# Knotted tori

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## 1 Introduction

By a torus we mean a product of spheres $S^p \times S^q$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}S^p \times S^q$, regarded as a closed manifold. Many interesting examples of embeddings are embeddings $S^p\times S^q\to\Rr^m$$S^p\times S^q\to\Rr^m$, i.e. knotted tori. For examples, see Hudson tori, [Alexander1924], [Kosinski1961], [Hudson1963], [Wall1965], [Tindell1969], [Boechat&Haefliger1970], [Boechat1971], [Milgram&Rees1971], [Lucas&Saeki2002], [Skopenkov2002]. Classifying knotted tori is a natural next step (after the classification of links [Haefliger1966a] and the classification of embeddings of highly-connected manifolds) towards the classification of embeddings of arbitrary manifolds. Since the general Knotting Problem is very hard, it is interesting to solve it for the important special 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 reveal 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, $\S$$\S$1, $\S$$\S$3]. We assume that $p\le q$$p\le q$. Denote

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

For definition of the embedded connected sumof embeddings, denote by $\#$$\#$, and for the corresponding action of the group $E^m_D(S^{p+q})$$E^m_D(S^{p+q})$ on the set $KT^m_{p,q,D}$$KT^m_{p,q,D}$, see e.g. [Skopenkov2016c, $\S$$\S$4].

## 2 Examples

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

Some of the first examples of knotted tori are the Hudson tori. See also examples of knotted 4-dimensional tori in 7-space [Skopenkov2016f, $\S$$\S$2].

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})$$\pi_q(V_{m-q,p+1})$ is isomorphic to the group of smooth maps $S^q\to V_{m-q,p+1}$$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}$$\varphi:S^q\times S^p\to\partial D^{m-q}$. Define the smooth embedding $\tau(\varphi)$$\tau(\varphi)$ as the composition

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Here $\subset$$\subset$ is the standard inclusion. Clearly, $\tau$$\tau$ is well-defined and, for $m\ge2p+q+3$$m\ge2p+q+3$, is a homomorphism.

Define the embedded connected sum' or local knotting' map

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Clearly,
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$\phantom{}_{\rm i}\#$ is well-defined and, for $m\ge2p+q+3$$m\ge2p+q+3$, is a homomorphism.

See construction of another map in [Skopenkov2015, $\S$$\S$3, definition of $\sigma^*$$\sigma^*$].

## 3 Classification just below the stable range

From the Haefliger-Zeeman Isotopy Theorem it follows that $KT^m_{p,q}=0$$KT^m_{p,q}=0$ for $p\le q$$p\le q$ and $m\ge p+2q+2$$m\ge p+2q+2$, provided that $m\ge p+q+3$$m\ge p+q+3$ or $2m\ge3(p+q)+4$$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.

For the next theorem, the Whitney invariant $W$$W$ is defined in [Skopenkov2016e, $\S$$\S$5].

Theorem 3.1. (a) The Whitney invariant

$\displaystyle W:KT^{p+2q+1}_{p,q}\to\Zz_{\varepsilon(q)}$

is an isomorphism for $1\le p\le q-2$$1\le p\le q-2$.

(b) The Whitney invariants

$\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{\varepsilon(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{\varepsilon(q)}\oplus \Zz_{\varepsilon(q)}$

are bijective for $q\ge2$$q\ge2$.

(c) The Whitney invariant $W^{6l}_{2l-1,2l}$$W^{6l}_{2l-1,2l}$ is surjective and for any $u\in\Zz$$u\in\Zz$ there is a 1-1 correspondence $W^{-1}(u)\to\Zz_u$$W^{-1}(u)\to\Zz_u$.

Theorem 3.1 follows from Theorems 6.2 and 6.3 of [Skopenkov2016e] (parts (a) and (b) also follow from Theorem 5.1 below). The Hudson torus $\Hud(1)$$\Hud(1)$ generates $KT^{p+2q+1}_{p,q}$$KT^{p+2q+1}_{p,q}$ for $1\le p\le q-2$$1\le p\le q-2$ (this holds by [Skopenkov2016e, Theorem 6.2] because $W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$$W(\Hud(1),\Hud(0))=1\in\Zz_{\varepsilon(q)}$).

