Extension theory of infinite symmetric products

Jerzy Dydak

Displaying similar documents to “Extension theory of infinite symmetric products”

Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of ℝⁿ

Anders Nilsson, Peter Wingren (2007)

Studia Mathematica

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A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(d i m_{H} (U), {\underset{̲}{d i m}}_{B} (U), {\bar{d i m}}_{B} (U)) = (r, s, t)$ . Moreover, $2^{- n} \leq H^{r} (K) \leq 2 n^{r / 2}$ .

Strong Cohomological Dimension

Jerzy Dydak, Akira Koyama (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that $I n d_{G} X = d i m_{G} X$ if X is a separable metric ANR and G is a countable Abelian group. Hence $d i m_{ℤ} X = d i m X$ for any separable metric ANR X.

Stable cohomotopy groups of compact spaces

Sławomir Nowak (2003)

Fundamenta Mathematicae

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We show that one can reduce the study of global (in particular cohomological) properties of a compact Hausdorff space X to the study of its stable cohomotopy groups $π_{s}^{k} (X)$ . Any cohomology functor on the homotopy category of compact spaces factorizes via the stable shape category ShStab. This is the main reason why the language and technique of stable shape theory can be used to describe and analyze the global structure of compact spaces. For a given Hausdorff compact space X, there exists...

Module-valued functors preserving the covering dimension

Jan Spěvák (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove a general theorem about preservation of the covering dimension $dim$ by certain covariant functors that implies, among others, the following concrete results. If $G$ G is a pathwise connected...

Character formula for wreath products of compact groups with the infinite symmetric group

Takeshi Hirai, Etsuko Hirai (2006)

Banach Center Publications

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Let $G =_{\infty} (T)$ be the wreath product of a compact group T with the infinite symmetric group $_{\infty}$ . We study the characters of factor representations of finite type of G, and give a formula which expresses all the characters explicitly.

Levi's forms of higher codimensional submanifolds

Andrea D&#039;Agnolo, Giuseppe Zampieri (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $X ≅ C^{n}$ , let $M$ be a $C^{2}$ hypersurface of $X$ , $S$ be a $C^{2}$ submanifold of $M$ . Denote by $L_{M}$ the Levi form of $M$ at $z_{0} \in S$ . In a previous paper [3] two numbers $s^{\pm} (S, p)$ , $p \in {({\dot{T}}_{S}^{*} X)}_{z_{0}}$ are defined; for $S = M$ they are the numbers of positive and negative eigenvalues for $L_{M}$ . For $S \subset M$ , $p \in S \times_{M} \dot{T}^{*}_{S} X)$ , we show here that $s^{\pm} (S, p)$ are still the numbers of positive and negative eigenvalues for $L_{M}$ when restricted to $T_{z_{0}}^{C} S$ . Applications to the concentration in degree for microfunctions at the boundary are given.

Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin (2002)

Fundamenta Mathematicae

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The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{α}}_{α \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $d i m (X_{α} \cap X_{β}) = n$ . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $c l c_{ℤ}^{n + 1}$ , where n ≥ 1, and G is an Abelian group. Let ${X_{α}}_{α \in J}$ be an uncountable family of closed subsets of X. If $d i m_{G} X = d i m_{G} X_{α} = n$ for all α ∈ J, then $d i m_{G} (X_{α} \cap X_{β}) = n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi...

Maps with dimensionally restricted fibers

Vesko Valov (2011)

Colloquium Mathematicae

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We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber $f^{- 1} (y)$ belongs to a class S of spaces, then there exists an $F_{σ}$ -set A ⊂ X such that A ∈ S and $d i m f^{- 1} (y) ∖ A = 0$ for all y ∈ Y. Here, S can be one of the following classes: (i) M: e-dim M ≤ K for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = M: dim M ≤ n, then dim f ∆ g ≤ 0 for almost all $g \in C (X,^{n + 1})$ .

