Realizability of A(3, A(Z/2)^3)

Determine whether the graded Stanley–Reisner ring A(3, A(Z/2)^3), defined as SR(Δ^2 * A(Z/2)^3, φ) where φ assigns degree 4 to the vertices of Δ^2 and degree 6 to the vertices of the graph A(Z/2)^3, is realizable as the integral cohomology ring H*(X; Z) of some topological space X.

Background

The paper studies realizability (Steenrod’s problem) for a class of Stanley–Reisner rings A(n, L) with generators in degrees 4 and 6, where A(n, L) = SR(Δ^{n−1} * L, φ) and φ assigns degrees 4 to Δ^{n−1} vertices and 6 to the vertices of L. A necessary condition for realizability is the existence of an unstable Steenrod algebra action on the mod-2 reduction, which can be constructed when the span chromatic number s_2χ(L_1) ≤ n. Conversely, realizability is guaranteed when the ordinary chromatic number χ(L_1) ≤ n by constructing spaces via a decomposition criterion.

For L equal to the representing graph A(Z/2)^3, the authors show s_2χ(A(Z/2)^3) = 3, but χ(A(Z/2)^3) > 3. Thus the sufficient condition χ(L_1) ≤ n for realizability (Theorem 8.1/8.4) does not apply, while an A_2 action does exist (by Theorem 7.7). This leaves unresolved whether A(3, A(Z/2)^3) can actually be realized as H*(X; Z) for some space.