A new index for polytopes

A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized Dehn-Sommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual. Margaret M. Bayer was supported in part by a National Science Foundation grant, by a Northeastern University Junior Research Fellowship, and by the Institute for Mathematics and Its Applications.