Characterization of local dimension functions of subsets of $ℝ^{d}$

L. Olsen (2005)

Colloquium Mathematicae

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For a subset $E \subseteq ℝ^{d}$ and $x \in ℝ^{d}$ , the local Hausdorff dimension function of E at x is defined by $d i m_{H, l o c} (x, E) = l i m_{r ↘ 0} d i m_{H} (E \cap B (x, r))$ where $d i m_{H}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.

Birational positivity in dimension $4$

Behrouz Taji (2014)

Annales de l’institut Fourier

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In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of $Ω^{p}$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

A study of universal elements in classes of bases of topological spaces

Dimitris N. Georgiou, Athanasios C. Megaritis, Inderasan Naidoo, Fotini Sereti (2021)

Commentationes Mathematicae Universitatis Carolinae

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The universality problem focuses on finding universal spaces in classes of topological spaces. Moreover, in “Universal spaces and mappings” by S. D. Iliadis (2005), an important method of constructing such universal elements in classes of spaces is introduced and explained in details. Simultaneously, in “A topological dimension greater than or equal to the classical covering dimension” by D. N. Georgiou, A. C. Megaritis and F. Sereti (2017), new topological dimension is introduced and...

$ℰ_{B}$ -dimension function of bitopological spaces

M. Jelić (1987)

Matematički Vesnik

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On the Separation Dimension of $K_{ω}$

Yasunao Hattori, Jan van Mill (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that $t r t K_{ω} > ω + 1$ , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.

Hurewicz-Serre theorem in extension theory

M. Cencelj, J. Dydak, A. Mitra, A. Vavpetič (2008)

Fundamenta Mathematicae

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The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that $X τ K (H_{k} (L), k)$ for all k ≥ 1, then $X τ K (π_{k} (F), k)$ and $X τ K (π_{k} (L), k)$ for all k ≥ $.$ Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈...

Universal acyclic resolutions for arbitrary coefficient groups

Michael Levin (2003)

Fundamenta Mathematicae

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We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective $U V^{n - 1}$ -map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that $d i m_{G} X \leq k \leq n$ we have $d i m_{G} Z \leq k$ and r is G-acyclic.

Szpilrajn type theorem for concentration dimension

Jozef Myjak, Tomasz Szarek (2002)

Fundamenta Mathematicae

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Let X be a locally compact, separable metric space. We prove that $d i m_{T} X = i n f d i m_{L} X^{'} : X^{'} i s h o m e o m o r p h i c t o X$ , where $d i m_{L} X$ and $d i m_{T} X$ stand for the concentration dimension and the topological dimension of X, respectively.

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$ -subset $A_{k}$ of X such that $d i m A_{k} \leq k$ and the restriction $f | (X ∖ A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

Appendix to the article of T. Peternell: the Kodaira dimension of Kummer threefolds

Frédéric Campana, Thomas Peternell (2001)

Bulletin de la Société Mathématique de France

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We prove that Kummer threefolds $T / G$ with algebraic dimension $0$ have Kodaira dimension 0.

Some results on Poincaré sets

Min-wei Tang, Zhi-Yi Wu (2020)

Czechoslovak Mathematical Journal

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It is known that a set $H$ of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if ${dim}_{ℋ} (X_{H}) = 0$ , where $X_{H} : = \{x = \sum_{n = 1}^{\infty} \frac{x_{n}}{2^{n}} : x_{n} \in {0, 1}, x_{n} x_{n + h} = 0 for all n \geq 1, h \in H\}$ and ${dim}_{ℋ}$ denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set $X_{H}$ by replacing $2$ with $b > 2$ . It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.

Cofiniteness of torsion functors of cofinite modules

Reza Naghipour, Kamal Bahmanpour, Imaneh Khalili Gorji (2014)

Colloquium Mathematicae

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Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules $T o r_{i}^{R} (N, M)$ are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules $T o r_{i}^{R} (N, H_{I}^{j} (M))$ are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules $T o r_{i}^{R} (N, M)$ are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it...

Scalar differential concomitants of first order of a symmetric connexion $Γ_{j k}^{i}$ in the two-dimensional space and their applications

Michał Lorens (1980)

Annales Polonici Mathematici

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