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On a class of toric manifolds arising from simplicial complexes

Ivan Limonchenko, Marinko Timotijević, Rade Živaljević

TL;DR

This work builds a bridge between combinatorial spheres and toric geometry by introducing Bier spheres and a canonical complete regular fan $\Sigma_K$ to produce toric manifolds $X_K$ with dimension $2m-2$. It proves a topological Dehn-Sommerville relation for Bier spheres via Poincaré duality on $X_K$, and demonstrates that many Bier spheres give toric spaces that are not quasitoric when the spheres are nonpolytopal. The authors classify canonical moment-angle manifolds of Lusternik-Schnirelmann category $\le 2$ and provide an orientability criterion for the canonical real toric manifolds, all while outlining a pathway to understand when a Delzant polytope arises from $\mathrm{Bier}(K)$ and when polytopal Bier spheres yield Delzant polytopes. Overall, the paper broadens the class of toric spaces associated to combinatorial spheres and showcases how topological methods illuminate classical combinatorial invariants.

Abstract

Given an arbitrary abstract simplicial complex $K$ on $[m]:=\{1,2,\ldots, m\}$, different from the simplex $Δ_{[m]}$ 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 $Σ_K$. 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.

On a class of toric manifolds arising from simplicial complexes

TL;DR

This work builds a bridge between combinatorial spheres and toric geometry by introducing Bier spheres and a canonical complete regular fan to produce toric manifolds with dimension . It proves a topological Dehn-Sommerville relation for Bier spheres via Poincaré duality on , and demonstrates that many Bier spheres give toric spaces that are not quasitoric when the spheres are nonpolytopal. The authors classify canonical moment-angle manifolds of Lusternik-Schnirelmann category and provide an orientability criterion for the canonical real toric manifolds, all while outlining a pathway to understand when a Delzant polytope arises from and when polytopal Bier spheres yield Delzant polytopes. Overall, the paper broadens the class of toric spaces associated to combinatorial spheres and showcases how topological methods illuminate classical combinatorial invariants.

Abstract

Given an arbitrary abstract simplicial complex on , different from the simplex with vertices, we introduce and study a canonical -dimensional toric manifold , associated to the canonical -dimensional complete regular fan . 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 . Finally, we classify the canonical real and complex moment-angle manifolds of Lusternik-Schnirelmann category and prove a criterion for orientability of the canonical real toric manifolds.

Paper Structure

This paper contains 4 sections, 8 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Abstract simplicial complexes and rational polyhedral fans
  3. Regular realizations for Bier spheres
  4. Applications

Key Result

Theorem 2.6

Let $K\neq\Delta_{[m]}$ be a simplicial complex on $[m]$ with $m\geq 3$. Then $\mathrm{Bier}(K)$ has a geometric realization as a starshaped sphere in the hyperplane $H_0:=\{x\in{\mathbb{R}}^m\,|\,\langle u,x\rangle=0\}$, where $u$ is the sum of the standard basis vectors $e_i, 1\leq i\leq m$ in ${\

Figures (2)

  • Figure 1: The canonical complete regular fans $\Sigma_{\varnothing_{[3]}}$ and $\Sigma_{\varnothing_{[4]}}$.
  • Figure 2: The canonical complete regular fan $\Sigma_K$ for the complex $K=\langle\{1\},\{2,3\}\rangle$.

Theorems & Definitions (32)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6: Zivaljevic19Zivaljevic21
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Example 3.2
  • ...and 22 more