Knotted tori

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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&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]. Denote

\displaystyle KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).

2 Examples

One of the first examples were Hudson tori.

An S^p-parametric connected sum group structure on KT^m_{p,q} is constructed for m\ge2p+q+3 in [Skopenkov2006], [Skopenkov2015a].

Let us construct a map

\displaystyle \tau:\pi_q(V_{m-q,p+1})\to KT^m_{p,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

Tex syntax error
Here
Tex syntax error
is the projection onto the second factor and \subset is the standard inclusion.

Clearly, \tau is well-defined and, for m\ge2p+q+3, is a homomorphism.

Define the `embedded connected sum' or `local knotting' map}

Tex syntax error
Clearly,
Tex syntax error
is well-defined and, for m\ge2p+q+3, is a homomorphism.

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

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, provided that 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.

We have the following table for 2m\ge3q+6 and for 2m\ge3q+7, for the PL and smooth categories, respectively.

\displaystyle \begin{array}{c|c|c|c|c|c|c|c|c} m                         &\ge2q+3 &2q+2 &2q+1      &2q &2q-1     &2q-2&2q-3\\ \hline KT^m_{1,q},\ q\text{ even}&0       &\Z   &2         &2^2&2^2      &24  &0\\ \hline KT^m_{1,q},\ q\text{ odd} &0       &2    &\Z\oplus2 &4  &2\oplus24&2   &0 \end{array}

Here n is short for \Z_n. The table follows from the theorems below.

We also have |KT^{10}_{1,5}|=4 and KT^{11}_{1,6}\cong\Z_2\oplus\Z\oplus E^{11}_D(S^7), of which E^{11}_D(S^7) is rank one infinite [Skopenkov2015a].

There is a finiteness criterion for KT^m_{D,p,q} when m\ge2p+q+3 [Skopenkov2015, Theorem 1.4]. The formulation is not so short but effective. This criterion is a corollary of Theorem 3.4 below.

Theorem 3.1. There are isomorphisms, or, for p\in\{q,q-1\}, 1-1 correspondences

\displaystyle KT^{p+2q+1}_{p,q,PL}\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 KT^{p+2q+1}_{p,q,D}\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.

We have 2l-1+2\cdot2l+1=6l; a description of KT^{6l}_{2l-1,2l,D} is given in [Skopenkov2016e, end of \S6.3].

Theorem 3.1 can be generalized as follows.

Theorem 3.2. (a) If m\ge2p+q+3 and 2m\ge3q+2p+4, then

\displaystyle \tau\oplus_i\#:\pi_q(V_{m-q,p+1})\oplus E^m(S^{p+q})\to KT^m_{p,q}

is an isomorphism.

(b) 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 KT^m_{p,q}\to\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. For the general case see [Skopenkov2002, Corollary 1.5.a]. For m\ge2p+q+3 there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for m<2p+q+2 no proof of Theorem 3.2 without referring to `the deleted product method' is known.

For m\ge2p+q+2 (which is automatic for p\le q and 2m\ge3p+3q+4) we have \pi_p(V_{m-p,q+1})=0 and the 1-1 correspondence of Theorem 3.2.b is \tau.

Let KT^m_{p,q,\#} be the quotient set of KT^m_{p,q,D} by the embedded connected sum action and q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#} the quotient map. For m\ge2p+q+3 a group structure on KT^m_{p,q,\#} is well-defined by q_{\#}f+q_{\#}f':=q_{\#}(f+f'), f,f'\in KT^m_{p,q,D}. The following result reduces description of KT^m_{p,q,D} to description of E^m_D(S^{p+q}) and of KT^m_{p,q,\#}, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of \S1].

Lemma 3.3 (Smoothing). For m\ge2p+q+3 we have KT^m_{p,q,D}\cong KT^m_{p,q,\#}\oplus E^m_D(S^{p+q}).

The isomorphism of Lemma 3.3 is q_{\#}\oplus\overline{\sigma}, where \overline{\sigma} is `surgery of S^p\times*'.

