On the Axiomatic Systems of Steenrod Homology Theory of Compact Spaces

Anzor Beridze, Leonard Mdzinarishvili

Published 2017-03-15Version 1

The Steenrod homology theory on the category of compact metric pairs was axiomatically described by J.Milnor. In [Mil] the uniqueness theorem is proved using the Eilenberg-Steenrod axioms and as well as relative homeomorphism and clusres axioms. J. Milnor constructed the homology theory on the category $Top^2_C$ of compact Hausdorff pairs and proved that on the given category it satisfies nine axioms - the Eilenberg-Steenrod, relative homeomorphis and cluster axioms (see theorem 5 in [Mil]). Besides, he proved that constructed homology theory satisfies partial continuity property on the subcategory $Top^2_{CM}$ (see theorem 4 in [Mil]) and the universal coefficient formula on the category $Top^2_C$ (see Lemma 5 in [Mil]). On the category of compact Hausdorff pairs, different axiomatic systems were proposed by N. Berikashvili [B1], [B2], H.Inasaridze and L. Mdzinarishvili [IM], L. Mdzinarishvili [M] and H.Inasaridze [I], but there was not studied any connection between them. The paper studies this very problem. In particular, in the paper it is proved that any homology theory in Inasaridze sense is the homology theory in the Berikashvili sense, which itself is the homology theory in the Mdzinarishvili sense. On the other hand, it is shown that if a homology theory in the Mdzinarishvili sense is exact functor of the second argument, then it is the homology in the Inasaridze sense.