Cohomology ring

Question

There exists a construction of the cohomology ring using only the Eilenberg–Steenrod axioms? I'm not able to find a reference where the theory is developed only with the axioms (I mean all of then, not generalized cohomology).

Answer 1

A ring structure isn't part of a generic generalized cohomology theory and its not derivable from the axioms.

One way in which it enters the picture would be for topological theories represented by ring spectra. Another, more algebraic way, would be for theories $E$ that have a Kunneth morphism $\mu:E^*(X)\otimes E^*(X)\rightarrow E^*(X\times X)$, but this is certainly not going to be true for all theories.

These conditions are in fact more or less the same as for spectra the Kunneth theorem is a special case of the universal coefficient theorem, which requires some sort of action maps between the involved spectra (Adams Stable Homotopy and Generalised Homology III.13).

In the case of singular cohomology one can lift the calculations to cochains and use the algebraic Kunneth theorems to derive the topological versions. You can see T. tom Dieck's Algebraic Topology 9.7 for a discussion of the Eilenberg-Zilber theorem in this context.