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?
Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
This article reviews the Knotting Problem. After making general remarks we establish notation and conventions, record the dimension ranges where no knotting is possible and make some comments on codimension 1 and 2 embeddings. The unknotting results and the results on the pages below record all known isotopy classification results for embeddings of manifolds into Euclidean spaces.
1.1 The Knotting Problem and classification of embeddings
The Knotting Problem is part of the problem on isotopy classification of embeddings. These problems are known to be hard:
- There are only a few cases in which there are classification results. However, the statements (but not the proofs!) are simple and accessible to non-specialists.
- For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in ), a complete readily calculable classification is neither known nor expected. (Note that 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.)
- In some cases there are algebraic reductions, but some of them `reduce geometric problems to algebraic problems which are even harder to solve' [Wall1970], cf. [Goodwillie&Weiss1999]. (Note that the approach of [Goodwillie&Weiss1999] does give explicit results on higher homotopy groups of the space of embeddings [Weiss].)
1.2 Sphere and Euclidean space
Classifications of embeddings into and are the same for compact manifolds. This holds because if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic (in spite of the existence of orientation-preserving diffeomorphisms not isotopic to the identity). Indeed, since and are isotopic, by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an ambient one [Hirsch1976], Theorem 1.3, and are isotopic.
1.3 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 .
1.4 Links for information about embeddings
Embeddings just below the stable range: classification
Knots, i.e. embeddings of spheres
Embeddings of highly-connected manifolds: classification
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
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- [Weiss] M. Weiss, private communication
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