Bullough–Dodd model: Difference between revisions

Revision as of 19:10, 20 April 2013

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 $(ℝ^{n}, ℝ^{d})$ , with a standard inclusion of $ℝ^{d}$ as a subspace of $ℝ^{n}$ . That is, there exists a homeomorphism $U \to R^{n}$ such that the image of $U \cap N$ coincides with $ℝ^{d}$ .

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 $(ℝ_{+}^{n}, ℝ^{d})$ , where $ℝ_{+}^{n}$ is a standard half-space and $ℝ^{d}$ is included as a standard subspace of its boundary. In more detail, we can set $ℝ_{+}^{n} = {y \in ℝ^{n} : y_{n} \geq 0}$ and $ℝ^{d} = {y \in ℝ^{n} : y_{d + 1} = \dots = y_{n} = 0}$ .

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

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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 (mathematics)|category]] of topological manifolds, locally flat submanifolds play a role similar to that of [[Submanifold#Embedded submanifolds|embedded submanifolds]] in the category of [[smooth manifolds]].
+Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' &lt; ''n'').  If <math>x \in N,</math> we say ''N'' is '''locally flat''' at ''x'' if there is a neighborhood <math> U \subset M</math> of ''x'' such that the [[topological pair]] <math>(U, U\cap N)</math> is [[homeomorphic]] to the pair <math>(\mathbb{R}^n,\mathbb{R}^d)</math>, with a standard inclusion of <math>\mathbb{R}^d</math> as a subspace of <math>\mathbb{R}^n</math>. That is, there exists a homeomorphism <math>U\to R^n</math> such that the [[image (mathematics)|image]] of <math>U\cap N</math> coincides with <math>\mathbb{R}^d</math>.
+The above definition assumes that, if ''M'' has a [[Boundary (topology)|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 <math>U\subset M</math> of ''x'' such that the topological pair <math>(U, U\cap N)</math> is homeomorphic to the pair <math>(\mathbb{R}^n_+,\mathbb{R}^d)</math>, where <math>\mathbb{R}^n_+</math> is a standard [[Half-space (geometry)|half-space]] and <math>\mathbb{R}^d</math> is included as a standard subspace of its boundary. In more detail, we can set
+<math>\mathbb{R}^n_+ = \{y \in \mathbb{R}^n\colon y_n \ge 0\}</math> and <math>\mathbb{R}^d = \{y \in \mathbb{R}^n\colon y_{d+1}=\cdots=y_n=0\}</math>.
+We call ''N'' '''locally flat''' in ''M'' if ''N'' is locally flat at every point. Similarly, a map <math>\chi\colon N\to M</math> is called '''locally flat''', even if it is not an embedding, if every ''x'' in ''N'' has a neighborhood ''U'' whose image <math>\chi(U)</math> 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'' &minus; 1, then ''N'' is [[collared]]; that is, it has a neighborhood which is homeomorphic to ''N'' &times; [0,1] with ''N'' itself corresponding to ''N'' &times; 1/2 (if ''N'' is in the interior of ''M'') or ''N'' &times; 0 (if ''N'' is in the boundary of ''M'').
+==See also==
+*[[Neat submanifold]]
+==References==
+* Brown, Morton (1962), Locally flat imbeddings of topological manifolds. ''Annals of Mathematics'', Second series, Vol. 75 (1962), pp. 331-341.
+[[Category:Topology]]
+[[Category:Geometric topology]]
+{{topology-stub}}

Bullough–Dodd model: Difference between revisions

Revision as of 19:10, 20 April 2013

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