Pseudomanifold

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.

Figure 1: 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]

(pure) X = |K| is the union of all n-simplices.
Every (n–1)-simplex 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 σ = σ₀, σ₁, …, σ_k = σ' such that the intersection σ_i ∩ σ_i+1 is an (n−1)-simplex for all i = 0, ..., k−1.

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]
If we require Condition 2 to hold only for (n−1)-simplexes in sequences of n-simplexes 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 n-simplexes satisfying Condition 2.[5]

Decomposition

Strongly connected n-complexes can always be assembled from n-simplexes gluing just two of them at (n−1)-simplexes. However, in general, construction by gluing can lead to non-pseudomanifoldness (see Figure 2).

Figure 2: Gluing a manifold along manifold edges (in green) may create non-pseudomanifold edges (in red). A decomposition is possible cutting (in blue) at a singular edge

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).

Figure 3: The non pseudomanifold surface on the left can be decomposed into an orientable manifold (central) or into a non-orientable one (on the right).

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).

Figure 4: Two 3-pseudomanifolds with singularities (in red) that cannot be broken into manifold parts only by cutting at singularities.

For n≥3, there are n-complexes that cannot be decomposed, even into pseudomanifold parts, only by cutting at singularities.[5]

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.[3]

(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.[4]

(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.[4]
Triangulable, compact, connected, homology manifolds over Z are examples of pseudomanifolds.[4]
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.[6]
Combinatorial n-complexes defined by gluing two n-simplexes at a (n-1)-face are not always n-pseudomanifolds. Gluing can induce non-pseudomanifoldness.[5]

References

Seifert, H.; Threlfall, W. (1980), Textbook of Topology, Academic Press Inc., ISBN 0-12-634850-2
Spanier, H. (1966), Algebraic Topology, McGraw-Hill Education, ISBN 0-07-059883-5
Brasselet, J. P. (1996). "Intersection of Algebraic Cycles". Journal of Mathematical Sciences. Springer New York. 82 (5): 3625–3632. doi:10.1007/bf02362566. S2CID 122992009.
D. V. Anosov (2001) [1994], "Pseudo-manifold", Encyclopedia of Mathematics, EMS Press, retrieved August 6, 2010
F. Morando. Decomposition and Modeling in the Non-Manifold domain (PhD). pp. 139–142. arXiv:1904.00306v1.
Baez, John C; Christensen, J Daniel; Halford, Thomas R; Tsang, David C (2002-08-22). "Spin foam models of Riemannian quantum gravity". Classical and Quantum Gravity. IOP Publishing. 19 (18): 4627–4648. arXiv:gr-qc/0202017. doi:10.1088/0264-9381/19/18/301. ISSN 0264-9381.

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[1] Seifert, 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

[BRAS-3] Brasselet, J. P. (1996). "Intersection of Algebraic Cycles". Journal of Mathematical Sciences. Springer New York. 82 (5): 3625–3632. doi:10.1007/bf02362566. S2CID 122992009.

[ANO-4] D. V. Anosov (2001) [1994], "Pseudo-manifold", Encyclopedia of Mathematics, EMS Press, retrieved August 6, 2010

[MRN-5] F. Morando. Decomposition and Modeling in the Non-Manifold domain (PhD). pp. 139–142. arXiv:1904.00306v1.

[Baez_Christensen_Halford_Tsang_pp._4627–4648-6] Baez, John C; Christensen, J Daniel; Halford, Thomas R; Tsang, David C (2002-08-22). "Spin foam models of Riemannian quantum gravity". Classical and Quantum Gravity. IOP Publishing. 19 (18): 4627–4648. arXiv:gr-qc/0202017. doi:10.1088/0264-9381/19/18/301. ISSN 0264-9381.