The following result reduces description of KT^m_{p,q,D} to description of objects which are easier to calculate, at least in some cases, see [Skopenkov2015a, \S1.3] for methods of their calculations.

Denote by TG the torsion subgroup of an abelian group G. Abelian group structures on E^m(D^p\times S^q) for m\ge q+3 is defined analogously to the well-known case p=0. The sum operation on E^m(D^p\times S^q) is `connected sum of q-spheres together with normal p-framings' or `D^p-parametric connected sum'. Define E^m_U(S^q\sqcup S^n) \subset E^m_D(S^q\sqcup S^n) to be the subgroup of links all whose components are unknotted. Let \lambda=\lambda^m_{q,n}:E^m_D(S^q\sqcup S^n)\to\pi_q(S^{m-n-1}) be the linking coefficient. Denote K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n).

Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a]. Assume that m\ge 2p+q+3.

(a) KT^m_{D,p,q}\otimes\Q\cong[\pi_q(V_{m-q,p+1})\oplus E^m_D(S^q)\oplus K^m_{q,p+q}\oplus E^m_D(S^{p+q})]\otimes\Q.

(b) |KT^m_{D,p,q}|=|E^m_D(D^{p+1}\times S^q)|\cdot|K^m_{q,p+q}|\cdot|E^m_D(S^{p+q})| (more precisely, whenever one part is finite, the other is finite and they are equal).

(c) |TKT^m_{D,p,q}|=|TE^m_D(D^{p+1}\times S^q)|\cdot|TK^m_{q,p+q}|\cdot|TE^m_D(S^{p+q})|, unless m=6k+p and q=4k-1 for some k.

The above results were obtained using more `theoretical' results [Skopenkov2015, Theorem 1.6], [Skopenkov2015a, Theorem 1.2], see also [Cencelj&Repovš&Skopenkov2008, Theorem 2.1].

