High codimension links

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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 Figure 5.
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 Figure 5.
[[Image:33.jpg|thumb|450px|Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere]]
[[Image:33.jpg|thumb|450px|Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere]]
The map $v^*$ is an isomorphism for $m\ge q+2$.
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The map $v^*$ is a 1--1 correspondence 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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.)
(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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.)
<!-- [[Media:33.pdf|figure 5]]. the example showing that MA does not display pdf figures: [[Image:33.pdf|thumb|450px|Figure 5]] -->
<!-- [[Media:33.pdf|figure 5]]. the example showing that MA does not display pdf figures: [[Image:33.pdf|thumb|450px|Figure 5]] -->
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== Linked manifolds ==
== Linked manifolds ==
<wikitex>;
<wikitex>;
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<!--Let us state the analogues of Theorem \ref{t:lkmany} for spheres replaced by highly-connected manifolds.
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We start with the simplest case when no high-connectivity needs to be mentioned. -->
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In this section we state some analogues of Theorem \ref{t:lkmany} where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.
Let us state the analogues of Theorem \ref{t:lkmany} for spheres replaced by highly-connected manifolds.
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{{beginthm|Theorem}}\label{t:man} Assume that $N_1,\ldots,N_s$ are closed manifolds and $m=2\dim N_i+1>\dim N_i+2$ for every $i$. Then for an embedding $f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m$ and every $1\le i<j\le s$ one defines the linking coefficient $\lambda_{ij}(f)$, see Remark \ref{lkrem}.e. We have $\lambda_{ij}(f)\in\Z$ if both $N_i$ and $N_j$ are orientable, and $\lambda_{ij}(f)\in\Z_2$ otherwise. Then the collection of pairwise linking coefficients
We start with the simplest case when no high-connectivity needs to be mentioned.
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{{beginthm|Theorem}}\label{t:man} Assume that $N_1,\ldots N_s$ are closed manifolds and $m=2\dim N_i+1>\dim N_i+2$ for every $i$. Then for an embedding $f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m$ and every $1\le i<j\le s$ one defines the linking coefficient $\lambda_{ij}(f)$, see Remark \ref{lkrem}.e. We have $\lambda_{ij}(f)\in\Z$ if both $N_i$ and $N_j$ are orientable, and $\lambda_{ij}(f)\in\Z_2$ otherwise. Then the collection of pairwise linking coefficients
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$$\bigoplus\limits_{1\le i<j\le s}\lambda_{ij} :
$$\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}$$
E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}$$
is well-defined and is a 1-1 correspondence, where $t$ of $N_1,\ldots N_s$ are orientable and $s-t$
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is well-defined and is a 1-1 correspondence, where $t$ of $N_1,\ldots, N_s$ are orientable and $s-t$
are not.
are not.
{{endthm}}
{{endthm}}
The general case is presented only for the simplest case of two oriented components of the same dimension.
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We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.
{{beginthm|Theorem}}\label{t:mang} For $i=1,2$ let $N_i$ be a closed $n$-dimensional homologically $(2n-m+1)$-connected orientable manifold. Then for an embedding $f:N_1\sqcup N_2\to\Rr^m$ one can define the invariant $\alpha(f)\in\pi_{2n-m+1}^S$ analogously to Definition \ref{d:alpha}. Then
{{beginthm|Theorem}}\label{t:mang} For $i=1,2$ let $N_i$ be a closed $n$-dimensional homologically $(2n-m+1)$-connected orientable manifold. Then for an embedding $f:N_1\sqcup N_2\to\Rr^m$ one can define the invariant $\alpha(f)\in\pi_{2n-m+1}^S$ analogously to Definition \ref{d:alpha}. Then
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{{endthm}}
{{endthm}}
These results are corollaries of \cite[the Haefliger-Weber Theorem 5.4]{Skopenkov2006}. Apparently Theorem \ref{t:man} has an easier proof using the Whitney trick or surgery. Although I have not seen these results in the literature (see though \cite[Proposition 1.2]{Skopenkov2000} for the ''link map'' analogue), I presume they are folklore.
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Theorems \ref{t:man} and \ref{t:mang} are corollaries of \cite[the Haefliger-Weber Theorem 5.4]{Skopenkov2006}; the calculations are analogous to the construction of the 1-1 correspondence $[S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m}$ in Definition \ref{d:alpha}. Apparently Theorem \ref{t:man} has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see \cite[Proposition 1.2]{Skopenkov2000} for the ''link map'' analogue), I presume they are folklore.
Now we present an extension of Theorems \ref{t:man}, \ref{t:mang} to the case when $m=2\dim N_i$, $2m=3n+3$
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Now we present an extension of Theorems \ref{t:man} and \ref{t:mang} to a case where $m = 6$ and $n = 3$. In particular, for the results below $m=2\dim N_i$, $2m=3n+3$
and $N_i$ are only $(2n-m)$-connected.
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and the manifolds $N_i$ are only $(2n-m)$-connected. An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$ is ''Brunnian'' if its restriction to each component is isotopic to the standard embedding. <!--For any triple of integers $k,l,n$ such that $l-n$ is even, one can explicitly construct \cite{Avvakumov2016} a Brunnian embedding $f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds. --> For each triple of integers $k,l,n$ such that $l-n$ is even, Avvakumov has constructed a Brunnian embedding $f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$, which appears in the next result \cite{Avvakumov2016}.
An embedding $(S^2\times S^1)\sqcup S^3\to\Rr^6$ is ''Brunnian'' if its restriction to each component is isotopic to the standard embedding. For any triple of integers $k,l,n$ such that $l-n$ is even, one can explicitly construct \cite{Avvakumov2016} a Brunnian embedding $f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6$ so that the following theorem holds.
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{{beginthm|Theorem}}\label{t:avv} \cite{Avvakumov2016}
{{beginthm|Theorem}}\label{t:avv} \cite{Avvakumov2016}
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The proof uses M. Skopenkov's classification of embeddings $S^3\sqcup S^3\to\Rr^6$ (Theorem \ref{belmethae} for $m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$ is not trivial.
The proof uses M. Skopenkov's classification of embeddings $S^3\sqcup S^3\to\Rr^6$ (Theorem \ref{belmethae} for $m=2p=2q=6$). The following corollary shows that the relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $S^3\sqcup S^3\to\Rr^6$ is not trivial.
{{beginthm|Corollary}}\label{t:avv} \cite{Avvakumov2016}, cf. \cite[Corollary 3.5.b]{Skopenkov2016t}
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{{beginthm|Corollary}}\label{c:avv} \cite{Avvakumov2016}, cf. \cite[Corollary 3.5.b]{Skopenkov2016t}
There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$ is isotopic to $f\#g'$ but $g$ is not isotopic to $g'$.
There exist embeddings $f:(S^2\times S^1)\sqcup S^3\to\Rr^6$ and $g,g':S^3\sqcup S^3\to\Rr^6$ such that the componentwise embedded connected sum $f\#g$ is isotopic to $f\#g'$ but $g$ is not isotopic to $g'$.
{{endthm}}
{{endthm}}
For an unpublished generalization to Corollary \ref{t:avv} see \cite{Avvakumov2017}.
