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| {{Expert-subject|Mathematics|date=February 2009}}
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| In [[mathematics]], the term '''simplicial manifold''' commonly refers to either of two different types of objects, which combine attributes of a [[simplex]] with those of a [[manifold]]. Briefly; a simplex is a generalization of the concept of a [[triangle]] into forms with more, or fewer, than two dimensions. Accordingly, a 3-simplex is the figure known as a [[tetrahedron]]. A manifold is simply a space which appears to be [[Euclidean space|Euclidean]] (following the laws of ordinary geometry, or more generally a flat [[Pseudo-Riemannian manifold|Pseudo-Riemannian]] space) in a given [[Neighborhood (mathematics)|local neighborhood]], though it can be greatly more complicated overall. The combination of these concepts gives us two useful definitions.
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| ==A manifold made out of simplices==
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| A simplicial manifold is a [[simplicial complex]] for which the [[geometric realization]] is [[homeomorphic]] to a [[topological manifold]]. This can mean simply that a [[neighborhood (mathematics)|neighborhood]] of each vertex (i.e. the set of [[simplices]] that contain that point as a vertex) is [[homeomorphic]] to a ''n''-dimensional [[ball (mathematics)|ball]].
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| A manifold made from simplices can be locally flat, or can approximate a smooth curve, just as a large [[geodesic dome]] appears relatively flat over small areas, and approximates a [[Sphere|hemisphere]] over its full extent. One can generalize this concept to more dimensions and other kinds of curved surfaces which makes it useful in various kinds of [[Computer simulation|simulations]].
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| This notion of simplicial manifold is important in [[Regge calculus]] and [[Causal dynamical triangulation]]s as a way to discretize [[spacetime]] by [[Triangulation (geometry)|triangulating]] it. A simplicial manifold with a metric is called a [[piecewise linear space]].
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| ==A simplicial object built from manifolds==
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| A simplicial manifold is also a [[simplicial object]] in the [[category (mathematics)|category]] of [[manifold]]s. This is a special case of a [[simplicial space]] in which, for each ''n'', the space of ''n''-simplices is a manifold.
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| For example, if ''G'' is a [[Lie group]], then the [[nerve (category theory)|simplicial nerve]] of ''G'' has the manifold <math>G^n</math> as its space of ''n''-simplices. More generally, ''G'' can be a [[Lie groupoid]].
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| [[Category:Structures on manifolds]]
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| [[Category:Simplicial sets]] | |
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| {{Geometry-stub}}
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