Universal simplicial complexes inspired by toric topology

Let k be the field F_p or the ring Z. We study combinatorial and topological properties of the universal simplicial complexes 
X(k^n) and K(k^n) whose simplices are certain unimodular subsets of k n. As a main result we show that X(k^n), K(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(Z^n) and K(F_2^n) 
are (n − 2)-connected simplicial complexes.
We discuss applications of these universal simplicial complexes to toric topology and number theory.