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Strong shape theory

100%
Dydak J., Segal J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction..................................................................................................................................... 5 2. Terminology and notation.................................................................................................................... 6 3. Proper maps on contractible telescopes.......................................................................................... 8 4. The strong shape category.................................................................................................................. 13 5. Semi-equivalences............................................................................................................................... 19 6. Geometric characterization of maps inducing strong shape equivalences............................... 21 7. Some classes of maps which induce strong shape equivalence............................................... 27 8. Concluding remarks............................................................................................................................. 35 References.................................................................................................................................................. 38
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A counterexample concerning products in the shape category

93%
Dydak J., Mardešić S.
Fundamenta Mathematicae
|
2005
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tom 186
|
nr 1
39-54
EN
We exhibit 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.
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Covering maps for locally path-connected spaces

75%
Brodskiy N., Dydak J., Labuz B., Mitra A.
Fundamenta Mathematicae
|
2012
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tom 218
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nr 1
13-46
EN
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 with generalized regular covering maps introduced by Fischer and Zastrow. If X is path-connected, then every Peano covering map is equivalent to the projection X̃/H → X, where H is a subgroup of the fundamental group of X and X̃ equipped with the topology introduced in Spanier's Algebraic Topology. The projection X̃/H → X is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on X̃ called the lasso topology. Then the fundamental group π₁(X) as a subspace of X̃ with the lasso topology becomes a topological group. Also, one has a characterization of X̃/H → X having the unique path lifting property if H is a normal subgroup of π₁(X). Namely, H must be closed in π₁(X) with the lasso topology. Such groups include π(𝓤,x₀) (𝓤 being an open cover of X) and the kernel of the natural homomorphism π₁(X,x₀) → π̌₁(X,x₀).
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Hurewicz-Serre theorem in extension theory

75%
Cencelj M., Dydak J., Mitra A., Vavpetič A.
Fundamenta Mathematicae
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2008
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tom 198
|
nr 2
113-123
EN
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 $XτK(H_{k}(L),k)$ for all k ≥ 1, then $XτK(π_{k}(F),k)$ and $XτK(π_{k}(L),k)$ for all k ≥ $. Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈ ANR. Suppose L is a nilpotent CW complex. If XτSP(L), then XτL in the following cases: (a) H₁(L) is finitely generated. (b) H₁(L) is a torsion group.
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Coarse structures and group actions

75%
Brodskiy N., Dydak J., Mitra A.
Colloquium Mathematicum
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2008
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tom 111
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nr 1
149-158
EN
The main results of the paper are: Proposition 0.1. A group G acting coarsely on a coarse space (X,𝓒) induces a coarse equivalence g ↦ g·x₀ from G to X for any x₀ ∈ X. Theorem 0.2. Two coarse structures 𝓒₁ and 𝓒₂ on the same set X are equivalent if the following conditions are satisfied: (1) Bounded sets in 𝓒₁ are identical with bounded sets in 𝓒₂. (2) There is a coarse action ϕ₁ of a group G₁ on (X,𝓒₁) and a coarse action ϕ₂ of a group G₂ on (X,𝓒₂) such that ϕ₁ commutes with ϕ₂. They generalize the following two basic results of coarse geometry: Proposition 0.3 (Shvarts-Milnor lemma [5, Theorem 1.18]). A group G acting properly and cocompactly via isometries on a length space X is finitely generated and induces a quasi-isometry equivalence g ↦ g·x₀ from G to X for any x₀ ∈ X. Theorem 0.4 (Gromov [4, p. 6]). Two finitely generated groups G and H are quasi-isometric if and only if there is a locally compact space X admitting proper and cocompact actions of both G and H that commute.
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Algebraic properties of quasi-finite complexes

66%
Cencelj M., Dydak J., Smrekar J., Vavpetič A., Virk Ž.
Fundamenta Mathematicae
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2007
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tom 197
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nr 1
67-80
EN
A countable CW complex K is quasi-finite (as defined by A. Karasev) if for every finite subcomplex M of K there is a finite subcomplex e(M) such that any map f: A → M, where A is closed in a separable metric space X satisfying XτK, has an extension g: X → e(M). Levin's results imply that none of the Eilenberg-MacLane spaces K(G,2) is quasi-finite if G ≠ 0. In this paper we discuss quasi-finiteness of all Eilenberg-MacLane spaces. More generally, we deal with CW complexes with finitely many nonzero Postnikov invariants. Here are the main results of the paper: Theorem 0.1. Suppose K is a countable CW complex with finitely many nonzero Postnikov invariants. If π₁(K) is a locally finite group and K is quasi-finite, then K is acyclic. Theorem 0.2. Suppose K is a countable non-contractible CW complex with finitely many nonzero Postnikov invariants. If π₁(K) is nilpotent and K is quasi-finite, then K is extensionally equivalent to S¹.
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