Simplicial manifold

In physics, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of Regge calculus. These objects combine attributes of a simplex with those of a manifold. There is no standard usage of this term in mathematics, and so the concept can refer to a triangulation in topology, or a piecewise linear manifold, or one of several different functors from either the category of sets or the category of simplicial sets to the category of manifolds.

Not to be confused with Symplectic manifold.

A manifold made out of simplices

A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold. This is essentially the concept of a triangulation in topology. This can mean simply that a neighborhood of each vertex (i.e. the set of simplices that contain that point as a vertex) is homeomorphic to a n-dimensional ball.

A simplicial object built from manifolds

A simplicial manifold is also a simplicial object in the category of manifolds. This is a special case of a simplicial space in which, for each n, the space of n-simplices is a manifold.

For example, if G is a Lie group, then the simplicial nerve of G has the manifold ${\displaystyle G^{n}}$ as its space of n-simplices. More generally, G can be a Lie groupoid.