Universal simplicial complexes inspired by toric topology

TL;DR

This paper investigates the topological and combinatorial properties of universal simplicial complexes derived from unimodular subsets over fields and rings, revealing their homotopy types and applications in toric topology and number theory.

Contribution

It generalizes existing results by showing these complexes are homotopy equivalent to wedges of spheres and explores their applications in related mathematical fields.

Findings

Homotopy equivalence to wedges of spheres

Generalization of Davis and Januszkiewicz's connectivity results

Applications to toric topology and number theory

Abstract

Let $\mathbf{k}$ be the field $\mathbb{F}_p$ or the ring $\mathbb{Z}$. We study combinatorial and topological properties of the universal simplicial complexes $X(\mathbf{k}^n)$ and $K(\mathbf{k}^n)$ whose simplices are certain unimodular subsets of $\mathbf{k}^n$. As a main result we show that $X(\mathbf{k}^n)$, $K(\mathbf{k}^n)$ and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that $K(\mathbb{Z}^n)$ and $K(\mathbb{F}_2^n)$ are $(n-2)$-connected simplicial complexes. We discuss applications of these universal simplicial complexes to toric topology and number theory.