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] [Haefliger1966C].
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/1/8/b/18b1d7e000cc51571b57f92b03227c8f.png)
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].
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006].
4 Classification below the metastable range
1 Borromean rings
The following example shows that the collection of pairwise linking coefficients is not injective for and
-dimensional links with more than two components in
.
Borromean rings example 4.1. The higher-dimensional Borromean rings
form a non-trivial embedding whose restrictions to 2-componented sublinks are trivial
[Haefliger1962], 4.1, [Haefliger1962T].
Denote coordinates in by
.
The higher-dimensional 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]. They are distinguished from the standard embedding by the Massey invariant.
2 The Whitehead link
3 Classification
See The Haefliger Trefoil knot
5 Further discussion
6 References
- [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
- [Haefliger1966C] Template:Haefliger1966C
- [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.
- [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/1/8/b/18b1d7e000cc51571b57f92b03227c8f.png)
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].
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006].
4 Classification below the metastable range
1 Borromean rings
The following example shows that the collection of pairwise linking coefficients is not injective for and
-dimensional links with more than two components in
.
Borromean rings example 4.1. The higher-dimensional Borromean rings
form a non-trivial embedding whose restrictions to 2-componented sublinks are trivial
[Haefliger1962], 4.1, [Haefliger1962T].
Denote coordinates in by
.
The higher-dimensional 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]. They are distinguished from the standard embedding by the Massey invariant.
2 The Whitehead link
3 Classification
See The Haefliger Trefoil knot
5 Further discussion
6 References
- [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
- [Haefliger1966C] Template:Haefliger1966C
- [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.
- [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