Simplicial manifold

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 (following the laws of ordinary geometry, or more generally a flat Pseudo-Riemannian space) in a given local neighborhood, though it can be greatly more complicated overall. The combination of these concepts gives us two useful definitions.

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 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 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 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 simulations.

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 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 as its space of n-simplices. More generally, G can be a Lie groupoid.