Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear inequalities.)

Is there a criterion (or a set of criteria) to tell whether $K$ is homeomorphic to a manifold? (Purity? Shellability? Something else?)

References?