Knotted tori

Revision as of 13:02, 19 March 2010

Many interesting examples of embeddings are embeddings $<wikitex> Many interesting examples of embeddings are embeddings $S^p\times S^q\to\Rr^m$, i.e. ''knotted tori''. See [[Embeddings just below the stable range: classification#Examples|Hudson tori]], \cite{Alexander1924}, \cite{Milgram&Rees1971}, \cite{Kosinski1961}, \cite{Hudson1963}, \cite{Wall1965}, \cite{Tindell1969}, \cite{Boechat&Haefliger1970}, \cite{Boechat1971}, \cite{Milgram&Rees1971}, \cite{Lucas&Saeki2002}, \cite{Skopenkov2002}. The classification of knotted tori is a natural next step (after the link theory \cite{Haefliger1966a}) and the [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|classification of embeddings of highly-connected manifolds]]) towards classification of embeddings of ''arbitrary'' manifolds. The classification of knotted tori gives some insight or even precise information concerning arbitrary manifolds and reveals new interesting relations to algebraic topology. Since the general [[High codimension embeddings: classification#Introduction|Knotting Problem]] is very hard, it is very interesting to solve it for the important particular case of knotted tori. See \cite{Skopenkov2006a}, \cite{Cencelj&Repov\v s&Skopenkov2007}, \cite{Cencelj&Repovš&Skopenkov2008}. <!--For notation and conventions throughout this page see [[High_codimension_embeddings|high codimension embeddings]].--> </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]] {{Stub}}S^p\times S^q\to\Rr^m$, 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&Repov\v s&Skopenkov2007], [Cencelj&Repovš&Skopenkov2008] give some insight or even precise information concerning arbitrary manifolds (cf. [Skopenkov2007], [Skopenkov2010]) and reveals new interesting relations to algebraic topology.

References

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• [Cencelj&Repov\v s&Skopenkov2007] Template:Cencelj&Repov\v s&Skopenkov2007
• [Cencelj&Repovš&Skopenkov2008] M. Cencelj, D. Repovš and M. Skopenkov, Classification of knotted tori in the 2-metastable dimension, Mat. Sbornik, 203:11 (2012), 1654-1681. Available at the arXiv:0811.2745.
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• [Lucas&Saeki2002] L. A. Lucas and O. Saeki, Embeddings of $S^p\times S^q\times S^r$ in $S^{p+q+r+1}$, Pacific J. Math. 207 (2002), no.2, 447–462. MR1972255 (2004c:57045) Zbl 1058.57022
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• [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 $\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
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