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 $x\in N,$ 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).

References

Brown, Morton (1962), Locally flat imbeddings of topological manifolds. Annals of Mathematics, Second series, Vol. 75 (1962), pp. 331-341.
Mazur, Barry. On embeddings of spheres. Bulletin of the American Mathematical Society, Vol. 65 (1959), no. 2, pp. 59-65. http://projecteuclid.org/euclid.bams/1183523034.

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Local flatness

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References