The Generalized Baues Problem
Victor Reiner · 1999
We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to triangulations, zonotopal tilings, monotone paths in linear programming, oriented matroid Grassmannians, singularities, and homotopy theory. Included are several open questions and problems. 1. Introduction The generalized Baues problem, or GBP for short, is a question arising in the work of Billera and Sturmfels [1992, p. 545] on fiber polytopes; see also [Billera et al. 1994, §3]. The question asks whether certain partially ordered sets whose elements are subdivisions of polytopes, endowed with a certain topology [Björner 1995], have the homotopy type of spheres. Cases are known [Rambau and Ziegler 1996] where this fails to be true, but the general question of when it is true or false remains an exciting subject of current research. The goal of this survey is to review the motivation for fiber polytopes and the GBP, and discuss recent progress on the GBP and the open questions remaining. Some recommended summary sources on this subject are the introductory chapters in the doctoral theses [Rambau 1996; Richter-Gebert 1992], Lecture 9 in [Ziegler 1995], and the paper [Sturmfels 1991]. The articles [Billera et al. 1990; 1993], though not discussed in the text, are nonetheless also relevant to the GBP. Before diving into the general setting of fiber polytopes and the GBP, it is worthwhile to ponder three motivating classes of examples.