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Unstable algebras over an operad II 操作数 II 上的不稳定数组
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2024-02-21 DOI: 10.4310/hha.2024.v26.n1.a4
Sacha Ikonicoff
{"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}
引用次数: 0
Magnitude, homology, and the Whitney twist 振幅、同调和惠特尼扭转
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2024-02-21 DOI: 10.4310/hha.2024.v26.n1.a7
Emily Roff
{"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}
引用次数: 0
Homotopy theory of spectral sequences 谱序列的同调理论
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2024-02-21 DOI: 10.4310/hha.2024.v26.n1.a5
Muriel Livernet, Sarah Whitehouse
{"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}
引用次数: 0
An elementary proof of the chromatic Smith fixed point theorem 色度史密斯定点定理的基本证明
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2024-02-21 DOI: 10.4310/hha.2024.v26.n1.a8
William Balderrama, Nicholas J. Kuhn
{"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}
引用次数: 0
On finite domination and Poincaré duality 论有限支配和泊恩卡对偶性
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2024-01-24 DOI: 10.4310/hha.2024.v26.n1.a3
John R. Klein
{"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}
引用次数: 0
Polynomial generators of $mathbf{MSU}^ast [1/2]$ related to classifying maps of certain formal group laws 与某些形式群法的分类映射有关的$mathbf{MSU}^ast [1/2]$ 的多项式生成器
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2024-01-24 DOI: 10.4310/hha.2024.v26.n1.a1
Malkhaz Bakuradze
{"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}
引用次数: 0
Independence complexes of $(n times 6)$-grid graphs $(n times 6)$网格图的独立复数
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2024-01-24 DOI: 10.4310/hha.2024.v26.n1.a2
Takahiro Matsushita, Shun Wakatsuki
{"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}
引用次数: 0
The cohomology of free loop spaces of rank $2$ flag manifolds 秩$2$标志流形的自由循环空间的上同调
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-22 DOI: 10.4310/hha.2023.v25.n2.a15
Matthew Burfitt, Jelena Grbić
{"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}
引用次数: 1
Erratum to “From loop groups to 2-groups” “从循环组到2组”的勘误
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-22 DOI: 10.4310/hha.2023.v25.n2.e18
John C. Baez, Alissa S. Crans, Urs Schreiber, Danny Stevenson
{"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}
引用次数: 0
Lifespan functors and natural dualities in persistent homology 持久同源中的寿命函子与自然对偶
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-22 DOI: 10.4310/hha.2023.v25.n2.a13
Ulrich Bauer, Maximilian Schmahl
{"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}
引用次数: 4
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