Let h1∗, h2∗ be cohomology
theories defined on the category of finite CW-complexes, and suppose that
h1∗(point), h2∗(point) are both countable. Then by Brown’s [5] Representability
Theorem, there are Ω-spectra Z1, Z2 such that h1∗(X) = [X,Zi]∗, the graded
group of homotopy classes of maps of X into the terms of the spectrum. If
we exercise some care in the choice of Z1, we shall see that every stable
cohomology operation φ : h1∗(X) → h2∗(X) defined for X finite extends to a map
φ : Z1→ Z2 of spectra. We shall examine the question: How many choices, up to
homotopy, are there for φ, given φ? As an intermediate question, we shall also
investigate: How many extensions are there to infinite CW-complexes are there of
φ?
