Obstructions to Shellability

We consider a simplicial complex generaliztion of a result of Billera and Meyers that every nonshellable poset contains the smallest nonshellable poset as an induced subposet. We prove that every nonshellable 2-dimensional simplicial complex contains a nonshellable induced subcomplex with less than 8 vertices. We also establish CL-shellability of interval orders and as a consequence obtain a formula for the Betti numbers of any interval order. A recent result of Billera and Meyers [BM] implies that every nonshellable poset contains as an induced subposet the 4 element poset Q consisting of two disjoint 2 element chains. (Throughout this paper shellability refers to the general notion of nonpure shellability introduced in [BW2].) Note that Q is the nonshellable poset with the fewest number of elements. Of course a shellable poset can also contain Q; eg., the lattice of subsets of a 3 element set. So the condition of not containing Q as an induced subposet is only sufficient for shellability; it does not characterize shellability. It is however a well-known characterization of a class of posets called interval orders and the question of whether all interval orders are shellable is what Billera and Meyers were considering in the first place. In this note we suggest a way to generalize the poset result to general simplicial complexes. We also give a simple proof of the poset result and prove the stronger result that any poset that does not contain Q as an induced subposet is CL-shellable. This yields a recursive formula for the Betti numbers of the poset. We assume familiarity with the general theory of shellability [BW2] [BW3]. All notation and terminology used here is defined in [BW2 ]and [BW3]. The most simple minded conjecture one could make is that every nonshellable simplicial complex contains the induced subcomplex consisting of edges {a, b} and {c, d}, where a, b, c, d are distinct vertices. A simple counterexample is given by the 5 vertex simplicial complex consisting of facets {a, b, c}, {c, d, e}, {a, d}. Indeed the situation for simplicial complexes turns out to be much more complicated than it is for posets. The most natural thing to do next is to look for other “obstructions” to simplicial complex shellability. Is there a finite list? Below we see that the answer is no. Define Research supported in part by NSF grant DMS 9311805 at the University of Miami and by NSF grant DMS-9022140 at MSRI.