Pseudomanifold

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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 forms a pseudomanifold.

Contents

Figure 1: A pinched torus Pinched torus.jpg
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]

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

Implications of the definition

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 Nonpmnonclok3.svg
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). M3foldok.svg
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.

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

Examples

(Note that a pinched torus is not a normal pseudomanifold, since the link of a vertex is not connected.)

(Note that real algebraic varieties aren't always pseudomanifolds, since their singularities can be of codimension 1, take xy=0 for example.)

See also

References

  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
  3. 1 2 Brasselet, J. P. (1996). "Intersection of Algebraic Cycles". Journal of Mathematical Sciences. 82 (5). Springer New York: 3625–3632. doi:10.1007/bf02362566. S2CID   122992009.
  4. 1 2 3 4 5 D. V. Anosov (2001) [1994], "Pseudo-manifold", Encyclopedia of Mathematics , EMS Press , retrieved August 6, 2010
  5. 1 2 3 F. Morando. Decomposition and Modeling in the Non-Manifold domain (PhD). pp. 139–142. arXiv: 1904.00306v1 .
  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. 19 (18). IOP Publishing: 4627–4648. arXiv: gr-qc/0202017 . Bibcode:2002CQGra..19.4627B. doi:10.1088/0264-9381/19/18/301. ISSN   0264-9381.