Pseudo-manifold

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 $z^2=x^2+y^2$ 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:

# (''pure'') 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

Strongly connected n-complexes can always be assembled from gluing just two of them at . However, in general, construction by gluing can lead to non-pseudomanifoldness (see Figure 2).
Nevertheless it is always possible to decompose a non-pseudomanifold surface into manifold parts cutting only at singular edges and vertices (see Figure 2 in blue). For some surfaces several non-equivalent options are possible (see Figure 3).

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 (see Figure 4).

* 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 (see Figure 1) is an example of an orientable, compact 2-dimensional pseudomanifold.
(Note that a pinched torus is not a normal pseudomanifold, since the link of a vertex is not connected.)
* Complex algebraic varieties (even with singularities) are examples of pseudomanifolds.
(Note that real algebraic varieties aren't always pseudomanifolds, since their singularities can be of codimension 1, take xy=0 for example.)
* 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.

See also

*Stratified space

References

{{Reflist

Topological spaces
Manifolds