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{{Stub}} == Introduction == <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}. A 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. Since the general [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|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 \cite{Skopenkov2006a}, \cite{Cencelj&Repovš&Skopenkov2007}, \cite{Cencelj&Repovš&Skopenkov2008}, \cite{Skopenkov2015}, \cite{Skopenkov2015a} give some insight or even precise information concerning arbitrary manifolds (cf. \cite{Skopenkov2007}, \cite{Skopenkov2010}, \cite{Skopenkov2014}) and reveals new interesting relations to algebraic topology. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$3]{Skopenkov2016c}. Denote $$KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).$$ </wikitex> == Examples == <wikitex>; 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 $$\phantom{}_{\rm i}\#:E^m(S^{p+q})\to KT^m_{p,q}\quad\text{by}\quad \phantom{}_{\rm i}\#(g):=0\#g=[{\rm i}]\#g.$$ Clearly, $\phantom{}_{\rm i}\#$ is well-defined and, for $m\ge2p+q+3$, is a homomorphism. See construction of another map in \cite[$\S$3, definition of $\sigma^*$]{Skopenkov2015}. </wikitex> == Reduction to classification modulo knots == <wikitex>; 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 $q_{\#}f+q_{\#}f':=q_{\#}(f+f')$, $f,f'\in KT^m_{p,q,D}$. For $m\ge2p+q+3$ the map $\overline{\sigma}:KT^m_{p,q,D}\to E^m(S^{p+q})$ constructed by `embedded surgery of $S^p\times*$' is well-defined \cite{Skopenkov2015a}. 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 $\S$1]{Crowley&Skopenkov2008}. {{beginthm|Lemma|(Smoothing)}}\label{t:smo} For $m\ge2p+q+3$ the map $$q_{\#}\oplus\overline{\sigma}:KT^m_{p,q,D}\cong KT^m_{p,q,\#}\oplus E^m_D(S^{p+q})$$ is an isomorphism. {{endthm}} </wikitex> == Classification == <wikitex>; 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 $2m\ge3(p+q)+4$ 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]]. We have the following table for $2m\ge3q+6$ and for $2m\ge3q+7$, 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 $$KT^{p+2q+1}_{p,q,PL}\to\left\{\begin{array}{cc} \Zz_{\varepsilon(q)} & \quad 1\le p<q \\ \Zz_{\varepsilon(q)}\oplus\Zz_{\varepsilon(q)}&\quad 2\le p=q\end{array}\right. \qquad\text{and} \qquad KT^{p+2q+1}_{p,q,D}\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.$$ {{endthm}} <!-- Theorem \lc8 follows from Theorem \wi8.b (as well as from Theorem \lc9 below). In the PL case of Theorem \lc8 for $p=q$ we only have a 1--1 correspondence of sets (because Group Structure Theorem \lc7 does not give a group structure for such dimensions). A description of $KT^{6k}_{2k-1,2k,DIFF}$ is given after Theorem \wi9. [HH63, Hu63, Vr77] the above-defined map $\tau$--> We have $2l-1+2\cdot2l+1=6l$; a description of $KT^{6l}_{2l-1,2l,D}$ is given in \cite[end of $\S$6.3]{Skopenkov2016e}. Theorem \ref{kt1} can be generalized as follows. {{beginthm|Theorem}}\label{kt} (a) If $m\ge2p+q+3$ and $2m\ge3q+2p+4$, 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 $2m\ge3q+2p+4$ and $2m\ge3q+3p+4$, 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 $2m\ge3p+3q+4$) we have $\pi_p(V_{m-p,q+1})=0$ and the 1-1 correspondence of Theorem \ref{kt}.b is $\tau$. 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[$\S$1.3]{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}. </wikitex> == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]

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