Simplicial manifold

In mathematics, the term simplicial manifold commonly refers to either of two different types of objects.

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 means 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.

This notion of simplicial manifold is important in Regge calculus and causal dynamical triangulations as a way to discretize spacetime by triangulating it. A simplicial manifold with a metric is called a piecewise linear space.

A simplicial object built from manifolds

A simplicial manifold is 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 $G^{n}$ as its space of n-simplices. More generally, G can be a Lie groupoid.