Generalized Steenrod homology theories are strong shape invariant

Abstract

It is shown that a reduced homology theory on the category of pointed compact metric spaces is strong shape invariant if and only if its homology functors hn satisfy the quotient exactness axiom, which means that for each pointed compact metric pair (X, A, a0) the natural sequence hn(A, a0) → hn(X, a0) → hn(X/A, *) is exact. As a consequence, all generalized Steenrod homology theories are strong shape invariant.