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 <math>z^2=x^2+y^2</math> forms a pseudomanifold.
thumb|{{center|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.<ref>{{Citation|first1=H.|last1=Seifert|first2=W.|last2=Threlfall|title=Textbook of Topology|publisher=Academic Press Inc.|year=1980|isbn=0-12-634850-2|url-access=registration|url=https://archive.org/details/seifertthrelfall0000seif}}</ref><ref>{{Citation|first=H.|last=Spanier|title=Algebraic Topology|publisher=McGraw-Hill Education|year=1966|isbn=0-07-059883-5}}</ref>
== Definition ==
A topological space ''X'' endowed with a triangulation ''K'' is an ''n''-dimensional pseudomanifold if the following conditions hold:<ref name="BRAS">{{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}}</ref>
# (''pure'') {{nowrap|1=''X'' = {{!}}''K''{{!}}}} is the union of all ''n''-simplices. # Every {{nowrap|1=(''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 {{nowrap|1=σ = σ<sub>0</sub>, σ<sub>1</sub>, ..., σ<sub>''k''</sub> = σ'}} such that the intersection {{nowrap|1=σ<sub>''i''</sub> ∩ σ<sub>''i''+1</sub>}} is an {{nowrap|1=(''n''−1)-simplex}} for all ''i'' = 0, ..., ''k''−1.
=== Implications of the definition ===
*Condition 2 means that ''X'' is a '''non-branching''' simplicial complex.<ref name="ANO">{{SpringerEOM|author=D. V. Anosov|title=Pseudo-manifold|accessdate=August 6, 2010}}</ref> *Condition 3 means that ''X'' is a '''strongly connected''' simplicial complex.<ref name="ANO"/> *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.<ref name="MRN">{{cite thesis|type=PhD |arxiv=1904.00306v1|author=F. Morando|title=Decomposition and Modeling in the Non-Manifold domain|pages=139–142}}</ref>
=== Decomposition ===
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). thumb|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 edgeNevertheless 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). thumb|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). thumb|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.<ref name="MRN" />
==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.<ref name="BRAS"/> (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.<ref name="ANO"/> (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.<ref name="ANO"/> * Triangulable, compact, connected, homology manifolds over '''Z''' are examples of pseudomanifolds.<ref name="ANO"/> * 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.<ref name="Baez Christensen Halford Tsang pp. 4627–4648">{{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 }}</ref> * 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.<ref name="MRN" />
== See also == *Stratified space
== References ==
{{Reflist}}
Category:Topological spaces Category:Manifolds