Steenrod problem and some graded Stanley-Reisner rings (Q6593011)

The \textit{Steenrod problem}, also known as the \textit{realization problem}, is a classical problem in algebraic topology, asking: which commutative graded algebras can be realized as the cohomology rings of topological spaces? Although it has been solved for special cases of rational algebras and polynomial rings, the problem remains notoriously challenging in general.\N\NThe goal of this paper is to investigate the realization problem for graded Stanley-Reisner rings which are defined as follows. Let \(K\) be a simplicial complex on the vertex set \(V=\{1,\ldots,m\}\), and let \(\phi\colon V\to 2\mathbb{Z}_{>0}\) be a function. Then the \textit{graded Stanley-Reisner ring} \(\text{SR}(K,\phi)\) associated with the pair \((K,\phi)\) is defined to be the quotient ring\N\[\N\text{SR}(K,\phi)=\mathbb{Z}[x_1,\ldots,x_m]/I_K,\N\]\Nwhere each generator \(x_i\) has degree \(\phi(i)\), and \(I_K\) is the ideal generated by those monic monomials \(x_{i_1}\ldots x_{i_k}\) such that \(\{i_1,\ldots,i_k\}\) is not contained in \(K\). Moreover, let\N\[\NP_{\max}(K)=\{\sigma_{i_1}\cap\cdots\cap\sigma_{i_k}\mid \sigma_{i_j}\text{ is a maximal simplex in }K\}\N\]\Nbe the collection of simplices formed by intersecting the maximal simplices in \(K\).\N\NThe main result of the paper is Theorem 1.1, which gives sufficient and necessary conditions for the realization of \(\text{SR}(K,\phi)\) under an assumption on the pair \((K,\phi)\). Here I restate the theorem with the assumption replaced by an equivalent condition.\N\N\textbf{Theorem 1.1}: Let \(K\) be a simplical complex with the vertex set \(V\) and let \(\phi\colon V\to2\mathbb{Z}_{>0}\) be a function. Suppose the pair \((K,\phi)\) satisfies the condition:\N\begin{itemize}\N\item if any two vertices \(x,y\in V\) have the same degree \(\phi(x)=\phi(y)=2^i\) for some \(i\geq2\), then the edge \(xy\) is not contained in \(K\).\N\end{itemize}\NThen the following are equivalent:\N\begin{itemize}\N\item there is a topological space \(X\) such that \(H^*(X;\mathbb{Z})\cong\text{SR}(K,\phi)\);\N\item For all \(\sigma\in P_{\max}(K)\), the multiset \(\{\phi(x)\mid x\in \sigma\}\) is equal to \(\{2,2,\ldots,2\}\), \(\{2,2,\ldots,2, 4,6,\ldots,2n\}\), or \(\{2,2,\ldots,2,4, 8,\ldots,4n\}\) for some \(n\).\N\end{itemize}\NThe author also proves the realization of \(\text{SR}(K,\phi)\) for a wider class of \((K,\phi)\).\N\N\textbf{Theorem 1.3:} Let \(K\) be a simplical complex with the vertex set \(V\) and let \(\phi\colon V\to2\mathbb{Z}_{>0}\) be a function. Suppose there is a decomposition \(V=\bigsqcup_i A_i\) such that for all \(i\) and \(\sigma\in P_{\max}(K)\), the multiset \(\{\phi(x)\mid x\in\sigma\cap A_i\}\) is equal to \(\{2,2,\ldots,2\}\), \(\{2,2,\ldots,2, 4,6,\ldots,2n\}\), or \(\{2,2,\ldots,2,4, 8,\ldots,4n\}\) for some \(n\). Then there is a topological space \(X\) such that \(H^*(X;\mathbb{Z})\cong\text{SR}(K,\phi)\).\N\NIn Sections 2, 3 and 4 the author proves Theorem 1.3 and constructs realizing spaces as homotopy colimits of products of \(BSU(n),BSP(n')\) and \(\mathbb{CP}^{\infty}\). In Sections 5, 6 and 7, he proves a necessary condition and hence the main theorem.