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A new class of affine $K(π,1)$ arrangements

Katherine Goldman, Jingyin Huang

TL;DR

The paper introduces a new class of affine hyperplane arrangements (admissible types built from B_n, D_n, skewed A_n and products) for which the complexified complements are K(π,1). By endowing Falk complexes with an injective metric and proving a local-to-global criterion via bowtie-free and upward-flag poset structures, it extends Falk’s 2D approach to higher dimensions. It constructs explicit infinite families H_{k,n} and K_{k,n}, demonstrates K(π,1) results for dimensions n≤4 unconditionally (and in general under a conjecture on spherical Artin groups for higher n), and provides a detailed verification framework for the skewed A_n case through a meticulous analysis of subdivided Coxeter/Artin complexes. The work yields new contractible Falk models underpinning asphericity and reveals connections to affine Artin groups via the Salvetti complex and quotient constructions.

Abstract

We show that a certain class of affine hyperplane arrangements are $K(π,1)$ by endowing their Falk complexes with an injective metric. This gives new examples of infinite $K(π,1)$ arrangements in dimension $n>2$.

A new class of affine $K(π,1)$ arrangements

TL;DR

The paper introduces a new class of affine hyperplane arrangements (admissible types built from B_n, D_n, skewed A_n and products) for which the complexified complements are K(π,1). By endowing Falk complexes with an injective metric and proving a local-to-global criterion via bowtie-free and upward-flag poset structures, it extends Falk’s 2D approach to higher dimensions. It constructs explicit infinite families H_{k,n} and K_{k,n}, demonstrates K(π,1) results for dimensions n≤4 unconditionally (and in general under a conjecture on spherical Artin groups for higher n), and provides a detailed verification framework for the skewed A_n case through a meticulous analysis of subdivided Coxeter/Artin complexes. The work yields new contractible Falk models underpinning asphericity and reveals connections to affine Artin groups via the Salvetti complex and quotient constructions.

Abstract

We show that a certain class of affine hyperplane arrangements are by endowing their Falk complexes with an injective metric. This gives new examples of infinite arrangements in dimension .

Paper Structure

This paper contains 16 sections, 54 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperplane arrangements and their dual polyhedra
  4. The Salvetti complex
  5. Injective metric spaces
  6. Coxeter complexes and Artin complexes
  7. The Falk complexes and their metric
  8. Falk complexes
  9. Links of Falk complexes
  10. Metrizing Falk complexes
  11. Falk complexes for admissible arrangements
  12. Some examples of admissible arrangements
  13. The skewed A_n arrangement
  14. The Coxeter complex
  15. The Artin complex
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $\mathcal{A}$ be an admissible affine arrangement in $\mathbb R^n$ which is invariant under the action of a discrete translation subgroup $\mathbb Z^n$ of $\mathbb R^n$ (this does not have to be the usual embedding of $\mathbb Z^n$). Suppose $n\le 4$. Then $\mathcal{A}$ is a $K(\pi,1)$ arrangeme

Figures (1)

  • Figure 1: Coxeter diagram of type $D_n$.

Theorems & Definitions (95)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 2.1: s87
  • Lemma 2.2
  • Lemma 2.3
  • ...and 85 more