Strong Homology, Derived Limits, and Set Theory

TL;DR

This paper explores the set-theoretic foundations of strong homology's additivity, revealing its dependence on higher derived limits and showing that under certain axioms, strong homology is not additive even on simple spaces.

Contribution

It identifies the set-theoretic content influencing strong homology's additivity and demonstrates its non-additivity under the Proper Forcing Axiom.

Findings

Higher derived limits govern strong homology additivity.

Strong homology is not additive under the Proper Forcing Axiom.

Additivity depends on set-theoretic assumptions.

Abstract

We consider the question of the additivity of strong homology. This entails isolating the set-theoretic content of the higher derived limits of an inverse system indexed by the functions from $\mathbb{N}$ to $\mathbb{N}$. We show that this system governs, at a certain level, the additivity of strong homology over sums of arbitrary cardinality. We show in addition that, under the Proper Forcing Axiom, strong homology is not additive, not even on closed subspaces of $\mathbb{R}^4$.