The class of (d − 1)-dimensional Buchsbaum* simplicial complexes is studied. It is shown that the rank-selected subcomplexes of a (completely) balanced Buchsbaum* simplicial complex are also Buchsbaum*. Using this result, lower bounds on the h-numbers of balanced Buchsbaum* simplicial complexes are established. In addition, sharp lower bounds on the h-numbers of flag m-Buchsbaum* simplicial complexes are derived, and the case of equality is treated.

In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. As a key ingredient of our work, we introduce the class of cointerval simplicial complexes and investigate their combinatorial and topological properties. As a concrete illustration of these structural results, we introduce and study nonnesting monomial ideals, an interesting family of combinatorially defined ideals.

Abstract. Suppose a group G acts properly on a simplicial complex Γ. Let l be the number of G-invariant vertices and p 1 , p 2 , . . . pm be the sizes of the G-orbits having size greater than 1. Then Γ must be a subcomplex of Λ = ∆ l−1 * ∂∆ p 1 −1 * . . . * ∂∆ pm −1 . A result of Novik gives necessary conditions on the face numbers of Cohen-Macaulay subcomplexes of Λ. We show that these conditions are also sufficient, and thus provide a complete characterization of the face numbers of these complexes.

A common generalization of two theorems on the face numbers of Cohen-Macaulay (CM, for short) simplicial complexes is established: the first is the theorem of Stanley (necessity) and Björner-Frankl-Stanley (sufficiency) that characterizes all possible face numbers of abalanced CM complexes, while the second is the theorem of Novik (necessity) and Browder (sufficiency) that characterizes the face numbers of CM subcomplexes of the join of the boundaries of simplices.