High codimension links
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Contents |
1 Introduction
For notation and conventions throughout this page see high codimension embeddings.
`Embedded connected sum' defines a commutative group structure on for
.
See Figure 3.3. of [Skopenkov2006], [Haefliger1966] [Haefliger1966a].
2 General position and the Hopf linking
General Position Theorem 2.1.
For each -manifold
and
, every two embeddings
are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2.
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure 2.1.a of [Skopenkov2006].
For arbitrary
(including
) the image of the Hopf Linking is the union of two
-spheres:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..](/images/math/1/b/f/1bf21a0c192a704c36d5acdafcf28ed8.png)
This embedding is distinguished from the standard embedding by the linking coefficient.
3 The Zeeman construction and linking coefficient
The following table was obtained by Zeeman around 1960:
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &2q+2 &2q+1 &2q &2q-1 &2q-2 &2q-3 &2q-4 \\ \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/b/2/2/b22cac6dff110756c80daca3082b8178.png)
1 Construction of the Zeeman map ![\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)](/images/math/2/f/9/2f905654e1b2410eb01a0f263eceaefc.png)
Take
Define embedding
on
to be the standard embedding into
.
Take any map
.
Define embedding
on
to be the composition
![\displaystyle S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,](/images/math/a/2/9/a29aef5baaba94d8741863f8a2bbe06c.png)
where is the equatorial inclusion and the latter inclusion is the standard.
See Figure 3.2 of [Skopenkov2006].
Clearly,
is well-defined and is a homomorphism.
2 Definition of linking coefficient
for ![m\ge q+3](/images/math/9/2/1/9217752e4d784a6c1079aac45fccd82e.png)
Fix orientations of ,
,
and
.
Take an embedding
.
Take an embedding
such that
intersects
transversally at exactly one point with positive sign (see Figure 3.1 of [Skopenkov2006]).
Then the restriction
of
to
is a homotopy equivalence.
(Indeed, since , the complement
is simply-connected.
By Alexander duality
induces isomorphism in homology.
Hence by Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/0/5/c/05c1bffa3c231c4e8f9df10db3009149.png)
Remark 3.1.
(a) Clearly, is indeed independent of
.
Clearly,
is a homomorphism.
(b) For there is a simpler alternative `homological' definition.
That definition works for
as well.
(c) Analogously one can define for
.
(d) This definition works for if
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
.
So
is surjective and
is injective.
3 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 3.2.
If , then both
and
are isomorphisms for
and for
,
in the PL and DIFF cases, respectively.
The surjectivity of (=the injectivity of
) follows from
.
The injectivity of
(=the surjectivity of
) is proved in [Haefliger1962T], [Zeeman1962]
(or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).
An analogue of this result holds for links with many components: the collection of pairwise linking
coefficients is bijective for and
-dimensional links in
.
4 Alpha-invariant
By Freudenthal Suspension Theorem is an isomorphism for
.
The stable suspension of the linking coefficient can be described alternatively as follows.
For an embedding
define a map
![\displaystyle \widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.](/images/math/8/8/a/88ae206e439a2fc6abf78f5b5a853d36.png)
See Figure 3.1 of [Skopenkov2006].
For define the
-invariant by
![\displaystyle \alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.](/images/math/8/3/f/83faec70806146f2d1f254e1efe2193d.png)
The second isomorphism in this formula is given by the Freudenthal Suspension Theorem.
The map is the quotient map.
See Figure 3.4 of [Skopenkov2006].
The map
is an isomorphism for
.
(For this follows by general position and for
by the cofibration Barratt-Puppe exact sequence of pair
and by the existence of a retraction
.)
We have by Lemma 5.1 of [Kervaire1959L].
Hence
.
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006], \S5.
4 Classification below the metastable range
1 Higher-dimensional Borromean rings
Let us present an example of non-injectivity of the collection of pairwise linking coefficients.
Borromean rings example 4.1.
The Borromean rings
is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial
[Haefliger1962], 4.1, [Haefliger1962T].
Denote coordinates in by
.
