Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $mathbb Z^{mathbb N}$ to $G$ factors through the projection to some finite product $mathbb Z^n$. Various authors have proposed generalizations to non-commutative groups, resulting in a plethora of similar but not completely equivalent concepts. In the first part of this work we present a unified treatment of these concepts and examine how are they related. In the second part of the paper we study slender groups in the context of co-small objects in certain categories, and give several new applications including the proof that certain homology groups of Barratt-Milnor spaces are cotorsion groups and a universal coefficients theorem for v{C}ech cohomology with coefficients in a slender group.
经典地说，如果从可数积$mathbb Z^{mathbb N}$到$G$的所有同态通过投影到某个有限积$mathbb Z^ N $，则一个阿贝尔群$G$是细长的。许多作者提出了对非交换群的推广，导致了大量相似但不完全等价的概念。在这项工作的第一部分，我们提出了这些概念的统一处理，并检查它们是如何相关的。在第二部分中，我们研究了特定范畴中共小对象下的细长群，给出了若干新的应用，包括证明Barratt-Milnor空间的某些同调群是扭转群，以及在细长群中v{C}ech上同调与系数的一个普适系数定理。
{"title":"Inverse limit slender groups","authors":"G. Conner, W. Herfort, Curtis Kent, Peter Pavesic","doi":"10.4064/fm118-12-2022","DOIUrl":"https://doi.org/10.4064/fm118-12-2022","url":null,"abstract":"Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $mathbb Z^{mathbb N}$ to $G$ factors through the projection to some finite product $mathbb Z^n$. Various authors have proposed generalizations to non-commutative groups, resulting in a plethora of similar but not completely equivalent concepts. In the first part of this work we present a unified treatment of these concepts and examine how are they related. In the second part of the paper we study slender groups in the context of co-small objects in certain categories, and give several new applications including the proof that certain homology groups of Barratt-Milnor spaces are cotorsion groups and a universal coefficients theorem for v{C}ech cohomology with coefficients in a slender group.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43981804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. A rotated odometer is an infinite interval exchange transformation (IET) obtained as a composition of the von Neumann-Kakutani map and a finite IET of intervals of equal length. In this paper, we consider rotated odometers for which the finite IET is of intervals of length 2 − N , for some N ≥ 1. We show that every such system is measurably isomorphic to a Z -action on a rooted tree, and that the unique minimal aperiodic subsystem of this action is always measurably isomorphic to the action of the adding machine. We discuss the applications of this work to the study of group actions on binary trees.
{"title":"Rotated odometers and actions on rooted trees","authors":"H. Bruin, O. Lukina","doi":"10.4064/fm74-10-2022","DOIUrl":"https://doi.org/10.4064/fm74-10-2022","url":null,"abstract":". A rotated odometer is an infinite interval exchange transformation (IET) obtained as a composition of the von Neumann-Kakutani map and a finite IET of intervals of equal length. In this paper, we consider rotated odometers for which the finite IET is of intervals of length 2 − N , for some N ≥ 1. We show that every such system is measurably isomorphic to a Z -action on a rooted tree, and that the unique minimal aperiodic subsystem of this action is always measurably isomorphic to the action of the adding machine. We discuss the applications of this work to the study of group actions on binary trees.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49211075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Our main object of interest is the following notion: we say that a topological space space $X$ is in FinBW($mathcal{I}$), where $mathcal{I}$ is an ideal on $omega$, if for each sequence $(x_n)_{ninomega}$ in $X$ one can find an $Anotinmathcal{I}$ such that $(x_n)_{nin A}$ converges in $X$. We define an ideal $mathcal{BI}$ which is critical for FinBW($mathcal{I}$) in the following sense: Under CH, for every ideal $mathcal{I}$, $mathcal{BI}notleq_Kmathcal{I}$ ($leq_K$ denotes the Katv{e}tov preorder of ideals) iff there is an uncountable separable space in FinBW($mathcal{I}$). We show that $mathcal{BI}notleq_Kmathcal{I}$ and $omega_1$ with the order topology is in FinBW($mathcal{I}$), for all $bf{Pi^0_4}$ ideals $mathcal{I}$. We examine when FinBW($mathcal{I}$)$setminus$FinBW($mathcal{J}$) is nonempty: we prove under MA($sigma$-centered) that for $bf{Pi^0_4}$ ideals $mathcal{I}$ and $mathcal{J}$ this is equivalent to $mathcal{J}notleq_Kmathcal{I}$. Moreover, answering in negative a question of M. Hruv{s}'ak and D. Meza-Alc'antara, we show that the ideal $text{Fin}timestext{Fin}$ is not critical among Borel ideals for extendability to a $bf{Pi^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.
