engleski

Finite-sheeted covering maps over 2-dimensional connected, compact abelian groups

Let $G$ be a 2-dimensional connected, compact abelian group, $\widehat{; ; G}; ; $ its Pontrjagin dual, and $s$ be a positive integer. We prove that a classification of $s$-sheeted covering maps over $G$ and that of $s$-sheeted covering homomorphisms from $G$ are reduced to a classification of $s$-index torsionfree supergroups of $\widehat{; ; G}; ; $ and that of $s$-index torsionfree subgroups of $\widehat{; ; G}; ; $. Using results of torsionfree abelian group of rank two we demonstrate its consequences. In a way of proofs we shall prove the following for a connected compact group $Y$: 1. Every finite-sheeted covering map from a connected space over $Y$ is equivalent to a covering homomorphism from a compact, connected group. Moreover, if $Y$ is abelian the domain of the homomorphism is abelian. 2. Let $f:X\rightarrow Y$ and $f^{; ; \prime}; ; :X^{; ; \prime}; ; \rightarrow Y$ be finite-sheeted covering homomorphisms over $Y$. Then $f$ and $f^{; ; \prime}; ; $ are equivalent if and only the two homomorphisms are equivalent as topological homomorphisms.

Compact group; compact abelian group; 2-dimensional; finite-sheeted covering map

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