些有待解决的问题。
{"title":"An elementary proof of the chromatic Smith fixed point theorem","authors":"William Balderrama, Nicholas J. Kuhn","doi":"10.4310/hha.2024.v26.n1.a8","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a8","url":null,"abstract":"A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if $A$ is a finite abelian $p$-group of rank $r$, then any finite $A$-space $X$ which is acyclic in the $n$th Morava $K$-theory with $n geqslant r$ will have its subspace $X^A$ of fixed points acyclic in the $(n-r)$th Morava Ktheory. This is a chromatic homotopy version of P. A. Smith’s classical theorem that if $X$ is acyclic in mod p homology, then so is $X^A$. The main purpose of this paper is to give an elementary proof of this new theorem that uses minimal background, and follows, as much as possible, the reasoning in standard proofs of the classical theorem. We also give a new fixed point theorem for finite dimensional, but possibly infinite, $Atextrm{-CW}$ complexes, which suggests some open problems.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924622","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 : 2024-01-24DOI: 10.4310/hha.2024.v26.n1.a3
John R. Klein
The object of this paper is to show that non-homotopy finite Poincaré duality spaces are plentiful. Let $π$ be a finitely presented group. Assuming that the reduced Grothendieck group $widetilde{K}_0 (mathbb{Z} [pi])$ has a non-trivial $2$-divisible element, we construct a finitely dominated Poincaré space $X$ with fundamental group $π$ such that $X$ is not homotopy finite. The dimension of $X$ can be made arbitrarily large. Our proof relies on a result which says that every finitely dominated space possesses a stable Poincaré duality thickening.
{"title":"On finite domination and Poincaré duality","authors":"John R. Klein","doi":"10.4310/hha.2024.v26.n1.a3","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a3","url":null,"abstract":"The object of this paper is to show that non-homotopy finite Poincaré duality spaces are plentiful. Let $π$ be a finitely presented group. Assuming that the reduced Grothendieck group $widetilde{K}_0 (mathbb{Z} [pi])$ has a non-trivial $2$-divisible element, we construct a finitely dominated Poincaré space $X$ with fundamental group $π$ such that $X$ is not homotopy finite. The dimension of $X$ can be made arbitrarily large. Our proof relies on a result which says that every finitely dominated space possesses a stable Poincaré duality thickening.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560962","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 : 2024-01-24DOI: 10.4310/hha.2024.v26.n1.a1
Malkhaz Bakuradze
This paper presents a commutative complex oriented cohomology theory that realizes the Buchstaber formal group law $F_B$ localized away from $2$. It is shown that the restriction of the classifying map of $F_B$ on the special unitary cobordism ring localized away from $2$ defines a four parameter genus, studied by Hoehn and Totaro.
{"title":"Polynomial generators of $mathbf{MSU}^ast [1/2]$ related to classifying maps of certain formal group laws","authors":"Malkhaz Bakuradze","doi":"10.4310/hha.2024.v26.n1.a1","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a1","url":null,"abstract":"This paper presents a commutative complex oriented cohomology theory that realizes the Buchstaber formal group law $F_B$ localized away from $2$. It is shown that the restriction of the classifying map of $F_B$ on the special unitary cobordism ring localized away from $2$ defines a four parameter genus, studied by Hoehn and Totaro.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560848","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 : 2024-01-24DOI: 10.4310/hha.2024.v26.n1.a2
Takahiro Matsushita, Shun Wakatsuki
We determine the homotopy types of the independence complexes of the $(n times 6)$-square grid graphs. In fact, we show that these complexes are homotopy equivalent to wedges of spheres.
我们确定了$(n times 6)$正方形网格图的独立复数的同调类型。事实上，我们证明了这些复数等同于球的楔形。
{"title":"Independence complexes of $(n times 6)$-grid graphs","authors":"Takahiro Matsushita, Shun Wakatsuki","doi":"10.4310/hha.2024.v26.n1.a2","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a2","url":null,"abstract":"We determine the homotopy types of the independence complexes of the $(n times 6)$-square grid graphs. In fact, we show that these complexes are homotopy equivalent to wedges of spheres.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139561199","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 : 2023-11-22DOI: 10.4310/hha.2023.v25.n2.a15
Matthew Burfitt, Jelena Grbić
By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray–Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank $2$ complete flag manifolds are $SU(3)/T^2$, $Sp(2)/T^2$, $mathit{Spin}(4)/T^2$, $mathit{Spin}(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of the rank $2$ complete flag manifolds.
通过将Gröbner基理论应用于谱序列代数，我们开发了一种新的计算方法，适用于任何Leray-Serre谱序列，其中基空间的上同调是有限生成多项式代数的商。我们通过推导标志流形的自由环空间的上同调来证明这一过程，对以前关于自由环空间拓扑的知识进行了重要的扩展。完备标志流形是李群与其最大环面之商。标志流形的秩是李群的最大环面的维数。等级2美元完成标志集合管是SU (3) / T ^ 2美元,Sp (2) / T ^ 2美元,美元 mathit{旋转}(4)/ T ^ 2美元,美元 mathit{旋转}(5)/ T ^ 2美元和G_2 / T ^ 2美元。本文计算了秩$2$完备标志流形的自由环空间的上同调。
{"title":"The cohomology of free loop spaces of rank $2$ flag manifolds","authors":"Matthew Burfitt, Jelena Grbić","doi":"10.4310/hha.2023.v25.n2.a15","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a15","url":null,"abstract":"By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray–Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank $2$ complete flag manifolds are $SU(3)/T^2$, $Sp(2)/T^2$, $mathit{Spin}(4)/T^2$, $mathit{Spin}(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of the rank $2$ complete flag manifolds.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520627","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 : 2023-11-22DOI: 10.4310/hha.2023.v25.n2.e18
John C. Baez, Alissa S. Crans, Urs Schreiber, Danny Stevenson
There were a number of sign errors in our paper “From loop groups to 2-groups” $href{https://dx.doi.org/10.4310/HHA.2007.v9.n2.a4 }{[textit{Homology Homotopy Appl.};textbf{9};textrm{(2007), 101–135}]}$. Here we explain how to correct those errors.
{"title":"Erratum to “From loop groups to 2-groups”","authors":"John C. Baez, Alissa S. Crans, Urs Schreiber, Danny Stevenson","doi":"10.4310/hha.2023.v25.n2.e18","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.e18","url":null,"abstract":"There were a number of sign errors in our paper “From loop groups to 2-groups” $href{https://dx.doi.org/10.4310/HHA.2007.v9.n2.a4 }{[textit{Homology Homotopy Appl.};textbf{9};textrm{(2007), 101–135}]}$. Here we explain how to correct those errors.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520650","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 : 2023-11-22DOI: 10.4310/hha.2023.v25.n2.a13
Ulrich Bauer, Maximilian Schmahl
We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent (co)homology, generalizing previous results in terms of barcodes. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for the efficient computation of barcodes for images, kernels, and co-kernels of such morphisms.
{"title":"Lifespan functors and natural dualities in persistent homology","authors":"Ulrich Bauer, Maximilian Schmahl","doi":"10.4310/hha.2023.v25.n2.a13","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a13","url":null,"abstract":"We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent (co)homology, generalizing previous results in terms of barcodes. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for the efficient computation of barcodes for images, kernels, and co-kernels of such morphisms.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520625","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：