Simplicial complexes of small codimension

TL;DR

This paper proves that Buchsbaum and ${ m CM}_t$ simplicial complexes of small codimension have large depth, with detailed results for codimension 2, and shows that ${ m CM}_t$ is a topological invariant.

Contribution

It establishes a link between small codimension and large depth for Buchsbaum and ${ m CM}_t$ complexes, and proves ${ m CM}_t$ invariance under topology.

Findings

Buchsbaum complexes of small codimension have large depth.

${ m CM}_t$ property is a topological invariant.

Detailed results for codimension 2 case.

Abstract

We show that a Buchsbaum simplicial complex of small codimension must have large depth. More generally, we achieve a similar result for ${\rm CM}_t$ simplicial complexes, a notion generalizing Buchsbaum-ness, and we prove more precise results in the codimension 2 case. Along the paper, we show that the ${\rm CM}_t$ property is a topological invariant of a simplicial complex.