Alexander Duality for Projections of Polytopes

An affine projection π : P p → Qq of convex polytopes induces an inclusion map of the face posets i : F(Q) → F(P ). We define an order-preserving map of posets h : F(P ) → Suspp−qF(Q) such that for any filter J of Suspp−qF(Q), the map h restricts to a homotopy equivalence between the order complexes of h−1(J) and J . As applications we prove (1) A conjecture of Stanley [11] concerning the relation between the homotopy type of two complexes. (2) A conjecture of Reiner [10] which says the order complex of F(P ) − i(F(Q)) has the homotopy type of a (p− q − 1)-sphere. (3) The non-face posets of a class of regular cell complexes have the homotopy type of spheres, thereby answering a question raised by Reiner [9] and Edelman and Reiner [4].