Embeddings in Euclidean space: an introduction to their classification
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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. |