The Steenrod problem of realizing polynomial cohomology rings

In this paper, we completely classify which graded polynomial
R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R=Z, our result states that the only polynomial cohomology rings over Z that can occur are tensor products of copies of $H^*(CP^\infty;Z)\cong Z[x_2]$,
$H^*(BSU(n);Z)\cong Z[x_4, x_6, \ldots, x_{2n}]$, and
$H^*(BSp(n);Z)\cong Z[x_4, x_8, \ldots, x_{4n}]$, confirming an old conjecture. Our classification extends Notbohm's solution for $R=F_p$, p odd. Odd degree generators, excluded above, only occur if R is an $F_2$-algebra and in that case the recent classification of
2-compact groups by the authors can be used instead of the present paper. Our proofs are short and rely on the general theory of
p-compact groups, but not on classification results for these.