# 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 ${\displaystyle z^{2}=x^{2}+y^{2}}$ 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, …, σk = σ' such that the intersection σi ∩ σi+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 ${\displaystyle \geq 2}$ is a pseudomanifold.

## Examples

(Note that a pinched torus is not a normal psedomanifold, since the link of a vertex is not connected.)

## 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. ^ a b 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.