# Cohen-Macaulay versus shellable simplicial complexes

There are some discussion of shellable simplicial complexes here Testing simplicial complexes for shellability. My question is the following:

Assume that $\Delta$ is a pure simplicial complex on a finite vertex set $V$. Assume that $\Delta$ is Cohen-Macaulay, i.e., it has at most top-dimensional homology. (There are examples of CM but non-shellable simplicial complexes). Moreover, assume that the symmetric group $S_V$ acts transitively on $\Delta$. Could this possibly imply that the simplicial complex $\Delta$ is shellable?

• What do you mean by "the symmetric group $S_V$ acts transitively on $\Delta$"? Do you mean that the automorphism group of $\Delta$ acts transitively on $V$? – Richard Stanley Dec 3 '13 at 18:09
• Yes, the automorphism group of $\Delta$ does act transitively on $\Delta$. But maybe something stronger than that, but I'm not sure how to phrase it. An example that I have in mind is the matching complex of a complete graph. – Thanh Vu Dec 3 '13 at 18:54
• Take care that you want also links of faces to have at most top homology for Cohen-Macaulayness. – Christian Stump Dec 3 '13 at 19:33
• If $S_V$ acts transitively on $\Delta$, then I suppose $\Delta$ is some skeleton of a simplex, hence shellable. But that's almost surely not what you have in mind. – Russ Woodroofe Dec 5 '13 at 1:16

The Dissertation of Frank Lutz (http://page.math.tu-berlin.de/~lutz/dissertation.ps‎) has examples of Cohen-Macaulay vertex-transitive simplicial complexes that are not shellable. In particular, there is such a triangulation of Poincaré dodecahedral space $X$ with 17 vertices. This triangulation is Cohen-Macaulay since $X$ is a topological 3-manifold with the same homology groups as a 3-sphere. It is not shellable because it is not simply-connected.