On the Axiomatic Systems of Singular Cohomology Theory

TL;DR

This paper explores various axiomatic systems characterizing singular cohomology, extending Milnor's axioms and examining their interrelations to deepen understanding of cohomology theories.

Contribution

It provides new axiomatic characterizations of singular cohomology and investigates the relationships among different axiomatic systems.

Findings

Additional axiomatic characterizations of singular cohomology are proposed.

Connections between various axiomatic systems are analyzed.

The study extends Milnor's axiomatic framework for cohomology theories.

Abstract

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. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomorphic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support \cite{3}. In the paper \cite{6}, L. Mdzinarishvili proposed {\it partially compact support property} for a cohomology theory and gave another axiomatic characterization of the singular cohomology theory \cite{6}. 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.