The topological structure of 3-pseudomanifolds

A 3-pseudomanifold (briefly 3-pm) is a finite connected simplicial 3-complex in which the link of every vertex is a closed 2-manifold. Such a link issingular if it is not a sphere. It is proved that for a preassigned list Σ of closed 2-manifolds (other than spheres), there is a 3-pm in which the list of singular links is precisely Σ, iff the number of the non-orientable members in Σ with odd genus is even. Close relationship is found between 3-pms and 3-manifolds with boundary. This yields a simple proof for the 2-dimensional case of Pontrjagin-Thom’s theorem (i.e., necessary and sufficient condition for a 2-manifold to bound a 3-manifold). The concept of a 3-pm is generalized to higher dimensions.