The Borromean rings are the three spheres given by the following three systems of equations:
![\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..](/images/math/b/3/f/b3f574425ee3eab2ba05dc7a32c1a793.png)
See Figures 3.5 and 3.6 of [Skopenkov2006]. This embedding is distinguished from the standard embedding by the Massey invariant.
2 Higher-dimensional Whitehead link
Let us present an example of non-injectivity of the linking coefficient.
Whitehead link example 4.2. The Whitehead link
is a non-trivial embedding whose linking coefficient
is trivial.
The Whitehead link is obtained from Borromean rings by joining two components with a tube.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
.
(It would be interesting to find or write a published proof of this fact.)
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962T], \S3.
This example shows that the dimension restriction is sharp in Theorem 3.2.
This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared.
3 Classification
See The Haefliger Trefoil knot.
Let .
Some information on this group.
The Haefliger Theorem 4.3.
(a) [Haefliger1966a] If , then
![\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.](/images/math/8/6/7/86756c51d19aa743079191fa1a8ed0d5.png)
(c) [Haefliger1966a], Theorem 10.7, [Skopenkov2009]
If and
, then
![\displaystyle E^m_{PL}(S^p\sqcup S^q)\cong\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/8/5/6/8566e8ee361f46a4c6dca50671d135ce.png)
The isomorphism in (b) is given by the sum of
-invariant and the
-invariant [Skopenkov2009].
Part (b) implies that .
This isomorphism is defined for
,
by map
![\displaystyle \lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l).](/images/math/1/6/1/161df3a5e454c564f55da25ba89bec67.png)
This map is injective for ,
; the image of this map is
[Haefliger1962T].
Part (b) shows that
is in general not injective.
5 Further discussion
See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011].
6 References
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Haefliger1962] A. Haefliger, Knotted
-spheres in
-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962T] Template:Haefliger1962T
- [Haefliger1966] A. Haefliger, Differential embeddings of
in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [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
- [Kervaire1959L] Template:Kervaire1959L
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [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.
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
![m-3\ge p,q](/images/math/4/f/b/4fbb02dd450593b466c83ce047a60181.png)
2 General position and the Hopf linking
General Position Theorem 2.1.
For each -manifold
and
, every two embeddings
are isotopic.
The restriction in Theorem 2.1 is sharp for non-connected manifolds.
Example: the Hopf linking 2.2.
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf Linking is shown in Figure 2.1.a of [Skopenkov2006].
For arbitrary
(including
) the image of the Hopf Linking is the union of two
-spheres:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..](/images/math/1/b/f/1bf21a0c192a704c36d5acdafcf28ed8.png)
This embedding is distinguished from the standard embedding by the linking coefficient.
3 The Zeeman construction and linking coefficient
The following table was obtained by Zeeman around 1960:
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &2q+2 &2q+1 &2q &2q-1 &2q-2 &2q-3 &2q-4 \\ \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/b/2/2/b22cac6dff110756c80daca3082b8178.png)
1 Construction of the Zeeman map ![\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)](/images/math/2/f/9/2f905654e1b2410eb01a0f263eceaefc.png)
Take
Define embedding
on
to be the standard embedding into
.
Take any map
.
Define embedding
on
to be the composition
![\displaystyle S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,](/images/math/a/2/9/a29aef5baaba94d8741863f8a2bbe06c.png)
where is the equatorial inclusion and the latter inclusion is the standard.
See Figure 3.2 of [Skopenkov2006].
Clearly,
is well-defined and is a homomorphism.
2 Definition of linking coefficient
for ![m\ge q+3](/images/math/9/2/1/9217752e4d784a6c1079aac45fccd82e.png)
Fix orientations of ,
,
and
.
Take an embedding
.
Take an embedding
such that
intersects
transversally at exactly one point with positive sign (see Figure 3.1 of [Skopenkov2006]).
Then the restriction
of
to
is a homotopy equivalence.
(Indeed, since , the complement
is simply-connected.
By Alexander duality
induces isomorphism in homology.
Hence by Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/0/5/c/05c1bffa3c231c4e8f9df10db3009149.png)
Remark 3.1.
(a) Clearly, is indeed independent of
.
Clearly,
is a homomorphism.
