Abstrakt
We say that a pure simplicial complex K of dimension d satisfies the removal-collapsibility condition if K is either empty or K becomes collapsible after removing (beta) over tilde (d)(K; Z(2)) facets, where (beta) over tilde (d)(K; Z(2)) denotes the dth reduced Betti number. In this paper, we show that if the link of each face of a pure simplicial complex K (including the link of the empty face which is the whole K) satisfies the removal-collapsibility condition, then the second barycentric subdivision of K is vertex decomposable and in particular shellable.
This is a higher-dimensional generalization of a result of Hachimori, who proved that if the link of each vertex of a pure 2-dimensional simplicial complex K is connected and K becomes simplicially collapsible after removing (chi) over tilde (K) facets, where (chi) over tilde (K) denotes the reduced Euler characteristic, then the second barycentric subdivision of 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.