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

$$\operatorname{depth}\mathbb{k}[\Delta]\ge \dim \mathbb{k}[\Delta]-1=d$$

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

$$\operatorname{depth}\mathbb{k}[\Delta]=\dim\mathbb{k}[\Delta]=4,$$

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

$$\pi_1(\Gamma)\cong \pi_1(\Delta)\neq 1.$$

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.