{"title":"Unboring ideals","authors":"A. Kwela","doi":"10.4064/fm44-2-2023","DOIUrl":"https://doi.org/10.4064/fm44-2-2023","url":null,"abstract":"Our main object of interest is the following notion: we say that a topological space space $X$ is in FinBW($mathcal{I}$), where $mathcal{I}$ is an ideal on $omega$, if for each sequence $(x_n)_{ninomega}$ in $X$ one can find an $Anotinmathcal{I}$ such that $(x_n)_{nin A}$ converges in $X$. We define an ideal $mathcal{BI}$ which is critical for FinBW($mathcal{I}$) in the following sense: Under CH, for every ideal $mathcal{I}$, $mathcal{BI}notleq_Kmathcal{I}$ ($leq_K$ denotes the Katv{e}tov preorder of ideals) iff there is an uncountable separable space in FinBW($mathcal{I}$). We show that $mathcal{BI}notleq_Kmathcal{I}$ and $omega_1$ with the order topology is in FinBW($mathcal{I}$), for all $bf{Pi^0_4}$ ideals $mathcal{I}$. We examine when FinBW($mathcal{I}$)$setminus$FinBW($mathcal{J}$) is nonempty: we prove under MA($sigma$-centered) that for $bf{Pi^0_4}$ ideals $mathcal{I}$ and $mathcal{J}$ this is equivalent to $mathcal{J}notleq_Kmathcal{I}$. Moreover, answering in negative a question of M. Hruv{s}'ak and D. Meza-Alc'antara, we show that the ideal $text{Fin}timestext{Fin}$ is not critical among Borel ideals for extendability to a $bf{Pi^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45338880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this short note, we introduce a generalization of the canonical base property, called transfer of internality on quotients. A structural study of groups definable in theories with this property yields as a consequence infinitely many new uncountably categorical additive covers of the complex numbers without the canonical base property.
{"title":"A (possibly new) structure without the canonical base property","authors":"M. Loesch","doi":"10.4064/fm156-3-2022","DOIUrl":"https://doi.org/10.4064/fm156-3-2022","url":null,"abstract":"In this short note, we introduce a generalization of the canonical base property, called transfer of internality on quotients. A structural study of groups definable in theories with this property yields as a consequence infinitely many new uncountably categorical additive covers of the complex numbers without the canonical base property.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42156364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular, if the fusion system has trivial center, or if its focal subgroup is the entire Sylow group, then this factorization is unique (up to the ordering of the factors). This result was motivated by questions about automorphism groups of products of fusion systems.
{"title":"A Krull–Remak–Schmidt theorem for fusion systems","authors":"B. Oliver","doi":"10.4064/fm160-5-2022","DOIUrl":"https://doi.org/10.4064/fm160-5-2022","url":null,"abstract":"We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular, if the fusion system has trivial center, or if its focal subgroup is the entire Sylow group, then this factorization is unique (up to the ordering of the factors). This result was motivated by questions about automorphism groups of products of fusion systems.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45286833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is a modified chapter of the author’s Ph.D. thesis. We introduce the notions of sequentially approximated types and sequentially approximated Keisler measures. As the names imply, these are types which can be approximated by a sequence of realized types and measures which can be approximated by a sequence of “averaging measures” on tuples of realized types. We show that both generically stable types (in arbitrary theories) and Keisler measures which are finitely satisfiable over a countable model (in NIP theories) are sequentially approximated. We also introduce the notion of a smooth sequence in a measure over a model and give an equivalent characterization of generically stable measures (in NIP theories) via this definition. In the last section, we take the opportunity to generalize the main result of [8].
{"title":"Sequential approximations for types and Keisler measures","authors":"K. Gannon","doi":"10.4064/fm133-12-2021","DOIUrl":"https://doi.org/10.4064/fm133-12-2021","url":null,"abstract":"This paper is a modified chapter of the author’s Ph.D. thesis. We introduce the notions of sequentially approximated types and sequentially approximated Keisler measures. As the names imply, these are types which can be approximated by a sequence of realized types and measures which can be approximated by a sequence of “averaging measures” on tuples of realized types. We show that both generically stable types (in arbitrary theories) and Keisler measures which are finitely satisfiable over a countable model (in NIP theories) are sequentially approximated. We also introduce the notion of a smooth sequence in a measure over a model and give an equivalent characterization of generically stable measures (in NIP theories) via this definition. In the last section, we take the opportunity to generalize the main result of [8].","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41492256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main goal of this paper is to prove that the space of directed loops on the final precubical set is homotopy equivalent to the “total” configuration space of points on the plane; by “total” we mean that any finite number of points in a configuration is allowed. We also provide several applications: we define new invariants of precubical sets, prove that directed path spaces on any precubical complex have the homotopy types of CW-complexes and construct certain presentations of configuration spaces of points on the plane as nerves of categories.
