Alexander duality for projections of polytopes
An affine projection \(P\to Q\) of convex polytopes induces an inclusion of face posets \(i:F(Q)\to F(P)\). The present paper shows that in a combinatorial sense, this map behaves much like an equatorial embedding of a lower-dimensional sphere into a higher-dimensional sphere. Specifically, there is a poset map from \(F(P)\) to the appropriate suspension of \(F(Q)\) that restricts to a homotopy equivalence between any filter and its preimage. In particular, this shows \(F(P)-i(F(Q))\) has the homotopy type of a sphere, answering a conjecture of Reiner. There are also less obvious consequences. Taking \(P\) to be a simplex with the same vertex set as \(Q\), one sees the non-face poset of \(Q\) has the homotopy type of a sphere. The main result also proves a conjecture of Stanley that shows two posets arising in certain local cohomology computations have the same homotopy type.
scientific article; zbMATH DE number 1838823