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For an unpublished generalization of Theorem \ref{t:avv} and Corollary \ref{c:avv} see \cite{Avvakumov2017}.
</wikitex>
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For $l\ge4$ see \cite[$\S$6]{Haefliger1962t}. The following proof for $l=2$ and general remark are intended for specialists.
For $l\ge4$ see \cite[$\S$6]{Haefliger1962t}. The following proof for $l=2$ and general remark are intended for specialists.
For $l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$,
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For $l=2$ there is an exact sequence $\pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3)$,
where $\lambda_{12}$ is $\lambda_{12,PL}$ not $\lambda_{12,D}$ \cite[Corollary 10.3]{Haefliger1966a}.
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where $\lambda_{12}$ is $\lambda_{12,PL}$ <!-- not $\lambda_{12,D}$ --> \cite[Corollary 10.3]{Haefliger1966a}.
We have $\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$ and $\Sigma$ is the reduction modulo 2.
We have $\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$ and $\Sigma$ is the reduction modulo 2.
By the exactness, $\text{im}\lambda_{21}=2\Z$.
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By the exactness of the previous sequence, <!--$\text{im} \, \lambda_{21}= 2\Z$. --> $ \lambda_{21}(\ker \lambda_{12}) = 2\Z$.
By (a) $\ker\lambda_{12}\cong\Z$.
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By (a) $\ker \lambda_{12} \cong \Z$.
Hence $\lambda_{21}|_{\ker\lambda_{12}}$ is injective.
Hence $\lambda_{21}|_{\ker\lambda_{12}}$ is injective.
We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by \cite[Proposition 10.2]{Haefliger1966a}.
We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by \cite[Proposition 10.2]{Haefliger1966a}.
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== Classification in codimension at least 3 ==
== Classification in codimension at least 3 ==
<wikitex>;\label{s:cl3}
<wikitex>;\label{s:cl3}
In this section we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$.
In this section we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$.
For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in
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For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$ was obtained in
\cite[Theorem 1.9]{Crowley&Ferry&Skopenkov2011}. Corollaries \cite[$\S$1.2]{Crowley&Ferry&Skopenkov2011} one are necessary and sufficient conditions on $(n)$ when $E^m_{PL}(S^{(n)})$ 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 \ref{d:hclink} and Theorem \ref{thm:hclink}.
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\cite[Theorem 1.9]{Crowley&Ferry&Skopenkov2011}. In particular, \cite[$\S$1.2]{Crowley&Ferry&Skopenkov2011} contains necessary and sufficient conditions on $(n)$ which determine when $E^m_{PL}(S^{(n)})$ is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition \ref{d:hclink} and Theorem \ref{thm:hclink} below.
<!--, as well as an effective procedure for computing the rank of the group $E^m_{PL}(S^{(n)})$.-->
<!--, as well as an effective procedure for computing the rank of the group $E^m_{PL}(S^{(n)})$.-->
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(a) calculate ${\rm rk}E^m_{PL}(S^{(n)})$.
(a) calculate ${\rm rk}E^m_{PL}(S^{(n)})$.
(b) find out whether $E^m_{PL}(S^{(n)})$ is finite.
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(b) determine whether $E^m_{PL}(S^{(n)})$ is finite.
{{endthm}}
{{endthm}}
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$$ \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.$$
$$ \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.$$
We briefly define the groups and homomorphisms in this sequence, refering to \cite[1.4]{Haefliger1966a} for details.
We briefly define the groups and homomorphisms in this sequence, refering to \cite[1.4]{Haefliger1966a} for details.
In the above sequence the $s$-tuples $m-n_1,\ldots,m-n_s$ are the same for different terms.
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We first note that in the sequence above the $s$-tuples $m-n_1,\ldots,m-n_s$ are the same for different terms.
Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$.
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Denote $W :=\vee_{i=1}^s S^{m-n_i-1}$. <!W_m^{(n)}-->
For any $j=1,\ldots,s$ and positive integer $k$ 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.
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For any $j=1,\ldots,s$ and positive integer $k$ denote by $p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1})$ the homomorphism induced by the collapse map onto to the $j$-component of the wedge. Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$ and $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.
Denote $(n-1):=(n_1 -1, \ldots, n_s-1)$.
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Denote $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$.
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Analogous to Definition \ref{dl} there is a canonical homotopy equivalence $C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$ 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.
Analogous to Definition \ref{dl} there is a canonical homotopy equivalence $C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$ 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 $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$ is a generalisation of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$.
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It can also be shown that the homotopy class of a push off of the $j$th component in the complement of the link gives a well-defined map $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$ is a generalisation of the linking coefficient. Finally, 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 $w:=\oplus_{j=1}^s w_j$.
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 $w:=\oplus_{j=1}^s w_j$.
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Part (b) follows because $\mu \otimes \Qq = 0$ \cite[Lemma 1.3]{Crowley&Ferry&Skopenkov2011}, and so the Haefliger link sequence after tensoring with the rational numbers $\Qq$ splits into short exact sequences.
Part (b) follows because $\mu \otimes \Qq = 0$ \cite[Lemma 1.3]{Crowley&Ferry&Skopenkov2011}, and so the Haefliger link sequence after tensoring with the rational numbers $\Qq$ splits into short exact sequences.
In general, the computation of the objects of Theorem \ref{thm:hclink} is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem \ref{belmethae} for $3m\ge2p+2q+7$ (note that Theorem \ref{belmethae} has a direct proof easier than proof of Theorem \ref{thm:hclink}.a). For (b) the computations constitute most of the non-trivial paper \cite{Crowley&Ferry&Skopenkov2011}. Hence Theorem \ref{thm:hclink} is not a readily calculable classification in general. <!-- $$ 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} (\oplus_{j=1}^s \mathop{Ker} p_{n_j,j}) \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.$$-->
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In general, the computation of the groups and homomorphisms appearing in Theorem \ref{thm:hclink} is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem \ref{belmethae} for $3m\ge2p+2q+7$ (note that Theorem \ref{belmethae} has a direct proof easier than proof of Theorem \ref{thm:hclink}.a). For (b) the computations constitute most of \cite{Crowley&Ferry&Skopenkov2011}. Hence Theorem \ref{thm:hclink} is not a readily calculable classification in general. <!-- $$ 0 \to E^m_{PL}(S^n) \otimes \Qq \xrightarrow{~\lambda \otimes \Qq~} (\oplus_{j=1}^s \mathop{Ker} p_{n_j,j}) \otimes \Qq \xrightarrow{~w \otimes \Qq~} \Pi^{m-2}_{n-1}\otimes \Qq \to 0.$$-->
</wikitex>
</wikitex>

