Classification theorems for cohomology rings of finite \(H\)-spaces (Q1890275)

Let \(X\) be a simply-connected mod 3 finite \(H\)-space such that \(H_ \ast(X; {\mathbb F}_ 3)\) and \(K(2)_\ast(X)\) or \(K(3)_\ast(X)\) (where \(K(n)\) is the \(n\)-th periodic Morava \(K\)-theory at the prime 3) are associative. The author derives certain results on the relationship between the primitives and the indecomposables in the cohomology algebra \(H^\ast(X;{\mathbb F}_ 3)\). If \(K(2)_\ast(X)\) is associative, these results imply that \(H^\ast(X;{\mathbb F}_ 3)\) is isomorphic (as an algebra) to the mod~3 cohomology algebra of a finite product of the exceptional Lie groups \(F_ 4\), \(E_ 8\), and odd dimensional spheres different from \(S^ 1\).