zbMATH — the first resource for mathematics

On the axiomatic systems of singular cohomology theory. (English) Zbl 1436.55004
Summary: On the category of pairs of topological spaces having the homotopy type of a CW complex the singular (co)homology theory was axiomatically studied by J. W. Milnor [Pac. J. Math. 12, 337–341 (1962; Zbl 0114.39604)]. In particular, Milnor gave an 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 [S. Eilenberg and N. Steenrod, Foundations of algebraic topology. Princeton: University Press, (1952; Zbl 0047.41402)]. In the paper [Trans. A. Razmadze Math. Inst. 172, No. 2, 265–275 (2018; Zbl 1397.55003)], 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. [A. Beridze and L. Mdzinarishvili, Topology Appl. 249, 73–82 (2018; Zbl 1447.55003)]).
55N07 Steenrod-Sitnikov homologies
55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
Full Text: DOI
[1] Berikashvili, N. A., Axiomatics of the Steenrod-Sitnikov homology theory on the category of compact Hausdorff spaces, Topology (Moscow, 1979). Topology (Moscow, 1979), Tr. Mat. Inst. Steklova, 154, 24-37 (1983), (Russian) · Zbl 0532.55008
[2] Beridze, Anzor; Mdzinarishvili, Leonard, On the axiomatic systems of Steenrod homology theory of compact spaces, Topol. Appl., 249, 73-82 (2018) · Zbl 1447.55003
[3] Eilenberg, Samuel; Steenrod, Norman, Foundations of Algebraic Topology (1952), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 0047.41402
[4] Huber, Martin; Meier, Willi, Cohomology theories and infinite CW-complexes, Comment. Math. Helv., 53, 2, 239-257 (1978) · Zbl 0432.55002
[5] Inassaridze, Hvedri, On the Steenrod homology theory of compact spaces, Mich. Math. J., 38, 3, 323-338 (1991) · Zbl 0762.55002
[6] Mdzinarishvili, L., The uniqueness theorem for cohomologies on the category of polyhedral pairs, Trans. A. Razmadze Math. Inst., 172, 2, 265-275 (2018) · Zbl 1397.55003
[7] Mdzinarishvili, L. D., On homology extensions, Glas. Mat. Ser. III, 21(41), 2, 455-482 (1986) · Zbl 0643.55006
[8] Milnor, J., On axiomatic homology theory, Pac. J. Math., 12, 337-341 (1962) · Zbl 0114.39604
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.