4 References

, $\S]{Skopenkov2016c}. Denote $$KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).$$ == Examples == ; One of the first examples were [[Embeddings just below the stable range: classification#Hudson_tori|Hudson tori]]. An [[Parametric_connected_sum#Applications|$S^p$-parametric connected sum]] group structure on $KT^m_{p,q}$ is constructed for $m\ge2p+q+3$ in \cite{Skopenkov2006}, \cite{Skopenkov2015a}. Let us construct a map $$\tau:\pi_q(V_{m-q,p+1})\to KT^m_{p,q}.$$ 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 $$S^p\times S^q\overset{\varphi\times{\rm pr}_2}\to\partial D^{m-q}\times S^q\overset{\subset}\to D^{m-q}\times S^q\overset{{\rm i}_{m,q}}\to\Rr^m.$$ Here ${\rm pr}_2$ is the projection onto the second factor and $\subset$ is the standard inclusion. Clearly, $\tau$ is well-defined and, for $m\ge2p+q+3$, is a homomorphism. Define the `embedded connected sum' or `local knotting' map} $$\phantom{}_{\rm i}\#:E^m(S^{p+q})\to KT^m_{p,q}\quad\text{by}\quad \cs(g):=0\#g=[{\rm i}]\#g.$$ Clearly, $\phantom{}_{\rm i}\#$ is well-defined and, for $m\ge2p+q+3$, is a homomorphism. See construction of another map in \cite[$\S, definition of $\sigma^*$]{Skopenkov2015}. == Classification == ; From [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|the Haefliger-Zeeman Isotopy Theorem]] it follows that $E^m(S^p\times S^q)=0$ for $p\le q$ and $m\ge p+2q+2$, provided that $m\ge p+q+3$ or m\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]]. We have the following table for m\ge3q+6$ and for m\ge3q+7$, for the PL and smooth categories, respectively. $$\begin{array}{c|c|c|c|c|c|c|c|c} m &\ge2q+3 &2q+2 &2q+1 &2q &2q-1 &2q-2&2q-3\ \hline KT^m_{1,q},\ q\text{ even}&0 &\Z &2 &2^2&2^2 &24 &0\ \hline KT^m_{1,q},\ q\text{ odd} &0 &2 &\Z\oplus2 &4 &2\oplus24&2 &0 \end{array}$$ Here $n$ is short for $\Z_n$. The table follows from the theorems below. We also have $|KT^{10}_{1,5}|=4$ and $KT^{11}_{1,6}\cong\Z_2\oplus\Z\oplus E^{11}_D(S^7)$, of which $E^{11}_D(S^7)$ is rank one infinite \cite{Skopenkov2015a}. There is a finiteness criterion for $KT^m_{D,p,q}$ when $m\ge2p+q+3$ \cite[Theorem 1.4]{Skopenkov2015}. The formulation is not so short but effective. This criterion is a corollary of Theorem \ref{t:cornum} below. {{beginthm|Theorem}}\label{kt1} There are isomorphisms, or, for $p\in\{q,q-1\}$, 1-1 correspondences $$KT^{p+2q+1}_{p,q,PL}\to\left\{\begin{array}{cc} \Zz_{\varepsilon(q)} & \quad 1\le p We have l-1+2\cdot2l+1=6l$; a description of $KT^{6l}_{2l-1,2l,D}$ is given in \cite[end of $\S.3]{Skopenkov2016e}. Theorem \ref{kt1} can be generalized as follows. {{beginthm|Theorem}}\label{kt} (a) If $m\ge2p+q+3$ and m\ge3q+2p+4$, then $$\tau\oplus_i\#:\pi_q(V_{m-q,p+1})\oplus E^m(S^{p+q})\to KT^m_{p,q}$$ is an isomorphism. (b) If m\ge3q+2p+4$ and m\ge3q+3p+4$, in the PL and DIFF categories respectively, then there is a 1-1 correspondence $$KT^m_{p,q}\to\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).$$ {{endthm}} This follows for $m\ge 2q+3$ from [[Embeddings_just_below_the_stable_range:_classification#Classification_further_below_the_stable_range|the Becker-Glover Theorem 5.3]]. For the general case see \cite[Corollary 1.5.a]{Skopenkov2002}. For $m\ge2p+q+3$ there is an alternative direct proof \cite{Skopenkov2006}, \cite{Skopenkov2015a}, but for $m<2p+q+2$ no proof of Theorem \ref{kt} without referring to `the deleted product method' is known. For $m\ge2p+q+2$ (which is automatic for $p\le q$ and m\ge3p+3q+4$) we have $\pi_p(V_{m-p,q+1})=0$ and the 1-1 correspondence of Theorem \ref{kt}.b is $\tau$. Let $KT^m_{p,q,\#}$ be the quotient set of $KT^m_{p,q,D}$ by the embedded connected sum action and $q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#}$ the quotient map. For $m\ge2p+q+3$ a group structure on $KT^m_{p,q,\#}$ is well-defined by $q_{\#}f+q_{\#}f':=q_{\#}(f+f')$, $f,f'\in KT^m_{p,q,D}$. The following result reduces description of $KT^m_{p,q,D}$ to description of $E^m_D(S^{p+q})$ and of $KT^m_{p,q,\#}$, cf. \cite{Schmidt1971}, \cite[end of $\SS^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]. Denote

\displaystyle KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).

2 Examples

One of the first examples were Hudson tori.

An S^p-parametric connected sum group structure on KT^m_{p,q} is constructed for m\ge2p+q+3 in [Skopenkov2006], [Skopenkov2015a].

Let us construct a map

\displaystyle \tau:\pi_q(V_{m-q,p+1})\to KT^m_{p,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

Tex syntax error
Here
Tex syntax error
is the projection onto the second factor and \subset is the standard inclusion.

Clearly, \tau is well-defined and, for m\ge2p+q+3, is a homomorphism.

