Knotted tori

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1 Introduction

Many interesting examples of embeddings are embeddings ${{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[$\SS^p\times S^q\to\Rr^m$ , 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, $\S$ 1, $\S$ 3]. We assume that $p\le q$ . Denote

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

2 Examples

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

One of the first examples were Hudson tori.

Let us construct a map

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

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

$\displaystyle S^p\times S^q\overset{\varphi\times{\rm pr}_2}\to\partial D^{m-q}\times S^q\overset{\subset}\to D^{m-q}\times S^q\overset{{\rm i}_{m,q}}\to\Rr^m.$

Here

Tex syntax error

${\rm pr}_2$ is the projection onto the second factor and $\subset$ $\subset$ 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

$\displaystyle \phantom{}_{\rm i}\#:E^m_D(S^{p+q})\to KT^m_{p,q,D}\quad\text{by}\quad \phantom{}_{\rm i}\#(g):=0\#g=[{\rm i}_{m,q}]\#g.$

Clearly,

Tex syntax error

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

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

3 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.

Tex syntax error

$KT^m_{p,q,\#}:=KT^m_{p,q,D}/\phantom{}_{\rm i}\#(E^m_D(S^{p+q}))$ . 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 $m\ge2p+q+3$ 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 $m\ge2p+q+3$ $m\ge2p+q+3$ 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,

Tex syntax error

$\overline{\sigma}\circ\phantom{}_{\rm i}\#={\rm id}$ .

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 $\S$ 1].

For $m\ge2p+q+3$ the map

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

is an isomorphism.

4 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.

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

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 4.5 below.

There are isomorphisms, or, for $p\in\{q,q-1\}$ , 1-1 correspondences

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

The isomorphisms and 1-1 correspondences are given by the Whitney invariant [Skopenkov2016e].

We have $2l-1+2\cdot2l+1=6l$ ; a description of $KT^{6l}_{2l-1,2l,D}$ is given in [Skopenkov2016e, end of $\S$ 6.3].

Theorem 4.1 can be generalized as follows.

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

$\displaystyle \tau:\pi_q(V_{m-q,p+1})\to KT^m_{p,q,PL}\quad\text{and}\quad \tau\oplus_i\#:\pi_q(V_{m-q,p+1})\oplus E^m_D(S^{p+q})\to KT^m_{p,q,D}$ $\displaystyle \tau:\pi_q(V_{m-q,p+1})\to KT^m_{p,q,PL}\quad\text{and}\quad \tau\oplus_i\#:\pi_q(V_{m-q,p+1})\oplus E^m_D(S^{p+q})\to KT^m_{p,q,D}$

are isomorphisms.

(b) If $q\le2p$ , then

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

is a 1-1 correspondence.

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

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

This follows for $m\ge 2q+3$ 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 configuration space of distinct pairs. For $m\ge2p+q+3$ there is an alternative direct proof of (a) [Skopenkov2006], [Skopenkov2015a], but for $m\le2p+q+2$ no proof of Theorem 4.2.(b)(c) without referring to `the Haefliger-Wu invariant' is known.

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

Assume that $m\ge 2p+q+3$ .

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

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

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

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 $K^m_{q,n}:=\ker\lambda\cap E^m_U(S^q\sqcup S^n)$ .

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

(b) For $m\ge2p+q+3$

$\displaystyle KT^m_{p,q}\cong E^m_D(D^{p+1}\times S^q)\oplus K^m_{q,p+q}\oplus E^m_D(S^{p+q}).$ $\displaystyle KT^m_{p,q}\cong E^m_D(D^{p+1}\times S^q)\oplus K^m_{q,p+q}\oplus E^m_D(S^{p+q}).$

For a discussion see [Skopenkov2015a, Remark 1.9].

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

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].

