Facial enumeration in polytopes, spheres and other complexes

If P is a d-dimensional convex polytope and S = {i(,0),...,i(,k)} (L-HOOK EQ) {0,1,...,d-1}, then f(,S)(P) is the number of chains of faces of P of the form (SLASHCIRC) (L-HOOK) F(,i(,0)) (L-HOOK) ... (L-HOOK) F(,i(,k)) (L-HOOK) P, where dim F(,j) = j. More generally we can allow P to be any poset and count chains in P having specified rank sets. Letting f(,i) = f(,{i})(P), we call f(P) = (f(,0),...,f(,d-1)) the f-vector of P. The set of f-vectors of simplicial d-polytopes has been characterized by Billera, Lee and Stanley. This thesis approaches the problem of describing the f-vectors of arbitrary d-polytopes by investigating the chain numbers f(,S)(P). The order complex of the lattice of faces of a polytope is an example of a completely balanced Gorenstein complex. We extend methods of Stanley on Cohen-Macaulay complexes to Gorenstein complexes and obtain relations on the f(,S)(P). In particular we find a set of linear relations generalizing the Dehn-Sommerville equations. A second proof of these comes out of an exant sequence of rings associated with the complex. The resulting relations, as well as Euler's formula, are then applied to the interval between two elements of P. This gives a large number of equations satisfied by the chain numbers. Some bounds on the dimension of the affine span of {(f(,S)(P))(,S(L-HOOK EQ){0,...,d-1}):P is a d-polytope} are developed. The thesis conjectures that the hyperplane of R('d) determined by the Dehn-Sommerville equations bound the space of f-vectors of all d-polytopes. Several facts supporting this conjecture are proved.