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.  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) 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
- Normal Cone
- Homotopy Type
- Relative Interior
- Oriented Matroids
- Local Coherence