Toric arrangements and Bloch-Kato pro-p groups

2025/07/22 by Emanuele Delucchi, Delucchi, Emanuele, Ettore Marmo +1
Mathematics · #12F10 #20E18 #20F36 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR) #Primary: 52C35 #Secondary: 06A07

paper · pdf · doi:10.48550/arxiv.2507.16428

openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a purely combinatorial obstruction for the Bloch-Kato property within the class of fundamental groups of complement manifolds of toric arrangements (i.e., arrangements of hypersurfaces in the complex torus). As a stepping stone we obtain a combinatorial obstruction for the cohomology of a supersolvable arrangement to be generated in degree 1. Our result allows us to prove that - for all prime numbers p, the pro-p completion of the pure braid group on k strands has the Bloch-Kato property if and only if k≤ 3; - for all prime numbers p, the pro-p completion of the pure mapping class group of the sphere S² with k punctures has the Bloch-Kato property if and only if k≤ 4.

Citations

• Higher dimensional algebraic fiberings for pro-p groups
• Supersolvable posets and fiber-type abelian arrangements
• A Tutte polynomial for toric arrangements

Related

• Monodromy of supersolvable toric arrangements
• On the geometry of toric arrangements
• The homotopy type of toric arrangements
• Supersolvable posets and fiber-type abelian arrangements
• The integer cohomology of toric Weyl arrangements