Lifting amalgamated sums and other colimits of groups and topological groups

Abstract

Suppose a group H is given as a free product with amalgamationdetermined by groups A0, A1, A2 and homomorphisms α1: A0 → A1, α2: A0 → A2. Thus H may be described as the quotient of the free product A * A2 by the relations i1 α1 (α0) = i2α2 (α0) for all α0 ∈ A0, where i1, i2 are the two injections of A1, A2 into A1 * A2. We do not assume that α1, α2 are injective, so the canonical homomorphisms α′i: Ai → H, i = 0,1,2, also need not be injective.