Revision as of 11:15, 17 May 2019

This page has been accepted for publication in the Bulletin of the Manifold Atlas.

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Contents

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 \Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

Tex syntax error
see Figure 3. We have
Tex syntax error
. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
Tex syntax error
.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

, $\S]{Skopenkov2016c}. The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem \ref{lkhaze} below): $$\begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \ \hline |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ [[Image:EmbeddedConnectedSum.jpg|thumb|350px|Figure 1: Embedded connected sum]] A component-wise version of [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded_connected_sum|embedded connected sum]] \cite[$\S]{Skopenkov2016c} defines a commutative group structure on the set $E^m(S^{(n)})$ for $m-3\ge n_i$ \cite{Haefliger1966}, \cite{Haefliger1966a}, \cite[Group Structure Lemma 2.2 and Remark 2.3.a]{Skopenkov2015}, see Figure 1. The ''standard embedding'' $S^k\to D^m$ is defined by $(x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0)$. Fix $s$ pairwise disjoint $m$-discs $D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m$. The ''standard embedding'' $S^{(n)}\to S^m$ is defined by taking the union of the compositions of the standard embeddings $S^{n_k} \to D^m_k$ with the fixed inclusions $D^m_k \to S^m$. == Examples == ; [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Recall that]] for any $q$-manifold $N$ and $m\ge2q+2$, every two embeddings $N\to\Rr^m$ are isotopic \cite[General Position Theorem 2.1]{Skopenkov2016c}, \cite[General Position Theorem 2.1]{Skopenkov2006}. The following example shows that the restriction $m\ge2q+2$ is sharp for non-connected manifolds. {{beginthm|Example|(The Hopf Link)}}\label{hopf} For every positive integer $q$ there is an embedding $S^q\sqcup S^q\to\Rr^{2q+1}$, which is not isotopic to the standard embedding. {{endthm}} [[Image:HopfLink-and-TrivialLink.jpg|thumb|350px|Figure 2: The Hopf link (a) and the trivial link (b)]] For $q=1$ the Hopf link is shown in Figure 2. For all $q$ the image of the Hopf link is the union of two $q$-spheres which can be described as follows: * either the spheres are $\partial D^{q+1}\times0$ and up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

