High codimension links
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1 Introduction
Most of 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.
On this page we describe readily calculable classifications of embeddings of closed disconnected manifolds into up to isotopy, and more generally for spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings
, where
are spheres (or even closed manifolds) and
for every
, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].
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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
10 References
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-spheres in
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in
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, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
10 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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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
10 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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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
10 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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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
10 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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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
10 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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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
10 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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![N_1\sqcup\ldots\sqcup N_s\to S^m](/images/math/8/2/2/822da9a4dd55b38205b8cada87309644.png)
![N_1,\ldots, N_s](/images/math/9/3/a/93abda1b7f4a337f5b8a2ba7cd39fbb2.png)
![m-3\ge\dim N_i](/images/math/2/f/8/2f8f4116266d20f8c14512d29ffe5054.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.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.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
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)
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 Figure 1.
The standard embedding is defined by
. Fix
pairwise disjoint
-discs
. The standard embedding
is defined by taking the union of the compositions of the standard embeddings
with the fixed inclusions
.
2 Examples
Recall that for any -manifold
and
, every two embeddings
are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1].
The following example shows that the restriction
is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link).
For every positive integer there is an embedding
, which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all
the image of the Hopf link is the union of two
-spheres which can be described as follows:
- either the spheres are
and
in
;
- or they are given as the sets of points in
satisfying the following 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 (see 3).
Analogously for any one constructs an embedding
, which is not isotopic to the standard embedding. The image is the union of two spheres which can be described as follows:
- either the spheres are
and
in
.
- or they are given as the points in
satisfying the following 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 also distinguished from the standard embedding by the linking coefficient (see 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' 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)
For a map representing an element of
let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
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; see Figure 4.
Then the restriction of
to
is a homotopy equivalence.
(Indeed, since , by general position 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 well-defined, i.e. is independent of the choices of
and of the representative
of
.
One can check that
is a homomorphism.
(b) Analogously one can define for
, by exchanging
and
in the above definition.
(c) Clearly, for the Zeman map
. So
is surjective and
is injective.
(d) For there is a simpler alternative definitions using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for
. The stable suspension 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 the Gauss map (see Figure 4)
![\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/5/3/d/53debfbd4a5d88f0c6d79b4fce1d64a3.png)
The second isomorphism in this formula is the suspension isomorphism.
The map is the quotient map, see Figure 5.
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
.)
One can easily check that is well-defined and for
is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1].
We have for
.
Hence , even though in general
as we explain in
8.
Note that the -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, each of the same dimension [Haefliger1962t, Theorem at the end of 5], [Haefliger1966a]. Let
be the
-tuple consisting entirely of some positive integer
.
Theorem 4.2. The collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}](/images/math/2/f/c/2fc47518d76bfa41e8e09d4418ba0637.png)
is a 1-1 correspondence for .
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.
Example 5.1 (Borromean rings).
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' 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)
See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings.
An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known triple linking number called the Massey number [Massey1968], for an elementary definition see [Skopenkov2017, 4.5 `Triple linking modulo 2' and
4.6 `Massey-Milnor and Sato-Levine numbers']. (Also, this embedding is not isotopic to the standard embedding 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.
Example 5.2 (Whitehead link). For every positive integer there is a non-trivial embedding
whose linking coefficient
is trivial.
Such an embedding is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3,
4]; see also the Wikipedia article on the Whitehead link. (For
the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.
We have because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details e.g. in [Skopenkov2006a]).
For
the Whitehead link is distinguished from the standard embedding by
equal to the Whitehead square
of the generator
.
This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14].
For
the Whitehead link 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.
For some results on links related to the Whitehead link [Skopenkov2015a,
2.5].
6 Linked manifolds
Let us state the analogues of Theorem 4.2 for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned.
Theorem 6.1. Assume that are closed manifolds and
for every
. Then for an embedding
and every
one defines the linking coefficient
, see Remark 3.2.e. We have
if both
and
are orientable, and
otherwise. Then the collection of pairwise linking coefficients
![\displaystyle \bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}](/images/math/b/8/2/b82670d5ce921a791eed0bad53065de5.png)
is well-defined and is a 1-1 correspondence, where of
are orientable and
are not.
The general case is presented only for the simplest case of two oriented components.
Theorem 6.2. For every let
be a closed
-dimensional
-connected orientabled manifold such that
. Then for an embedding
one can define the invariant
analogously to Definition 3.3. Then
![\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{n_1+n_2-m+1}^S](/images/math/f/4/9/f49b19c0c9de0d711360946537cd8156.png)
is well-defined and is a 1-1 correspondence.
