Lower Bounds for Buchsbaum* Complexes

TL;DR

This paper investigates Buchsbaum* simplicial complexes, establishing that their rank-selected subcomplexes are also Buchsbaum*, and derives lower bounds on their h-numbers, including special cases for flag m-Buchsbaum* complexes.

Contribution

It proves that rank-selected subcomplexes of balanced Buchsbaum* complexes are also Buchsbaum* and establishes sharp lower bounds on their h-numbers.

Findings

Rank-selected subcomplexes of balanced Buchsbaum* complexes are Buchsbaum*.

Lower bounds on h-numbers for balanced Buchsbaum* complexes.

Sharp lower bounds and equality cases for flag m-Buchsbaum* complexes.

Abstract

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.