On spherical Deligne complexes of type D_n

2024/05/18 by Jingyin Huang, Huang, Jingyin · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2405.11374

openalex publication_date 2024/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Δ be the Artin complex of the Artin group of type D_n. This complex is also called the spherical Deligne complex of type D_n. We show certain types of 6-cycles in the 1-skeleton of Δ either have a center, which is a vertex adjacent to each vertex of the 6-cycle, or a quasi-center, which is a vertex adjacent to three of the alternating vertices of the 6-cycle. This will be a key ingredient in proving K(π,1)-conjecture for several classes of Artin groups in a companion article. As a consequence, we also deduce that certain 2-dimensional relative Artin complex inside the D_n-type Artin complex, endowed with the induced Moussong metric, is CAT(1).

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

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