In the smooth category for $q$$q$ even $W^{3q}_{q-1,q}$$W^{3q}_{q-1,q}$ is not injective (by Theorem 3.1.c), $W^{3q+1}_{q,q}$$W^{3q+1}_{q,q}$ is not surjective [Boechat1971], [Skopenkov2016f], and $W^7_{2,2}$$W^7_{2,2}$ is not injective [Skopenkov2016f].

## 4 Reduction to classification modulo knots

Let $KT^m_{p,q,\#}$$KT^m_{p,q,\#}$ be the quotient set of $KT^m_{p,q,D}$$KT^m_{p,q,D}$ by the embedded connected sum action, i.e.
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$KT^m_{p,q,\#}:=KT^m_{p,q,D}/\phantom{}_{\rm i}\#(E^m_D(S^{p+q}))$. Let $q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#}$$q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#}$ be the quotient map.

For $m\ge2p+q+3$$m\ge2p+q+3$ a group structure on $KT^m_{p,q,\#}$$KT^m_{p,q,\#}$ is well-defined by $q_{\#}f+q_{\#}f':=q_{\#}(f+f')$$q_{\#}f+q_{\#}f':=q_{\#}(f+f')$, $f,f'\in KT^m_{p,q,D}$$f,f'\in KT^m_{p,q,D}$.

For $m\ge2p+q+3$$m\ge2p+q+3$ the map $\overline{\sigma}:KT^m_{p,q,D}\to E^m(S^{p+q})$$\overline{\sigma}:KT^m_{p,q,D}\to E^m(S^{p+q})$ constructed by embedded surgery of $S^p\times*$$S^p\times*$' is well-defined [Skopenkov2015a, $\S$$\S$3.3]. Clearly,
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$\overline{\sigma}\circ\phantom{}_{\rm i}\#={\rm id}$.

The following result reduces description of $KT^m_{p,q,D}$$KT^m_{p,q,D}$ to description of $E^m_D(S^{p+q})$$E^m_D(S^{p+q})$ and of $KT^m_{p,q,\#}$$KT^m_{p,q,\#}$, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of $\S$$\S$1].

Lemma 4.1 [Skopenkov2015a, Smoothing Lemma 1.1]. For $m\ge2p+q+3$$m\ge2p+q+3$ the map

$\displaystyle q_\#\oplus\overline{\sigma}:KT^m_{p,q,D}\to KT^m_{p,q,\#}\oplus E^m_D(S^{p+q})$
is an isomorphism.

## 5 Further classification

We have the following table for $2m\ge3q+6$$2m\ge3q+6$ and for $2m\ge3q+7$$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$$n$ is short for $\Z_n$$\Z_n$. We also have $|KT^{10}_{1,5}|=4$$|KT^{10}_{1,5}|=4$ and $KT^{11}_{1,6}\cong\Z_2\oplus\Z\oplus E^{11}_D(S^7)$$KT^{11}_{1,6}\cong\Z_2\oplus\Z\oplus E^{11}_D(S^7)$, of which $E^{11}_D(S^7)$$E^{11}_D(S^7)$ is rank one infinite. The table and the additional results follow from the theorems below, see [Skopenkov2015a].

There is a finiteness criterion for $KT^m_{D,p,q}$$KT^m_{D,p,q}$ when $m\ge2p+q+3$$m\ge2p+q+3$ [Skopenkov2015, Theorem 1.4]. The formulation is not so short but effective. This criterion is a corollary of Theorem 5.4 below.

Theorem 3.1.(a)(b) can be generalized as follows.

Theorem 5.1. (a) If $m\ge2p+q+3$$m\ge2p+q+3$ and $2m\ge3q+2p+4$$2m\ge3q+2p+4$, then

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are isomorphisms.

(b) If $q\le2p$$q\le2p$, then

$\displaystyle \tau:\pi_q(V_{2p+2,p+1})\to KT^{2p+q+2}_{p,q,PL}$
is a 1-1 correspondence.

