Realization of graded monomial ideal rings modulo torsion

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.Comment: 23 page

Weighted polyhedral products and Steenrod's problem

We construct a weighted version of polyhedral products and compute its cohomology in some special cases. This is applied to resolve Steenrod's cohomology realization problem in a case related to exterior algebras.Comment: 43 page