Knotted tori

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== Examples ==
== Examples ==
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One of the first examples were [[Embeddings just below the stable range: classification#Examples|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 $$S^p\times S^q\overset{\varphi\times pr_2}\to\partial D^{m-q}\times S^q\subset D^{m-q}\times S^q\subset\Rr^m.$$ Here $pr_2$ is the projection onto the second factor and $\subset$ are the standard inclusions.
One of the first examples were [[Embeddings just below the stable range: classification#Examples|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 $$S^p\times S^q\overset{\varphi\times pr_2}\to\partial D^{m-q}\times S^q\subset D^{m-q}\times S^q\subset\Rr^m.$$ Here $pr_2$ is the projection onto the second factor and $\subset$ are the standard inclusions.
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== Classification ==
== Classification ==
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From [[High codimension embeddings: classification#Unknotting theorems|the Haefliger-Zeeman Isotopy Theorem]] it follows that
From [[High codimension embeddings: classification#Unknotting theorems|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 [[Embeddings just below the stable range: classification#Examples|Hudson tori]].
$E^m(S^p\times S^q)=0$ 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 [[Embeddings just below the stable range: classification#Examples|Hudson tori]].
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{{beginthm|Theorem}}\label{kt1} There are 1-1 correspondences
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$$E^{p+2q+1}_{PL}(S^p\times S^q)\to\left\{\begin{array}{cc} \Zz_{(q)} & 1\le p<q \\
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\Zz_{(q)}\oplus\Zz_{(q)}&2\le p=q\end{array}\right.$$
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{{endthm}}
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<!--
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\qquad\text{and}
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\qquad E^{p+2q+1}(S^p\times S^q)_{DIFF}\cong\Zz_{(q)}\quad\text{for}\quad 1\le p\le q-2.$$
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Theorem \lc8 follows from Theorem \wi8.b (as well as from Theorem
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\lc9 below). In the PL case of Theorem \lc8 for $p=q$ we only have
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a 1--1 correspondence of sets (because Group Structure Theorem
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\lc7 does not give a group structure for such dimensions). A
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description of $KT^{6k}_{2k-1,2k,DIFF}$ is given after Theorem
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\wi9.
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[HH63, Hu63, Vr77]
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-->
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This result can be generalized as follows.
{{beginthm|Theorem}}\label{kt} If $2m\ge3q+2p+4$ and $2m\ge3q+3p+4$, in the PL and DIFF categories respectively, then there is a 1-1 correspondence $$E^m(S^p\times S^q)=\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).$$
{{beginthm|Theorem}}\label{kt} If $2m\ge3q+2p+4$ and $2m\ge3q+3p+4$, in the PL and DIFF categories respectively, then there is a 1-1 correspondence $$E^m(S^p\times S^q)=\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).$$

Revision as of 13:19, 10 April 2010

Contents

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&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.

For notation and conventions throughout this page see high codimension embeddings.

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.

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_{(q)} & 1\le p<q \\ \Zz_{(q)}\oplus\Zz_{(q)}&2\le p=q\end{array}\right.


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 [Becker&Glover1971], Corollary 1.3, 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.

4 References

This page has not been refereed. The information given here might be incomplete or provisional.

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