Universal simplicial complexes in Toric topology

We study combinatorial and topological properties of the universal complexes X(F_p^n)  and  K(F_p^n)  whose simplices are certain unimodular subsets of F_p^n.
 We calculate their f-vectors and the bigraded Betti numbers of their Tor-algebras, show that they are shellable, and find their applications in toric topology and number theory. 
We show that the Lusternick-Schnirelmann category of the moment angle complex of X(F_p^n) is n, provided p is an odd prime, and the Lusternick-Schnirelmann category 
of the moment angle complex of K(F_p^n) is [n/2]. Based on the universal complexes, we introduce the Buchstaber invariant  for a prime number p.