Realization of some Stanley-Reisner algebras and graph colorings

TL;DR

The paper extends the ST25 framework for realizing graph-related Stanley-Reisner algebras by introducing $\mathsf{B}(n, \mathsf{G})$, $\mathsf{B}_p(\mathbf{r}, \mathsf{G})$, $\mathsf{A}_p(\mathbf{s}, \mathsf{G})$, and $\mathsf{A}(\mathbf{s}, \mathsf{G})$, and by linking their $\mathcal{A}_p$-algebra structures to span-graph colorings. It proves lower bounds $s_p\chi(\mathsf{G}) \le n$ (or $r_1$, $s_1$) whenever these algebras admit $\mathcal{A}_p$-actions, establishing a tight connection between algebraic realizability and graph colorability, and defines topological chromatic invariants $\chi_{Top, \mathsf{A}}(\mathsf{G})$ and $\chi_{Top, \mathsf{B}}(\mathsf{G})$ that lie between $s_p\chi(\mathsf{G})$ and $\chi(\mathsf{G})$. The work also provides realizability results when the ordinary chromatic number satisfies $\chi(\mathsf{G}) \le n$, yielding upper bounds $\chi_{Top, \mathsf{A}}(\mathsf{G}), \chi_{Top, \mathsf{B}}(\mathsf{G}) \le \chi(\mathsf{G})$. Beyond these, it discusses generalizations to $\mathsf{A}(\mathbf{s}, \mathsf{G})$ and highlights obstructions and open questions in the odd-prime setting via Tak24-type criteria and explicit non-realizable examples.

Abstract

It is a classical problem in algebraic topology to decide whether a given graded $\mathbb{Z}$-algebra can be realized as the cohomology ring of a space. In this paper, we introduce families of Stanley-Reisner algebras depending on graphs, and relate their realizability to the span coloring of the graph.

Theorems & Definitions (50)

• Theorem 1.1: ST25, Theorem 7.8
• Theorem 1.2: ST25, Theorem 8.4
• Theorem 1.3: ST25, Theorem 8.6
• Definition 2.1
• Theorem 2.2
• Lemma 2.3: ST25, Lemma 6.1
• Lemma 2.4: ST25, Lemma 6.2
• Lemma 2.5
• proof
• proof : Proof of \ref{['thm:main3']}