Strong Homology, Derived Limits, and Set Theory

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 $\mathbb{R}^4$.