Universal Complexes in Toric Topology

Abstract

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 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(Fpn) is n, provided p is an odd prime, and the Lusternick–Schnirelmann category of the moment angle complex of K(Fpn) is [n2] . Based on the universal complexes, we introduce the Buchstaber invariant sp for a prime number p.