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see Figure 3. We have
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. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
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.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$; * or they are given as the sets of points in $\Rr^{2q+1}$ satisfying the following equations: $$\left\{\begin{array}{c} x_1=\dots=x_q=0\ x_{q+1}^2+\dots+x_{2q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.$$ This embedding is distinguished from the standard embedding by [[#The linking coefficient|the linking coefficient]] (see $\S$\ref{s:inv}). 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 which can be described as follows: * either the spheres are $\partial D^{p+1}\times0$ and up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

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see Figure 3. We have
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. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
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.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

\times\partial D^{q+1}$ in $\partial(D^{p+1}\times D^{q+1})\cong S^{p+q+1}$. * or they are given as the points in $\Rr^{p+q+1}$ satisfying the following equations: $$\left\{\begin{array}{c} x_1=\dots=x_p=0\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$ This embedding is also distinguished from the standard embedding by [[#The linking coefficient|the linking coefficient]] (see $\S$\ref{s:inv}). {{beginthm|Definition|(A link with prescribed [[#The linking coefficient|linking coefficient]])}}\label{dz} We define the `Zeeman' map [[Image:LinkingCoefficient.jpg|thumb|350px|Figure 3: A link with linking coefficient $\pm 1$]] $$\zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.$$ For a map $x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$ let $$\overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad S^p\overset{x\times{\rm i}}\to S^{m-q-1}\times S^q\overset{{\rm i}_{m,q}}\to S^m,$$ see Figure 3. We have $\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let $\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$. {{endthm}} One can easily check that $\zeta$ is well-defined and is a homomorphism.
== The linking coefficient == ;\label{s:inv} Here we define the linking coefficient and discuss is properties. Fix orientations of the standard spheres and balls. {{beginthm|Definition|(The linking coefficient)}}\label{dl} We define a map $$\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\quad\text{for}\quad m\ge q+3.$$ 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; see Figure 4. [[Image:GaussMap-and-MeridianDisc.jpg|thumb|450px|Figure 4: The disc $gD^{m-q}$ and Gauss map $\widetilde f$]] 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 $$\lambda[f]=\lambda_{12}[f]:=[S^p\xrightarrow{~f|_{S^p}~} S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ {{endthm}} {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda[f]$ is well-defined, i.e. is independent of the choices of $g,h',h$ and of the 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, $\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the [[#Examples|Zeman map]] $\zeta$. So $\lambda$ is surjective and $\zeta$ is injective. (d) For $m=p+q+1$ 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. [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|Remark 5.3.f]] of \cite{Skopenkov2016e}. {{endthm}} Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$. The stable suspension of the linking coefficient can be described as follows. {{beginthm|Definition|(The $\alpha$-invariant)}}\label{d:alpha} 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 the Gauss map (see Figure 4) $$\widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$$ For $p,q\le m-2$ define the $\alpha$-invariant by $$\alpha[f]=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$$ 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 Figure 5. [[Image:33.jpg|thumb|450px|Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere]] 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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.) {{endthm}} One can easily check that $\alpha$ is well-defined and for $p,q\le m-3$ is a homomorphism. {{beginthm|Lemma|\cite[Lemma 5.1]{Kervaire1959a}}} We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$. {{endthm}} Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$ as we explain in $\S$\ref{s:ebmr}. Note that the $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite[$\S]{Skopenkov2006}. == Classification in the metastable range == ; {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} (D) If \Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

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see Figure 3. We have
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. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
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.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

\le p\le q$, then both $\lambda$ and $\zeta$ are isomorphisms for $m\ge\frac{3q}2+2$ in the smooth category. (PL) If \Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

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see Figure 3. We have
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. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
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.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

