pseudomanifold

A topological space X endowed with a triangulation K is an n-dimensional pseudomanifold if the following conditions hold:{{cite journal |last1=Brasselet|first1=J. P.|year=1996 |title=Intersection of Algebraic Cycles |journal= Journal of Mathematical Sciences|publisher=Springer New York|volume= 82|issue= 5|pages=3625–3632|doi=10.1007/bf02362566|s2cid=122992009}}

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

• Condition 2 means that X is a non-branching simplicial complex.{{SpringerEOM|author=D. V. Anosov|title=Pseudo-manifold|accessdate=August 6, 2010}}
• Condition 3 means that X is a strongly connected simplicial complex.
• If we require Condition 2 to hold only for {{nowrap|1=(n−1)-simplexes}} in sequences of {{nowrap|1=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 {{nowrap|1=n-simplexes}} satisfying Condition 2.{{cite thesis|type=PhD |arxiv=1904.00306v1|author=F. Morando|title=Decomposition and Modeling in the Non-Manifold domain|pages=139–142}}

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

File:Nonpmnonclok3.svgNevertheless 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).

File:M3foldok.svg

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).

File:Pinchedpieaok.svg

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

• A pseudomanifold is called normal if the link of each simplex with codimension ≥ 2 is a pseudomanifold.

• 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.{{cite journal | last1=Baez | first1=John C | last2=Christensen | first2=J Daniel | last3=Halford | first3=Thomas R | last4=Tsang | first4=David C | title=Spin foam models of Riemannian quantum gravity | journal=Classical and Quantum Gravity | publisher=IOP Publishing | volume=19 | issue=18 | date=2002-08-22 | issn=0264-9381 | doi=10.1088/0264-9381/19/18/301 | pages=4627–4648| arxiv=gr-qc/0202017 | bibcode=2002CQGra..19.4627B }}
• Combinatorial n-complexes defined by gluing two {{nowrap|1=n-simplexes}} at a {{nowrap|1=(n-1)-face}} are not always n-pseudomanifolds. Gluing can induce non-pseudomanifoldness.

• Stratified space

{{Reflist}}

Category:Topological spaces

Category:Manifolds