Strong homology, derived limits, and set theory

Bergfalk, Jeffrey

doi:10.4064/fm140-4-2016

Cited by 9 publications

(21 citation statements)

References 3 publications

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“…To begin with, it fully disentangles the combinatorial and algebraic components of arguments whose hybridity had hitherto been an impediment to their comprehension. It thereby facilitates a much closer analysis of the set-theoretic content of these arguments, and this, indeed, is our work's main contribution: we show that the partition hypotheses PH n hold for all universally Baire partitions of powers of ω ω; this carries the corollary that for all n ∈ ω all universally Baire n-coherent families of functions indexed by ω ω are trivial, answering Questions 6 and 7.12 of [2] and [4], respectively, and generalizing a main result of [31]. Moreover, the associated trivializations can themselves be taken to have low complexity relative to the n-coherent family; in particular, in the presence of suitable large cardinal hypotheses, they are universally Baire.…”

Section: Introductionmentioning

confidence: 59%

“…In Section 3 we introduce refinements of the standard topological and Baire measurability structures on powers of ω ω and in Section 4 we show that the hypotheses PH n hold for partitions which are measurable with respect to these structures. We obtain as an immediate corollary a negative answer to the question, appearing in both [2] and [4], of whether a nontrivial n-coherent family may be analytic.…”

Section: Introductionmentioning

confidence: 90%

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A descriptive approach to higher derived limits

Bannister¹,

Bergfalk²,

Moore³

et al. 2022

Preprint

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We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set Ω of functions from N to N and prove that cocycles of this complexity are images of cochains of the roughly the same complexity. In the course of this work, we isolate a partition principle for powers D n+1 of products of directed sets D and show that whenever this principle holds, the corresponding derived limit lim n is additive; vanishing results for this limit are the typical corollary. The formulation of this partition hypothesis synthesizes and clarifies several recent advances in this area.

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Section: Introductionmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 90%

A descriptive approach to higher derived limits

Bannister¹,

Bergfalk²,

Moore³

et al. 2022

Preprint

Self Cite

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“…Standard resolutions convert this description to a more concrete general form, as before. The characterisations of lim 1 A and of lim A ( ≥ 1) in [22] and [4], respectively, then each involve one further conversion, via the long exact sequence…”

Section: Homological Backgroundmentioning

confidence: 99%

“…Moore further observed therein that 'it is entirely possible that it is a theorem of ZFC that either lim 1 A ≠ 0 or lim 2 A ≠ 0 .' The first set-theoretic computation of lim 2 A appeared some seven years later in [4]; here framings of lim A ( > 1) in terms of higher-dimensional coherence were given and applied to show that the Proper Forcing Axiom implies that lim 2 A ≠ 0. Here also Goblot's work [13] was applied to show that ≤ ℵ implies that lim A = 0 for all > .…”

Section: Introductionmentioning

confidence: 99%

Simultaneously vanishing higher derived limits

Bergfalk

Lambie-Hanson

2021

Forum of Mathematics, Pi

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In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

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“…The study of the triviality of coherent families of functions indexed by ω ω dates to [1]; there, Mardešić and Prasolov showed that the additivity of strong homology implies that every coherent family is trivial. Higher-dimensional variants of these notions were introduced in [10]; these encoded the behaviors of the higher derived limits associated to the system A described above. In this section, we further generalize these notions to s-coherence and s-triviality for arbitrary Ω-systems G, and we analyze their relationship to these systems' associated higher derived limits.…”

Section: Higher Coherence and The Derived Limit Lim Smentioning

confidence: 99%

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

Bannister¹,

Bergfalk²,

Moore³

2020

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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 [1] 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 [2] 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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Strong homology, derived limits, and set theory

Cited by 9 publications

References 3 publications

A descriptive approach to higher derived limits

A descriptive approach to higher derived limits

Simultaneously vanishing higher derived limits

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

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