\le p\le q$, then both $\lambda$ and $\zeta$ are isomorphisms for $m\ge\frac p2+q+2$ in the PL category. {{endthm}} 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 \cite{Haefliger1962t}, \cite{Zeeman1962} (or follows from \cite[the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]{Skopenkov2006}). An analogue of this result holds for links with many components, each of the same dimension \cite[Theorem at the end of $\S]{Haefliger1962t}, \cite{Haefliger1966a}. Let $(q) = (q, \dots, q)$ be the $s$-tuple consisting entirely of some positive integer $q$. {{beginthm|Theorem}}\label{t:lkmany} The collection of pairwise linking coefficients $$\bigoplus\limits_{1\le i ==Examples beyond the metastable range== ;\label{s:ebmr} We present an example of the ''non-injectivity of the collection of pairwise linking coefficients'', which shows that the dimension restriction is sharp in Theorem \ref{t:lkmany}. {{beginthm|Example|(Borromean rings)}}\label{belmetbor} The embedding defined below is a non-trivial embedding $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose restrictions to each 2-component sublink is trivial \cite[4.1]{Haefliger1962}, \cite{Haefliger1962t}. Denote coordinates in $\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$. The (higher-dimensional) [[Wikipedia:Borromean_rings|`Borromean rings']] are the three spheres given by the following three systems of equations: $$\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.$$ [[Image:Borromean.jpg|thumb|400px|Figure 6: The Borromean rings]] See Figure 6. The required embedding is any embedding whose image consists of the Borromean rings. {{endthm}} An embedding with image the Borromean rings is distinguished from the standard embedding by the well-known ''triple linking number'' called the ''Massey number'' \cite{Massey1968}, for an elementary definition see \cite[$\S.5 `Triple linking modulo 2' and $\S.6 `Massey-Milnor and Sato-Levine numbers']{Skopenkov2017}. (Also, this embedding is not isotopic to the standard embedding because joining the three components with two tubes, i.e. `linked analogue' of [[High_codimension_embeddings#Embedded_connected_sum|embedded connected sum]] of the components \cite{Skopenkov2016c}, yields a non-trivial knot \cite{Haefliger1962}, cf. [[3-manifolds_in_6-space#Examples|the Haefliger Trefoil knot]] \cite{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 \ref{lkhaze}. {{beginthm|Example|(Whitehead link)}}\label{belmetwhi} For every positive integer $l$ 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. [[Image:Borromean-to-Whitehead.jpg|thumb|400px|Figure 7: The Borromean rings and Whitehead link for $l = 1$]] 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 [[Embeddings_just_below_the_stable_range:_classification#Embedded_connected_sum|embedded connected sum]] of the components \cite[$\S, $\S]{Skopenkov2016c}; see also the [[Wikipedia:Whitehead_link|Wikipedia article on the Whitehead link]]. (For $l=1$ the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7. {{endthm}} 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 \cite{Skopenkov2006a}). For $l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)$ equal to the Whitehead square $[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$. This fact should be well-known, but I do not know a published proof except \cite[the Whitehead link Lemma 2.14]{Skopenkov2015a}. For $l=1,3,7$ the Whitehead link is distinguished from the standard embedding by more complicated invariants \cite{Skopenkov2006a}, \cite[$\S]{Haefliger1962t}. This example (in higher dimensions, i.e. for $l>1$) seems to have been discovered by Whitehead, in connection with Whitehead product. For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link \cite[$\S.5]{Skopenkov2015a}. == Linked manifolds == ; Let us state the analogues of Theorem \ref{t:lkmany} for spheres replaced by highly-connected manifolds. We start with the simplest case when no high-connectivity needs to be mentioned. {{beginthm|Theorem}}\label{t:man} Assume that $N_1,\ldots N_s$ are closed manifolds and $m=2\dim N_i+1>\dim N_i+2$ for every $i$. Then for an embedding $f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m$ and every \Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

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see Figure 3. We have
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. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
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.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

\le i == Reduction to the case with unknotted components == ; In this section we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$. Define $E^m_U(S^{(n)}) \subset E^m_D(S^{(n)})$ to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding $S^{n_i} \to S^m$. We remark that $E^m_U(S^{(n)})\cong E^m_{PL}(S^{(n)})$ \cite[$\S\S$ 2.4, 2.6 and 9.3]{Haefliger1966a}, \cite[$\S\Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

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see Figure 3. We have
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. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
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.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

.5]{Crowley&Ferry&Skopenkov2011}. Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$-tuple of the isotopy classes of its components: $$ r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$ Take $s$ pairwise disjoint $m$-discs in $S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define $$j \colon \bigoplus_{i=1}^s E^m_D(S^{n_i}) \to E^m_D(S^{(n)})\quad\text{by}\quad j([f_1], \ldots, [f_s]):=[(g\circ\sqcup_{i=1}^s f_i):S^{(n)}\to S^m].$$ Then $j$ is a right inverse of the restriction homomorphism $r$, i.e. $r\circ j=\mathrm{id}$. The unknotting homomorphism $u$ is defined to be the homomorphism $$ u = \mathrm{id} - j \circ r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)}).$$ Informally, the homomorphism $u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$ are small and are close to the components. {{beginthm|Theorem}}\label{dpl} \cite{Haefliger1966a} For $n_1,\ldots,n_s\le m-3$, the homomorphism $$u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})$$ is an isomorphism. {{endthm}} For [[Knots,_i.e._embeddings_of_spheres|some information on the groups $E^m_D(S^q)$]] see \cite{Skopenkov2016s}, \cite[$\S.3]{Skopenkov2006}.
==Classification beyond the metastable range== ;\label{s:ebmr} {{beginthm|Theorem}}\label{belmethae} \cite[Theorem 10.7]{Haefliger1966a}, \cite[Theorem 1.1]{Skopenkov2009}, \cite[Theorem 1.1]{Skopenkov2006b} If $p\le q\le m-3$ and m\ge2p+2q+6$, then there is a homomorphism $\beta$ for which the following map is an isomorphism $$\lambda_{12}\oplus\beta:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).$$ {{endthm}} The map $\beta$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in \cite{Haefliger1966} and \cite{Haefliger1966a}. For alternative geometric (and presumably equivalent) definitons of $\beta$ see \cite[$\S]{Skopenkov2009}, \cite[$\S]{Skopenkov2006b}, cf. \cite[$\S]{Skopenkov2007} and \cite[$\S.3]{Crowley&Skopenkov2016}. For a historical remark see \cite[the second paragraph in p. 2]{Skopenkov2009}. {{beginthm|Remark}}\label{r:bel} (a) Theorem \ref{belmethae} implies that for any $l\ge2$ we have an isomorphism $$\lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{\varepsilon(l)}.$$ (b) For any $l\not\in\{1,3,7\}$, the map $$\lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l)$$ is injective and its image is $\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$. For $l\ge4$ see \cite[$\S]{Haefliger1962t}. The following proof for $l=2$ and general remark are intended for specialists. For $l=2$ there is an exact sequence $\pi_5(S^3)\to\ker\lambda_{12}\overset{\lambda_{21}}\to\pi_3(S^2)\overset{\Sigma}\to\pi_4(S^3)$, where $\lambda_{12}$ is $\lambda_{12,PL}$ not $\lambda_{12,D}$ \cite[Corollary 10.3]{Haefliger1966a}. We have $\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$ and $\Sigma$ is the reduction modulo 2. By the exactness, $\text{im}\lambda_{21}=2\Z$. By (a) $\ker\lambda_{12}\cong\Z$. Hence $\lambda_{21}|_{\ker\lambda_{12}}$ is injective. We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by \cite[Proposition 10.2]{Haefliger1966a}. So the formula of (b) follows. Analogously to \cite[Theorem 3.5]{Skopenkov2009} using geometric definitons of $\beta$ \cite[$\S]{Skopenkov2009}, \cite[$\S]{Skopenkov2006b} and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ \cite{Koschorke&Sanderson1977} one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow. (c) For any $l\in\{1,3,7\}$ the map $\lambda_{12}\oplus\lambda_{21}$ in (b) above is not injective \cite[$\S]{Haefliger1962t}. (d) For any $l\ge4$, $l\ne7$ we have an isomorphism $$E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)^3\oplus\Z_{\varepsilon(l)}^4$$ which is the sum of 3 pairwise invariants of (a) and (b) above, and the Massey number ($\S$\ref{s:ebmr}). This follows from \cite[$\S]{Haefliger1962t}. We conjecture that this result holds also for $l=2,3,7$. {{endthm}} == Classification in codimension at least 3 == ;\label{s:cl3} In this section we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$. For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$ is obtained in \cite[Theorem 1.9]{Crowley&Ferry&Skopenkov2011}. Corollaries \cite[$\S\Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

