manifoldlike

Manifoldlike is a term used in topology and geometric topology to describe spaces that resemble manifolds in some precise sense. The usage is varied, and there is no single universally accepted definition. In many contexts, a space is called manifoldlike if it is locally modeled on manifolds, meaning that every point has a neighborhood homeomorphic to an open set in some topological manifold. This local modeling mirrors the defining property of manifolds and yields many similar local invariants, such as local dimension and local homology.

In other contexts, particularly in shape theory and the study of continua, manifoldlike refers to a global property: a space is manifoldlike if its large-scale or “shape” is the same as that of a manifold. A compact metric space X may be called manifoldlike if it has the same shape as a manifold, i.e., there exists a manifold M such that X is shape equivalent to M. This allows non-manifold spaces to be analyzed via manifold techniques.

Manifoldlike spaces are contrasted with genuine manifolds, which satisfy strict axioms such as being locally Euclidean, second countable, and locally connected. The term is also used informally to describe spaces that are close to manifolds in other senses, such as stratified spaces or orbifolds that possess manifold-like strata.

See also: manifold, topological manifold, shape theory, continuum, stratified space.