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 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:- is the union of all n-simplices.
- Every is a face of exactly one or two n-simplices for n > 1.
- For every pair of n-simplices σ and σ' in K, there is a sequence of n-simplices such that the intersection is an for all i = 0,..., k−1.
Implications of the definition
- Condition 2 means that X is a non-branching simplicial complex.
- Condition 3 means that X is a strongly connected simplicial complex.
- If we require Condition 2 to hold only for in sequences of in Condition 3, we obtain an equivalent definition only for n=2. For n≥3 there are examples of combinatorial non-pseudomanifolds that are strongly connected through sequences of satisfying Condition 2.
Decomposition
Nevertheless it is always possible to decompose a non-pseudomanifold surface into manifold parts cutting only at singular edges and vertices. For some surfaces several non-equivalent options are possible.
On the other hand, in higher dimension, for n>2, the situation becomes rather tricky.
- In general, for n≥3, n-pseudomanifolds cannot be decomposed into manifold parts only by cutting at singularities.
- For n≥3, there are n-complexes that cannot be decomposed, even into pseudomanifold parts, only by cutting at singularities.
Related definitions
- A pseudomanifold is called normal if the link of each simplex with codimension ≥ 2 is a pseudomanifold.
Examples
- A pinched torus is an example of an orientable, compact 2-dimensional pseudomanifold.
- Complex algebraic varieties are examples of pseudomanifolds.
- Thom spaces of vector bundles over triangulable compact manifolds are examples of pseudomanifolds.
- Triangulable, compact, connected, homology manifolds over Z are examples of pseudomanifolds.
- Complexes obtained gluing two 4-simplices at a common tetrahedron are a proper superset of 4-pseudomanifolds used in spin foam formulation of loop quantum gravity.
- Combinatorial n-complexes defined by gluing two at a are not always n-pseudomanifolds. Gluing can induce non-pseudomanifoldness.