We study combinatorial and topological properties of the universal complexes
X(Fpn) and K(Fpn) whose simplices are certain
unimodular subsets of Fpn. We calculate their f-vectors
and their Tor-algebras, show that they are shellable but not shifted, and find
their applications in toric topology and number theory. We showed that the
Lusternick-Schnirelmann category of the moment angle complex of
X(Fpn) is n, provided p is an odd prime, and the
Lusternick-Schnirelmann category of the moment angle complex of
K(Fpn) is [2n]. Based on the universal complexes, we
introduce the Buchstaber invariant sp for a prime number p