Define the `embedded connected sum' or `local knotting' map}

Tex syntax error
Clearly,
Tex syntax error
is well-defined and, for m\ge2p+q+3, is a homomorphism.

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

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, provided that 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.

We have the following table for 2m\ge3q+6 and for 2m\ge3q+7, for the PL and smooth categories, respectively.

\displaystyle \begin{array}{c|c|c|c|c|c|c|c|c} m                         &\ge2q+3 &2q+2 &2q+1      &2q &2q-1     &2q-2&2q-3\\ \hline KT^m_{1,q},\ q\text{ even}&0       &\Z   &2         &2^2&2^2      &24  &0\\ \hline KT^m_{1,q},\ q\text{ odd} &0       &2    &\Z\oplus2 &4  &2\oplus24&2   &0 \end{array}

Here n is short for \Z_n. The table follows from the theorems below.

We also have |KT^{10}_{1,5}|=4 and KT^{11}_{1,6}\cong\Z_2\oplus\Z\oplus E^{11}_D(S^7), of which E^{11}_D(S^7) is rank one infinite [Skopenkov2015a].

There is a finiteness criterion for KT^m_{D,p,q} when m\ge2p+q+3 [Skopenkov2015, Theorem 1.4]. The formulation is not so short but effective. This criterion is a corollary of Theorem 3.4 below.

Theorem 3.1. There are isomorphisms, or, for p\in\{q,q-1\}, 1-1 correspondences

\displaystyle KT^{p+2q+1}_{p,q,PL}\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 KT^{p+2q+1}_{p,q,D}\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.

We have 2l-1+2\cdot2l+1=6l; a description of KT^{6l}_{2l-1,2l,D} is given in [Skopenkov2016e, end of \S6.3].

Theorem 3.1 can be generalized as follows.

Theorem 3.2. (a) If m\ge2p+q+3 and 2m\ge3q+2p+4, then

\displaystyle \tau\oplus_i\#:\pi_q(V_{m-q,p+1})\oplus E^m(S^{p+q})\to KT^m_{p,q}

is an isomorphism.

(b) 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 KT^m_{p,q}\to\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. For the general case see [Skopenkov2002, Corollary 1.5.a]. For m\ge2p+q+3 there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for m<2p+q+2 no proof of Theorem 3.2 without referring to `the deleted product method' is known.

For m\ge2p+q+2 (which is automatic for p\le q and 2m\ge3p+3q+4) we have \pi_p(V_{m-p,q+1})=0 and the 1-1 correspondence of Theorem 3.2.b is \tau.

Let KT^m_{p,q,\#} be the quotient set of KT^m_{p,q,D} by the embedded connected sum action and q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#} the quotient map. For m\ge2p+q+3 a group structure on KT^m_{p,q,\#} is well-defined by q_{\#}f+q_{\#}f':=q_{\#}(f+f'), f,f'\in KT^m_{p,q,D}. The following result reduces description of KT^m_{p,q,D} to description of E^m_D(S^{p+q}) and of KT^m_{p,q,\#}, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of \S1].

Lemma 3.3 (Smoothing). For m\ge2p+q+3 we have KT^m_{p,q,D}\cong KT^m_{p,q,\#}\oplus E^m_D(S^{p+q}).

The isomorphism of Lemma 3.3 is q_{\#}\oplus\overline{\sigma}, where \overline{\sigma} is `surgery of S^p\times*'.

The following result reduces description of KT^m_{p,q,D} to description of objects which are easier to calculate, at least in some cases, see [Skopenkov2015a, \S1.3] for methods of their calculations.

Denote by TG the torsion subgroup of an abelian group G. Abelian group structures on E^m(D^p\times S^q) for m\ge q+3 is defined analogously to the well-known case p=0. The sum operation on E^m(D^p\times S^q) is `connected sum of q-spheres together with normal p-framings' or `D^p-parametric connected sum'. Define E^m_U(S^q\sqcup S^n) \subset E^m_D(S^q\sqcup S^n) to be the subgroup of links all whose components are unknotted. Let \lambda=\lambda^m_{q,n}:E^m_D(S^q\sqcup S^n)\to\pi_q(S^{m-n-1}) be the linking coefficient. Denote K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n).

Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a]. Assume that m\ge 2p+q+3.

(a) KT^m_{D,p,q}\otimes\Q\cong[\pi_q(V_{m-q,p+1})\oplus E^m_D(S^q)\oplus K^m_{q,p+q}\oplus E^m_D(S^{p+q})]\otimes\Q.

(b) |KT^m_{D,p,q}|=|E^m_D(D^{p+1}\times S^q)|\cdot|K^m_{q,p+q}|\cdot|E^m_D(S^{p+q})| (more precisely, whenever one part is finite, the other is finite and they are equal).

(c) |TKT^m_{D,p,q}|=|TE^m_D(D^{p+1}\times S^q)|\cdot|TK^m_{q,p+q}|\cdot|TE^m_D(S^{p+q})|, unless m=6k+p and q=4k-1 for some k.

The above results were obtained using more `theoretical' results [Skopenkov2015, Theorem 1.6], [Skopenkov2015a, Theorem 1.2], see also [Cencelj&Repovš&Skopenkov2008, Theorem 2.1].

4 References

]{Crowley&Skopenkov2008}. {{beginthm|Lemma|(Smoothing)}}\label{t:smo} For $m\ge2p+q+3$ we have $KT^m_{p,q,D}\cong KT^m_{p,q,\#}\oplus E^m_D(S^{p+q})$. {{endthm}} The isomorphism of Lemma \ref{t:smo} is $q_{\#}\oplus\overline{\sigma}$, where $\overline{\sigma}$ is `surgery of $S^p\times*$'. The following result reduces description of $KT^m_{p,q,D}$ to description of objects which are easier to calculate, at least in some cases, see \cite[$\SS^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]. Denote

\displaystyle KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).

2 Examples

One of the first examples were Hudson tori.

An S^p-parametric connected sum group structure on KT^m_{p,q} is constructed for m\ge2p+q+3 in [Skopenkov2006], [Skopenkov2015a].

Let us construct a map

\displaystyle \tau:\pi_q(V_{m-q,p+1})\to KT^m_{p,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

Tex syntax error
Here
Tex syntax error
is the projection onto the second factor and \subset is the standard inclusion.

Clearly, \tau is well-defined and, for m\ge2p+q+3, is a homomorphism.

Define the `embedded connected sum' or `local knotting' map}

Tex syntax error
Clearly,
Tex syntax error
is well-defined and, for m\ge2p+q+3, is a homomorphism.

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

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, provided that 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.

We have the following table for 2m\ge3q+6 and for 2m\ge3q+7, for the PL and smooth categories, respectively.

\displaystyle \begin{array}{c|c|c|c|c|c|c|c|c} m                         &\ge2q+3 &2q+2 &2q+1      &2q &2q-1     &2q-2&2q-3\\ \hline KT^m_{1,q},\ q\text{ even}&0       &\Z   &2         &2^2&2^2      &24  &0\\ \hline KT^m_{1,q},\ q\text{ odd} &0       &2    &\Z\oplus2 &4  &2\oplus24&2   &0 \end{array}

Here n is short for \Z_n. The table follows from the theorems below.

We also have |KT^{10}_{1,5}|=4 and KT^{11}_{1,6}\cong\Z_2\oplus\Z\oplus E^{11}_D(S^7), of which E^{11}_D(S^7) is rank one infinite [Skopenkov2015a].

There is a finiteness criterion for KT^m_{D,p,q} when m\ge2p+q+3 [Skopenkov2015, Theorem 1.4]. The formulation is not so short but effective. This criterion is a corollary of Theorem 3.4 below.

