Bulletin of the Polish Academy of Sciences Mathematics, 2008
We characterize strong cohomological dimension of separable metric spaces in terms of extension o... more 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 IndG X = dimG X if X is a separable metric ANR and G is a countable Abelian group. Hence dim Z X = dim X for any separable metric ANR X.
A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bi... more A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. In we gave characterizations of monomorphisms (resp. epimorphisms) in arbitrary pro-categories, pro-C, where C has direct sums (resp. weak push-outs). In this paper we introduce the notions of strong monomorphism and strong epimorphism. Part of their significance is that they are preserved by functors. These notions and their characterizations lead us to important classical properties and problems in shape and pro-homotopy. For instance, strong epimorphisms allow us to give a categorical point of view of uniform movability and to introduce a new kind of movability, the sequential movability. Strong monomorphisms are connected to a problem of K.Borsuk regarding a descending chain of retracts of ANRs. If f : X → Y is a bimorphism in the pointed shape category of topological spaces, we prove that f is a weak isomorphism and f is an isomorphism provided Y is sequentially movable and X or Y is the suspension of a topological space. If f : X → Y is a bimorphism in the pro-category pro-H0 (consisting of inverse systems in H0, the homotopy category of pointed connected CW complexes) we show that f is an isomorphism provided Y is sequentially movable.
It is well-known that a paracompact space X is of covering dimension n if and only if any map f f... more It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to
Extension Theory can be defined as studying extensions of maps from topological spaces to metric ... more Extension Theory can be defined as studying extensions of maps from topological spaces to metric simplicial complexes or CW complexes. One has a natural notion of an absolute (neighborhood) extensor K of X. It is shown that several concepts of set-theoretic topology can be naturally introduced using ideas of Extension Theory. Also, it is shown that several results of set-theoretic topology have a natural interpretation and simple proofs in Extension Theory. Here are sample results.
The dimension algebra of graded groups is introduced. With the help of known geometric results of... more The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of those equivalence classes: \linebreak 1. For pointed CW complexes $K$ and $L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). \linebreak 2. For pointed compact spaces $X$ and $Y$, $\dim(\cal H^{-\ast}(X))=\dim(\cal H^{-\ast}(Y))$ if and only if $X$ and $Y$ are of the same dimension type (i.e., $\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$). Dranishnikov's version of Hurewicz Theorem in extension theory becomes $\dim(\pi_\ast(K))=\dim(H_\ast(K))$ for all simply connected $K$. The concept of cohomological dimension $\...
Transactions of the American Mathematical Society, 1993
The main result of the first part of the paper is a generalization of the classical result of Men... more The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : dim(A ∪ B) ≤ dim A + dim B + 1.
We prove existence of extension dimension for paracompact spaces. Here is the main result of the ... more We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper:
The following theorem is the main result of the paper: Theorem. Let G be an Abelian group and m &... more The following theorem is the main result of the paper: Theorem. Let G be an Abelian group and m > 0. Let G be a countable family of countable Abelian groups and let D : G → Z+ be a function. The following conditions are equivalent: 1.(1) For any CW complex P and any aϵHm(P;G) − {0} there is a compactum X and a map π : X → P such that dim X = m, dimHX ⩽ D(H) for each H ϵ G and a ϵ im(Ȟm(X ; G) → Ȟm(P;G)).2.(2) H̃k(K(H,D(H));G) = 0 for all k < m and all H ϵ G.As an application, we prove the existence of compacta realizing dimension functions, a result due to A.N. Dranishnikov.
The purpose of this note is to introduce a class of compacta, called movable in the sense of nsha... more The purpose of this note is to introduce a class of compacta, called movable in the sense of nshape. We investigate its properties and relation to n-movability. Also, we give a negative solution to a question of A. Chigogidze concerning a characterization of hereditarily n-shape equivalences. 2004 Published by Elsevier B.V.