5 References

[Alexander1924] J. W. Alexander, On the subdivision of 3-space by polyhedron, Proc. Nat. Acad. Sci. USA, 10, (1924) 6–8. Zbl 50.0659.01
[Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension $4$ dans $\Rr^7$ , (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
[Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension $4k$ dans $\Rr^{6k+1}$ , Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
[Cencelj&Repovš&Skopenkov2007] M. Cencelj, D. Repovš and M. Skopenkov, Homotopy type of the complement of an immersion and classification of embeddings of tori., Russ. Math. Surv.62 (2007), no.5, 985-987. Zbl 1141.57009
[Cencelj&Repovš&Skopenkov2008] M. Cencelj, D. Repovš and M. Skopenkov, Classification of knotted tori in the 2-metastable dimension, Mat. Sbornik, 203:11 (2012), 1654-1681. Available at the arXiv:0811.2745.
[Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
[Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
[Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
[Kosinski1961] A. Kosinski, On Alexander's theorem and knotted tori, In: Topology of 3-Manifolds, Prentice-Hall, Englewood Cliffs, Ed. M.~K.~Fort, N.J., 1962, 55--57. Cf. Fort1962.
[Lucas&Saeki2002] L. A. Lucas and O. Saeki, Embeddings of $S^p\times S^q\times S^r$ in $S^{p+q+r+1}$ , Pacific J. Math. 207 (2002), no.2, 447–462. MR1972255 (2004c:57045) Zbl 1058.57022
[Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
[Schmidt1971] R. Schultz, On the inertia groups of a product of spheres, Trans. AMS, 156 (1971), 137–153.
[Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
[Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
[Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$ , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
[Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
[Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
[Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
[Tindell1969] R. Tindell, Extending homeomorphisms of $S\sp p \times S\sp q$ ., Proc. Am. Math. Soc. 22 (1969), 230-232. MRMR0248852 (40 #2102) Zbl 0177.26802
[Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602

, $\S]{Skopenkov2016c}. We assume that $p\le q$. Denote $$KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).$$ == Examples == ; 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}. One of the first examples were [[Embeddings just below the stable range: classification#Hudson_tori|Hudson tori]]. 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 $$\phantom{}_{\rm i}\#:E^m_D(S^{p+q})\to KT^m_{p,q,D}\quad\text{by}\quad \phantom{}_{\rm i}\#(g):=0\#g=[{\rm i}_{m,q}]\#g.$$ 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}. == Reduction to classification modulo knots == ; Let $KT^m_{p,q,\#}$ be the quotient set of $KT^m_{p,q,D}$ by the embedded connected sum action, i.e. $KT^m_{p,q,\#}:=KT^m_{p,q,D}/\phantom{}_{\rm i}\#(E^m_D(S^{p+q}))$. Let $q_\#:KT^m_{p,q,D}\to KT^m_{p,q,\#}$ be 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[$\S.3]{Skopenkov2015a}. Clearly, $\overline{\sigma}\circ\phantom{}_{\rm i}\#={\rm id}$. 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, $\S$ 1, $\S$ 3]. We assume that $p\le q$ . Denote

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

2 Examples

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

One of the first examples were Hudson tori.

Let us construct a map

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

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

$\displaystyle S^p\times S^q\overset{\varphi\times{\rm pr}_2}\to\partial D^{m-q}\times S^q\overset{\subset}\to D^{m-q}\times S^q\overset{{\rm i}_{m,q}}\to\Rr^m.$

Here

Tex syntax error

${\rm pr}_2$ is the projection onto the second factor and $\subset$ $\subset$ 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

$\displaystyle \phantom{}_{\rm i}\#:E^m_D(S^{p+q})\to KT^m_{p,q,D}\quad\text{by}\quad \phantom{}_{\rm i}\#(g):=0\#g=[{\rm i}_{m,q}]\#g.$

Clearly,

Tex syntax error

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

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

3 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 $m\ge2p+q+3$ 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 $m\ge2p+q+3$ $m\ge2p+q+3$ 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 $KT^m_{p,q,D}$ to description of $E^m_D(S^{p+q})$ and of $KT^m_{p,q,\#}$ , cf. [Schmidt1971], [Crowley&Skopenkov2008, end of $\S$ 1].

For $m\ge2p+q+3$ the map

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

is an isomorphism.

4 Classification