{"title":"Configuration spaces and directed paths\u0000on the final precubical set","authors":"Jakub Paliga, Krzysztof Ziemia'nski","doi":"10.4064/fm114-9-2021","DOIUrl":"https://doi.org/10.4064/fm114-9-2021","url":null,"abstract":"The main goal of this paper is to prove that the space of directed loops on the final precubical set is homotopy equivalent to the “total” configuration space of points on the plane; by “total” we mean that any finite number of points in a configuration is allowed. We also provide several applications: we define new invariants of precubical sets, prove that directed path spaces on any precubical complex have the homotopy types of CW-complexes and construct certain presentations of configuration spaces of points on the plane as nerves of categories.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43836606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop techniques that lay out a basis for generalizations of the famous Thurston's Topological Characterization of Rational Functions for an infinite set of marked points and branched coverings of infinite degree. Analogously to the classical theorem we consider the Thurston's $sigma$-map acting on a Teichm"uller space which is this time infinite-dimensional -- and this leads to a completely different theory comparing to the classical setting. We demonstrate our techniques by giving an alternative proof of the result by Markus F"orster about the classification of exponential functions with the escaping singular value.
我们开发了一种技术，为著名的Thurston有理函数的拓扑表征的推广奠定了基础，该表征适用于无限次的标记点和分支覆盖的无限集。与经典定理类似，我们考虑瑟斯顿映射作用于泰希姆乌勒空间，这一次是无限维的，这导致了一个完全不同的理论，与经典的设置相比。我们通过给出Markus F orster关于带转义奇异值的指数函数分类结果的另一种证明来证明我们的技术。
{"title":"Infinite-dimensional Thurston theory and transcendental dynamics I: infinite-legged spiders","authors":"K. Bogdanov","doi":"10.4064/fm82-11-2022","DOIUrl":"https://doi.org/10.4064/fm82-11-2022","url":null,"abstract":"We develop techniques that lay out a basis for generalizations of the famous Thurston's Topological Characterization of Rational Functions for an infinite set of marked points and branched coverings of infinite degree. Analogously to the classical theorem we consider the Thurston's $sigma$-map acting on a Teichm\"uller space which is this time infinite-dimensional -- and this leads to a completely different theory comparing to the classical setting. We demonstrate our techniques by giving an alternative proof of the result by Markus F\"orster about the classification of exponential functions with the escaping singular value.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47008368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a completely metrizable space X, let par(X) denote the smallest possible size of a partition of X into Polish spaces, and cov(X) the smallest possible size of a covering of X with Polish spaces. Observe that cov(X) ≤ par(X) for every X, because every partition of X is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality cov(X) < par(X) can hold for some completely metrizable space X. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if cov(X) < par(X) for any completely metrizable X, then 0 exists.
{"title":"Covering versus partitioning with Polish spaces","authors":"W. Brian","doi":"10.4064/fm28-5-2022","DOIUrl":"https://doi.org/10.4064/fm28-5-2022","url":null,"abstract":"Given a completely metrizable space X, let par(X) denote the smallest possible size of a partition of X into Polish spaces, and cov(X) the smallest possible size of a covering of X with Polish spaces. Observe that cov(X) ≤ par(X) for every X, because every partition of X is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality cov(X) < par(X) can hold for some completely metrizable space X. We also prove that using large cardinals is necessary for obtaining this strict inequality, because if cov(X) < par(X) for any completely metrizable X, then 0 exists.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45123151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Thurston introduced invariant (quadratic) laminations in his 1984 preprint as a vehicle for understanding the connected Julia sets and the parameter space of quadratic polynomials. Important ingredients of his analysis of the angle doubling map σ 2 on the unit circle S 1 were the Central Strip Lemma, non-existence of wandering polygons, the transitivity of the first return map on vertices of periodic polygons, and the non-crossing of minors of quadratic invariant laminations. We use Thurston’s methods to prove similar results for unicritical laminations of arbitrary degree d and to show that the set of so-called minors of unicritical laminations themselves form a Unicritical Minor Lamination UML d . In the end we verify the Fatou conjecture for the unicritical laminations and extend the Lavaurs algorithm onto UML d .
{"title":"Unicritical laminations","authors":"Sourav Bhattacharya, A. Blokh, D. Schleicher","doi":"10.4064/fm18-2-2022","DOIUrl":"https://doi.org/10.4064/fm18-2-2022","url":null,"abstract":". Thurston introduced invariant (quadratic) laminations in his 1984 preprint as a vehicle for understanding the connected Julia sets and the parameter space of quadratic polynomials. Important ingredients of his analysis of the angle doubling map σ 2 on the unit circle S 1 were the Central Strip Lemma, non-existence of wandering polygons, the transitivity of the first return map on vertices of periodic polygons, and the non-crossing of minors of quadratic invariant laminations. We use Thurston’s methods to prove similar results for unicritical laminations of arbitrary degree d and to show that the set of so-called minors of unicritical laminations themselves form a Unicritical Minor Lamination UML d . In the end we verify the Fatou conjecture for the unicritical laminations and extend the Lavaurs algorithm onto UML d .","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43515155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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