Face numbers of pseudomanifolds with isolated singularities

Abstract

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower and upper bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the f -vector can be completely characterized are described. We also show that the Hilbert function of a generic Artinian reduction of the face ring of a simplicial complex with isolated singularities minus the h-vector of is a PL-topological invariant.

Face enumeration on simplicial complexes

“…It is worth adding that the "in particular" part (i.e., the statement on f -vectors) of Theorem 7 was extended to all odd-dimensional Eulerian pseudomanifolds with isolated singularities in [29]; moreover, Novik and Swartz [74], proved the UBT for all Eulerian pseudomanifolds with so-called homologically isolated singularities as long as n ≥ 3d − 4. However, Klee's conjecture [42] that the UBC holds for all Eulerian complexes remains wide open.…”

The upper bound theorem for flag homology 5-manifolds

“…By Theorem 3.1 in [12] we have that f 3 (∆) = f 1 (∆) − f 0 (∆) + χ(∆), and 2χ(∆) = v∈∆ β 1 (lk(v)). Also since any vertex link in ∆ is a simplicial 2-manifold, so f 2 (lk(v)) = 2f 0 (lk(v))− 4 + 2β 1 (lk(v)).…”

Ext and local cohomology modules of face rings of simplicial posets

“…Our Theorem 4.4 detailing the A-module structure of H i m (A P ) has the potential to extend many results about simplicial complexes relying upon Gräbe's theorem to the simplicial poset case. For example, many of the theorems for face rings of simplicial complexes with isolated singularities found in [MNS11], [NS12], and [Saw] appear likely to be true in this greater generality.…”