Finitistic spaces form a natural class containing compact and finite-dimensional spaces. Introduc... more Finitistic spaces form a natural class containing compact and finite-dimensional spaces. Introduced and investigated by fixed-point theorists, finitistic spaces found an application in cohomological dimension theory. In the paper, two characterizations of paracompact, finitistic spaces are proved. These characterizations allow to create a mechanism of generalizing results of finite dimension theory. As an application we obtain results on compact group actions on paracompact spaces which were previously known for compact Lie group actions.
A b s t r a c t In the paper one exhibits a metric continuum X and a polyhedron P such that the C... more A b s t r a c t In the paper one exhibits a metric continuum X and a polyhedron P such that the Cartesian product X × P fails to be the product of X and P in the shape category of topological spaces. 1
Let $X$ be a Hausdorff compact space and $C(X)$ be the algebra of all continuous complex-valued f... more Let $X$ be a Hausdorff compact space and $C(X)$ be the algebra of all continuous complex-valued functions on $X$, endowed with the supremum norm. We say that $C(X)$ is (approximately) $n$-th root closed if any function from $C(X)$ is (approximately) equal to the $n$-th power of another function. We characterize the approximate $n$-th root closedness of $C(X)$ in terms of
The famous \v{S}varc-Milnor Lemma says that a group $G$ acting properly and cocompactly via isome... more The famous \v{S}varc-Milnor Lemma says that a group $G$ acting properly and cocompactly via isometries on a length space $X$ is finitely generated and induces a quasi-isometry equivalence $g\to g\cdot x_0$ for any $x_0\in X$. We redefine the concept of coarseness so that the proof of the Lemma is automatic.
Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions o... more Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov \cite{Dr$_1$} and Dranishnikov-Keesling-Uspienskij \cite{DKU}.
We define Peano covering maps and prove basic properties analogous to classical covers. Their dom... more We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical
James \cite{Jam} introduced uniform covering maps as an analog of covering maps in the topologica... more James \cite{Jam} introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut \cite{BP3} introduced a theory of covers for uniform spaces generalizing their results for topological groups \cite{BP1}-\cite{BP2}. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal
Bulletin of the Polish Academy of Sciences Mathematics, 2008
We characterize strong cohomological dimension of separable metric spaces in terms of extension o... more 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 IndG X = dimG X if X is a separable metric ANR and G is a countable Abelian group. Hence dim Z X = dim X for any separable metric ANR X.
A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bi... more A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. In we gave characterizations of monomorphisms (resp. epimorphisms) in arbitrary pro-categories, pro-C, where C has direct sums (resp. weak push-outs). In this paper we introduce the notions of strong monomorphism and strong epimorphism. Part of their significance is that they are preserved by functors. These notions and their characterizations lead us to important classical properties and problems in shape and pro-homotopy. For instance, strong epimorphisms allow us to give a categorical point of view of uniform movability and to introduce a new kind of movability, the sequential movability. Strong monomorphisms are connected to a problem of K.Borsuk regarding a descending chain of retracts of ANRs. If f : X → Y is a bimorphism in the pointed shape category of topological spaces, we prove that f is a weak isomorphism and f is an isomorphism provided Y is sequentially movable and X or Y is the suspension of a topological space. If f : X → Y is a bimorphism in the pro-category pro-H0 (consisting of inverse systems in H0, the homotopy category of pointed connected CW complexes) we show that f is an isomorphism provided Y is sequentially movable.
It is well-known that a paracompact space X is of covering dimension n if and only if any map f f... more It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to
Extension Theory can be defined as studying extensions of maps from topological spaces to metric ... more Extension Theory can be defined as studying extensions of maps from topological spaces to metric simplicial complexes or CW complexes. One has a natural notion of an absolute (neighborhood) extensor K of X. It is shown that several concepts of set-theoretic topology can be naturally introduced using ideas of Extension Theory. Also, it is shown that several results of set-theoretic topology have a natural interpretation and simple proofs in Extension Theory. Here are sample results.
