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 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]
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, …, σk = σ' such that the intersectionσi ∩ σi+1 is an (n−1)-simplex for all i = 0, ..., k−1.
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]
DecompositionEdit
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 definitionsEdit
A pseudomanifold is called normal if the link of each simplex with codimension ≥ 2 is a pseudomanifold.
ExamplesEdit
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.)
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]
^Seifert, H.; Threlfall, W. (1980), Textbook of Topology, Academic Press Inc., ISBN0-12-634850-2
^Spanier, H. (1966), Algebraic Topology, McGraw-Hill Education, ISBN0-07-059883-5
^ abBrasselet, J. P. (1996). "Intersection of Algebraic Cycles". Journal of Mathematical Sciences. Springer New York. 82 (5): 3625–3632. doi:10.1007/bf02362566. S2CID 122992009.
^ abcF. 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.
August 26, 2023
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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 displaystyle z 2 x 2 y 2 forms a pseudomanifold Figure 1 A pinched torusA 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 Contents 1 Definition 1 1 Implications of the definition 1 2 Decomposition 2 Related definitions 3 Examples 4 See also 5 ReferencesDefinition EditA 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 gt 1 For every pair of n simplices s and s in K there is a sequence of n simplices s s0 s1 sk s such that the intersection si si 1 is an n 1 simplex for all i 0 k 1 Implications of the definition Edit 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 Edit 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 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 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 gt 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 EditA pseudomanifold is called normal if the link of each simplex with codimension 2 is a pseudomanifold Examples EditA 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 See also EditStratified spaceReferences Edit 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 a b 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 a b c d e D V Anosov 2001 1994 Pseudo manifold Encyclopedia of Mathematics EMS Press retrieved August 6 2010 a b c 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 Retrieved from https en wikipedia org w index php title Pseudomanifold amp oldid 1139280058, wikipedia, wiki, book, books, library,
