engleski

On covering maps over the product of two solenoids

For a sequence p=(p_{;i};) of positive integers p_{;i};, let Σ (p) denote a solenoid obtained by the sequence p. We consider finite-sheeted covering maps over the product Σ (p)×Σ (r) of two solenoids and in some particular cases examine under which conditions total spaces are homeomorphic to the base space Σ (p)×Σ (r). We show that a compact connected 2-dimensional abelian group X covers a Klein bottle weak solenoidal space (i.e. the inverse limit of an inverse sequence, where each term is a Klein bottle K and each bonding map is a covering map over K) if and only if X covers the product Σ (p)×Σ (r) of two solenoids and at least one of the sequences p and r consists of odd integers. This enables us to answer the following question: Does every compact connected 2-dimensional abelian group X cover Klein bottle weak solenoidal spaces ? We answer the question in the negative. Moreover, we give an example of a group that covers groups with any finite number of sheets but does not cover any Klein bottle weak solenoidal space.

covering map; compact abelian group; solenoid; Klein bottle; weak solenoidal space

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