Strong Homology and the Proper Forcing Axiom

Abstract

This paper concerns applications of set theory to the problem of calculating the strong homology of certain subsets of Euclidean spaces. We prove the set theoretic result that it is consistent that every almost coinciding family indexed by $^\omega \omega$ is trivial (e.g., the proper forcing axiom implies this). This result, combined with results of S. Mardešić and A. Prasalov, show that the statement "the $k$-dimensional strong homology of ${Y^{(k + 1)}}$ (the discrete sum of countably many copies of the $(k + 1)$-dimensional Hawaiian earring) is trivial" is consistent with and independent of the usual axioms of set theory.