Universal complexes in toric topology

TL;DR

This paper investigates the combinatorial and topological properties of universal complexes over finite fields, computes their algebraic invariants, and explores their applications in toric topology and number theory.

Contribution

It introduces and analyzes universal complexes in toric topology, computes their algebraic invariants, and defines the Buchstaber invariant based on these complexes.

Findings

Universal complexes are shellable but not shifted.

Calculated f-vectors and Tor-algebras of the complexes.

Determined Lusternick-Schnirelmann categories for associated moment angle complexes.

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 $\mathbf f$-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 $p$.