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 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
- Thom spaces of vector bundles over triangulable compact manifolds are examples of pseudomanifolds.
- Triangulable, compact, connected, homology manifolds over Z are examples of pseudomanifolds.
