Buchsbaum* complexes

TL;DR

This paper introduces Buchsbaum* complexes, a new class of simplicial complexes that generalize known structures like triangulations of orientable homology manifolds and doubly Cohen-Macaulay complexes, with various characterizations and properties.

Contribution

The paper defines Buchsbaum* complexes, explores their properties, characterizations, and constructions, expanding the understanding of their combinatorial and topological features.

Findings

Buchsbaum* complexes are doubly Buchsbaum.

Various constructions produce Buchsbaum* complexes.

Enumerative and graph theoretic properties are characterized.

Abstract

A class of simplicial complexes, which we call Buchsbaum* over a field, is introduced. Buchsbaum* complexes generalize triangulations of orientable homology manifolds as well as doubly Cohen-Macaulay complexes. By definition, the Buchsbaum* property depends only on the geometric realization and the field. Characterizations in terms of simplicial and local cohomology are given. It is proved that Buchsbaum* complexes are doubly Buchsbaum. Enumerative and graph theoretic properties of Buchsbaum* complexes are investigated. It is shown that various constructions, among them one which generalizes convex ear decompositions, yield Buchsbaum* simplicial complexes.