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2012 | 218 | 1 | 13-46
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Covering maps for locally path-connected spaces

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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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  • Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
autor
  • Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
autor
  • Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
autor
  • Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-1-2
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