Online Talk

In this talk, we present 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 the bigraded Betti numbers of their Tor-algebras, show that they are shellable, 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 $p$. We investigate the mod $p$ Buchstaber invariant of the skeleta of simplices for a prime number $p$ and compare them for different values of $p$. For $p=2$, the invariant is the real Buchstaber invariant. Our findings reveal that these values are generally distinct. Additionally, we determine or estimate the mod $p$ Buchstaber invariants of some universal simplicial complexes $X(\mathbb{F}_p^n)$. The talk is based on joint research with with Ale\v{s} Vavpeti\'{c} and Aleksandar Vu\v{c}i\'{c}.