A generalization of $k$-Cohen-Macaulay complexes

TL;DR

This paper introduces a new class of simplicial complexes called $k$-${ m CM}_t$, generalizing Cohen-Macaulay and Buchsbaum complexes, with characterizations based on homology vanishing and link properties, extending known results.

Contribution

It defines the $k$-${ m CM}_t$ class, provides characterizations, and generalizes existing results for Cohen-Macaulay and Buchsbaum complexes.

Findings

Characterizations of ${ m CM}_t$ complexes via homology vanishing.

Links of faces in $k$-${ m CM}_t$ complexes are $k$-${ m CM}_{t-1}$.

Skeletons of $k$-${ m CM}_t$ complexes retain $k$-${ m CM}_t$ properties.

Abstract

For a positive integer $k$ and a non-negative integer $t$ a class of simplicial complexes, to be denoted by $k$-${\rm CM}_t$, is introduced. This class generalizes two notions for simplicial complexes: being $k$-Cohen-Macaulay and $k$-Buchsbaum. In analogy with the Cohen-Macaulay and Buchsbaum complexes, we give some characterizations of ${\rm CM}_t(=$1-${\rm CM}_t)$ complexes, in terms of vanishing of some homologies of its links and, in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We show that a complex is $k$-${\rm CM}_t$ if and only if the links of its nonempty faces are $k$-${\rm CM}_{t-1}$. We prove that for an integer $s\le d$, the $(d-s-1)$-skeleton of a $(d-1)$-dimensional $k$-${\rm CM}_t$ complex is $(k+s)$-${\rm CM}_t$. This result generalizes Hibi's result for Cohen-Macaulay complexes and Miyazaki's result for Buchsbaum complexes.