Embeddings in Euclidean space: an introduction to their classification
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greater than 2. | greater than 2. | ||
See more in [[Wikipedia:Knot_theory|knot theory]].) | See more in [[Wikipedia:Knot_theory|knot theory]].) | ||
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− | == Notation and conventions | + | == Notation and conventions used in the links below == |
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For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[smooth]] or [[piecewise-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. | For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[smooth]] or [[piecewise-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. | ||
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For an embedding $f:N\to\Rr^m$ denote by | For an embedding $f:N\to\Rr^m$ denote by | ||
* $C_f$ 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 spherical normal bundle of $f$. |
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== Links to specific results and examples == | == Links to specific results and examples == | ||
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+ | [[Some general remarks]] | ||
[[Hudson tori|Example of Hudson tori]] | [[Hudson tori|Example of Hudson tori]] | ||
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+ | [[Classification just below the stable range]] | ||
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+ | [[Examples of embeddings of 3-manifolds into the 6-space]] | ||
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+ | [[Classification of embeddings of 3-manifolds in the 6-space]] | ||
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+ | [[Definition of the Kreck invariant for 3-manifolds in the 6-space]] | ||
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+ | [[Classification for m=2n-1]] | ||
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+ | [[Examples of embeddings of 4-manifolds into the 7-space]] | ||
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+ | [[Classification of embeddings of 4-manifolds into the 7-space]] | ||
[[Links, i.e. embeddings of non-connected manifolds]] | [[Links, i.e. embeddings of non-connected manifolds]] | ||
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[[Embeddings of manifolds with boundary]] | [[Embeddings of manifolds with boundary]] | ||
+ | [[Some open problems]] | ||
+ | For more information see e.g. \cite{Skopenkov2006}. | ||
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== References == | == References == | ||
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[[Category:Manifolds]] | [[Category:Manifolds]] | ||
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Revision as of 14:03, 14 February 2010
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
Contents |
1 Introduction
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. This is done in particular for codimension 1 and 2 embeddings.
(There is an extensive study of codimension 2 embeddings not directly aiming at complete classification. Almost none is said here about that. So in the title of this article we only mention embeddings of codimension greater than 2. See more in knot theory.)
2 Notation and conventions used in the links below
For a manifold let or denote the set of smooth or piecewise-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 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 spherical normal bundle of .
3 Links to specific results and examples
Classification just below the stable range
Examples of embeddings of 3-manifolds into the 6-space
Classification of embeddings of 3-manifolds in the 6-space
Definition of the Kreck invariant for 3-manifolds in the 6-space
Examples of embeddings of 4-manifolds into the 7-space
Classification of embeddings of 4-manifolds into the 7-space
Links, i.e. embeddings of non-connected manifolds
Embeddings of manifolds with boundary
For more information see e.g. [Skopenkov2006].
4 References
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
This page has not been refereed. The information given here might be incomplete or provisional. |