Pseudomanifold
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 
forms a pseudomanifold.
A pseudomanifold can be considered 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]
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Definition
A topological space X endowed with a triangulation K is an n-dimensional pseudomanifold if the following conditions hold:[3]
- X = |K| is the union of all n-simplices.
- Every (n – 1)-simplex 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 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]
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
- ^ Steifert, H.; Threlfall, W. (1980), Textbook of Topology, Academic Press Inc., ISBN 0126348502
- ^ Spanier, H. (1966), Algebraic Topology, McGraw-Hill Education, ISBN 0070598835
- ^ a b Brasselet, J. P. (1996). "Intersection of Algebraic Cycles". Journal of Mathematical Sciences (Springer New York) 82 (5): 3625 − 3632. http://www.springerlink.com/content/ju28j2wqm174hx10.
- ^ a b c d e D. V. Anosov. "Pseudo-manifold". http://eom.springer.de/p/p075720.htm. Retrieved August 6, 2010.