Local Flatness

In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds.

Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If we say N is locally flat at x if there is a neighborhood of x such that the topological pair is homeomorphic to the pair, with a standard inclusion of as a subspace of . That is, there exists a homeomorphism such that the image of coincides with .

The above definition assumes that, if M has a boundary, x is not a boundary point of M. If x is a point on the boundary of M then the definition is modified as follows. We say that N is locally flat at a boundary point x of M if there is a neighborhood of x such that the topological pair is homeomorphic to the pair, where is a standard half-space and is included as a standard subspace of its boundary. In more detail, we can set and .

We call N locally flat in M if N is locally flat at every point. Similarly, a map is called locally flat, even if it is not an embedding, if every x in N has a neighborhood U whose image is locally flat in M.

Local flatness of an embedding implies strong properties not shared by all embeddings. Brown (1962) proved that if d = n − 1, then N is collared; that is, it has a neighborhood which is homeomorphic to N × with N itself corresponding to N × 1/2 (if N is in the interior of M) or N × 0 (if N is in the boundary of M).