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Projections of polytopes and the generalized baues conjecture

Abstract

Associated with every projection π:P→π(P) of a polytopeP is a partially ordered set of all “locally coherent strings”: the families of proper faces ofP that project to valid subdivisions of π(P), partially ordered by the natural inclusion relation. The “Generalized Baues Conjecture” posed by Billeraet al. [4] asked whether this partially ordered set always has the homotopy type of a sphere of dimension dim(P—dim(π(P))−1. We show that this is true in the cases when dim(π(P))=1 (see[4]) and when dim(P)—dim(π(P))≤2, but fails in general. For an explicit counterexample we produce a nondegenerate projection of a five-dimensional, simplicial, 2-neighborly polytopeP with 10 vertices and 42 facets to a hexagon π(P)⊆ℝ2. The construction of the counterexample is motivated by a geometric analysis of the relation between the fibers in an arbitrary projection of polytopes.

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The first author was supported by the Deutsche Forschungsgemeinschaft (DFG), Grant We 1265/2-1. The second author was supported by a “Gerhard-Hess-Forschungsförderpreis” of the Deutsche Forschungsgemeinschaft (DFG).

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Rambau, J., Ziegler, G.M. Projections of polytopes and the generalized baues conjecture. Discrete Comput Geom 16, 215–237 (1996). https://doi.org/10.1007/BF02711510

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Keywords

  • Normal Cone
  • Homotopy Type
  • Relative Interior
  • Oriented Matroids
  • Local Coherence