Non-vanishing higher derived limits

Abstract

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups $\mathbf{A}$ indexed by elements of ${\omega }^{\omega }$. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits ${\lim }^n\mathbf{A}$ must vanish, for $n>0$. They also proved that under the Continuum Hypothesis ${\lim }^1\mathbf{A}\ne 0$. The question whether ${\lim }^n\mathbf{A}$ vanishes, for $n>0$, has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under PFA ${\lim }^1\mathbf{A}=0$. Bergfalk show that it is consistent that ${\lim }^2\mathbf{A}$ does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that ${\lim }^n\mathbf{A}=0$, for all $n$. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Henson. We complete the picture by showing that, for any $n>0$, it is relatively consistent with ZFC that ${\lim }^n\mathbf{A}\ne 0$.