Universal acyclic resolutions for finitely generated coefficient groups

Universal acyclic resolutions for finitely generated coefficient groups

We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dim ≤ n and a surjective UV n−1-map r : Z −→ X having the property that: for every finitely generated abelian group G and every integer k ≥ 2 such that dimG X ≤ k ≤ n we have dimG Z ≤ k and r is G-acyclic, or equivalently: for every simply connected CW-complex K with finitely generated homotopy groups such th...

Universal acyclic resolutions for arbitrary coefficient groups

We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dim ≤ n + 1 and a surjective UV n−1-map r : Z −→ X such that for every abelian group G and every integer k ≥ 2 such that dimG X ≤ k ≤ n we have dimG Z ≤ k and r is G-acyclic.

Acyclic resolutions for arbitrary groups

We prove that for every abelian group G and every compactum X with dimG X ≤ n ≥ 2 there is a G-acyclic resolution r : Z −→ X from a compactum Z with dimG Z ≤ n and dimZ ≤ n + 1 onto X.

Isomorphism of finitely generated solvable groups is weakly universal

We show that the isomorphism relation for finitely generated solvable groups of class 3 is a weakly universal countable Borel equivalence relation. This improves on previous results. The proof uses a modification of a construction of Neumann and Neumann. Elementary arguments show that isomorphism of finitely generated metabelian or nilpotent groups can not achieve this Borel complexity. In this...

Finitely Generated Abelian Groups