High codimension embeddings: classification
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Introduction
This page is intended not only for specialists in embeddings, but also for mathematicians from other areas who want to apply or to learn the theory of embeddings.
According to [Zeeman93, p. 3], the classical problems of topology are the following.
- Homeomorphism Problem: Classify -manifolds.
- Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
- Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The 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 some of the dimension ranges where no knotting is possible, introduce connected sum and make some comments on codimension 1 and 2 embeddings. At the time of writing, the unknotting theorems and the results on the pages listed below in Section 4 record all known complete readily calculable isotopy classification results for embeddings of closed manifolds into Euclidean spaces which are known to the author.
The Knotting Problem is known to be hard:
- There are only a few cases, see 4, in which there are complete readily calculable classification results. (However, the statements, although 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 at the time of writing. (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 6. See more in Wikipedia article on knot theory.)
- In some cases `geometric problems are reduced to algebraic problems which are even harder to solve' [Wall1999]. E.g. so far an interesting approach of [Weiss96], [Goodwillie&Weiss1999] did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
The list of known results in the rest of this page shows that the complete classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for . The lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For known information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.1 (Readily calculable classification). By readily calculable classification I mean a classification in terms of homology of a manifold (and certain structures in homology like intersection, characteristic classes etc). A readily calculable classification is also a reduction to calculation of stable homotopy groups of spheres (or to another standard algebraic problem involving only homology of our manifold, which problem is solved in particular cases, although could be unsolved generally). An important feature of a useful classification is accessibility of statement to general mathematical audience which is only familiar with basic notions of the area; this in turn is an approximation to beauty. Another important feature is whether the classification gives an algorithm and how complex the algorithm is.
Remark 1.2 (Sphere and Euclidean space). (a) Every two embeddings are isotopic (this intuitively clear assertion is Schöenfliess Theorem) [Rushing1973]. On the other hand, embeddings given by and are not isotopic. This example is generalized to (locally flat) embeddings and [Rushing1973].
(b) For classifications of embeddings of -manifolds into and into are the same.
Let us prove this for the smooth category. It suffices to show that 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). For showing that assume that and are isotopic. Then 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.
[edit] 2 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.
Denote by the Stiefel manifold of -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold denote .
A closed manifold is called homologically -connected, if is connected and for each . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for each .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the space of unit length vectors identified with .
[edit] 3 Unknotting theorems
General Position Theorem 3.1 ([Hirsch1976], [Rourke&Sanderson1972]). For each -manifold and , every two embeddings are isotopic (i.e. ).
The restriction in Theorem 3.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h] [Skopenkov2006, Figure 2.1.a].
Theorem 3.2 ([Wu1958], [Wu1958a] and [Wu1959]). For each connected -manifold , and , every two embeddings are isotopic (i.e. ).
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.
Unknotting Spheres Theorem 3.3 ([Zeeman1960], [Stallings1963], [Gluck1963], [Haefliger1961], [Adachi1993]). For , or even for a homology -sphere, or in the PL or smooth category, respectively, every two embeddings are isotopic (i.e. ).
This result is also true for in the topological locally flat category [Rushing1973], [Scharlemann1977].
Knots in codimension 2 and the higher-dimensional analogue of the Haefliger trefoil knot show that the dimension restrictions are sharp (even for ) in the Unknotting Spheres Theorem 3.3.
Theorems 3.2 and 3.3 may be generalized as follows.
The Haefliger-Zeeman Unknotting Theorem 3.4. For , and each closed homologically -connected -manifold , every two embeddings are isotopic (i.e. ).
The Haefliger-Zeeman Unknotting Theorem 3.4 was proved directly in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969] for homotopically -connected manifolds. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds.
So can be called a `stable range for -connected manifolds'.
Note that if , then every closed homologically -connected -manifold is a homology sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 3.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 3.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 3.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979].
[edit] 4 References to information on the classification of embeddings
Below we list references to information about the classification of embeddings of manifolds into Euclidean space. The first list is structured by the dimension of the source manifold and the target Euclidean space:
Information structured by the `complexity' of the source manifold:
For more information see e.g. [Skopenkov2006].
[edit] 5 Embedded connected sum
Suppose that is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for (embedded) connected sum defines a group structure on [Haefliger1966], and an action of on .
[edit] 6 Codimension 2 embeddings
In this subsection let be a closed connected -manifold.
Using connected sums we can produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . However, note that there are embeddings and such that is not isotopic to but is isotopic to [Viro1973].
One can also apply Artin's spinning construction [Artin1928] for .
Thus description of is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of hardness of this problem.
On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a], [Cappell&Shaneson1974] and [Ranicki1998].
