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

TL;DR

This paper demonstrates, under certain set-theoretic assumptions, that strong homology is additive and compactly supported for locally compact separable metric spaces, extending previous results and addressing longstanding questions in algebraic topology.

Contribution

It proves the consistency of strong homology's additivity and compact support in a broad class of spaces, building on and generalizing prior work involving pro-abelian groups and set-theoretic assumptions.

Findings

Strong homology is additive under a weakly compact cardinal assumption.

The results extend to a wider class of pro-abelian groups.

Addresses the failure of strong homology properties under the Continuum Hypothesis.

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 Marde\v{s}i\'{c} and Prasolov 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 which establishes the consistency, relative to a weakly compact cardinal, of $\mathrm{lim}^s \mathbf{A} = 0$ for all $s \geq 1$ for a certain pro-abelian group $\mathbf{A}$; we show that that work's arguments carry implications for the vanishing and additivity of the $\mathrm{lim}^s$ functors over a substantially more general class of pro-abelian groups indexed by $\mathbb{N}^{\mathbb{N}}$.