PARTITION AND COHEN-MACAULAY EXTENDERS

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Problem: Reconstructed Conjecture 7.2: for every finite pure simplicial complex $\Delta$ of dimension $d$, if

for every field $\mathbb{k}$, then there exists a pure $d$-dimensional simplicial complex $\Gamma\supseteq\Delta$ such that $\Gamma$ is shellable and the relative complex $(\Gamma,\Delta)$ is shellable.

Result: The conjecture is false.

Let $\Delta$ be any finite triangulation of the Poincaré homology $3$-sphere. Such a triangulation exists. Since $\Delta$ is an integral homology $3$-sphere and every vertex link is a $2$-sphere, Reisner’s criterion implies that $\mathbb{k}[\Delta]$ is Cohen–Macaulay over every field $\mathbb{k}$. Hence

so in particular $\operatorname{depth}\mathbb{k}[\Delta]\ge 3=\dim\mathbb{k}[\Delta]-1$.

But $\pi_1(\Delta)$ is the binary icosahedral group, hence nontrivial.

Suppose, for contradiction, that $\Delta$ has a shelling extender $\Gamma$. Then $(\Gamma,\Delta)$ is shellable. In a relative shelling of a $3$-dimensional pair, each added tetrahedron is attached along either a $2$-ball or a $2$-sphere in its boundary. These attaching spaces are connected and simply connected. By van Kampen, attaching such tetrahedra does not change the fundamental group. Therefore

On the other hand, every pure shellable $3$-complex is homotopy equivalent to a wedge of $3$-spheres, hence is simply connected. Thus $\pi_1(\Gamma)=1$, a contradiction.

Therefore no shelling extender exists for this $\Delta$, despite the depth hypothesis holding over all fields.

Citation: The conjecture is from Doolittle–Goeckner–Lazar, “Partition and Cohen-Macaulay Extenders,” Conjecture 7.2. Standard ingredients used: Reisner’s Cohen–Macaulay criterion and the classical fact that pure shellable complexes have the homotopy type of wedges of top-dimensional spheres.