(b) For there is a simpler alternative `homological' definition.
That definition works for
as well.
(c) Analogously one can define for
.
(d) This definition works for if
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
.
So
is surjective and
is injective.
3 Classification in the `metastable' range
The Haefliger-Zeeman Theorem 3.2.
If , then both
and
are isomorphisms for
and for
,
in the PL and DIFF cases, respectively.
The surjectivity of (=the injectivity of
) follows from
.
The injectivity of
(=the surjectivity of
) is proved in [Haefliger1962T], [Zeeman1962]
(or follows from the Haefliger-Weber Theorem 5.4 and Deleted Product Lemma 5.3.a of [Skopenkov2006]).
An analogue of this result holds for links with many components: the collection of pairwise linking
coefficients is bijective for and
-dimensional links in
.
4 Alpha-invariant
By Freudenthal Suspension Theorem is an isomorphism for
.
The stable suspension of the linking coefficient can be described alternatively as follows.
For an embedding
define a map
![\displaystyle \widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.](/images/math/8/8/a/88ae206e439a2fc6abf78f5b5a853d36.png)
See Figure 3.1 of [Skopenkov2006].
For define the
-invariant by
![\displaystyle \alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.](/images/math/8/3/f/83faec70806146f2d1f254e1efe2193d.png)
The second isomorphism in this formula is given by the Freudenthal Suspension Theorem.
The map is the quotient map.
See Figure 3.4 of [Skopenkov2006].
The map
is an isomorphism for
.
(For this follows by general position and for
by the cofibration Barratt-Puppe exact sequence of pair
and by the existence of a retraction
.)
We have by Lemma 5.1 of [Kervaire1959L].
Hence
.
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006], \S5.
4 Classification below the metastable range
1 Higher-dimensional Borromean rings
Let us present an example of non-injectivity of the collection of pairwise linking coefficients.
Borromean rings example 4.1.
The Borromean rings
is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial
[Haefliger1962], 4.1, [Haefliger1962T].
Denote coordinates in by
.
The Borromean rings are the three spheres given by the following three systems of equations:
![\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..](/images/math/b/3/f/b3f574425ee3eab2ba05dc7a32c1a793.png)
See Figures 3.5 and 3.6 of [Skopenkov2006]. This embedding is distinguished from the standard embedding by the Massey invariant.
2 Higher-dimensional Whitehead link
Let us present an example of non-injectivity of the linking coefficient.
Whitehead link example 4.2. The Whitehead link
is a non-trivial embedding whose linking coefficient
is trivial.
The Whitehead link is obtained from Borromean rings by joining two components with a tube.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
.
(It would be interesting to find or write a published proof of this fact.)
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962T], \S3.
This example shows that the dimension restriction is sharp in Theorem 3.2.
This example seems to be discovered by Whitehead, in connection with Whitehead product. It would be interesting to find a publication where it first appeared.
3 Classification
See The Haefliger Trefoil knot.
Let .
Some information on this group.
The Haefliger Theorem 4.3.
(a) [Haefliger1966a] If , then
![\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.](/images/math/8/6/7/86756c51d19aa743079191fa1a8ed0d5.png)
(c) [Haefliger1966a], Theorem 10.7, [Skopenkov2009]
If and
, then
![\displaystyle E^m_{PL}(S^p\sqcup S^q)\cong\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/8/5/6/8566e8ee361f46a4c6dca50671d135ce.png)
The isomorphism in (b) is given by the sum of
-invariant and the
-invariant [Skopenkov2009].
Part (b) implies that .
This isomorphism is defined for
,
by map
![\displaystyle \lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l).](/images/math/1/6/1/161df3a5e454c564f55da25ba89bec67.png)
This map is injective for ,
; the image of this map is
[Haefliger1962T].
Part (b) shows that
is in general not injective.
5 Further discussion
See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011].
6 References
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Haefliger1962] A. Haefliger, Knotted
-spheres in
-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962T] Template:Haefliger1962T
- [Haefliger1966] A. Haefliger, Differential embeddings of
in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [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
- [Kervaire1959L] Template:Kervaire1959L
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [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.
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069