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.

We describe classification of embeddings S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m for m-3\ge n_i.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, \S1, \S3].

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}

For an s-tuple (n):=(n_1,\ldots,n_s) denote S^{(n)}:=S^{n_1}\sqcup\ldots\sqcup S^{n_s}. Although S^{(n)} is not a manifold when n_1,\ldots,n_s are not all equal, embeddings S^{(n)}\to S^m and (ambient) isotopy between such embeddings are defined analogously to the case of manifolds. Denote by E^m(S^{(n)}) the set of embeddings S^{(n)}\to S^m up to isotopy.

A component-wise version of embedded connected sum [Skopenkov2016c, \S5] defines a commutative group structure on the set E^m(S^{(n)}) for m-3\ge n_i [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], see [Skopenkov2006, Figure 3.3].

2 Examples

Recall that for any q-manifold N and m\ge2q+2, every two embeddings N\to\Rr^m are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction m\ge2q+2 is sharp for non-connected manifolds.

Example 2.1 (The Hopf Link). For any n there is an embedding S^q\sqcup S^q\to\Rr^{2q+1} which is not isotopic to the standard embedding.

For q=1 the Hopf link is shown e.g. in [Skopenkov2006, Figure 2.1.a]. For arbitrary q (including q=1) the image of the Hopf link is the union of two q-spheres:

  • either \partial D^{q+1}\times0 and 0\times\partial D^{q+1} in \partial(D^{q+1}\times D^{q+1})\cong S^{2q+1};
  • or given by the following equations in \Rr^{2q+1}:
\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.

This embedding is distinguished from the standard embedding by the linking coefficient (\S3).

Analogously for any p,q one constructs an embedding S^p\sqcup S^q\to\Rr^{p+q+1} which is not isotopic to the standard embedding. The image is the union of two spheres:

  • either \partial D^{p+1}\times0 and 0\times\partial D^{q+1} in \partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}.
  • or given by the following equations in \Rr^{p+q+1}:
\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.

This embedding is distinguished from the standard embedding also by the linking coefficient (\S3).

Definition 2.2 (The Zeeman map). We define a map

\displaystyle \zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.

Denote by i_{m,p}:S^p\to S^m the equatorial inclusion. For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

\displaystyle \overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times i_{q,p}}\to S^{m-q-1}\times S^q\overset{i}\to S^m,

where i is the natural `standard embedding' defined in [Skopenkov2015a, \S2.1], see [Skopenkov2006, Figure 3.2]. We have \overline\zeta_x(S^p)\cap i_{m,q}(S^q)\subset i(S^{m-q-1}\times S^p)\cap i(0_{m-q}\times S^q) =\emptyset. Let \zeta[x]:=[\overline\zeta_x\sqcup i_{m,q}].

One can easily check that \zeta 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.

Take an embedding f:S^p\sqcup S^q\to S^m representing an element [f]\in E^m(S^p\sqcup S^q). Take an embedding g:D^{m-q}\to S^m such that g(D^{m-q}) intersects f(S^q) transversely at exactly one point with positive sign [Skopenkov2006, Figure 3.1]. Then the restriction h':S^{m-q-1}\to S^m-fS^q of g to \partial D^{m-q} is a homotopy equivalence.

(Indeed, since m\ge q+3, by general position the complement S^m-fS^q is simply-connected. By Alexander duality, h' induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems h' is a homotopy equivalence.)

Let h be a homotopy inverse of h'. 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}).

Remark 3.2. (a) Clearly, \lambda[f] is well-defined, i.e. is independent of the choises of g,h',h and of representative f of [f]. One can check that \lambda is a homomorphism.

(b) Analogously one can define \lambda_{21}[f]\in\pi_q(S^{m-p-1}) for m\ge p+3, by exchanging p and q in the above definition.

(c) Clearly, \lambda\zeta=\id\pi_p(S^{m-q-1}) (see Definition 2.2; this also holds for m=q+2, see (d) and (e) below). So \lambda is surjective and \zeta is injective.

(d) This definition extends to the case m=q+2, provided S^m-fS^q is simply-connected.

(e) For m=p+q+1 or m=q+2 there are simpler alternative definitions using homological ideas. These definitions can be generalized to the case where the components are closed orientable manifolds, cf. Remark 5.3.f of [Skopenkov2016e].

By the Freudenthal Suspension Theorem \Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m} is an isomorphism for m\ge\frac p2+q+2. The stabilization of the linking coefficient can be described as follows.

Definition 3.3 (The \alpha-invariant). We define a map E^m(S^p\sqcup S^q)\to\pi^S_{p+q+1-m} for p,q\le m-2. Take an embedding f:S^p\sqcup S^q\to S^m representing an element [f]\in E^m(S^p\sqcup S^q). Define a map [Skopenkov2006, Figure 3.1]

\displaystyle \widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.

For p,q\le m-2 define the \alpha-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}.

The second isomorphism in this formula is the suspension isomorphism. The map v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q} is the quotient map. See [Skopenkov2006, Figure 3.4]. The map v^* is an isomorphism for m\ge q+2. (For m\ge q+3 this follows by general position and for m=q+2 by the cofibration Barratt-Puppe exact sequence of pair (S^p\times S^q,S^p\vee S^q) and by the existence of a retraction \Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q).)

One can check that \alpha is well-defined and is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12}.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}.

Note that \alpha-invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, \S5].

4 Classification in the metastable range

The Haefliger-Zeeman Theorem 4.1. (D) If 1\le p\le q, then both \lambda and \zeta are isomorphisms for m\ge\frac{3q}2+2 in the smooth category.

(PL) If 1\le p\le q, then both \lambda and \zeta are isomorphisms for m\ge\frac p2+q+2 in the PL category.

