nonmanifold

Nonmanifold is a term used in topology and geometry to describe spaces that fail to be manifolds. A manifold is a space in which every point has a neighborhood that is topologically equivalent to Euclidean space of some dimension. When a space includes points whose neighborhoods do not resemble any Euclidean space, or neighborhoods that resemble the boundary of a space rather than a full space, it is called nonmanifold. In those contexts, the distinction between manifolds and nonmanifolds helps formalize local structure.

In polyhedral geometry, computer graphics, and related fields, nonmanifold structures arise when multiple patches meet at a single edge or vertex in incompatible ways. A classic example is an edge that is shared by more than two faces. At such an edge, a small neighborhood may look like several sheets glued together rather than a single surface, so the space is not locally Euclidean. A vertex where the incident faces form an inconsistent fan, or where multiple distinct sheets meet, can also be nonmanifold. In three-dimensional meshes, an edge that is incident to more than two tetrahedra or polygons is a common nonmanifold feature; a vertex whose surrounding faces do not form a disk-like neighborhood is another.

Formally, in a d-dimensional simplicial complex intended to model a d-manifold, the link of every k-simplex should be a sphere of dimension d−k−1 (or a disk for boundary components). If this condition fails for any simplex, the point or region is nonmanifold.

Nonmanifold features can complicate geometric processing and simulation, and are often detected by examining incidence counts and neighborhood topology. Common repair methods include splitting edges or duplicating vertices to restore manifoldness where needed.