[edit] 7 Codimension 1 embeddings
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence . So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
It is known that
- (Schoenfliess theorem) [Rushing1973].
- (Alexander theorem); a counterexample to the topological version is the Alexander Horned Sphere [Rushing1973].
- The description of is equivalent to the PL Schoenfliess problem and therefore is very hard for [Rushing1973].
- for [Smale1962a].
- for [Kosinski1961], [Wall1965], [Lucas&Neto&Saeki1996], cf. [Goldstein1967].
Let be the set of PL locally flat embeddings up to PL locally flat isotopy. Note that could admit complete readily calculable classification even when does not [Goldstein1967].
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
[edit] 8 References
- [Adachi1993] M. Adachi, Embeddings and immersions, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001
- [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.
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Cappell&Shaneson1974] S. E. Cappell and J. L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277–348. MR0339216 (49 #3978) Zbl 0279.57011
- [Gluck1963] H. Gluck, Unknotting in ., Bull. Am. Math. Soc. 69 (1963), 91-94. MR0142114 (25 #5507) Zbl 0108.36503
- [Goldstein1967] R. Z. Goldstein, Piecewise linear unknotting of in , Michigan Math. J. 14 (1967), 405–415. MR0220299 (36 #3365) Zbl 0157.54801
- [Goodwillie&Weiss1999] T. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory, II, Geometry and Topology, 3 (1999), 103-118. MR1694812 (2000c:57055a) Zbl 0927.57028
- [Gordon1972] C. Gordon, Embedding piecewise linear manifolds with boundary., Proc. Camb. Philos. Soc. 72 (1972), 21-25. MR0295359 (45 #4425) Zbl 0236.57009
- [Hacon1968] D. Hacon, Embeddings of in in the metastable range., Topology 7 (1968), 1-10. MR0222903 (36 #5953) Zbl 0153.53902
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Hirsch1976] M. W. Hirsch, Differential topology., Graduate Texts in Mathematics, No. 33. Springer-Verlag., New York-Heidelberg, 1976. MR0448362 (56 #6669) Zbl 0356.57001
- [Hudson1967] J. Hudson, Piecewise linear embeddings, Ann. of Math. (2) 85 (1967) 1–31. MR0215308 (35 #6149) Zbl 0153.25601
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Hudson1972] J. Hudson, Embeddings of bounded manifolds., Proc. Camb. Philos. Soc. 72 (1972), 11-20. MR0298679 (45 #7728) Zbl 0241.57006
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Kearton1979] C. Kearton, Obstructions to embedding and isotopy in the metastable range, Math. Ann. 243 (1979), 103-113. MR0543720 (82k:57012) Zbl 0401.57033
- [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. \cite{Fort1962}.
- [Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
- [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
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
- [Rourke&Sanderson1972] C. Rourke and B. Sanderson, Introduction to piecewise-linear topology., Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 69. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 1972. MR0665919 (83g:57009) Zbl 0254.57010
- [Rushing1973] T. Rushing, Topological embeddings., {Pure and Applied Mathematics, 52. New York-London: Academic Press. XIII, 316 p. \$ 18.50 }, 1973. MRMR0348752 (50 #1247) Zbl 0295.57003
- [Scharlemann1977] M. Scharlemann, Isotopy and cobordism of homology spheres in spheres., J. Lond. Math. Soc., II. Ser. 16 (1977), 559-567. MRMR0464246 (57 #4180) Zbl 0375.57003
- [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.
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, http://arxiv.org/abs/1502.04470
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, submitted to Boll. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, submitted to Boll. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, submitted to Boll. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, submitted to Boll. Man. Atl.
- [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
- [Stallings1963] J. Stallings, On topologically unknotted spheres., (1963). MRMR0149458 (26 #6946) Zbl 0121.18202
- [Viro1973] O. J. Viro, Local knotting of sub-manifolds, Mat. Sb. (N.S.) 90(132) (1973), 173–183, 325. MR0370606 (51 #6833)
- [Vrabec1977] J. Vrabec, Knotting a -connected closed -manifold in , Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [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
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Weiss] M. Weiss, private communication
- [Weiss96] M. Weiss, Calculus of Embeddings, Bull. Amer. Math. Soc. 33 (1996), 177-187
- [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. I, Sci. Sinica 7 (1958), 251–297. MR0099026 (20 #5471) Zbl 0183.28302
- [Wu1959] W. Wu, On the realization of complexes in euclidean spaces. III, Sci. Sinica 8 (1959), 133–150. MR0098365 (20 #4825b) Zbl 0207.53104
- [Zeeman1960] E. Zeeman, Unknotting spheres, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520)
- [Zeeman93] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://www.mccme.ru/circles/oim/foto2014/l_ht.html