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

2020/08/30 by Nathaniel Bannister, Bannister, Nathaniel, Jeffrey Bergfalk +3 · 2 citations
Mathematics · Medicine · #03E35 #03E75 #55N07 #55N40 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Neurological and metabolic disorders

paper · pdf · doi:10.48550/arxiv.2008.13089

openalex publication_date 2020/08/30 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

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 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 lim^s A = 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 lim^s functors over a substantially more general class of pro-abelian groups indexed by ℕ^ℕ.

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