# Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

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?

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Some references can be found at this question mathoverflow.net/questions/71400/… –  Mark Grant Jul 16 '13 at 6:30
In fact, Mark's reference completely addresses this question. That question asks about finite simplicial complexes, but notice that a finite simplicial complex on $N$ vertices can always be realized as a subcomplex of the boundary of the $N-1$ dimensional simplex. –  David Speyer Jul 16 '13 at 22:50