Theorem 3.1. There are isomorphisms, or, for p\in\{q,q-1\}, 1-1 correspondences

\displaystyle KT^{p+2q+1}_{p,q,PL}\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 KT^{p+2q+1}_{p,q,D}\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.

We have 2l-1+2\cdot2l+1=6l; a description of KT^{6l}_{2l-1,2l,D} is given in [Skopenkov2016e, end of \S6.3].

Theorem 3.1 can be generalized as follows.

Theorem 3.2. (a) If m\ge2p+q+3 and 2m\ge3q+2p+4, then

\displaystyle \tau\oplus_i\#:\pi_q(V_{m-q,p+1})\oplus E^m(S^{p+q})\to KT^m_{p,q}

is an isomorphism.

(b) 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 KT^m_{p,q}\to\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. For the general case see [Skopenkov2002, Corollary 1.5.a]. For m\ge2p+q+3 there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for m<2p+q+2 no proof of Theorem 3.2 without referring to `the deleted product method' is known.

For m\ge2p+q+2 (which is automatic for p\le q and 2m\ge3p+3q+4) we have \pi_p(V_{m-p,q+1})=0 and the 1-1 correspondence of Theorem 3.2.b is \tau.

Let KT^m_{p,q,\#} be the quotient set of KT^m_{p,q,D} by the embedded connected sum action and q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#} the quotient map. For m\ge2p+q+3 a group structure on KT^m_{p,q,\#} is well-defined by q_{\#}f+q_{\#}f':=q_{\#}(f+f'), f,f'\in KT^m_{p,q,D}. The following result reduces description of KT^m_{p,q,D} to description of E^m_D(S^{p+q}) and of KT^m_{p,q,\#}, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of \S1].

Lemma 3.3 (Smoothing). For m\ge2p+q+3 we have KT^m_{p,q,D}\cong KT^m_{p,q,\#}\oplus E^m_D(S^{p+q}).

The isomorphism of Lemma 3.3 is q_{\#}\oplus\overline{\sigma}, where \overline{\sigma} is `surgery of S^p\times*'.

The following result reduces description of KT^m_{p,q,D} to description of objects which are easier to calculate, at least in some cases, see [Skopenkov2015a, \S1.3] for methods of their calculations.

Denote by TG the torsion subgroup of an abelian group G. Abelian group structures on E^m(D^p\times S^q) for m\ge q+3 is defined analogously to the well-known case p=0. The sum operation on E^m(D^p\times S^q) is `connected sum of q-spheres together with normal p-framings' or `D^p-parametric connected sum'. Define E^m_U(S^q\sqcup S^n) \subset E^m_D(S^q\sqcup S^n) to be the subgroup of links all whose components are unknotted. Let \lambda=\lambda^m_{q,n}:E^m_D(S^q\sqcup S^n)\to\pi_q(S^{m-n-1}) be the linking coefficient. Denote K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n).

Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a]. Assume that m\ge 2p+q+3.

(a) KT^m_{D,p,q}\otimes\Q\cong[\pi_q(V_{m-q,p+1})\oplus E^m_D(S^q)\oplus K^m_{q,p+q}\oplus E^m_D(S^{p+q})]\otimes\Q.

(b) |KT^m_{D,p,q}|=|E^m_D(D^{p+1}\times S^q)|\cdot|K^m_{q,p+q}|\cdot|E^m_D(S^{p+q})| (more precisely, whenever one part is finite, the other is finite and they are equal).

(c) |TKT^m_{D,p,q}|=|TE^m_D(D^{p+1}\times S^q)|\cdot|TK^m_{q,p+q}|\cdot|TE^m_D(S^{p+q})|, unless m=6k+p and q=4k-1 for some k.

The above results were obtained using more `theoretical' results [Skopenkov2015, Theorem 1.6], [Skopenkov2015a, Theorem 1.2], see also [Cencelj&Repovš&Skopenkov2008, Theorem 2.1].

