Chromatic numbers, Buchstaber numbers and chordality of Bier spheres

Abstract

We describe all the Bier spheres of dimension $d$ with chromatic number equal to $d+1$ and prove that all other $d$-dimensional Bier spheres have chromatic number equal to $d+2$, for any integer $d\geq 0$. Then we prove a general formula for complex and mod $p$ Buchstaber numbers of a Bier sphere $\mathrm{Bier}(K)$, for each prime $p\in\mathbb{N}$ in terms of the $f$-vector of the underlying simplicial complex $K$. Finally, we classify all chordal Bier spheres and obtain their canonical realizations as boundaries of stacked polytopes.

Figures (7)

• Figure 1: The only cycles that are 1-dimensional Bier spheres.
• Figure 2: The complexes $G_4$ and $\Gamma_6$ are self-dual, $G_4$ has a ghost vertex. The indices in the notation of a complex indicate the number of vertices in the corresponding Bier sphere.
• Figure 3: The boundary of an octahedron is a suspension and therefore a weak suspension, but the boundaries of a tetrahedron and an icosahedron are not even weak suspensions.
• Figure 4: On the left it is the simplicial complex $K_5$ and its Alexander dual $K_5^\vee$. On the right it is the simplicial complex $K_6$ (there is no face $\{2,3,4,5\}$) and its Alexander dual $K_6^\vee$.
• Figure 5: On the left is the graph $\Gamma_6$, which is self-dual, and its Bier sphere, which is the $6$-cycle. On the right is a cone over $\Gamma_6$, which is also self-dual, and its Bier sphere, which is a suspension over $\mathrm{Bier}(\Gamma_6)$.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Definition 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Definition 2.5
• Example 2.6