Face Numbers of Polytopes, Posets, and Complexes

Abstract

A key tool that combinatorialists use to study simplicial complexes and polytopes is the {\bf $f$-vector} (or face vector), which records the number of faces of each dimension. In order to better understand the face numbers, relations involving both equalities and inequalities on $f$-vectors have been extensively studied. In this dissertation we discuss the author's contributions to these topics. The classical Dehn--Sommerville relations assert that the $h$-vector of an Eulerian simplicial complex is symmetric. In Chapter 2, we establish three generalizations of the Dehn--Sommerville relations: one for the $h$-vectors of pure simplicial complexes, another one for the flag $h$-vectors of balanced simplicial complexes and graded posets, and yet another one for the toric $h$-vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of ``errors coming from the links." For simplicial complexes, this further extends Klee's semi-Eulerian relations. In Chapters 3 and 4, we change our focus from equalities to inequalities on $f$-vectors. In 1967, Gr\"unbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least \[\phi_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 } \] $k$-faces. In Chapter 3, we prove this conjecture and also characterize the cases in which equality holds. In Chapter 4, several extensions of Gr\"unbaum's conjecture are established. Specifically, it is proved that every lattice with diamond property and $d+s\leq 2d$ atoms has at least $\phi_k(s)$ elements of rank $k+1$. Furthermore, in the case of lattices that are face lattices of strongly regular CW complexes representing normal pseudomanifolds with up to $2d$ vertices, a characterization of equality cases is given. Finally, sharp lower bounds on the number of $k$-faces of strongly regular CW complexes representing normal pseudomanifolds with $2d+1$ vertices are obtained. These bounds are given by the face numbers of certain polytopes with $2d+1$ vertices.