Knotted tori

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[edit] 1 Introduction

Many interesting examples of embeddings are embeddings 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&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, \S1, \S3].

[edit] 2 Examples

One of the first examples were Hudson tori. Let us construct a map \tau:\pi_q(V_{m-q,p+1})\to E^m(S^p\times S^q). Recall that \pi_q(V_{m-q,p+1}) is isomorphic to the group of smooth maps S^q\to V_{m-q,p+1} up to smooth homotopy. The latter maps can be considered as smooth maps \varphi:S^q\times S^p\to\partial D^{m-q}. Define the smooth embedding \tau(\varphi) as the composition
\displaystyle S^p\times S^q\overset{\varphi\times pr_2}\to\partial D^{m-q}\times S^q\subset D^{m-q}\times S^q\subset\Rr^m.
Here pr_2 is the projection onto the second factor and \subset are the standard inclusions.

See another construction in [Skopenkov2015], \S3, definition of \sigma^*.

An S^p-parametric connected sum group structure on E^m(S^p\times S^q) is constructed for m\ge2p+q+3 in [Skopenkov2006], [Skopenkov2015a].

[edit] 3 Classification

From the Haefliger-Zeeman Isotopy Theorem it follows that E^m(S^p\times S^q)=0 for p\le q and m\ge p+2q+2 (and m\ge p+q+3 or 2m\ge3(p+q)+4 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

\displaystyle E^{p+2q+1}_{PL}(S^p\times S^q)\to\left\{\begin{array}{cc} \Zz_{\varepsilon(q)} & \quad 1\le p<q \\ \Zz_{\varepsilon(q)}\oplus\Zz_{\varepsilon(q)}&\quad 2\le p=q\end{array}\right. \qquad\text{and} \qquad E^{p+2q+1}_{DIFF}(S^p\times S^q)\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.

This result can be generalized as follows.

Theorem 3.2. If 2m\ge3q+2p+4 and 2m\ge3q+3p+4, in the PL and DIFF categories respectively, then there is a 1-1 correspondence
\displaystyle E^m(S^p\times S^q)=\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).

This follows for m\ge 2q+3 from the Becker-Glover Theorem 5.3, and for general case from [Skopenkov2002], Corollary 1.5.a. Note that \pi_p(V_{m-p,q+1})=0 for m\ge2p+q+2 (which is automatic for p\le q and 2m\ge3p+3q+4), so the 1--1 correspondence is \tau. For m\ge2p+q+2 there is an alternative direct proof [Skopenkov2006], but for m<2p+q+2 no proof of Theorem 3.2 without referring to 'the deleted product method' is known.

[edit] 4 References

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