Tex syntax error
see Figure 3. We have
Tex syntax error
. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
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.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

.2]{Crowley&Ferry&Skopenkov2011} one are necessary and sufficient conditions on $(n)$ when $E^m_{PL}(S^{(n)})$ 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 \ref{d:hclink} and Theorem \ref{thm:hclink}. {{beginthm|Theorem}}\label{t:cfs} \cite{Crowley&Ferry&Skopenkov2011} There are algorithms which for integers $m,n_1,\ldots,n_s>0$ (a) calculate ${\rm rk}E^m_{PL}(S^{(n)})$. (b) find out whether $E^m_{PL}(S^{(n)})$ is finite. {{endthm}} {{beginthm|Definition|(The Haefliger link sequence)}}\label{d:hclink} In \cite{Haefliger1966a} Haefliger defined a long exact sequence of abelian groups $$ \ldots \to \Pi_{m-1}^{(n)} \xrightarrow{~\mu~} E^m_{PL}(S^{(n)}) \xrightarrow{~\lambda~} \oplus_{j=1}^s \ker p_{n_j,j} \xrightarrow{~w~} \Pi_{m-2}^{(n-1)} \xrightarrow{~\mu~} E^{m-1}_{PL}(S^{(n-1)})\to \ldots~.$$ We briefly define the groups and homomorphisms in this sequence, refering to \cite[1.4]{Haefliger1966a} for details. In the above sequence the $s$-tuples $m-n_1,\ldots,m-n_s$ are the same for different terms. Denote $W=W_m^{(n)}:=\vee_{i=1}^s S^{m-n_i-1}$. For any $j=1,\ldots,s$ and positive integer $k$ 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 $\Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i})$. Analogous to Definition \ref{dl} there is a canonical homotopy equivalence $C_f \sim W$. It can be shown that each component of a link $S^{(n)} \to S^m$ 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 $\lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}$. In fact, the map $\lambda_j$ is a generalisation of the linking coefficient. Finally, 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 given in \cite[1.5]{Haefliger1966a}. {{endthm}} {{beginthm|Theorem}}\label{thm:hclink} (a) \cite[Theorem 1.3]{Haefliger1966a} The Haefliger link sequence is exact. (b) \cite{Crowley&Ferry&Skopenkov2011} The map $\lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq)$ is an isomorphism. {{endthm}} Part (b) follows because $\mu \otimes \Qq = 0$ \cite[Lemma 1.3]{Crowley&Ferry&Skopenkov2011}, and so the Haefliger link sequence after tensoring with the rational numbers $\Qq$ splits into short exact sequences. In general, the computation of the objects of Theorem \ref{thm:hclink} is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem \ref{belmethae} for m\ge2p+2q+7$ (note that Theorem \ref{belmethae} has a direct proof easier than proof of Theorem \ref{thm:hclink}.a). For (b) the computations constitute most of the non-trivial paper \cite{Crowley&Ferry&Skopenkov2011}. Hence Theorem \ref{thm:hclink} is not a readily calculable classification in general.
== References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\Rr^m up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings N_1\sqcup\ldots\sqcup N_s\to S^m, where N_1,\ldots, N_s are spheres (or even closed manifolds) and m-3\ge\dim N_i for every i, under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].

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 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.

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}
Figure 1: Embedded connected sum

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 Figure 1.

The standard embedding S^k\to D^m is defined by (x_1, \ldots, x_{k+1})\mapsto(x_1, \ldots, x_{k+1}, 0, \ldots, 0). Fix s pairwise disjoint m-discs D^m_1\sqcup\ldots\sqcup D^m_s\subset S^m. The standard embedding S^{(n)}\to S^m is defined by taking the union of the compositions of the standard embeddings S^{n_k} \to D^m_k with the fixed inclusions D^m_k \to S^m.

