Knotted tori
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[edit] 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].
[edit] 2 Examples
See another construction in [Skopenkov2015], \S3, definition of .
An -parametric connected sum group structure on is constructed for in [Skopenkov2006], [Skopenkov2015a].
[edit] 3 Classification
From the Haefliger-Zeeman Isotopy Theorem it follows that for and (and 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, and for general case from [Skopenkov2002], Corollary 1.5.a. Note that for (which is automatic for and ), so the 1--1 correspondence is . For there is an alternative direct proof [Skopenkov2006], but for no proof of Theorem 3.2 without referring to 'the deleted product method' is known.
[edit] 4 References
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