$3$-pseudomanifolds with preassigned links

Abstract:

A 3-pseudomanifold is a finite connected simplicial 3-complex $\mathcal {K}$ such that every triangle in $\mathcal {K}$ belongs to precisely two 3-simplices of $\mathcal {K}$, the link of every edge in $\mathcal {K}$ is a circuit, and the link of every vertex in $\mathcal {K}$ is a closed 2-manifold. It is proved that for every finite set $\sum$ of closed 2-manifolds, there exists a 3-pseudomanifold $\mathcal {K}$ such that the link of every vertex in $\mathcal {K}$ is homeomorphic to some $S \in \sum$, and every $S \in \sum$ is homeomorphic to the link of some vertex in $\mathcal {K}$.