High codimension links
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Contents |
1 Introduction
This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
2].
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [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] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [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
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [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
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For an -tuple
denote
. Although
is not a manifold when
are not all equal, embeddings
and isotopy between such embeddings are defined analogously to the case of manifolds. Denote by
the set of embeddings
up to isotopy.
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set
for
[Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
![\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}](/images/math/c/b/0/cb0115bc8758c70947d4c589d18b8f99.png)
2 Examples
Recall that for each -manifold
and
, any two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 3.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For each there is an embedding
which is not isotopic to the standard embedding.
For the Hopf link is shown in [Skopenkov2006, Figure 2.1.a].
For arbitrary
(including
) the image of the Hopf link is the union of two
-spheres:
- either
and
in
;
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_q=0\\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.](/images/math/6/8/2/6829fdbed4157dc4c7d8f47f04d5d2a0.png)
This embedding is distinguished from the standard embedding by the linking coefficient (3).
Analogously for each one constructs an embedding
which is not isotopic to the standard embedding. The image is the union of two spheres:
- either
and
in
.
- or given by equations:
![\displaystyle \left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.](/images/math/6/e/f/6efc58d4d378417a14bddb51f191fe8f.png)
This embedding is distinguished from the standard embedding also by the linking coefficient (3).
Definition 2.2 (The Zeeman map). We define a map
![\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.](/images/math/0/9/a/09ad75c13f73655afdc8792313e1e10c.png)
Denote by the equatorial inclusion.
For a map
representing an element of
let
![\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,](/images/math/b/b/6/bb61a5b51c36f7416677c11da556647c.png)
where is the standard embedding [Skopenkov2006, Figure 3.2].
We have
.
Let
.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
![\displaystyle \lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.](/images/math/8/c/c/8cc1a4f0c392389024d67a262842c987.png)
Take an embedding representing an element
.
Take an embedding
such that
intersects
transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1].
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 the Hurewicz and Whitehead theorems
is a homotopy equivalence.)
Let be a homotopy inverse of
.
Define
![\displaystyle \lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).](/images/math/7/0/2/702421c61c119e9f17cfea8b085f374a.png)
Remark 3.2.
(a) Clearly, is indeed independent of
.
One can check that
is a homomorphism.
(b) For or
there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].
(c) Analogously one can define for
, by exchanging
and
in the above definition.
(d) This definition extends to the case when
is simply-connected
(or, equivalently for
, if the restriction of
to
is unknotted).
(e) Clearly, , even for
(see Definition 2.2). So
is surjective and
is injective.
By the Freudenthal Suspension Theorem is an isomorphism for
.
The stabilization of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map
for
.
Take an embedding
representing an element
.
Define a map [Skopenkov2006, Figure 3.1]
![\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)
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 [Skopenkov2006, Figure 3.4].
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
.)
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have .
Hence .
Note that -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006,
5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1.
(D) If , then both
and
are isomorphisms for
in the smooth category.
(PL) If , then both
and
are isomorphisms for
in the PL category.
The surjectivity of (or the injectivity of
) follows from
.
The injectivity of
(or the surjectivity of
) is proved in [Haefliger1962t], [Zeeman1962]
(or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).
An analogue of this result holds for links with many components [Haefliger1966a].
Theorem 4.2.
The collection of pairwise linking coefficients is bijective for and
-dimensional links in
.
5 Examples beyond the metastable range
We present an example of the non-injectivity of the collection of pairwise linking coefficients, which shows that the dimension restriction is sharp in Theorem 4.2.
Borromean rings example 5.1.
The embedding defined below is a non-trivial embedding whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].
Denote coordinates in by
.
The (higher-dimensional) Borromean rings [Skopenkov2006, Figures 3.5 and 3.6], i
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/4/d/a/4da01a49e79810e7d1e951b83b8654f1.png)
The required embedding is any embedding whose image consists of Borromean rings.
This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because joining the three components with two tubes, i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c], yields a non-trivial knot [Haefliger1962], cf. the Haefliger Trefoil knot [Skopenkov2016t]).
For this and other results of this section are parts of low-dimensional link theory and so were known well before given references.
Next we present an example of the non-injectivity of the linking coefficient, which shows that the dimension restriction is sharp in Theorem 4.1.
