Embeddings in Euclidean space: an introduction to their classification
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].
1 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
2 References
This page has not been refereed. The information given here might be incomplete or provisional. |