Local flatness

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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 x\in N, we say N is locally flat at x if there is a neighborhood U\subset M of x such that the topological pair (U,U\cap N) is homeomorphic to the pair ({\mathbb  {R}}^{n},{\mathbb  {R}}^{d}), with a standard inclusion of {\mathbb  {R}}^{d} as a subspace of {\mathbb  {R}}^{n}. That is, there exists a homeomorphism U\to R^{n} such that the image of U\cap N coincides with {\mathbb  {R}}^{d}.

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 U\subset M of x such that the topological pair (U,U\cap N) is homeomorphic to the pair ({\mathbb  {R}}_{+}^{n},{\mathbb  {R}}^{d}), where {\mathbb  {R}}_{+}^{n} is a standard half-space and {\mathbb  {R}}^{d} is included as a standard subspace of its boundary. In more detail, we can set {\mathbb  {R}}_{+}^{n}=\{y\in {\mathbb  {R}}^{n}\colon y_{n}\geq 0\} and {\mathbb  {R}}^{d}=\{y\in {\mathbb  {R}}^{n}\colon y_{{d+1}}=\cdots =y_{n}=0\}.

We call N locally flat in M if N is locally flat at every point. Similarly, a map \chi \colon N\to M is called locally flat, even if it is not an embedding, if every x in N has a neighborhood U whose image \chi (U) 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 × [0,1] 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).

See also

References

  • Brown, Morton (1962), Locally flat imbeddings of topological manifolds. Annals of Mathematics, Second series, Vol. 75 (1962), pp. 331-341.
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