4 References

.3]{Skopenkov2015a} for methods of their calculations. Denote by $TG$ the torsion subgroup of an abelian group $G$. Abelian group structures on $E^m(D^p\times S^q)$ for $m\ge q+3$ is defined analogously to the well-known case $p=0$. The sum operation on $E^m(D^p\times S^q)$ is `connected sum of $q$-spheres together with normal $p$-framings' or `$D^p$-parametric connected sum'. Define $E^m_U(S^q\sqcup S^n) \subset E^m_D(S^q\sqcup S^n)$ to be the subgroup of links all whose components are unknotted. Let $\lambda=\lambda^m_{q,n}:E^m_D(S^q\sqcup S^n)\to\pi_q(S^{m-n-1})$ be the [[High_codimension_links#The_linking_coefficient|linking coefficient]]. Denote $K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n)$. {{beginthm|Theorem|\cite[Corollary 1.7]{Skopenkov2015}, \cite{Skopenkov2015a}}}\label{t:cornum} Assume that $m\ge 2p+q+3$. (a) $KT^m_{D,p,q}\otimes\Q\cong[\pi_q(V_{m-q,p+1})\oplus E^m_D(S^q)\oplus K^m_{q,p+q}\oplus E^m_D(S^{p+q})]\otimes\Q$. (b) $|KT^m_{D,p,q}|=|E^m_D(D^{p+1}\times S^q)|\cdot|K^m_{q,p+q}|\cdot|E^m_D(S^{p+q})|$ (more precisely, whenever one part is finite, the other is finite and they are equal). (c) $|TKT^m_{D,p,q}|=|TE^m_D(D^{p+1}\times S^q)|\cdot|TK^m_{q,p+q}|\cdot|TE^m_D(S^{p+q})|$, unless $m=6k+p$ and $q=4k-1$ for some $k$. {{endthm}} The above results were obtained using more `theoretical' results \cite[Theorem 1.6]{Skopenkov2015}, \cite[Theorem 1.2]{Skopenkov2015a}, see also \cite[Theorem 2.1]{Cencelj&Repovš&Skopenkov2008}.
== References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]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]. Denote

\displaystyle KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).

2 Examples

One of the first examples were Hudson tori.

An S^p-parametric connected sum group structure on KT^m_{p,q} is constructed for m\ge2p+q+3 in [Skopenkov2006], [Skopenkov2015a].

Let us construct a map

\displaystyle \tau:\pi_q(V_{m-q,p+1})\to KT^m_{p,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

Tex syntax error
Here
Tex syntax error
is the projection onto the second factor and \subset is the standard inclusion.

Clearly, \tau is well-defined and, for m\ge2p+q+3, is a homomorphism.

Define the `embedded connected sum' or `local knotting' map}

Tex syntax error
Clearly,
Tex syntax error
is well-defined and, for m\ge2p+q+3, is a homomorphism.

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

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, provided that 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.

We have the following table for 2m\ge3q+6 and for 2m\ge3q+7, for the PL and smooth categories, respectively.

\displaystyle \begin{array}{c|c|c|c|c|c|c|c|c} m                         &\ge2q+3 &2q+2 &2q+1      &2q &2q-1     &2q-2&2q-3\\ \hline KT^m_{1,q},\ q\text{ even}&0       &\Z   &2         &2^2&2^2      &24  &0\\ \hline KT^m_{1,q},\ q\text{ odd} &0       &2    &\Z\oplus2 &4  &2\oplus24&2   &0 \end{array}

Here n is short for \Z_n. The table follows from the theorems below.

We also have |KT^{10}_{1,5}|=4 and KT^{11}_{1,6}\cong\Z_2\oplus\Z\oplus E^{11}_D(S^7), of which E^{11}_D(S^7) is rank one infinite [Skopenkov2015a].

There is a finiteness criterion for KT^m_{D,p,q} when m\ge2p+q+3 [Skopenkov2015, Theorem 1.4]. The formulation is not so short but effective. This criterion is a corollary of Theorem 3.4 below.

