Alexander duality for projections of polytopes

An affine projection π : Pp→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) ecture of Stanley (Invent. Math. 68 (1982) 175) concerning the relation between the homotopy type of two complexes. ecture of Reiner (pers. comm. 1999) which says the order complex of F(P)−i(F(Q)) has the homotopy type of a (p−q−1)-sphere. n-face posets of a class of regular cell complexes have the homotopy type of spheres, thereby answering a question raised by Reiner On some instances of the generalized Baues problem, unpublished manuscript, 1998 (http://www.math.umn.edu/~reiner/Papers/papers.html) and Edelman and Reiner (Discrete Comput. Geom. 23 (1) (2000) 1).