Universal complexes in toric topology

We study combinatorial and topological properties of the universal complexes $X({\mathbb{F}}_p^n)$ and $K({\mathbb{F}}_p^n)$ whose simplices are certain unimodular subsets of ${\mathbb{F}}_p^n$. We calculate their $\mathbf{f}$-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 $X({\mathbb{F}}_p^n)$ is $n$, provided $p$ is an odd prime, and the Lusternick-Schnirelmann category of the moment angle complex of $K({\mathbb{F}}_p^n)$ is $[\frac{n}{2}]$. Based on the universal complexes, we introduce the Buchstaber invariant $s_p$ for a prime number $p$.