On the topology of the Baues poset of polyhedral subdivisions

Given an affine projection π : P→Q of convex polytopes, let ω(P, π) be the refinement poset of proper polyhedral subdivisions of Q which are induced by π, in the sense of Billera and Sturmfels. Let ωcoh(P, π) be the spherical subposet of π-coherent subdivisions. It is proved here that the inclusion of the latter poset into the former induces injections in homology and homotopy. In particular, the poset ω(P, π) is homotopically nontrivial. As a corollary, the equivalence of the weak and strong forms of the generalized Baues problem of Billera, Kapranov and Sturmfels is established. As special cases, these results apply to the refinement poset of proper polyhedral subdivisions of a point configuration and to the extension poset of a realizable oriented matroid.