Embeddings in Euclidean space: an introduction to their classification
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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
Knots, i.e. embeddings of spheres
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
Embeddings of highly-connected manifolds
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. |