Non-vanishing higher derived limits

Abstract

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups [Formula: see text] indexed by elements of [Formula: see text]. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits [Formula: see text] must vanish, for [Formula: see text]. They also proved that under the Continuum Hypothesis [Formula: see text]. The question whether [Formula: see text] vanishes, for all [Formula: see text], has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under the Proper Forcing Axiom (PFA) [Formula: see text]. Bergfalk showed that it is consistent that [Formula: see text] does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with [Formula: see text] that [Formula: see text], for all [Formula: see text]. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Hanson. We complete the picture by showing that, for any [Formula: see text], it is relatively consistent with [Formula: see text] that [Formula: see text].