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Graph colouring and Steenrod's problem for Stanley-Reisner rings

Donald Stanley, Masahiro Takeda

TL;DR

The paper establishes a deep link between graph-theoretic span colourings and the Steenrod realization problem for Stanley-Reisner algebras with generators in degrees $4$ and $6$, via the construction of $A(n,L)$ and $A(n,G)$ and a representing graph $A_{oldsymbol{k}^n}$. It proves that a mod-$2$ Steenrod action on $SR(K,oldsymbol{ ho})$ corresponds to an $n$-span colouring of the 1-skeleton, and that $A(n,G)$ admits an unstable $oldsymbol{ ext{A}}_2$-action if and only if the 2-colour span condition $s_2oldchi(G) leq n$, yielding a precise combinatorial criterion for realizability in the graph case. A constructive framework based on poset limits gives Steenrod actions from span colourings, while a topological invariant $ chi_{ ext{Top}}(G)$ bounds and aligns with classical chromatic numbers. The work also classifies realizable algebras with two degree-$4$ generators and extensions to arbitrary numbers of degree-$6$ generators, and closes with several open problems on refinements, stabilization, and core structures. The results illuminate how algebraic topology, combinatorics, and homotopy-theoretic realizability interact through span colourings and Steenrod structures, offering new tools for Steenrod-realizability questions in a combinatorial setting.

Abstract

It is a classical problem in algebraic topology asked by Steenrod which graded rings occur as the cohomology ring of a space. In this paper, we define an algebraic version of the graph colouring, span colouring, and observe the relation between span colourings and Steenrod's problem for graded Stanley-Reisner rings, in other words polynomial rings divided by an ideal generated by square-free monic monomials.

Graph colouring and Steenrod's problem for Stanley-Reisner rings

TL;DR

The paper establishes a deep link between graph-theoretic span colourings and the Steenrod realization problem for Stanley-Reisner algebras with generators in degrees and , via the construction of and and a representing graph . It proves that a mod- Steenrod action on corresponds to an -span colouring of the 1-skeleton, and that admits an unstable -action if and only if the 2-colour span condition , yielding a precise combinatorial criterion for realizability in the graph case. A constructive framework based on poset limits gives Steenrod actions from span colourings, while a topological invariant bounds and aligns with classical chromatic numbers. The work also classifies realizable algebras with two degree- generators and extensions to arbitrary numbers of degree- generators, and closes with several open problems on refinements, stabilization, and core structures. The results illuminate how algebraic topology, combinatorics, and homotopy-theoretic realizability interact through span colourings and Steenrod structures, offering new tools for Steenrod-realizability questions in a combinatorial setting.

Abstract

It is a classical problem in algebraic topology asked by Steenrod which graded rings occur as the cohomology ring of a space. In this paper, we define an algebraic version of the graph colouring, span colouring, and observe the relation between span colourings and Steenrod's problem for graded Stanley-Reisner rings, in other words polynomial rings divided by an ideal generated by square-free monic monomials.
Paper Structure (25 sections, 56 theorems, 42 equations, 2 figures)

This paper contains 25 sections, 56 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Span colourings
  3. Algebras with generators in degrees $4$ and $6$, $\bold{A(n,L)}$
  4. Outline of sections
  5. Acknowledgements
  6. Preliminaries
  7. The algebras $A(n, L)$ and $A(n,G)$
  8. Roughly equivalent versions of span colouring
  9. A graph representing span colourings
  10. Chromatic number of $A_{\mathbf{k}^n}$
  11. Span colouring and Steenrod algebra
  12. The Steenrod Algebra and its action on Stanley-Reisner rings
  13. Steenrod action on Stanley-Reisner and $P_{\max}(K)$
  14. Steenrod actions on Stanley-Reisner algebras generated in degrees $4$ and $6$
  15. When $A(n,G)$ has a Steenrod algebra action
  16. ...and 10 more sections

Key Result

Proposition 1.2

For every field $\mathbf{k}$ and every graph $G$, $clique(G)\leq s_{\mathbf{k}}\chi(G)\leq\chi(G)$.

Figures (2)

  • Figure 1:
  • Figure :

Theorems & Definitions (109)

  • Definition 1.1: Weak span colouring
  • Proposition 1.2: Propositions \ref{['important2']} and \ref{['important']}
  • Proposition 1.3: Proposition \ref{['chromatic_difference']}
  • Theorem 1.4: Theorem \ref{['construct_Steenrod_mod_str']}
  • Theorem 1.5: Theorem \ref{['sameSteen']}
  • Theorem 1.6: Corollary \ref{['realimpspan']}
  • Theorem 1.7: Theorem \ref{['color_theorem']}
  • Theorem 1.8: Theorem \ref{['topboundsfromgraphs']}
  • Example 2.1
  • Definition 2.2
  • ...and 99 more