Embeddings in Euclidean space: an introduction to their classification
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1 Introduction and restrictions
According to Zeeman, the classical problems of topology are the following.
-
When are two given spaces homeomorphic?
-
When does a given space embed into
?
-
When are two given embeddings isotopic?
This article concerns the Knotting Problem.
We recall all known {\it complete readily calculable} isotopy
classification results for of {\it closed connected}
manifolds into Euclidean spaces.
(Thus for 1- and 2- dimensional manifolds we only indicate that such results
are not available.)
We present constructions of embeddings and invariants.
See knot theory,
%\linebreak
and open
problems below.
Later we hope to add information for manifolds with boundary.
For more information see [Sk08].
\bigskip {\bf Notation and conventions.}
For a manifold let
or
denote the set of smooth
or PL embeddings
up to smooth or PL isotopy.
%The sign
or
between embeddings means that they are PL or
%smoothly isotopic.
If a category is omitted, then the result holds (or a definition or a
construction is given) in both categories.
All manifolds in this note are tacitly assumed to be compact.
Let be a closed
-ball in a closed connected
-manifold
.
Tex syntax error.
Tex syntax errorbe
![\Z](/images/math/6/4/5/64518e230a67ecb1ea9ed536d6ac4c9e.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax errorfor
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
We omit -coefficients from the notation of (co)ho\-mo\-lo\-gy
groups.
Tex syntax erroris an embedding, unless another meaning of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
%is explicitly given.
For an embeddingTex syntax errordenote by
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
Tex syntax errorto a tubular
neighborhood of and
the restriction of the normal bundle of
.
2 References
This page has not been refereed. The information given here might be incomplete or provisional. |