Shellings and sheddings induced by collapses

We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$ satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty or ${\mathbf K}$ becomes collapsible after removing $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$ facets, where $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$ denotes the $d$th reduced Betti number. In this paper, we show that if the link of each face of a pure simplicial complex ${\mathbf K}$ (including the link of the empty face which is the whole ${\mathbf K}$) satisfy the removal-collapsibility condition, then the second barycentric subdivision of ${\mathbf K}$ is vertex decomposable and in particular shellable. This is a higher dimensional generalization of a result of Hachimori, who proved that that if the link of each vertex of a pure 2-dimensional simplicial complex ${\mathbf K}$ is connected, and ${\mathbf K}$ becomes simplicially collapsible after removing $\tilde{\chi}({\mathbf K})$ facets, where $\tilde \chi ({\mathbf K})$ denotes the reduced Euler characteristic, then the second barycentric subdivision of ${\mathbf K}$ is shellable. For the proof, we introduce a new variant of decomposability of a simplicial complex, stronger than vertex decomposability, which we call star decomposability. This notion may be of independent interest.