Polyhedral Subdivisions and Projections of Polytopes Jörg Rambau Dissertation (Advisor: Günter M. Ziegler, TU Berlin)

   
The Generalized Baues Conjecture

Associated with every projection $\pi : P \to \pi (P)$ of a polytope $P$one has a partially ordered set of all ``locally coherent strings'': the families of proper faces of $P$ that project to valid subdivisions of $\pi (P)$, partially ordered by the natural inclusion relation.

The ``Generalized Baues Conjecture'' posed by Billera, Kapranov & Sturmfels [9] asked whether this partially ordered set always has the homotopy type of a sphere of dimension $\dim(P)-\dim(\pi (P))-1$. We show that this is true in the cases when $\dim(\pi (P)) = 1$(see [9]) and when $\dim(P) - \dim(\pi (P)) \le 2$, but fails in general.

For an explicit counterexample we produce a non-degenerate projection of a $5$-dimensional, simplicial, $2$-neighborly polytope $P$ with $10$ vertices and $42$ facets to a hexagon $\pi(P) \subseteq \mathbb{R} ^2$. The construction of the counterexample is motivated by a geometric analysis of the relation between the fibers in an arbitrary projection of polytopes.

This chapter is based on a joint work with ZIEGLER [61].