Embeddings in Euclidean space: an introduction to their classification
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and open problems below. | and open problems below. | ||
Later we hope to add information for manifolds with boundary. For more information see --> | Later we hope to add information for manifolds with boundary. For more information see --> | ||
− | and | + | and \cite{Skopenkov2006}. |
== Notation and conventions == | == Notation and conventions == |
Revision as of 13:34, 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 [Skopenkov2006].
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
- [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. |