Associated with every projection π:P \to π(P) of
a polytope P one has a partially ordered set of all
"locally coherent strings": the families of proper
faces of P that project to valid subdivisions of
π(P), partially ordered by the natural inclusion
relation. The "Generalized Baues Conjecture" posed by
Billera, Kapranov & Sturmfels
BilleraKapranovSturmfels1994} 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 BilleraKapranovSturmfels1994}) and when
\dim(P) - \dim(π(P)) \le 2, but fails otherwise.
For an explicit counterexample we produce a
non-degenerate projection of a 5-dimensional,
simplicial, 2-neighborly polytope P with 10
vertices and 42 facets to a hexagon `π(P) \subseteq
\R^2`. The construction of the counterexample is
motivated by a geometric analysis of the relation
between the fibers in an arbitrary projection of
polytopes.
