Embedding

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An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print.

You may view the version used for publication as of 12:03, 16 May 2013 and the changes since publication.

The user responsible for this page is Ulrich Koschorke. No other user may edit this page at present.

Contents

1 Definition

We work in a fixed category CAT of topological, piecewise linear, C^r-differentiable (1 \leq r \leq \infty ) or real analytic manifolds (second countable, Hausdorff, without boundary) and maps between them.

Let f : M^m \rightarrow N^n be such a map between manifolds of the indicated dimensions 1 \leq m < n.

Definition 1.1. We call f an embedding (and we write f : M \hookrightarrow N) if f is an immersion which maps M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_HmPMuq homeomorphically onto its image.

It follows that an embedding cannot have selfintersections. But even an injective immersion need not be an embedding; e. g. the figure six 6 is the image of a smooth immersion but not of an embedding. Note that in the topological and piecewise linear categories, CAT = TOP or PL, our definition yields locally flat embeddings. In these categories there are other concepts of embeddings - e.g. wild embeddings - which are not locally flat: the condition of local flatness is implied by our definition of immersion. Embeddings (and immersions) into familiar target manifolds such as \R^n may help to visualize abstractly defined manifolds. E. g. all smooth surfaces can be immersed into \R^3; but non-orientable surfaces (such as the projective plane and the Klein bottle) allow no embeddings into \R^3.

2 Existence of embeddings

Theorem 2.1 [Penrose&Whitehead&Zeeman1961]. For every compact m--dimensional PL-manifold M there exists a PL--embedding M \hookrightarrow \R^{2m}.

Remark 2.2. For a good exposition of Theorem 2.1 see also [Rourke&Sanderson1972a, p. 63].

Theorem 2.3 [Whitney1944]. For every closed m--dimensional C^{\infty}--manifold M there exists a C^{\infty}--embedding M \hookrightarrow \R^{2m}.

Remark 2.4. For more modern expositions see also [Adachi1993, p. 67ff] and [Prasolov2007, 22.1].

Similar existence results for embeddings M^m \hookrightarrow \R^N are valid also in the categories of real analytic maps and of isometrics (Nash) when N \gg 2m is sufficiently high.

3 Classification

In order to get a survey of all ``essentially distinct´´ embeddings f : M \hookrightarrow N it is meaningful to introduce equivalence relations such as (ambient) isotopy, concordance, bordism etc., and to aim at classifying embeddings accordingly. The difficulty of this task depends heavily on the choices of M and N and especially their dimensions: for more information please see the page on high codimension embeddings. Already for the most basic choices of M and N this may turn out to be a very difficult task. E.g. in the theory of knots (or links) where M is a 1-sphere (or a finite union of 1-spheres), and N = \R^{3} the multitude of possible knotting and linking phenomena is just overwhelming. Even classifying links up to the very crude equivalence relation `link homotopy´ is very far from having been achieved yet.

4 References

5 External links

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