The dimension algebra of graded groups is introduced. With the help of known geometric results of... more The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of those equivalence classes: \linebreak 1. For pointed CW complexes $K$ and $L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). \linebreak 2. For pointed compact spaces $X$ and $Y$, $\dim(\cal H^{-\ast}(X))=\dim(\cal H^{-\ast}(Y))$ if and only if $X$ and $Y$ are of the same dimension type (i.e., $\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$). Dranishnikov's version of Hurewicz Theorem in extension theory becomes $\dim(\pi_\ast(K))=\dim(H_\ast(K))$ for all simply connected $K$. The concept of cohomological dimension $\...
Transactions of the American Mathematical Society, 1993
The main result of the first part of the paper is a generalization of the classical result of Men... more The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : dim(A ∪ B) ≤ dim A + dim B + 1.
We prove existence of extension dimension for paracompact spaces. Here is the main result of the ... more We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper:
The following theorem is the main result of the paper: Theorem. Let G be an Abelian group and m &... more The following theorem is the main result of the paper: Theorem. Let G be an Abelian group and m > 0. Let G be a countable family of countable Abelian groups and let D : G → Z+ be a function. The following conditions are equivalent: 1.(1) For any CW complex P and any aϵHm(P;G) − {0} there is a compactum X and a map π : X → P such that dim X = m, dimHX ⩽ D(H) for each H ϵ G and a ϵ im(Ȟm(X ; G) → Ȟm(P;G)).2.(2) H̃k(K(H,D(H));G) = 0 for all k < m and all H ϵ G.As an application, we prove the existence of compacta realizing dimension functions, a result due to A.N. Dranishnikov.
The purpose of this note is to introduce a class of compacta, called movable in the sense of nsha... more The purpose of this note is to introduce a class of compacta, called movable in the sense of nshape. We investigate its properties and relation to n-movability. Also, we give a negative solution to a question of A. Chigogidze concerning a characterization of hereditarily n-shape equivalences. 2004 Published by Elsevier B.V.
Finitistic spaces form a natural class containing compact and finite-dimensional spaces. Introduc... more Finitistic spaces form a natural class containing compact and finite-dimensional spaces. Introduced and investigated by fixed-point theorists, finitistic spaces found an application in cohomological dimension theory. In the paper, two characterizations of paracompact, finitistic spaces are proved. These characterizations allow to create a mechanism of generalizing results of finite dimension theory. As an application we obtain results on compact group actions on paracompact spaces which were previously known for compact Lie group actions.
A b s t r a c t In the paper one exhibits a metric continuum X and a polyhedron P such that the C... more A b s t r a c t In the paper one exhibits a metric continuum X and a polyhedron P such that the Cartesian product X × P fails to be the product of X and P in the shape category of topological spaces. 1
Let $X$ be a Hausdorff compact space and $C(X)$ be the algebra of all continuous complex-valued f... more Let $X$ be a Hausdorff compact space and $C(X)$ be the algebra of all continuous complex-valued functions on $X$, endowed with the supremum norm. We say that $C(X)$ is (approximately) $n$-th root closed if any function from $C(X)$ is (approximately) equal to the $n$-th power of another function. We characterize the approximate $n$-th root closedness of $C(X)$ in terms of
The famous \v{S}varc-Milnor Lemma says that a group $G$ acting properly and cocompactly via isome... more The famous \v{S}varc-Milnor Lemma says that a group $G$ acting properly and cocompactly via isometries on a length space $X$ is finitely generated and induces a quasi-isometry equivalence $g\to g\cdot x_0$ for any $x_0\in X$. We redefine the concept of coarseness so that the proof of the Lemma is automatic.
Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions o... more Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov \cite{Dr$_1$} and Dranishnikov-Keesling-Uspienskij \cite{DKU}.
We define Peano covering maps and prove basic properties analogous to classical covers. Their dom... more We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical
James \cite{Jam} introduced uniform covering maps as an analog of covering maps in the topologica... more James \cite{Jam} introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut \cite{BP3} introduced a theory of covers for uniform spaces generalizing their results for topological groups \cite{BP1}-\cite{BP2}. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal
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Papers by Jerzy Dydak