Steenrod problem and some graded Stanley-Reisner rings

TL;DR

This paper investigates which graded Stanley-Reisner rings can be realized as the singular cohomology rings of topological spaces, providing necessary and sufficient conditions under certain assumptions.

Contribution

It establishes a characterization of realizable graded Stanley-Reisner rings as singular cohomology rings, addressing a classical problem in algebraic topology.

Findings

Provides necessary and sufficient conditions for realizability.

Connects algebraic properties of Stanley-Reisner rings to topological realizability.

Advances understanding of the Steenrod problem in the context of combinatorial rings.

Abstract

``What kind of ring can be represented as the singular cohomology ring of a space?'' is a classic problem in algebraic topology, posed by Steenrod. In this paper, we consider this problem when rings are the graded Stanley-Reisner rings, in other words the polynomial rings divided by an ideal generated by square-free monomials. Under some assumption, we give a necessary and sufficiently condition that a graded Stanley-Reisner ring is realizable.