Logic Seminar

Hanul JeonCornell University
Partitions and simultaneously vanishing higher derived limits, part 3

Monday, September 18, 2023 - 2:55pm
Malott 205

Let $\mathbf{A}$ be the inverse system of abelian groups $A_x = \bigoplus_{n=0}^\infty \bigoplus_{i < x(n)} \mathbb{Z}$, for $x \in \mathbb{N}^\mathbb{N}$, equipped with the canonical projection maps. The higher derived limits $\lim^s \mathbf{A}$ appear in Mardesic and Prasolov's calculations of the strong homology groups of a countable topological sum of $d$-dimensional Hawaiian earrings. In particular, in order for strong homology to be additive for the class of locally compact second countable spaces $\lim^s \mathbf{A}$ must vanish for all $s$. While Mardesic and Prasolov showed that $\lim^1 \mathbf{A} \ne 0$ if the Continuum Hypothesis holds, Bergfalk and Lambie-Hanson recently proved the consistency, relatively to a weakly compact cardinal, of $\lim^s \mathbf{A} = 0$ for all $s$. We will show that Bergfalk and Lambie-Hanson's result can be viewed as a combination of a consistency proof of a partition hypothesis and a ZFC proof that this partition hypothesis implies $\lim^s \mathbf{A} = 0$ for all $s$. This is a multipart talk: part 1 will give an introduction and overview of the results, part 2 will establish the consistency result, and part 3 will derive $\lim^s \mathbf{A}$ from the partition hypothesis. The result being presented is due to Bannister, Bergfalk, Moore, and Todorcevic.