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 z^2=x^2+y^2 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:

Implications of the definition

Decomposition

Strongly connected n-complexes can always be assembled from n-simplexes gluing just two of them at (n−1)-simplexes. 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.

Related definitions

Examples

(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.)