Pseudomanifold

A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities.

[1][2] A topological space X endowed with a triangulation K is an n-dimensional pseudomanifold if the following conditions hold:[3] Strongly connected n-complexes can always be assembled from n-simplexes gluing just two of them at (n−1)-simplexes.

Nevertheless it is always possible to decompose a non-pseudomanifold surface into manifold parts cutting only at singular edges and vertices (see Figure 2 in blue).

(Note that a pinched torus is not a normal pseudomanifold, since the link of a vertex is not connected.)

(Note that real algebraic varieties aren't always pseudomanifolds, since their singularities can be of codimension 1, take xy=0 for example.)

Figure 1: A pinched torus
Figure 2: Gluing a manifold along manifold edges (in green) may create non-pseudomanifold edges (in red). A decomposition is possible cutting (in blue) at a singular edge
Figure 3: The non pseudomanifold surface on the left can be decomposed into an orientable manifold (central) or into a non-orientable one (on the right).
Figure 4: Two 3-pseudomanifolds with singularities (in red) that cannot be broken into manifold parts only by cutting at singularities.