Whitehead link example 5.2. There is a non-trivial embedding whose linking coefficient
is trivial.
The (higher-dimensional) Whitehead link is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c].
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
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18].
For
the Whitehead link is is distinguished from the standard embedding by more complicated invariants
[Skopenkov2006a], [Haefliger1962t,
3].
This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
6 Classification in the 2-metastable range
Let .
For some information about this group see [Skopenkov2006,
3.3].
Theorem 6.1.
[Haefliger1966a] If , then
![\displaystyle E^m_D(S^n)\cong E^m_{PL}(S^n)\oplus \bigoplus_{i=1}^s C^{m-n_i}_{n_i}.](/images/math/6/4/e/64ef28e29f5b46e7c5b755d7cae3b4c2.png)
Thus is isomorphic to the kernel of the homomorphism
and to
the group of
-Brunnian links (i.e. of links whose restrictions to the components are unknotted).
The Haefliger Theorem 6.2.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009]
If and
, then there is a homomorphism
for which the following map is an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).](/images/math/d/c/1/dc19a7ae429cb4a280766454f3ed2bec.png)
The map and its right inverse
are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].
The case is called a `2-metastable range' for embeddings of
-polyhdera in
.
Remark 6.3.
(a) The Haefliger Theorem 6.2(b) implies that for each we have an isomorphism
![\displaystyle \lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{(l)}.](/images/math/1/5/c/15c0b5ace2d8461137d36356e2f29a95.png)
(b) When but
, the 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/6/c/b/6cb7dea060340fc473c8e7dcfe4bbffc.png)
is injective and its image is .
This is proved in [Haefliger1962t] and also follows from Theorem 6.2(b).
(c) For the map
in (b) above is not injective [Haefliger1962t].
(d) [Haefliger1962t] For each we have an isomorphism
![\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{(l)}^4.](/images/math/e/d/c/edc0f2bba4e8b235793094ea91a38d01.png)
which is the sum of 3 pairwise linking coefficients, 3 pairwise -invariants and triplewise Massey invariant.
7 Classification beyond the 2-metastable range
In this subsection we assume that is an
-tuple such that
. We describe an exact sequence of Haefliger [Haefliger1966a] which contains the group
.
Definition 7.1 (the Haefliger sequence). Let us define the following long sequence of abelian groups
Tex syntax error
Denote and
.
For an
-tuple
denote
.
For each
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![f \colon S^n\to S^m](/images/math/2/e/3/2e3b86d414c11abe48f8d3cd64137d6b.png)
![C_f](/images/math/1/6/7/167f6972258fe5040b8a18821795ddda.png)
![W_n^m](/images/math/6/e/7/6e7eca58e98532f5759b4995aeb121d0.png)
![C_f \sim W_n^m](/images/math/f/8/d/f8d39a257204d7553a3fc22686ff9e2e.png)
Tex syntax error, see details in [Haefliger1966a, 1.4] (this is a generalization of linking coefficient). Define
![\lambda:= \oplus_{j=1}^s \lambda_j](/images/math/e/f/9/ef902edceb5115c1c01c3299283e3096.png)
Denote by the class of the identity.
![\iota_{n_j}](/images/math/0/d/d/0dd56f0b1fe4b1006906096ee699fda8.png)
Tex syntax error. Define
![w:=\oplus_{j=1}^s w_j](/images/math/1/1/4/114e1dc727b4fc9d3c3f29a49e2ab23d.png)
Definition of the homomorphism is sketched in [Haefliger1966a, 1.5].
Theorem 7.2. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011, Lemma 1.3] After tensoring with the rational numbers ,
, and so the Haefliger sequence in (a) above splits into short exact sequencs
![\displaystyle 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} \Lambda_n^m \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.](/images/math/3/2/b/32b85688b82d05d9849716ff29e6f1c2.png)
- An effective procedure for computing the rank of the group
can be found in [Crowley&Ferry&Skopenkov2011,
1.3].
- Necessary and sufficient conditions on
and
for when
is finite can be found in [Crowley&Ferry&Skopenkov2011,
1.2].
For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, 2.5], [Skopenkov2016k].
8 References
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-spheres in
-space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
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in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
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