Pseudomanifold

In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of formula_1 forms a pseudomanifold.
A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities. The concepts of orientability, orientation and degree of a mapping make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.
Definition.
A topological space "X" endowed with a triangulation "K" is an "n"-dimensional pseudomanifold if the following conditions hold:
Decomposition.
Strongly connected n-complexes can always assembled from gluing just two of them at . However, in general, construction by gluing can lead to non-pseudomanifoldness (see Figure 2). 
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). For some surfaces several non-equivalent options are possible (see Figure 3).
On the other hand, in higher dimension, for n>2, the situation becomes rather tricky. 
Examples.
(Note that real algebraic varieties aren't always pseudomanifolds, since their singularities can be of codimension 1, take xy=0 for example.) 