PROJECTIONS OF POLYTOPES AND THE GENERALIZED BAUES CONJECTURE

Associated with every projection pi: P-->pi(P) of a polytope P is a pa rtially ordered set of all ''locally coherent strings'': the families of proper faces of P that project to valid subdivisions of pi(P), part ially ordered by the natural inclusion relation. The ''Generalized Bau es Conjecture'' posed by Billera et al. [4] asked whether this partial ly ordered set always has the homotopy type of a sphere of dimension d im(P)-dim(pi(P))-1. We show that this is true in the cases when dim (p i(P))=1 (see [4]) and when dim(P)-dim(pi(P))less than or equal to 2, b ut fails in general. For an explicit counterexample we produce a nonde generate projection of a five-dimensional, simplicial, 2-neighborly po lytope P with 10 vertices and 42 facets to a hexagon pi(P)subset of or equal to R(2). The construction of the counterexample is motivated by a geometric analysis of the relation between the fibers in an arbitra ry projection of polytopes.