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 reviews the Knotting Problem. We establish notation and conventions, record the dimension ranges where there no knotting is possible and make some comments on codimension 1 and 2 embeddings. The unkotting results and the results on the pages below record all known
isotopy classification results for embeddings of
manifolds into Euclidean spaces. In particular, we do this for codimension 1 and 2 embeddings.
There is an extensive study of codimension 2 embeddings not directly aiming at complete classification. Almost nothing is said here about this. See more in knot theory.
1.1 Notation and conventions
The following notations and conventions will be widely used for pages about embeddings.
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
.
1.2 Links for information about embeddings
Embeddings just below the stable range
Embeddings of 3-manifolds in 6-space
Knots, i.e. embeddings of spheres
Embeddings of 4-manifolds in 7-space
Embeddings of highly-connected manifolds
Links, i.e. embeddings of non-connected manifolds
For more information see e.g. [Skopenkov2006].
2 Unknotting theorems
General Position Theorem 2.1.
For each -manifold
and
, every two embeddings
are isotopic (i.e.
).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking
shows (see Figure~2.1.a of [Skopenkov2006]).
Theorem 2.2.
For each connected -manifold
,
and
, every two
embeddings
are isotopic (i.e.
) [Wu1958], [Wu1958a] and [Wu1959].
All the three assumptions in this result are indeed necessary:
- the assumption
because of the existence of non-trivial knots
;
- the connectedness assumption because of the existence of the Hopf linking above;
- the assumption
because of the example of Hudson tori.
3 Embedded connected sum
Suppose that is a closed connected
-manifold and an embedding
(or an orientation of
, if
is orientable) is chosen.
If the images of embeddings
and
are contained in
disjoint cubes, then we can define embedded
.
We make connected summation along
and a path in
joining
to
.
If
, this operation is not well-defined, i.e. depends on the choice of the path.
If
, this operation is well-defined, i.e. is independent on the choice of the path.
Moreover, for
(embedded) connected sum defines a group structure on
[Haefliger1966], and an action
.
4 Codimension 2 embeddings
A description of and, more generally, of
is a
well-known very hard open problem.
Let
be a closed connected
-manifold.
Using connected sums we can produce an overwhelming multitude of embeddings
from the overwhelming multitude of embeddings
.
(Or using Artin's rotation construction [Artin1928]
.)
Thus description of
seems to be very hard open problem.
For
see e.g. [CappellShaneson], [Levine], [Ranicki].
It would be interesting to give a more formal explanation of why
the description of is hard, using known information that
the description of
is hard.
Note that
- there are embeddings
and
such that
is not isotopic to
but
is isotopic to
[Viro1973].
However, note that can be known even when
is unknown, see the next subsection.
See open problems on classification `modulo knots' below.
5 Codimension 1 embeddings
Every embedding extends to an embedding either
or
[Alexander1924].
Clearly, only the standard embedding extends to both.
When one proves that this extension respects isotopy, this gives a 1-1 correspondence
.
So description of
is as hopeless as that of
.
Thus description of
for
a sphere with handles is apparently
hopeless.
It is known that
-
for
[Smale1962a].
- The description of
is equivalent to the PL Schoenfliess problem and therefore is very hard for
.
-
for
[Goldstein1967], [Lucas&Neto&Saeki1996].
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
6 Conclusion
Thus complete classification of embeddings into of closed connected
-manifolds is non-trivial but accessible only for
or
for
.
7 References
- [Alexander1924] J. W. Alexander, On the subdivision of 3-space by polyhedron, Proc. Nat. Acad. Sci. USA, 10, (1924) 6–8. Zbl 50.0659.01
- [Artin1928] E. Artin, Zur Isotopie zwei-dimensionaler Flaechen im
, Abh. Math. Sem. Hamburg Univ. 4 (1928) 174–177.
- [CappellShaneson] Template:CappellShaneson
- [Goldstein1967] R. Z. Goldstein, Piecewise linear unknotting of
in
, Michigan Math. J. 14 (1967), 405–415. MR0220299 (36 #3365) Zbl 0157.54801
- [Haefliger1966] A. Haefliger, Differential embeddings of
in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Levine] Template:Levine
- [Lucas&Neto&Saeki1996] L. A. Lucas, O. M. Neto and O. Saeki, A generalization of Alexander's torus theorem to higher dimensions and an unknotting theorem for
embedded in
, Kobe J. Math. 13 (1996), no.2, 145–165. MR1442202 (98e:57041) Zbl 876.57045
- [Lucas&Saeki2002] L. A. Lucas and O. Saeki, Embeddings of
in
, Pacific J. Math. 207 (2002), no.2, 447–462. MR1972255 (2004c:57045) Zbl 1058.57022
- [Ranicki] Template:Ranicki
- [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.
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
- [Viro1973] O. J. Viro, Local knotting of sub-manifolds, Mat. Sb. (N.S.) 90(132) (1973), 173–183, 325. MR0370606 (51 #6833)
- [Wu1958] W. Wu, On the realization of complexes in euclidean spaces. I, Sci. Sinica 7 (1958), 251–297. MR0099026 (20 #5471) Zbl 0183.28302
- [Wu1958a] W. Wu, On the realization of complexes in euclidean spaces. II, Sci. Sinica 7 (1958), 365–387.
- [Wu1959] W. Wu, On the realization of complexes in euclidean spaces. III, Sci. Sinica 8 (1959), 133–150. MR0098365 (20 #4825b) Zbl 0207.53104
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