Pseudomanifold Explained

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:^[1]

1. (pure) is the union of all n-simplices.
2. Every is a face of exactly one or 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 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.^[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.^[1]

(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.^[3]
• Combinatorial n-complexes defined by gluing two at a are not always n-pseudomanifolds. Gluing can induce non-pseudomanifoldness.

See also

• Stratified space

Notes and References

1. Brasselet. J. P.. 1996 . Intersection of Algebraic Cycles . Journal of Mathematical Sciences. Springer New York. 82. 5. 3625–3632. 10.1007/bf02362566. 122992009.
2. PhD . 1904.00306v1. F. Morando. Decomposition and Modeling in the Non-Manifold domain. 139–142.
3. Baez . John C . Christensen . J Daniel . Halford . Thomas R . Tsang . David C . Spin foam models of Riemannian quantum gravity . Classical and Quantum Gravity . IOP Publishing . 19 . 18 . 2002-08-22 . 0264-9381 . 10.1088/0264-9381/19/18/301 . 4627–4648. gr-qc/0202017 .