On the axiomatic systems of singular cohomology theory

On the category of pairs of topological spaces having a homotopy type of CW complexes the singular (co)homology theory was axiomatically studied by J. Milnor [8]. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given categ...

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Bibliographic Details
Published in:Topology and its applications 2020-04, Vol.275, p.107014, Article 107014
Main Authors: Beridze, Anzor, Mdzinarishvili, Leonard
Format: Article
Language:English
Subjects:
Injective group
Nontrivial internal extension
Partially compact support
The uniqueness theorem
The universal coefficients formula
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Summary:On the category of pairs of topological spaces having a homotopy type of CW complexes the singular (co)homology theory was axiomatically studied by J. Milnor [8]. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomophic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support [3]. In the paper [6], L. Mdzinarishvili proposed partially compact support property for a cohomology theory and gave another axiomatic characterization of the singular cohomology theory. In this paper, we will give additional different axiomatic characterizations of the singular cohomology theory. Moreover, we will study connections of the mentioned axiomatic systems (cf. [2]).
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2019.107014