Pseudomanifold
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a pseudomanifold is a special type of
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. It looks like a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
at most of its points, but it may contain singularities. For example, the
cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...
of solutions of z^2=x^2+y^2 forms a pseudomanifold. A pseudomanifold can be regarded as a
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
realisation of the general idea of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
with singularities. The concepts of
orientability In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ...
, 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.


Definition

A topological space ''X'' endowed with a
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle m ...
''K'' is an ''n''-dimensional pseudomanifold if the following conditions hold: # (''pure'') is the union of all ''n''-
simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
. # Every is a
face The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...
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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of ''n''-simplices such that the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
is an for all ''i'' = 0, ..., ''k''−1.


Implications of the definition

*Condition 2 means that ''X'' is a non-branching
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
. *Condition 3 means that ''X'' is a strongly connected simplicial complex. *If we require Condition 2 to hold only for in sequences of 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 satisfying Condition 2.


Decomposition

Strongly connected n-complexes can always be assembled from gluing just two of them at . However, in general, construction by gluing can lead to non-pseudomanifoldness (see Figure 2). 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). 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). * For n≥3, there are n-complexes that cannot be decomposed, even into pseudomanifold parts, only by cutting at singularities.


Related definitions

*A pseudomanifold is called ''normal'' if the link of each simplex with
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...
≥ 2 is a pseudomanifold.


Examples

*A pinched torus (see Figure 1) is an example of an orientable,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
(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 space In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact s ...
s of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s over triangulable
compact manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...
s are examples of pseudomanifolds. * Triangulable, compact,
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, 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 In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structure ...
formulation of
loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based direc ...
. * Combinatorial n-complexes defined by gluing two at a are not always n-pseudomanifolds. Gluing can induce non-pseudomanifoldness.


See also

*
Stratified space In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset ...


References

{{Reflist Topological spaces Manifolds