Covering maps for locally path-connected spaces

• Autorzy: N. Brodskiy , J. Dydak , B. Labuz , A. Mitra
• Języki publikacji: EN

Abstrakty

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).

Kategorie tematyczne

• 54E15: Uniform structures and generalizations
• 55M10: Dimension theory
• 55Q52: Homotopy groups of special spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2012
• Tom: 218
• Numer: 1
• Strony: 13-46

Daty

wydano

2012

Twórcy

autor

N. Brodskiy

• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.

autor

J. Dydak

• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.

autor

B. Labuz

• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.

autor

A. Mitra

• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.

Identyfikatory

DOI

10.4064/fm218-1-2