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Projections of polytopes and the generalized baues conjecture


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.


  1. 1.

    J. F. Adams. On the cobar construction.Proceedings of the National Academy of Science, 42:409–412, 1956.

    MATH  Article  Google Scholar 

  2. 2.

    H. J. Baues. Geometry of loop spaces and the cobar construction.Memoirs of the American Mathematical Society, 25(230):171, 1980.

    MathSciNet  Google Scholar 

  3. 3.

    L. J. Billera, I. M. Gel'fand, and B. Sturmfels. Duality and minors of secondary polyhedra.Journal of Combinatorial Theory, Series B, 57:258–268, 1993.

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    L. J. Billera, M. M. Kapranov, and B. Sturmfels. Cellular strings on polytopes.Proceedings of the American Mathematical Society, 122:549–555, 1994.

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    L. J. Billera and B. Sturmfels. Fiber polytopes.Annals of Mathematics, 135:527–549, 1992.

    Article  MathSciNet  Google Scholar 

  6. 6.

    A. Björner. Essential chains and homotopy types of posets.Proceedings of the American Mathematical Society, 402:1179–1181, 1992.

    Article  Google Scholar 

  7. 7.

    A. Björner. Topological methods. In R. Graham, M. Grötschel, and L. Lovász, editors,Handbook of Combinatorics, pp. 1819–1872. North-Holland, Amsterdam, 1995.

    Google Scholar 

  8. 8.

    T. Christof. Porta—a polyhedron representation transformation algorithm v.1.2.1. Available from the ZIB electronic library eLib via or by anonymous ftp from, directory /pub/mathprog/polyth.

  9. 9.

    P. H. Edelman and V. Reiner. Visibility complexes and the Baues Problem for triangulations in the plane. Preprint 1995.

  10. 10.

    H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for regular triangulations.Proceedings of the 8th Annual ACM Symposium on Computational Geometry, Berlin, pp. 43–52. ACM Press, New York, 1992.

    Google Scholar 

  11. 11.

    B. Joe. Three-dimensional triangulations from local transformations.SIAM Journal of Scientific Statistical Computation, 10:718–741, 1989.

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    B. Joe. Construction of three-dimensional Delaunay triangulations using local transformations.Computer Aided Geometric Design, 8:123–142, 1991.

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    R. D. MacPherson. Combinatorial differential manifolds. In L. R. Goldberg and A. V. Phillips, editors,Topological Methods in Modern Mathematics: a Symposium in Honor of John Milnor's Sixtieth Birthday, Stony Brook, NY, 1991, pp. 203–221. Publish or Perish, Houston, TX, 1993.

    Google Scholar 

  14. 14.

    N. E. Mnëv and G. M. Ziegler. Combinatorial models for the finite-dimensional Grassmannians.Discrete & Computational Geometry, 10:241–250, 1993.

    MATH  Article  MathSciNet  Google Scholar 

  15. 15.

    A. Nabutovsky. Extremal triangulations of manifolds. Preprint, 1994.

  16. 16.

    B. Sturmfels. Fiber polytopes: a brief overview. In M. Yoshida, editor,Special Differential Equations, pp. 117–124. Kyushu University, Fukuoka, 1991.

    Google Scholar 

  17. 17.

    B. Sturmfels and G. M. Ziegler. Extension spaces of oriented matroids.Discrete & Computational Geometry 10:23–45, 1993.

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

    G. M. Ziegler.Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer-Verlag, New York, 1995. Updates, corrections, and more available at∼ziegler

    MATH  Google Scholar 

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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).

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