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.
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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See construction of another map in [Skopenkov2015, 3, definition of ].
An -parametric connected sum group structure on is constructed for in [Skopenkov2006], [Skopenkov2015a].
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.
Theorem 3.1. There are 1-1 correspondences
This result can be generalized as follows.
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 is .
4 References
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- [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
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