Every finite-dimensional compact connected abelian group has a canonical resolution as a quotient of a periodic locally compact group times a real vector space, and every morphism between such groups lifts to a product map.
Periodic Locally Compact Groups, a Study of Totally Disconnected Topological Groups; De Gruyter: Berlin, Germany, 2019
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Structure of Finite-Dimensional Protori
Every finite-dimensional compact connected abelian group has a canonical resolution as a quotient of a periodic locally compact group times a real vector space, and every morphism between such groups lifts to a product map.