These results are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]. Apparently Theorem 6.1 has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.
Now we present an extension of Theorems 6.1, 6.2 to the case when ,
and
are only
-connected.
An embedding
is Brunnian if its restriction to each component is isotopic to the standard embedding. For any triple of integers
such that
is even, one can explicitly construct [Avvakumov2016] a Brunnian embedding
so that the following theorem holds.
Theorem 6.3. [Avvakumov2016]
Any Brunnian embedding is isotopic to
for some integers
such that
is even. Two embeddings
and
are isotopic if and only if
and both
and
are divisible by
.
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for
). The following corollary shows that the relation between the embeddings
and
is not trivial.
Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b]
There exist embeddings and
such that the componentwise embedded connected sum
is isotopic to
but
is not isotopic to
.
See an unpublished generalization in [Avvakumov2017].
7 Reduction to the case with unknotted components
In this section we assume that is an
-tuple such that
.
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding
. We remark that
[Haefliger1966a,
2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011,
1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
![\displaystyle r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].](/images/math/2/5/d/25d2ec311c8371fe4ac42f34989c9ad0.png)
Take pairwise disjoint
-discs in
, i.e. take an embedding
. Define
![\displaystyle j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].](/images/math/3/2/9/329c5a3b46fae2defd6d32e45fb15014.png)
Then is a right inverse of the restriction homomorphism
, i.e.
.
The unknotting homomorphism
is defined to be the homomorphism
![\displaystyle u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).](/images/math/0/3/1/03189743eece251d863279810dbcb4e5.png)
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots
representing the elements of
inverse to the components, whose images
are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006,
3.3].
8 Classification beyond the metastable range
Theorem 8.1.
[Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1]
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].
For alternative geometric (and presumably equivalent) definitons of
see [Skopenkov2009,
3], [Skopenkov2006b,
5], cf. [Skopenkov2007,
2] and [Crowley&Skopenkov2016,
2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2.
(a) Theorem 8.1 implies that for any 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_{\varepsilon(l)}.](/images/math/0/2/a/02a96a307efc5c4248259e026e2ec527.png)
(b) For any , 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 .
For see [Haefliger1962t,
6]. The following proof for
and general remark are intended for specialists.
For there is an exact sequence
,
where
is
not
[Haefliger1966a, Corollary 10.3].
We have
,
and
is the reduction modulo 2.
By the exactness,
.
By (a)
.
Hence
is injective.
We have
by [Haefliger1966a, Proposition 10.2].
So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009,
3], [Skopenkov2006b,
5] and geomeric interpretation of the EHP sequence
[Koschorke&Sanderson1977] one can possibly prove that
. Then (b) would follow.
(c) For any the map
in (b) above is not injective [Haefliger1962t,
6].
(d) For any ,
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_{\varepsilon(l)}^4](/images/math/9/9/d/99d383b099b2098e36f4ce8515e4e39f.png)
which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number (8).
This follows from [Haefliger1962t,
6].
We conjecture that this result holds also for
.
9 Classification in codimension at least 3
In this section we assume that is an
-tuple such that
.
For this case a readily calculable classification of
is obtained in
[Crowley&Ferry&Skopenkov2011, Theorem 1.9]. Corollaries [Crowley&Ferry&Skopenkov2011,
1.2] one are necessary and sufficient conditions on
when
is finite. The answers are elementary but a bit technical. So we only state the following corollary and describe methods of its proof in Definition 9.2 and Theorem 9.3.
Theorem 9.1. [Crowley&Ferry&Skopenkov2011]
There are algorithms which for integers
Tex syntax error.
(b) find out whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a] Haefliger defined a long exact sequence of abelian groups
![\displaystyle \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.](/images/math/9/6/d/96d938c14fdf06907b0d7542e161f30b.png)
We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details.
In the above sequence the -tuples
are the same for different terms.
Denote
.
For any
and positive integer
denote by
the homomorphisms induced by the projection to the
-component of the wedge.
Denote
.
Denote
.
Analogous to Definition 3.1 there is a canonical homotopy equivalence . It can be shown that each component of a link
has a non-zero normal vector field and so each component of a link can be pushed off along such a vector field into the complement.
One can show that the homotopy class of a push off of one component in the complement of the entire link gives a well-defined map
. In fact, the map
is a generalisation of the linking coefficient. Finally, define
.
Taking the Whitehead product with the class of the identity in defines a homomorphism
. Define
.
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) [Crowley&Ferry&Skopenkov2011] The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers
splits into short exact sequences.
In general, the computation of the objects of Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of the non-trivial paper [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.
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