We characterize H-spaces which are p-torsion Postnikov pieces of finite type by a cohomological property together with a necessary acyclicity condition. When the mod p cohomology of an H-space is finitely generated as an algebra over the Steenrod algebra we prove that its homotopy groups behave like those of a finite complex.

Relying on the computation of the André-Quillen homology groups for unstable Hopf algebras, we prove that the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.

We consider the polynomial algebra H∗(CP∞ ×CP∞;Fp) as a module over the mod p Steenrod algebra, A(p), p being an odd prime. We give a minimal set of generators consisting of monomials and characterise all such ‘monomial bases’.

We describe algebraic obstruction theories for realizing an abstract (co)algebra K∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K∗.

Let G be an exceptional Lie group with a maximal torus T , and let Ap be the mod–p Steenrod algebra. Based on common properties in the Schubert presentation of the cohomology H∗(G/T ; Fp), we obtain a complete characterization for the Ap–algebra H ∗(G; Fp). 2000 Mathematical Subject Classification: 57T15; 14M15. Email addresses: [email protected]; [email protected]