Pro-homology and isomorphisms of pro-groups

Abstract

We extend the scope of some useful theorems of E. Dror and J. Stallings of which the following result of Stallings is easy to state: A homomorphism θ: A→B of nilpotent groups is an isomorphism if and only if θ induces an isomorphism and an epimorphism on the first and the second homology groups with integer coefficients, respectively. We extend and enlarge the algebraic framework to the category of pro-groups in a natural way so that an anaogue of these results of Dror and Stallings can be appropriately stated and proved. This is done in a natural manner so that our extended results cań be useful in pro-homotopy theory or shape theory which is analogous to the usefulness of the results of Dror and Stallings in the context of homotopy theory. As an application, we obtain a version of a Whitehead Theorem in pro-homotopy and shape theory which only involves homology pro-groups; a complete proof of this result will be given elsewhere.