Knotted tori
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Contents |
1 Introduction
Many interesting examples of embeddings are embeddings , 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, 1,
3]. Denote
![\displaystyle KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).](/images/math/5/1/d/51d15e1f4491e986c51af0482e4191fd.png)
2 Examples
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}.](/images/math/5/b/8/5b8c1559bb1d04447b03444502fd8439.png)
Recall that is isomorphic to the group of smooth maps
up to smooth homotopy. The latter maps can be considered as smooth maps
. Define the smooth embedding
as the composition
Tex syntax error
Tex syntax erroris the projection onto the second factor and
![\subset](/images/math/e/7/8/e78407b806f164e4860cb8443ebc5eda.png)
See construction of another map in [Skopenkov2015, 3, definition of
].
An -parametric connected sum group structure on
is constructed for
in [Skopenkov2006], [Skopenkov2015a].
3 Classification
From the Haefliger-Zeeman Isotopy Theorem it follows that for
and
, provided that
or
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 and for
, 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}](/images/math/2/f/0/2f0f34f47375238141335ccc24414086.png)
Here is short for
. The table follows from the theorems below.
We also have and
, of which
is rank one infinite [Skopenkov2015a].
Theorem 3.1. There are isomorphisms, or, for , 1-1 correspondences
![\displaystyle E^{p+2q+1}_{PL}(S^p\times S^q)\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 E^{p+2q+1}_{DIFF}(S^p\times S^q)\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.](/images/math/a/7/0/a70bf538b8d166036704ad2445f71802.png)
A description of is given in [Skopenkov2016e, end of
6.3].
Theorem 3.1 can be generalized as follows.
![2m\ge3q+2p+4](/images/math/d/5/b/d5bf0aed91f3f376d0e377465c027038.png)
![2m\ge3q+3p+4](/images/math/e/e/0/ee06e5c1762480a2137cb79da2ea60d8.png)
![\displaystyle E^m(S^p\times S^q)=\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).](/images/math/0/2/8/02841202f5ff6238d81a419acf0005d5.png)
This follows for from the Becker-Glover Theorem 5.3. For the general case see [Skopenkov2002, Corollary 1.5.a]. For
there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for
no proof of Theorem 3.2 without referring to `the deleted product method' is known.
For (which is automatic for
and
) we have
and the 1-1 correspondence of Theorem 3.2 is
.
Let be the quotient set of
by the embedded connected sum action and
the quotient map.
For
a group structure on
is well-defined by
,
.
The following result reduces description of
to description of
and of
, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of \S1].
Lemma 3.3 (Smoothing).
For we have
.
The isomorphism of Lemma 3.3 is , where
is `surgery of
'.
Denote by the torsion subgroup of an abelian group
. Abelian group structures on
for
is defined analogously to the well-known case
. The sum operation on
is `connected sum of
-spheres together with normal
-framings' or `
-parametric connected sum'. Define
to be the subgroup of links all whose components are unknotted. Let
be the linking coefficient. Denote
.
Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a].
Assume that .
(a) .
(b)
(more precisely, whenever one part is finite, the other is finite and they are equal).
(c) , unless
and
for some
.
For methods of calculation of the objects appearing in the right-hand side of the formulas from Theorem 3.4 see [Skopenkov2015a, \1.3].
4 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
dans
, (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
dans
, 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
in
, 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
, 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
., 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
A description of $KT^{6l}_{2l-1,2l,DIFF}$ is given in \cite[end of $\S.3]{Skopenkov2016e}. Theorem \ref{kt1} can be generalized as follows. {{beginthm|Theorem}}\label{kt} If m\ge3q+2p+4$ and m\ge3q+3p+4$, in the PL and DIFF categories respectively, then there is a 1-1 correspondence $$E^m(S^p\times S^q)=\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).$$ {{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} 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 \S1]{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 $S^p\times*$'. 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}} For methods of calculation of the objects appearing in the right-hand side of the formulas from Theorem \ref{t:cornum} 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,
1,
3]. Denote
2 Examples
One of the first examples were Hudson tori.
Let us construct a map
Recall that
is isomorphic to the group of smooth maps
up to smooth homotopy. The latter maps can be considered as smooth maps
. Define the smooth embedding
as the composition
HereTex syntax errorTex syntax erroris the projection onto the second factor andis the standard inclusion.
See construction of another map in [Skopenkov2015,
3, definition of
].
An
-parametric connected sum group structure on
is constructed for
in [Skopenkov2006], [Skopenkov2015a].
3 Classification
From the Haefliger-Zeeman Isotopy Theorem it follows that
for
and
, provided that
or
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
and for
, for the PL and smooth categories, respectively.
Here
is short for
. The table follows from the theorems below.
We also have
and
, of which
is rank one infinite [Skopenkov2015a].
Theorem 3.1. There are isomorphisms, or, for
, 1-1 correspondences
A description of
is given in [Skopenkov2016e, end of
6.3].
Theorem 3.1 can be generalized as follows.
