Embeddings in Euclidean space: an introduction to their classification
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== Notation and conventions == | == Notation and conventions == | ||
− | For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[smooth]] | + | For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[smooth]] or [[piecesise-linear]] (PL) embeddings $N\to\Rr^m$ up to smooth or PL isotopy. If a category is omitted, then the result holds (or a definition or a construction is given) in both categories. |
− | or [[piecesise-linear]] (PL) embeddings $N\to\Rr^m$ up to smooth or PL isotopy. | + | |
− | + | ||
− | construction is given) in both categories. | + | |
All manifolds in this note are tacitly assumed to be compact. | All manifolds in this note are tacitly assumed to be compact. | ||
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* $C_f$ the closure of the complement in $S^m\supset\Rr^m$ to a tubular neighborhood of $f(N)$ and | * $C_f$ the closure of the complement in $S^m\supset\Rr^m$ to a tubular neighborhood of $f(N)$ and | ||
*$\nu_f:\partial C_f\to N$ the restriction of the normal bundle of $f$. | *$\nu_f:\partial C_f\to N$ the restriction of the normal bundle of $f$. | ||
− | + | </wikitex> | |
== Links to specific results == | == Links to specific results == | ||
− | + | <wikitex>; | |
− | [[ | + | [[Hudson tori|Hudson tori]] |
− | + | ||
</wikitex> | </wikitex> |
Revision as of 13:22, 12 February 2010
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
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 isotopy classification results for embeddings of 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 more in knot theory and [Sk08].
Notation and conventions
For a manifold let or denote the set of smooth or piecesise-linear (PL) embeddings up to smooth or PL isotopy. 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 . Denote .
Let be for even and for odd.
We omit -coefficients from the notation of (co)homology groups.
For an embedding denote by
- the closure of the complement in to a tubular neighborhood of and
- the restriction of the normal bundle of .
2 Links to specific results
3 References
This page has not been refereed. The information given here might be incomplete or provisional. |