Projections of polytopes and the generalized Baues conjecture
With a projection \(\pi:P \to\pi (P)\) of polytopes a partially ordered set of all ``locally coherent strings'' is associated: Roughly speaking it consists of the families of proper faces of \(P\) that project to valid subdivisions of \(\pi(P)\), partially ordered by inclusion. Inspired by work of Baues, Billera stated the following generalized Baues conjecture: Conjecture: For every projection \(\pi:P \to\pi(P)\) of a \(d\)-polytope \(P\subset \mathbb{R}^d\) to a \(d'\)-polytope \(\pi (P) \subset \mathbb{R}^{d'}\), the poset \(\omega (P,\pi)\) of all locally \(\pi\)-coherent strings is homotopy equivalent to the \((d-d'-1)\)-sphere. Known results where the generalized Baues conjecture holds are: \(\bullet\) \(d'\leq 1\); \(\bullet\) \(P\) is a simplex and \(\dim (\pi(P)) \leq 2\); \(\bullet\) \(P\) is a hypercube and \(\dim (\pi(P)) \leq 2\) or \(\dim (P)- \dim (\pi(P)) \leq 3\); The authors establish: Theorem: The generalized Baues conjecture holds for \(d-d'\leq 2\) and is false in general for \(d'\geq 2\) and \(d-d'\geq 3\). As a counterexample they produce a nondegenerate projection of a 5-dimensional, simplicial, 2-neighborly polytope \(P\) with 10 vertices and 42 facets to a hexagon \(\pi(P) \subset \mathbb{R}^2\).
scientific article; zbMATH DE number 949785