Universal simplicial complexes inspired by toric topology

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.