Jonathan Browder: Simplicial posets, manifolds, and Buchsbaum rings

Simplicial posets describe regular CW-complexes whose closed cells are combinatorially simplices (these have variously been referred to also as Boolean cell complexes, simplicial cell complexes, and semi-simplicial complexes). The f-vector of a simplicial poset simply lists the number of simplices the poset has in each dimension; it is natural to ask what f-vectors are possible when the poset triangulates certain types of spaces. Algebraic tools have proven quite effective in addressing these questions, via associating to each simplicial poset a ring, called the face ring (generalizing the Stanley-Reisner ring of a simplicial complex), which reflects the topology of the underlying space in interesting ways. For example, the face ring of a simplicial poset that triangulates a manifold is always Buchsbaum. In this talk we will discuss some recent results on the face vectors of various classes of simplicial poset (spheres, balls, manifolds without boundary, etc), including a complete characterization of the face vectors of Buchsbaum simplicial posets with given Betti numbers.