Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege; Jerzy Dydak; Rolando Jiménez; Akira Koyama; Evgeny V. Shchepin

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Displaying similar documents to “Borsuk-Sieklucki theorem in cohomological dimension theory”

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.

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.

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...

The $G$ -graded identities of the Grassmann Algebra

Lucio Centrone (2016)

Archivum Mathematicum

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Let $G$ be a finite abelian group with identity element $1_{G}$ and $L = ⨁_{g \in G} L^{g}$ be an infinite dimensional $G$ -homogeneous vector space over a field of characteristic $0$ . Let $E = E (L)$ be the Grassmann algebra generated by $L$ . It follows that $E$ is a $G$ -graded algebra. Let $| G |$ be odd, then we prove that in order to describe any ideal of $G$ -graded identities of $E$ it is sufficient to deal with $G^{'}$ -grading, where $| G^{'} | \leq | G |$ , ${dim}_{F} L^{1_{G^{'}}} = \infty$ and ${dim}_{F} L^{g^{'}} < \infty$ if $g^{'} \neq 1_{G^{'}}$ . In the same spirit of the case $| G |$ odd, if $| G |$ is even it is sufficient to study only those $G$ -gradings...

Seeking a network characterization of Corson compacta

Ziqin Feng (2021)

Commentationes Mathematicae Universitatis Carolinae

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We say that a collection $𝒜$ of subsets of $X$ has property $(C C)$ if there is a set $D$ and point-countable collections $𝒞$ of closed subsets of $X$ such that for any $A \in 𝒜$ there is a finite subcollection $ℱ$ of $𝒞$ such that $A = D ∖ ⋃ ℱ$ . Then we prove that any compact space is Corson if and only if it has a point- $σ$ - $(C C)$ base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis”...

The canonical constructions of connections on total spaces of fibred manifolds

Włodzimierz M. Mikulski (2024)

Archivum Mathematicum

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We classify classical linear connections $A (Γ, Λ, Θ)$ on the total space $Y$ of a fibred manifold $Y \to M$ induced in a natural way by the following three objects: a general connection $Γ$ in $Y \to M$ , a classical linear connection $Λ$ on $M$ and a linear connection $Θ$ in the vertical bundle $V Y \to Y$ . The main result says that if $\dim (M) \geq 3$ and $\dim (Y) - \dim (M) \geq 3$ then the natural operators $A$ under consideration form the $17$ dimensional affine space.

Extension properties of Stone-Čech coronas and proper absolute extensors

A. Chigogidze (2013)

Fundamenta Mathematicae

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We characterize, in terms of X, the extensional dimension of the Stone-Čech corona βX∖X of a locally compact and Lindelöf space X. The non-Lindelöf case is also settled in terms of extending proper maps with values in $I^{τ} ∖ L$ , where L is a finite complex. Further, for a finite complex L, an uncountable cardinal τ and a $Z_{τ}$ -set X in the Tikhonov cube $I^{τ}$ we find a necessary and sufficient condition, in terms of $I^{τ} ∖ X$ , for X to be in the class AE([L]). We also introduce a concept of a proper absolute...

Continuous images of Lindelöf $p$ -groups, $σ$ -compact groups, and related results

Aleksander V. Arhangel'skii (2019)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that there exists a $σ$ -compact topological group which cannot be represented as a continuous image of a Lindelöf $p$ -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$ -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$ -group, then there exists a covering $γ$ of $Y$ by dyadic compacta such that $| γ | \leq 2^{ω}$ . We also show that if a homogeneous compact space $Y$ is...

Maximal non valuation domains in an integral domain

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring $R$ of an integral domain $S$ is called a maximal non valuation domain in $S$ if $R$ is not a valuation subring of $S$ , and for any ring $T$ such that $R \subset T \subset S$ , $T$ is a valuation subring of $S$ . For a local domain $S$ , the equivalence of an integrally closed maximal non VD in $S$ and a maximal non local subring of $S$ is established. The relation between $dim (R, S)$ and...

The cleanness of (symbolic) powers of Stanley-Reisner ideals

Somayeh Bandari, Ali Soleyman Jahan (2017)

Czechoslovak Mathematical Journal

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Let $Δ$ be a pure simplicial complex on the vertex set $[n] = {1, ..., n}$ and $I_{Δ}$ its Stanley-Reisner ideal in the polynomial ring $S = K [x_{1}, ..., x_{n}]$ . We show that $Δ$ is a matroid (complete intersection) if and only if $S / I_{Δ}^{(m)}$ ( $S / I_{Δ}^{m}$ ) is clean for all $m \in ℕ$ and this is equivalent to saying that $S / I_{Δ}^{(m)}$ ( $S / I_{Δ}^{m}$ , respectively) is Cohen-Macaulay for all $m \in ℕ$ . By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S / I^{m}$ or $S / I^{(m)}$ is not (pretty) clean for all integer $m \geq 3$ . If $dim (Δ) = 1$ , we also prove that $S / I_{Δ}^{(2)}$ ( $S / I_{Δ}^{2}$ ) is clean if and only...

Contracting endomorphisms and dualizing complexes

Saeed Nasseh, Sean Sather-Wagstaff (2015)

Czechoslovak Mathematical Journal

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We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$ . Our focus is on homological properties of contracting endomorphisms of $R$ , e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$ -finite and $C$ is a semidualizing $R$ -complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$ -complex; (ii) $C \sim 𝐑 {Hom}_{R} (^{n} R, C)$ for some $n > 0$ ; (iii) $G_{C} {-dim}^{n} R < \infty$ and $C$ is derived...

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials $P_{c} (z) = z^{d} + c$ for c ∈ ℂ. Denote by $J_{c}$ the Julia set of $P_{c}$ and let $ℳ_{d} = c | J_{c} i s c o n n e c t e d$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c ₀ \in \partial ℳ_{d}$ : those for which the critical point 0 is not recurrent by $P_{c ₀}$ and without parabolic cycles. The Hausdorff dimension of $J_{c}$ , denoted by $H D (J_{c})$ , does not depend continuously on c at such $c ₀ \in \partial ℳ_{d}$ ; on the other hand the function $c \mapsto H D (J_{c})$ is analytic in $ℂ - ℳ_{d}$ . Our first result asserts that there is still some...