Knotted tori
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
By a ``torus`` we mean a product of spheres , regarded as a closed manifold. Many interesting examples of embeddings are embeddings , i.e. knotted tori. For examples, see Hudson tori, [Alexander1924], [Kosinski1961], [Hudson1963], [Wall1965], [Tindell1969], [Boechat&Haefliger1970], [Boechat1971], [Milgram&Rees1971], [Lucas&Saeki2002], [Skopenkov2002]. Classifying knotted tori is a natural next step (after the classification of links [Haefliger1966a] and the classification of embeddings of highly-connected manifolds) towards the classification of embeddings of arbitrary manifolds. Since the general Knotting Problem is very hard, it is interesting to solve it for the important special case of knotted tori. Recent classification results for knotted tori [Skopenkov2006a], [Cencelj&Repovš&Skopenkov2007], [Cencelj&Repovš&Skopenkov2008], [Skopenkov2015], [Skopenkov2015a] give some insight or even precise information concerning arbitrary manifolds (cf. [Skopenkov2007], [Skopenkov2010], [Skopenkov2014]) and reveal new interesting relations to algebraic topology.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. We assume that . Denote
For definition of the embedded connected sumof embeddings, denote by , and for the corresponding action of the group on the set , see e.g. [Skopenkov2016c, 4].
2 Examples
An -parametric connected sum group structure on is constructed for in [Skopenkov2006], [Skopenkov2015a].
Some of the first examples of knotted tori are the Hudson tori. See also examples of knotted 4-dimensional tori in 7-space [Skopenkov2016f, 2].
Let us construct a map
Recall that is isomorphic to the group of smooth maps up to smooth homotopy. The latter maps can be considered as smooth maps . Define the smooth embedding as the composition
Tex syntax error
Here is the standard inclusion. Clearly, is well-defined and, for , is a homomorphism.
Define the `embedded connected sum' or `local knotting' map
Tex syntax error
Tex syntax erroris well-defined and, for , is a homomorphism.
See construction of another map in [Skopenkov2015, 3, definition of ].
3 Classification just below the stable range
From the Haefliger-Zeeman Isotopy Theorem it follows that for and , provided that or in the PL or smooth category, respectively. The dimension restriction in this result is sharp by the example of Hudson tori.
For the next theorem, the Whitney invariant is defined in [Skopenkov2016e, 5].
Theorem 3.1. (a) The Whitney invariant
is an isomorphism for .
(b) The Whitney invariants
are bijective for .
(c) The Whitney invariant is surjective and for any there is a 1-1 correspondence .
Theorem 3.1 follows from Theorems 6.2 and 6.3 of [Skopenkov2016e] (parts (a) and (b) also follow from Theorem 5.1 below). The Hudson torus generates for (this holds by [Skopenkov2016e, Theorem 6.2] because ).
In the smooth category for even is not injective (by Theorem 3.1.c), is not surjective [Boechat1971], [Skopenkov2016f], and is not injective [Skopenkov2016f].
4 Reduction to classification modulo knots
Tex syntax error. Let be the quotient map.
For a group structure on is well-defined by , .
For the map constructed by `embedded surgery of ' is well-defined [Skopenkov2015a, 3.3]. Clearly,Tex syntax error.
The following result reduces description of to description of and of , cf. [Schmidt1971], [Crowley&Skopenkov2008, end of 1].
Lemma 4.1 [Skopenkov2015a, Smoothing Lemma 1.1]. For the map
5 Further classification
We have the following table for and for , for the PL and smooth categories, respectively.
Here is short for . We also have and , of which is rank one infinite. The table and the additional results follow from the theorems below, see [Skopenkov2015a].
There is a finiteness criterion for when [Skopenkov2015, Theorem 1.4]. The formulation is not so short but effective. This criterion is a corollary of Theorem 5.4 below.
Theorem 3.1.(a)(b) can be generalized as follows.
Theorem 5.1. (a) If and , then
Tex syntax error
are isomorphisms.
(b) If , then
(c) If , then there is a 1-1 correspondence
This follows for from the Becker-Glover Theorem 5.3. For the general case see [Skopenkov2002, Corollary 1.5.a]. The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5]. For there is an alternative direct proof of (a) [Skopenkov2006], [Skopenkov2015a], but for no proof of Theorem 5.1.(b)(c) without referring to `the Haefliger-Wu invariant' is known.
For we have , so part (c) reduces to part (b) and the PL case of part (a).
Theorem 5.2 [Skopenkov2015a, Corollary 1.5.(b)(c)]. Assume that .
(a) If , then and have isomorphic subgroups with isomorphic quotients.
(b) If , then has a subgroup isomorphic to ( large), whose quotient and have isomorphic subgroups with isomorphic quotients.
The following conjecture and results reduce description of to description of objects which are easier to calculate, at least in some cases, see [Skopenkov2015a, 1.3] for methods of their calculations.
Abelian group structure on for is defined analogously to the well-known case . The sum operation on is `connected sum of -spheres together with normal -framings' or `-parametric connected sum'. Define to be the subgroup of links all whose components are unknotted. Let be the linking coefficient. Denote .
Conjecture 5.3. Assume that .
(a) If , thenTex syntax erroris an isomorphism.
(b) The map
Tex syntax error
is an isomorphism. Here is the `restriction' map induced by the inclusion and is defined in [Skopenkov2015, 3, definition of ].
For a discussion see [Skopenkov2015a, Remark 1.9 and footnote 7].
Denote by the torsion subgroup of an abelian group .
Theorem 5.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a, Corollary 1.4]. Assume that .
(a) .
(b) (more precisely, whenever one part is finite, the other is finite and they are equal).
(c) , unless and for some .
Theorems 5.2 and 5.4 were obtained using more `theoretical' results [Skopenkov2015, Theorem 1.6], [Skopenkov2015a, Theorem 1.2], see also [Cencelj&Repovš&Skopenkov2008, Theorem 2.1].
6 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
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension dans , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension dans , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Cencelj&Repovš&Skopenkov2007] M. Cencelj, D. Repovš and M. Skopenkov, Homotopy type of the complement of an immersion and classification of embeddings of tori., Russ. Math. Surv.62 (2007), no.5, 985-987. Zbl 1141.57009
- [Cencelj&Repovš&Skopenkov2008] M. Cencelj, D. Repovš and M. Skopenkov, A new invariant of higher-dimensional embeddings, Mat. Sbornik, 203:11 (2012), 1654-1681. Available at the arXiv:0811.2745.
- [Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [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.
- [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
- [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
- [Schmidt1971] R. Schultz, On the inertia groups of a product of spheres, Trans. AMS, 156 (1971), 137–153.
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [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.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Tindell1969] R. Tindell, Extending homeomorphisms of ., Proc. Am. Math. Soc. 22 (1969), 230-232. MRMR0248852 (40 #2102) Zbl 0177.26802
- [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