On the additivity of strong homology for locally compact separable metric spaces

Abstract

We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Mardešić and Prasolov [14] showing that the Continuum Hypothesis implies that a countable sum of Hawaiian earrings witnesses the failure of strong homology to possess either of these properties. Our results build directly on work of Lambie-Hanson and the second author [3] which establishes the consistency, relative to a weakly compact cardinal, of limsA = 0 for all s ≥ 1 for a certain pro-abelian group A; we show that that work’s arguments carry implications for the vanishing and additivity of the lims functors over a substantially more general class of pro-abelian groups indexed by ℕℕ.