Homological properties of quotient divisible abelian groups and compact groups dual to them

The paper is devoted to the study of homological properties of quotient divisible Abelian groups. These groups constitute an important class of groups that has been intensively studied in recent years. In the first part of the paper, we study the conditions for the vanishing of extension groups in which one of the arguments is a quotient divisible group. Under some additional assumptions, groups of homomorphisms from quotient divisible groups to reduced abelian groups are described. Some properties of the universality of quotient divisible Abelian groups are investigated. The second part of the work is devoted to the study of the homological properties of compact Abelian groups, which are dual in the sense of L. S. Pontryagin to quotient divisible groups. Such groups are called quotient toroidal. We study the conditions for the vanishing of extension groups in which one of the arguments is a quotient toroidal group. Some groups of continuous homomorphisms in which the second argument is a quotient toroidal group are described. In the last part of the paper, we study the conditions for the vanishing of extension groups of quotient divisible groups with the help of compact quotient toroidal ones. The fundamental group of the topological space of the quotient toroidal group is characterized.