Wyniki wyszukiwania

1

Covering maps for locally path-connected spaces

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Brodskiy N., Dydak J., Labuz B., Mitra A.

Fundamenta Mathematicae

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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).

2

On continuous extension of uniformly continuous functions and metrics

100%

Banakh T., Brodskiy N., Stasyuk I., Tymchatyn E.

Colloquium Mathematicum

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2009

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tom 116

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nr 2

191-202

EN

We prove that there exists a continuous regular, positive homogeneous extension operator for the family of all uniformly continuous bounded real-valued functions whose domains are closed subsets of a bounded metric space (X,d). In particular, this operator preserves Lipschitz functions. A similar result is obtained for partial metrics and ultrametrics.

3

Coarse structures and group actions

100%

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.

Ograniczanie wyników

2 Colloquium Mathematicum

1 Fundamenta Mathematicae

3 Brodskiy N.

2 Dydak J.

2 Mitra A.

1 Banakh T.

1 Labuz B.

1 Stasyuk I.

1 Tymchatyn E.

1 2012

1 2009

1 2008

Covering maps for locally path-connected spaces

On continuous extension of uniformly continuous functions and metrics

Coarse structures and group actions