Universal Complexes in Toric Topology

Abstract

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 $\textbf{f}$-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 $X({\mathbb {F}}_p^n)$ 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.