(c) If $2m\ge3q+2p+4$$2m\ge3q+2p+4$, then there is a 1-1 correspondence

$\displaystyle KT^m_{p,q,PL}\to\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).$

This follows for $m\ge 2q+3$$m\ge 2q+3$ from the Becker-Glover Theorem 5.3. For the general case see [Skopenkov2002, Corollary 1.5.a]. The 1-1 correspondence is constructed using the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, $\S$$\S$5]. For $m\ge2p+q+3$$m\ge2p+q+3$ there is an alternative direct proof of (a) [Skopenkov2006], [Skopenkov2015a], but for $m\le2p+q+2$$m\le2p+q+2$ no proof of Theorem 5.1.(b)(c) without referring to the Haefliger-Wu invariant' is known.

For $m\ge2p+q+2$$m\ge2p+q+2$ we have $\pi_p(V_{m-p,q+1})=0$$\pi_p(V_{m-p,q+1})=0$, so part (c) reduces to part (b) and the PL case of part (a).

Theorem 5.2 [Skopenkov2015a, Corollary 1.5.(b)(c)]. Assume that $m\ge 2p+q+3$$m\ge 2p+q+3$.

(a) If $2m\ge p+3q+4$$2m\ge p+3q+4$, then $KT^m_{p,q,\#}$$KT^m_{p,q,\#}$ and $\pi_q(V_{m-q,p+1})$$\pi_q(V_{m-q,p+1})$ have isomorphic subgroups with isomorphic quotients.

(b) If $2m\ge 3q+4$$2m\ge 3q+4$, then $KT^m_{p,q,\#}$$KT^m_{p,q,\#}$ has a subgroup isomorphic to $\pi_{p+2q+2-m}(V_{M+m-q-1,M})$$\pi_{p+2q+2-m}(V_{M+m-q-1,M})$ ($M$$M$ large), whose quotient and $\pi_q(V_{m-q,p+1})$$\pi_q(V_{m-q,p+1})$ have isomorphic subgroups with isomorphic quotients.

The following conjecture and results reduce description of $KT^m_{p,q,D}$$KT^m_{p,q,D}$ to description of objects which are easier to calculate, at least in some cases, see [Skopenkov2015a, $\S$$\S$1.3] for methods of their calculations.

Abelian group structure on $E^m(D^p\times S^q)$$E^m(D^p\times S^q)$ for $m\ge q+3$$m\ge q+3$ is defined analogously to the well-known case $p=0$$p=0$. The sum operation on $E^m(D^p\times S^q)$$E^m(D^p\times S^q)$ is connected sum of $q$$q$-spheres together with normal $p$$p$-framings' or $D^p$$D^p$-parametric connected sum'. Define $E^m_U(S^q\sqcup S^n) \subset E^m_D(S^q\sqcup S^n)$$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})$$\lambda=\lambda^m_{q,n}:E^m_D(S^q\sqcup S^n)\to\pi_q(S^{m-n-1})$ be the linking coefficient. Denote $K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n)$$K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n)$.

Conjecture 5.3. Assume that $m\ge2p+q+3$$m\ge2p+q+3$.

(a) If $2m\ge p+3q+4$$2m\ge p+3q+4$, then
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$\tau\oplus\phantom{}_{\rm i}\#:\pi_q(V_{m-q,p+1})\oplus E^m_D(S^{p+q})\to KT^m_{p,q,D}$ is an isomorphism.

(b) The map

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is an isomorphism. Here $r$$r$ is the restriction' map induced by the inclusion $S^p\times S^q\subset D^{p+1}\times S^q$$S^p\times S^q\subset D^{p+1}\times S^q$ and $\sigma$$\sigma$ is defined in [Skopenkov2015, $\S$$\S$3, definition of $\sigma^*$$\sigma^*$].

For a discussion see [Skopenkov2015a, Remark 1.9 and footnote 7].

Denote by $TG$$TG$ the torsion subgroup of an abelian group $G$$G$.

Theorem 5.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a, Corollary 1.4]. Assume that $m\ge 2p+q+3$$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$$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})|$$|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})|$$|TKT^m_{D,p,q}|=|TE^m_D(D^{p+1}\times S^q)|\cdot|TK^m_{q,p+q}|\cdot|TE^m_D(S^{p+q})|$, unless $m=6k+p$$m=6k+p$ and $q=4k-1$$q=4k-1$ for some $k$$k$.

Theorems 5.2 and 5.4 were obtained using more `theoretical' results [Skopenkov2015, Theorem 1.6], [Skopenkov2015a, Theorem 1.2], see also [Cencelj&Repovš&Skopenkov2008, Theorem 2.1].