Pseudomanifold

A pseudomanifold is a special type of topological space. It looks like a manifold at most of the points, but may contain singularities. For example, the cone of solutions of $z^{2}=x^{2}+y^{2}$ forms a pseudomanifold.

A pinched torus

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.^[1][2]

Definition

A topological space X endowed with a triangulation K is an n-dimensional pseudomanifold if the following conditions hold:^[3]

1. (pure) X = |K| is the union of all n-simplices.
2. Every (n – 1)-simplex is a face of exactly two n-simplices for n > 1.
3. For every pair of n-simplices σ and σ' in K, there is a sequence of n-simplices σ = σ₀, σ₁, …, σ_k = σ' such that the intersection σ_i ∩ σ_i+1 is an (n − 1)-simplex for all i.

Implications of the definition

• Condition 2 means that X is a non-branching simplicial complex.^[4]
• Condition 3 means that X is a strongly connected simplicial complex.^[4]

Related definitions

• A pseudomanifold is called normal if link of each simplex with codimension $\geq 2$ is a pseudomanifold.

Examples

• A pinched torus (see figure) is an example of an orientable, compact 2-dimensional pseudomanifold.^[3]
• Complex algebraic varieties (even with singularities) are examples of pseudomanifolds.^[4]
• Thom spaces of vector bundles over triangulable compact manifolds are examples of pseudomanifolds.^[4]
• Triangulable, compact, connected, homology manifolds over Z are examples of pseudomanifolds.^[4]

References

1. Steifert, H.; Threlfall, W. (1980), Textbook of Topology, Academic Press Inc., ISBN 0-12-634850-2
2. Spanier, H. (1966), Algebraic Topology, McGraw-Hill Education, ISBN 0-07-059883-5
3. Brasselet, J. P. (1996). "Intersection of Algebraic Cycles". Journal of Mathematical Sciences. Springer New York. 82 (5): 3625 − 3632. doi:10.1007/bf02362566.
4. D. V. Anosov. "Pseudo-manifold". Retrieved August 6, 2010.