Knotted tori
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1 Introduction
Many interesting examples of embeddings are embeddings , i.e. knotted tori. See Hudson tori, [Alexander1924], [Milgram&Rees1971], [Kosinski1961], [Hudson1963], [Wall1965], [Tindell1969], [Boechat&Haefliger1970], [Boechat1971], [Milgram&Rees1971], [Lucas&Saeki2002], [Skopenkov2002]. A classification of knotted tori is a natural next step (after the link theory [Haefliger1966a] and the classification of embeddings of highly-connected manifolds) towards classification of embeddings of arbitrary manifolds. Since the general Knotting Problem is very hard, it is very interesting to solve it for the important particular 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 reveals 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]. Denote
2 Examples
One of the first examples were Hudson tori.
An -parametric connected sum group structure on is constructed for in [Skopenkov2006], [Skopenkov2015a].
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
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Clearly, is well-defined and, for , is a homomorphism.
Define the `embedded connected sum' or `local knotting' map}
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See construction of another map in [Skopenkov2015, 3, definition of ].
3 Classification
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.
We have the following table for and for , for the PL and smooth categories, respectively.
Here is short for . The table follows from the theorems below.
We also have and , of which is rank one infinite [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 3.4 below.
Theorem 3.1. There are isomorphisms, or, for , 1-1 correspondences
We have ; a description of is given in [Skopenkov2016e, end of 6.3].
Theorem 3.1 can be generalized as follows.
Theorem 3.2. (a) If and , then
is an isomorphism.
(b) If and , in the PL and DIFF categories respectively, 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]. For there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for no proof of Theorem 3.2 without referring to `the deleted product method' is known.
For (which is automatic for and ) we have and the 1-1 correspondence of Theorem 3.2.b is .
Let be the quotient set of by the embedded connected sum action and the quotient map. For a group structure on is well-defined by , . The following result reduces description of to description of and of , cf. [Schmidt1971], [Crowley&Skopenkov2008, end of 1].
Lemma 3.3 (Smoothing). For we have .
The isomorphism of Lemma 3.3 is , where is `surgery of '.
The following result reduces 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.
Denote by the torsion subgroup of an abelian group . Abelian group structures 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 .
Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a]. Assume that .
(a) .
(b) (more precisely, whenever one part is finite, the other is finite and they are equal).
(c) , unless and for some .
The above results were obtained using more `theoretical' results [Skopenkov2015, Theorem 1.6], [Skopenkov2015a, Theorem 1.2], see also [Cencelj&Repovš&Skopenkov2008, Theorem 2.1].
4 References
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