Toric complexes and Artin kernels

TL;DR

This paper investigates the algebraic and topological properties of toric complexes derived from simplicial complexes, focusing on cohomology, homology, and Lie algebra structures of associated Artin kernels, under various conditions.

Contribution

It provides explicit computations of cohomology jumping loci, homology groups, and Lie algebras of Artin kernels, along with criteria for trivial -action and 1-formality results.

Findings

Computed cohomology jumping loci of toric complexes.

Determined homology groups of Artin kernels with field coefficients.

Established 1-formality of certain 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 \chi\colon G_L\to \Z, let T_L^\chi be the corresponding cover, with fundamental group the Artin kernel N_\chi. We compute the cohomology jumping loci of the toric complex T_L, as well as the homology groups of T_L^\chi with coefficients in a field \k, viewed as modules over the group algebra \k\Z. We give combinatorial conditions for H_{\le r}(T_L^\chi;\k) to have trivial \Z-action, allowing us to compute the truncated cohomology ring, H^{\le r}(T_L^\chi;\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.