All you need is $\mathbf{A}_\kappa$

TL;DR

This paper demonstrates that the vanishing of higher derived limits of the system \(\mathbf{A}_\kappa\) ensures the additivity of strong homology on certain locally compact metric spaces, providing a converse to a known theorem.

Contribution

It establishes a new connection between derived limits of \(\mathbf{A}_\kappa\) and the additivity of strong homology, extending previous results.

Findings

Vanishing of higher derived limits implies additivity of strong homology.

Provides a converse to Mardešić and Prasolov's theorem.

Links algebraic properties of \(\mathbf{A}_\kappa\) to topological homology.

Abstract

We show that the vanishing of higher derived limits of the system $\mathbf{A}_\kappa$ implies the additivity of strong homology on the class of locally compact metric spaces of weight at most $\kappa$, thereby establishing a converse to a theorem of Marde\v{s}i\'c and Prasolov.