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 which can occur, are tensor products of copies of H^(CP^\infty;Z) = Z[x_2], H^(BSU(n);Z) = Z[x_4,x_6,…,x_{2n}], and H^*(BSp(n):Z) = Z[x_4,x_8,…,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.

💡 Research Summary

The paper provides a complete solution to the Steenrod problem, which asks which graded polynomial algebras over a commutative ring R can be realized as the singular cohomology ring of a topological space. The authors work under mild hypotheses on R (e.g., Noetherian, bounded 2‑torsion) and focus primarily on the case where all generators have even degree. Their main achievement is a classification that holds for arbitrary such R, and in particular gives a definitive answer for the integral case R = ℤ.

The central theorem states that a graded polynomial R‑algebra
 A = R