A generalization of $k$-Cohen-Macaulay complexes

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.