Covering maps for locally path-connected spaces

N. Brodskiy, et al. "Covering maps for locally path-connected spaces." Fundamenta Mathematicae 218.1 (2012): 13-46. <http://eudml.org/doc/283364>.

@article{N2012,
abstract = { 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). },
author = {N. Brodskiy, J. Dydak, B. Labuz, A. Mitra},
journal = {Fundamenta Mathematicae},
keywords = {covering maps; locally path-connected spaces},
language = {eng},
number = {1},
pages = {13-46},
title = {Covering maps for locally path-connected spaces},
url = {http://eudml.org/doc/283364},
volume = {218},
year = {2012},
}

TY - JOUR
AU - N. Brodskiy
AU - J. Dydak
AU - B. Labuz
AU - A. Mitra
TI - Covering maps for locally path-connected spaces
JO - Fundamenta Mathematicae
PY - 2012
VL - 218
IS - 1
SP - 13
EP - 46
AB - 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).
LA - eng
KW - covering maps; locally path-connected spaces
UR - http://eudml.org/doc/283364
ER -