The surjectivity of \lambda (or the injectivity of \zeta) follows from \lambda\zeta=\id. The injectivity of \lambda (or the surjectivity of \zeta) 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 [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a].

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}

is bijective for m\ge\frac{3q}2+2.

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 S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} whose restrictions to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t].

Denote coordinates in \Rr^{3l} by (x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l). The (higher-dimensional) `Borromean rings' [Skopenkov2006, Figures 3.5 and 3.6] 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.

The required embedding is any embedding whose image consists of Borromean rings.

This embedding is distinguished from the standard embedding by the well-known triple linking number called Massey number [Massey1968], for an elementary definition see [Skopenkov2017, \S4.5 `Triple linking modulo 2' and \S4.6 `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 l=1 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). There is a non-trivial embedding w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} whose linking coefficient \lambda_{12}(w) is trivial.

The (higher-dimensional) `Whitehead link' is obtained from the Borromean rings embedding by joining two components with a tube, i.e. by `linked analogue' of embedded connected sum of the components [Skopenkov2016c, \S3, \S4].

We have \lambda_{12}(w)=0 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 l\ne1,3,7 the Whitehead link is distinguished from the standard embedding by \lambda_{21}(w)=[\iota_l,\iota_l]\ne0. This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, the Whitehead link Lemma 2.14]. For l=1,3,7 the Whitehead link is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962t, \S3].

This example (in higher dimensions, i.e. for l>1) seems to have been discovered by Whitehead, in connection with Whitehead product.

6 Reduction to unknotted components

Theorem 6.1. [Haefliger1966a] If n_1,\ldots,n_s\le m-3, then

\displaystyle E^m_D(S^{(n)})\cong E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i}),

where E^m_U(S^{(n)}) is the subgroup of E^m_D(S^{(n)}) formed by links whose restrictions to the components are unknotted.

The isomorphism of Theorem 6.1 is the sum of the restriction and the unknotting homomorphisms

\displaystyle  E^m_D(S^{(n)})\to\oplus_{i=1}^s E^m_D(S^{n_i})\quad\text{and}\quad E^m_D(S^{(n)})\to E^m_U(S^{(n)}).

The restriction homomorphism is defined to be the sum of homomorphisms induced by the restrictions to components. The unknotting homomorphism is defined by taking embedded connected sums of components with knots S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images are small and are close to the components. We have E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)}) [Haefliger1966a, \S\S 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, \S1.5].

For some information on the groups E^m_D(S^q) see [Skopenkov2006, \S3.3]. (See also foundational paper [Haefliger1966] on E^m_D(S^q) where this information is less complete and harder to find, and foundational paper [Levine1965] on different but related group.)

7 Classification in codimension 3

The Haefliger Theorem 7.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009] If p\le q\le m-3 and 3m\ge2p+2q+6, then there is a homomorphism \beta 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}).

The map \beta and its right inverse \pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12} are constructed in [Haefliger1966] and [Haefliger1966a], cf. [Skopenkov2009].

Remark 7.2. (a) The Haefliger Theorem 7.1 implies that for any l\ge2 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)}.

(b) When l \geq 2 but l \neq 3, 7, 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)

is injective and its image is \{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}. See [Haefliger1962t, \S6] for l\ge4; for specialists note that this also follows from (a) above and P\beta=\lambda_{12}+(-1)^l\lambda_{21}, where P:\pi_l(SO,SO_l)\to\pi_{2l-1}(S^l) is the map from the EHP sequence.

(c) For l\in\{1,3,7\} the map \lambda_{12}\oplus\lambda_{21} in (b) above is not injective [Haefliger1962t, \S6].

(d) For any l\ge4, l\ne7 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

which is the sum of 3 pairwise linking coefficients, 3 pairwise \beta-invariants and Massey number. This follows from [Haefliger1962t, \S6] and P\beta_{ij}=\lambda_{ij}+(-1)^l\lambda_{ji}. We conjecture that this result holds also for l=2,3,7.

Below in this subsection we assume that (n) is an s-tuple such that m\ge n_i+3.

Definition 7.3 (The Haefliger link sequence). We define the following long sequence of abelian groups

\displaystyle  \dots \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 \dots~.

In the above sequence s-tuples m-n_1,\ldots,m-n_s are the same for different terms. Denote W:=\vee_{i=1}^s S^{m-n_i-1}. For any j=1,\ldots,s and integer k>0 denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphisms induced by the projection to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1). Denote \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogously to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. The homotopy class of a push off of one component in the complement of the entire link gives then a map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}, see details in [Haefliger1966a, 1.4]. (This map \lambda_j is a generalization of linking coefficient.) Define \lambda:= \oplus_{j=1}^s \lambda_j.

Taking the Whitehead product with the class of the identity in \pi_{n_j}(S^{n_j}) defines a homomorphism w_j \colon \ker p_{n_j,j}\to \Pi_{m-2}^{(n-1)}. Define w:=\oplus_{j=1}^s w_j.

The definition of the homomorphism \mu is sketched in [Haefliger1966a, 1.5].

Theorem 7.4. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.

(b) [Crowley&Ferry&Skopenkov2011] The map \lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq) is an isomorphism.

Part (b) follows because \mu \otimes \Qq = 0 [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers \Qq splits into short exact sequences.

In [Crowley&Ferry&Skopenkov2011, \S1.2, \S1.3] one can find necessary and sufficient conditions on p and q when E^m_{PL}(S^p\sqcup S^q) is finite, as well as an effective procedure for computing the rank of the group E^m_{PL}(S^{(n)}).

For more results related to high codimension links we refer the reader to [Skopenkov2009], [Avvakumov2016], [Skopenkov2015a, \S2.5], [Skopenkov2016k].

8 References

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