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 every positive integer q there is an embedding S^q\sqcup S^q\to\Rr^{2q+1}, which is not isotopic to the standard embedding.

Figure 2: The Hopf link (a) and the trivial link (b)

For q=1 the Hopf link is shown in Figure 2. For all q the image of the Hopf link is the union of two q-spheres which can be described as follows:

  • either the spheres are \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 they are given as the sets of points in \Rr^{2q+1} 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.

This embedding is distinguished from the standard embedding by the linking coefficient (see \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 which can be described as follows:

  • either the spheres are \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 they are given as the points in \Rr^{p+q+1} 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.

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

Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map

Figure 3: A link with linking coefficient \pm 1
\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.

For a map x:S^p\to S^{m-q-1} representing an element of \pi_p(S^{m-q-1}) let

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see Figure 3. We have
Tex syntax error
. Let
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.

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; see Figure 4.

Figure 4: The disc gD^{m-q} and Gauss map \widetilde f

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 choices of g,h',h and of the 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}) for the Zeman map \zeta. So \lambda is surjective and \zeta is injective.

(d) For m=p+q+1 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 \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 stable suspension 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 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|}.

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 Figure 5.

Figure 5: Contraction of the meridian and the parallel of the torus yield the sphere

The map v^* is a 1--1 correspondence 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 easily check that \alpha is well-defined and for p,q\le m-3 is a homomorphism.

Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have \alpha=\pm\Sigma^{\infty}\lambda_{12} for p,q\le m-3.

Hence \Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}, even though in general \lambda_{12} \neq \pm \lambda_{21} as we explain in \S8.

Note that the \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, each of the same dimension [Haefliger1962t, Theorem at the end of \S5], [Haefliger1966a]. Let (q) = (q, \dots, q) be the s-tuple consisting entirely of some positive integer q.

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 a 1-1 correspondence 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' 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.
Figure 6: The Borromean rings

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, \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). For every positive integer l 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.

Figure 7: The Borromean rings and Whitehead link for l = 1

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, \S3, \S4]; see also the Wikipedia article on the Whitehead link. (For l=1 the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.

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) equal to the Whitehead square [\iota_l,\iota_l]\ne0 of the generator \iota_l\in\pi_l(S^l). 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.

For some results on links S^{2l-1}\sqcup S^{2l-1}\to\R^{3l} related to the Whitehead link [Skopenkov2015a, \S2.5].

6 Linked manifolds

In this section we state some analogues of Theorem 4.2 where spheres are replaced by general manifolds. We start with a simple case where no connectivity assumptions are made on the source manifolds.

Theorem 6.1. Assume that N_1,\ldots,N_s are closed manifolds and m=2\dim N_i+1>\dim N_i+2 for every i. Then for an embedding f:N_1\sqcup\ldots\sqcup N_s\to\Rr^m and every 1\le i<j\le s one defines the linking coefficient \lambda_{ij}(f), see Remark 3.2.e. We have \lambda_{ij}(f)\in\Z if both N_i and N_j are orientable, and \lambda_{ij}(f)\in\Z_2 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}

is well-defined and is a 1-1 correspondence, where t of N_1,\ldots, N_s are orientable and s-t are not.

We present the general case of the classification of embeddings of disjoint manifolds only for the case of two oriented components of the same dimension.

Theorem 6.2. For i=1,2 let N_i be a closed n-dimensional homologically (2n-m+1)-connected orientable manifold. Then for an embedding f:N_1\sqcup N_2\to\Rr^m one can define the invariant \alpha(f)\in\pi_{2n-m+1}^S analogously to Definition 3.3. Then

\displaystyle \alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S

is well-defined and is a 1-1 correspondence, provided 2m\ge3n+4.

Theorems 6.1 and 6.2 are corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4]; the calculations are analogous to the construction of the 1-1 correspondence [S^p\times S^q,S^{m-1}]\to\pi^S_{p+q+1-m} in Definition 3.3. Apparently Theorem 6.1 has easier proofs (using the Whitney trick or surgery). Although I have not seen these results in the literature (however see [Skopenkov2000, Proposition 1.2] for the link map analogue), I presume they are folklore.

Now we present an extension of Theorems 6.1 and 6.2 to a case where m = 6 and n = 3. In particular, for the results below m=2\dim N_i, 2m=3n+3 and the manifolds N_i are only (2n-m)-connected. An embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers k,l,n such that l-n is even, Avvakumov has constructed a Brunnian embedding f_{k,l,n}:(S^2\times S^1)\sqcup S^3\to\Rr^6, which appears in the next result [Avvakumov2016].

Theorem 6.3. [Avvakumov2016] Any Brunnian embedding (S^2\times S^1)\sqcup S^3\to\Rr^6 is isotopic to f_{k,l,n} for some integers k,l,n such that l-n is even. Two embeddings f_{k,l,n} and f_{k',l',n'} are isotopic if and only if k=k' and both l-l' and n-n' are divisible by 2k.

The proof uses M. Skopenkov's classification of embeddings S^3\sqcup S^3\to\Rr^6 (Theorem 8.1 for m=2p=2q=6). The following corollary shows that the relation between the embeddings (S^2\times S^1)\sqcup S^3\to\Rr^6 and S^3\sqcup S^3\to\Rr^6 is not trivial.

