engleski

On covering homomorphisms

Let Y be a connected group and let f:X→Y be a covering map from a connected space X. We say that f is a covering homomorphism if it is possible to define a multiplication ⋅ on X in such a way that X becomes a topological group and f becomes a homomorphism of topological groups. It is well-known that each covering map f:X→Y from a pathwise connected space X on a pathwise connected, locally pathwise connected group Y is a covering homomorphism. In 1972, R.H. Fox introduced a notion of on overlay map in order to extend the classical classification theorem for covering maps to arbitrary connected metric spaces. Using shape-theoretic techniques we show that a covering map f:X→Y on a compact connected group Y is a covering homomorphism if and only if f is an overlay map. In particular, each finite-sheeted covering map f:X→Y on a compact connected group Y is a covering homomorphism.

Covering map; covering homomorphism; topological group; overlay map

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