Toric complexes and Artin kernels

Abstract

A simplicial complex L on n vertices determines a subcomplex T_L of the n-torus, with fundamental group the right-angled Artin group G_L. Given an epimorphism χ\colon G_L\to \Z, let T_L^χbe the corresponding cover, with fundamental group the Artin kernel N_χ. We compute the cohomology jumping loci of the toric complex T_L, as well as the homology groups of T_L^χwith coefficients in a field \k, viewed as modules over the group algebra \k\Z. We give combinatorial conditions for H_{\le r}(T_L^χ;\k) to have trivial \Z-action, allowing us to compute the truncated cohomology ring, H^{\le r}(T_L^χ;\k). We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy \Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups.

34 pages

References in corpus (10)

• Topology and geometry of cohomology jump loci
• Algebraic invariants for right-angled Artin groups
• Moment-angle complexes, monomial ideals, and Massey products
• Chen Lie algebras
• Higher homotopy groups of complements of complex hyperplane arrangements
• The spectral sequence of an equivariant chain complex and homology with local coefficients
• Bieri-Neumann-Strebel-Renz invariants and homology jumping loci
• Algebraic invariants for Bestvina-Brady groups
• Formality, Alexander invariants, and a question of Serre
• The cohomology of Bestvina-Brady groups

Cited by in corpus (21)

• Topology and geometry of cohomology jump loci
• The spectral sequence of an equivariant chain complex and homology with local coefficients
• Geometric and algebraic aspects of 1-formality
• Fundamental groups, Alexander invariants, and cohomology jumping loci
• Combinatorial covers and vanishing of cohomology
• Resonance varieties and Dwyer-Fried invariants
• Abelian duality and propagation of resonance
• Geometric and homological finiteness in free abelian covers
• A Leray model for the Orlik-Solomon algebra
• Finite Galois covers, cohomology jump loci, formality properties, and multinets
• Homology of even Artin kernels
• Graphical splittings of Artin kernels
• Formality and finiteness in rational homotopy theory
• Infinitesimal finiteness obstructions
• Milnor fibrations of arrangements with trivial algebraic monodromy
• Module structure of the homology of right-angled Artin kernels
• Higher resonance schemes and Koszul modules of simplicial complexes
• Alexander invariants and cohomology jump loci in group extensions
• Sigma-invariants and tropical varieties
• Characteristic varieties for a class of line arrangements
• Coabelian ideals in $\mathbb{N}$-graded Lie algebras and applications to right angled Artin Lie algebras