Strong excision and strong shape invariance are equivalent for homology theories on the category of compact metric pairs (Q2427616)

In 1960 Milnor gave an axiomatic characterization of Steenrod homology as an ordinary homology theory on the category of compact metric pairs satisfying the strong excision axiom and the cluster axiom. In this paper the author explores the role of the strong excision axiom on its own and proves an interesting result: ``For \(CM^{2}\) the category of compact metric pairs and continuous maps of pairs, a homology theory \( (H_{n},\partial )\) on \(CM^{2}\) is strong shape invariant if and only if it satisfies the strong excision axiom.'' As an application he shows that the homology theory constructed by \textit{L. G. Brown, R. G. Douglas} and \textit{P. A. Fillmore} [Ann. Math. (2) 105, 265--324 (1977; Zbl 0376.46036)] satisfies the strong excision axiom, thus is strong shape invariant.