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Steenrod operations for $4$-dimensional toric orbifolds

Tseleung So

TL;DR

This work determines exact conditions for nontrivial mod-2 Steenrod actions on the cohomology of 4-dimensional toric orbifolds $X(P,\lambda)$ by tying the actions to determinant data $d_{ij}$ from the characteristic labels and to a $2$-local smooth vertex. The approach leverages a $q$-CW decomposition and degenerate toric spaces to reduce Steenrod computations to triangle cases, yielding explicit parity criteria: $Sq^1$ is nontrivial precisely when $g=\gcd(d_{ij})$ is even, while $Sq^2$ on $H^2$ is nontrivial iff $\prod_{i=1}^n\left(1-\frac{d_{i,n+1}d_{i,n+2}}{g}\right)\equiv 0\pmod{2}$ under a $2$-local smooth vertex. As corollaries, the paper provides a sharp stable splitting for $\Sigma X(P,\lambda)$, a spin criterion in the smooth (quasi-toric) case, partial cohomological rigidity results, and explicit gauge-group homotopy decompositions. By connecting combinatorial input $(P,\lambda)$ with the Steenrod algebra action, the results enable concrete invariants and homotopy classifications for a broad class of 4D toric orbifolds.

Abstract

We prove necessary and sufficient conditions for the existence of non-trivial Steenrod actions on the mod-$2$ cohomology of 4-dimensional toric orbifolds. As applications, the stable homotopy type and the gauge groups of a $4$-dimensional toric orbifold are determined, a partial solution to the cohomological rigidity problem for $4$-dimensional toric orbifolds is provided, and, in the smooth case, a combinatorial criterion is established for when the toric orbifold is spin.

Steenrod operations for $4$-dimensional toric orbifolds

TL;DR

This work determines exact conditions for nontrivial mod-2 Steenrod actions on the cohomology of 4-dimensional toric orbifolds by tying the actions to determinant data from the characteristic labels and to a -local smooth vertex. The approach leverages a -CW decomposition and degenerate toric spaces to reduce Steenrod computations to triangle cases, yielding explicit parity criteria: is nontrivial precisely when is even, while on is nontrivial iff under a -local smooth vertex. As corollaries, the paper provides a sharp stable splitting for , a spin criterion in the smooth (quasi-toric) case, partial cohomological rigidity results, and explicit gauge-group homotopy decompositions. By connecting combinatorial input with the Steenrod algebra action, the results enable concrete invariants and homotopy classifications for a broad class of 4D toric orbifolds.

Abstract

We prove necessary and sufficient conditions for the existence of non-trivial Steenrod actions on the mod- cohomology of 4-dimensional toric orbifolds. As applications, the stable homotopy type and the gauge groups of a -dimensional toric orbifold are determined, a partial solution to the cohomological rigidity problem for -dimensional toric orbifolds is provided, and, in the smooth case, a combinatorial criterion is established for when the toric orbifold is spin.
Paper Structure (8 sections, 13 theorems, 91 equations, 3 figures)

This paper contains 8 sections, 13 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary facts about toric orbifolds
  3. Combintorial construction of toric orbifolds
  4. The cohomology of 4-dimensional toric orbifolds
  5. $q$-CW structures of toric orbifolds
  6. Degenerate toric spaces
  7. Steenrod operations on the cohomology of toric orbifolds
  8. Proofs of Theorem \ref{['thm_main thm']} and corollaries

Key Result

Theorem 1.2

Let $X(P,\lambda)$ be a 4-dimensional toric orbifold associated with an $(n+2)$-gon $P$ and a characteristic function $\lambda$. Suppose $v_{n+2}\in P$ is a $2$-local smooth vertex. Then

Figures (3)

  • Figure 1: Labels of edges and vertices in polygon $P$
  • Figure 2: $q$-CW construction of $X(P,\lambda)$
  • Figure 3: Edge contraction $\rho_i$

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 24 more