On a class of toric manifolds arising from simplicial complexes

Abstract

Given an arbitrary abstract simplicial complex $K$ on $[m]:=\{1,2,\ldots, m\}$, different from the simplex $\Delta_{[m]}$ with $m$ vertices, we introduce and study a canonical $(2m-2)$-dimensional toric manifold $X_K$, associated to the canonical $(m-1)$-dimensional complete regular fan $\Sigma_K$. This construction yields an infinite family of toric manifolds that are not quasitoric and provides a topological proof of the Dehn-Sommerville relations for the associated Bier sphere $\mathrm{Bier}(K)$. Finally, we classify the canonical real and complex moment-angle manifolds of Lusternik-Schnirelmann category $\leq 2$ and prove a criterion for orientability of the canonical real toric manifolds.