Theorem 3.1. There are isomorphisms, or, for p\in\{q,q-1\}, 1-1 correspondences

\displaystyle KT^{p+2q+1}_{p,q,PL}\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 KT^{p+2q+1}_{p,q,D}\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.

We have 2l-1+2\cdot2l+1=6l; a description of KT^{6l}_{2l-1,2l,D} is given in [Skopenkov2016e, end of \S6.3].

Theorem 3.1 can be generalized as follows.

Theorem 3.2. (a) If m\ge2p+q+3 and 2m\ge3q+2p+4, then

\displaystyle \tau\oplus_i\#:\pi_q(V_{m-q,p+1})\oplus E^m(S^{p+q})\to KT^m_{p,q}

is an isomorphism.

(b) 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 KT^m_{p,q}\to\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. For the general case see [Skopenkov2002, Corollary 1.5.a]. For m\ge2p+q+3 there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for m<2p+q+2 no proof of Theorem 3.2 without referring to `the deleted product method' is known.

For m\ge2p+q+2 (which is automatic for p\le q and 2m\ge3p+3q+4) we have \pi_p(V_{m-p,q+1})=0 and the 1-1 correspondence of Theorem 3.2.b is \tau.

Let KT^m_{p,q,\#} be the quotient set of KT^m_{p,q,D} by the embedded connected sum action and q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#} the quotient map. For m\ge2p+q+3 a group structure on KT^m_{p,q,\#} is well-defined by q_{\#}f+q_{\#}f':=q_{\#}(f+f'), f,f'\in KT^m_{p,q,D}. The following result reduces description of KT^m_{p,q,D} to description of E^m_D(S^{p+q}) and of KT^m_{p,q,\#}, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of \S1].

Lemma 3.3 (Smoothing). For m\ge2p+q+3 we have KT^m_{p,q,D}\cong KT^m_{p,q,\#}\oplus E^m_D(S^{p+q}).

The isomorphism of Lemma 3.3 is q_{\#}\oplus\overline{\sigma}, where \overline{\sigma} is `surgery of S^p\times*'.

The following result reduces description of KT^m_{p,q,D} to description of objects which are easier to calculate, at least in some cases, see [Skopenkov2015a, \S1.3] for methods of their calculations.

Denote by TG the torsion subgroup of an abelian group G. Abelian group structures on E^m(D^p\times S^q) for m\ge q+3 is defined analogously to the well-known case p=0. The sum operation on E^m(D^p\times S^q) is `connected sum of q-spheres together with normal p-framings' or `D^p-parametric connected sum'. Define E^m_U(S^q\sqcup S^n) \subset E^m_D(S^q\sqcup S^n) to be the subgroup of links all whose components are unknotted. Let \lambda=\lambda^m_{q,n}:E^m_D(S^q\sqcup S^n)\to\pi_q(S^{m-n-1}) be the linking coefficient. Denote K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n).

Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a]. Assume that m\ge 2p+q+3.

(a) KT^m_{D,p,q}\otimes\Q\cong[\pi_q(V_{m-q,p+1})\oplus E^m_D(S^q)\oplus K^m_{q,p+q}\oplus E^m_D(S^{p+q})]\otimes\Q.

(b) |KT^m_{D,p,q}|=|E^m_D(D^{p+1}\times S^q)|\cdot|K^m_{q,p+q}|\cdot|E^m_D(S^{p+q})| (more precisely, whenever one part is finite, the other is finite and they are equal).

(c) |TKT^m_{D,p,q}|=|TE^m_D(D^{p+1}\times S^q)|\cdot|TK^m_{q,p+q}|\cdot|TE^m_D(S^{p+q})|, unless m=6k+p and q=4k-1 for some k.

The above results were obtained using more `theoretical' results [Skopenkov2015, Theorem 1.6], [Skopenkov2015a, Theorem 1.2], see also [Cencelj&Repovš&Skopenkov2008, Theorem 2.1].

4 References

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