Theorem 3.2. Ifand
, in the PL and DIFF categories respectively, then there is a 1-1 correspondence
This follows for
from the Becker-Glover Theorem 5.3. For the general case see [Skopenkov2002, Corollary 1.5.a]. For
there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for
no proof of Theorem 3.2 without referring to `the deleted product method' is known.
For
(which is automatic for
and
) we have
and the 1-1 correspondence of Theorem 3.2 is
.
Let
be the quotient set of
by the embedded connected sum action and
the quotient map. For
a group structure on
is well-defined by
,
. The following result reduces description of
to description of
and of
, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of \S1].
Lemma 3.3 (Smoothing). For
we have
.
The isomorphism of Lemma 3.3 is
, where
is `surgery of
'.
Denote by
the torsion subgroup of an abelian group
. Abelian group structures on
for
is defined analogously to the well-known case
. The sum operation on
is `connected sum of
-spheres together with normal
-framings' or `
-parametric connected sum'. Define
to be the subgroup of links all whose components are unknotted. Let
be the linking coefficient. Denote
.
Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a]. Assume that
.
(a)
.
(b)
(more precisely, whenever one part is finite, the other is finite and they are equal).
(c)
, unless
and
for some
.
For methods of calculation of the objects appearing in the right-hand side of the formulas from Theorem 3.4 see [Skopenkov2015a, \
1.3].
4 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
dans
, (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
dans
, 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
in
, 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
, 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
.3]{Skopenkov2015a}.
- [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
., 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
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. Denote
![\displaystyle KT^m_{p,q,CAT}:=E^m_{CAT}(S^p\times S^q).](/images/math/5/1/d/51d15e1f4491e986c51af0482e4191fd.png)
2 Examples
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}.](/images/math/5/b/8/5b8c1559bb1d04447b03444502fd8439.png)
Recall that is isomorphic to the group of smooth maps
up to smooth homotopy. The latter maps can be considered as smooth maps
. Define the smooth embedding
as the composition
Tex syntax error
Tex syntax erroris the projection onto the second factor and
![\subset](/images/math/e/7/8/e78407b806f164e4860cb8443ebc5eda.png)
See construction of another map in [Skopenkov2015, 3, definition of
].
An -parametric connected sum group structure on
is constructed for
in [Skopenkov2006], [Skopenkov2015a].
3 Classification
From the Haefliger-Zeeman Isotopy Theorem it follows that for
and
, provided that
or
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 and for
, 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}](/images/math/2/f/0/2f0f34f47375238141335ccc24414086.png)
Here is short for
. The table follows from the theorems below.
We also have and
, of which
is rank one infinite [Skopenkov2015a].
Theorem 3.1. There are isomorphisms, or, for , 1-1 correspondences
![\displaystyle E^{p+2q+1}_{PL}(S^p\times S^q)\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 E^{p+2q+1}_{DIFF}(S^p\times S^q)\to\Zz_{\varepsilon(q)}\quad\text{for}\quad 1\le p\le q-2.](/images/math/a/7/0/a70bf538b8d166036704ad2445f71802.png)
A description of is given in [Skopenkov2016e, end of
6.3].
Theorem 3.1 can be generalized as follows.
![2m\ge3q+2p+4](/images/math/d/5/b/d5bf0aed91f3f376d0e377465c027038.png)
![2m\ge3q+3p+4](/images/math/e/e/0/ee06e5c1762480a2137cb79da2ea60d8.png)
![\displaystyle E^m(S^p\times S^q)=\pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}).](/images/math/0/2/8/02841202f5ff6238d81a419acf0005d5.png)
This follows for from the Becker-Glover Theorem 5.3. For the general case see [Skopenkov2002, Corollary 1.5.a]. For
there is an alternative direct proof [Skopenkov2006], [Skopenkov2015a], but for
no proof of Theorem 3.2 without referring to `the deleted product method' is known.
For (which is automatic for
and
) we have
and the 1-1 correspondence of Theorem 3.2 is
.
Let be the quotient set of
by the embedded connected sum action and
the quotient map.
For
a group structure on
is well-defined by
,
.
The following result reduces description of
to description of
and of
, cf. [Schmidt1971], [Crowley&Skopenkov2008, end of \S1].
Lemma 3.3 (Smoothing).
For we have
.
The isomorphism of Lemma 3.3 is , where
is `surgery of
'.
Denote by the torsion subgroup of an abelian group
. Abelian group structures on
for
is defined analogously to the well-known case
. The sum operation on
is `connected sum of
-spheres together with normal
-framings' or `
-parametric connected sum'. Define
to be the subgroup of links all whose components are unknotted. Let
be the linking coefficient. Denote
.
Theorem 3.4 [Skopenkov2015, Corollary 1.7], [Skopenkov2015a].
Assume that .
(a) .
(b)
(more precisely, whenever one part is finite, the other is finite and they are equal).
(c) , unless
and
for some
.
For methods of calculation of the objects appearing in the right-hand side of the formulas from Theorem 3.4 see [Skopenkov2015a, \1.3].
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