Surfacebranched

Surfacebranched is a term used in mathematics to describe phenomena where branching occurs along a surface, i.e., along a two-dimensional subset of a higher-dimensional space. It is not a single universally defined concept, but a descriptive phrase used across different fields to indicate that the branching locus has codimension one rather than being a set of isolated points or a lower-dimensional subset.

In topology and geometry, surfacebranched often refers to maps between manifolds whose critical set—the set where the map fails to be a local diffeomorphism—forms a smoothly embedded hypersurface. In such contexts, the branching is described as occurring along a surface, and local models typically reflect folding or projection in directions normal to that surface. The precise meaning depends on the category (smooth, piecewise-linear, or analytic) and on additional structure such as orientation or regularity assumptions.

Related notions include branched coverings, where the branching locus is usually codimension two (points in two-dimensional domains, links in three-dimensional ones). Another closely related concept is branched surfaces in 3-manifold theory, which are two-dimensional objects used to study foliations and laminations. Surfacebranched descriptions can be helpful for emphasizing the geometry of where and how branching occurs, without committing to a single formal framework.

Because surfacebranched is an informal descriptor, exact definitions are context-dependent and rely on the chosen local models and regularity conditions. It appears in discussions of folding maps, hypersurface branching in smooth or PL categories, and studies of branched structures in geometry and topology.