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 a manifold denote .
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 . One can also apply Artin's spinning construction [Artin1928] . Thus description of seems to be very hard open problem. For see e.g. [Cappell&Shaneson], [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 [Goldstein1967] (here stands for locally flat). See open problems on classification `modulo knots' to appear 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 [Kosinski1961], [Wall1965], [Lucas&Neto&Saeki1996], cf. [Goldstein1967].
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.
- [Cappell&Shaneson] Template:Cappell&Shaneson
- [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
- [Kosinski1961] A. Kosinski, On Alexander's theorem and knotted tori, In: Topology of 3-Manifolds, Prentice-Hall, Englewood Cliffs, Ed. M.~K.~Fort, N.J., 1962, 55--57. Cf. Fort1962.
- [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)
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [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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