Corollary 6.4. [Avvakumov2016], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings f:(S^2\times S^1)\sqcup S^3\to\Rr^6 and g,g':S^3\sqcup S^3\to\Rr^6 such that the componentwise embedded connected sum f\#g is isotopic to f\#g' but g is not isotopic to g'.

For an unpublished generalization of Theorem 6.3 and Corollary 6.4 see [Avvakumov2017].

7 Reduction to the case with unknotted components

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

Define E^m_U(S^{(n)}) \subset E^m_D(S^{(n)}) to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding S^{n_i} \to S^m. We remark that 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].

Define the restriction homomorphism by mapping the isotopy class of a link to the ordered s-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].

Take s pairwise disjoint m-discs in S^m, i.e. take an embedding g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m. 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].

Then j is a right inverse of the restriction homomorphism r, i.e. r\circ j=\mathrm{id}. The unknotting homomorphism u 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)}).

Informally, the homomorphism u is obtained by taking embedded connected sums of components with knots h_i:S^{n_i}\to S^m representing the elements of E^m_D(S^{n_i}) inverse to the components, whose images h_i(S^{n_i}) are small and are close to the components.

Theorem 7.1. [Haefliger1966a] For n_1,\ldots,n_s\le m-3, the homomorphism

\displaystyle u \oplus r \colon E^m_D(S^{(n)}) \to E^m_U(S^{(n)})\oplus \bigoplus_{i=1}^s E^m_D(S^{n_i})

is an isomorphism.

For some information on the groups E^m_D(S^q) see [Skopenkov2016s], [Skopenkov2006, \S3.3].

8 Classification beyond the metastable range

Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] 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]. For alternative geometric (and presumably equivalent) definitons of \beta see [Skopenkov2009, \S3], [Skopenkov2006b, \S5], cf. [Skopenkov2007, \S2] and [Crowley&Skopenkov2016, \S2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].

Remark 8.2. (a) Theorem 8.1 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) For any l\not\in\{1,3,7\}, 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\}.

For l\ge4 see [Haefliger1962t, \S6]. The following proof for l=2 and general remark are intended for specialists.

For l=2 there is an exact sequence \pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3), where \lambda_{12} is \lambda_{12,PL} [Haefliger1966a, Corollary 10.3]. We have \pi_3(S^2)\cong\Zz, \pi_4(S^3)\cong\Zz_2 and \Sigma is the reduction modulo 2. By the exactness of the previous sequence, \lambda_{21}(\ker \lambda_{12}) = 2\Z. By (a) \ker \lambda_{12} \cong \Z. Hence \lambda_{21}|_{\ker\lambda_{12}} is injective. We have \Sigma\lambda_{21}=\Sigma\lambda_{12} by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.

Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of \beta [Skopenkov2009, \S3], [Skopenkov2006b, \S5] and geomeric interpretation of the EHP sequence \ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots [Koschorke&Sanderson1977] one can possibly prove that \pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}. Then (b) would follow.

(c) For any 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 invariants of (a) and (b) above, and the Massey number (\S8). This follows from [Haefliger1962t, \S6]. We conjecture that this result holds also for l=2,3,7.

9 Classification in codimension at least 3

In this section we assume that (n) is an s-tuple such that m\ge n_i+3. For this case a readily calculable classification of E^m_{PL}(S^{(n)})\otimes\Qq was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, \S1.2] contains necessary and sufficient conditions on (n) which determine when E^m_{PL}(S^{(n)}) is finite. These results are elementary but the precise formulations are technical. Hence we only state the following corollary and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.

Theorem 9.1. [Crowley&Ferry&Skopenkov2011] There are algorithms which for integers m,n_1,\ldots,n_s>0

(a) calculate
Tex syntax error
.

(b) determine whether E^m_{PL}(S^{(n)}) 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~.

We briefly define the groups and homomorphisms in this sequence, refering to [Haefliger1966a, 1.4] for details. We first note that in the sequence above the 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}. <!W_m^{(n)}--> For any j=1,\ldots,s and positive integer k denote by p_{k,j}\colon\pi_k(W)\to\pi_k(S^{m-n_j-1}) the homomorphism induced by the collapse map onto to the j-component of the wedge. Denote (n-1):=(n_1 -1, \ldots, n_s-1) and \Pi_{m-1}^{(n)}:=\ker(\oplus_{i=1}^s p_{m-1,i}).

Analogous to Definition 3.1 there is a canonical homotopy equivalence C_f \sim W. It can be shown that each component of a link S^{(n)} \to S^m 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. It can also be shown that the homotopy class of a push off of the jth component in the complement of the link gives a well-defined map \lambda_j:E^m_{PL}(S^{(n)})\to\ker p_{n_j,j}. In fact, the map \lambda_j is a generalisation of the linking coefficient. Finally, 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 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 \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 general, the computation of the groups and homomorphisms appearing in Theorem 9.3 is difficult. For (a) no computations are known except for those giving (b) and those giving Theorem 8.1 for 3m\ge2p+2q+7 (note that Theorem 8.1 has a direct proof easier than proof of Theorem 9.3.a). For (b) the computations constitute most of [Crowley&Ferry&Skopenkov2011]. Hence Theorem 9.3 is not a readily calculable classification in general.

10 References

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