High codimension links
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want | + | 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. |
− | 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 | |
+ | $S^{n_1}\sqcup\ldots\sqcup S^{n_s}\to S^m$ for $m-3\ge n_i$ (under some further restrictions),--> 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 \cite{Skopenkov2016k}. | ||
− | For a | + | 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 [[Isotopy|the case of manifolds]] \cite{Skopenkov2016i}. Denote by $E^m(S^{(n)})$ the set of embeddings $S^{(n)}\to S^m$ up to isotopy. |
+ | |||
+ | For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$3]{Skopenkov2016c}. | ||
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem \ref{lkhaze} below): | The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem \ref{lkhaze} below): | ||
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|E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 | |E^m(S^q\sqcup S^q)| &1 &\infty &2 &2 &24 &1 &1 | ||
\end{array}$$ | \end{array}$$ | ||
− | + | [[Image:EmbeddedConnectedSum.jpg|thumb|350px|Figure 1: Component-wise 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$5]{Skopenkov2016c} defines a commutative group structure on the set $E^m(S^{(n)})$ for $m-3\ge n_i$ \cite[2.5]{Haefliger1966a}, \cite[Group Structure Lemma 2.2 and Remark 2.3.a]{Skopenkov2015}, \cite[$\S$1]{Avvakumov2016}, \cite[$\S$1.4]{Avvakumov2017}, see Figure 1. | |
− | A component-wise version of [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Embedded_connected_sum|embedded connected sum]] \cite[$\S$5]{Skopenkov2016c} defines a commutative group structure on the set $E^m(S^{(n)})$ for $m-3\ge n_i$ \cite | + | |
+ | 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 $D^m_k \to S^m$. <!-- {\rm i}_k $g\circ({\rm i}_{n_1}\sqcup\ldots\sqcup{\rm i}_s)$. in $S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$ We define the standard embedding $S^{(n)}\to S^m$ as follows. --> | ||
</wikitex> | </wikitex> | ||
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{{beginthm|Example|(The Hopf Link)}}\label{hopf} | {{beginthm|Example|(The Hopf Link)}}\label{hopf} | ||
− | For | + | (a) 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. |
− | + | ||
− | For $q=1$ the Hopf link is shown | + | [[Image:HopfLink-and-TrivialLink.jpg|thumb|350px|Figure 2: The Hopf link (a) and the trivial link (b)]] |
− | For | + | 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 $\partial D^{q+1}\times0$ and $0\times\partial D^{q+1}$ in $\partial(D^{q+1}\times D^{q+1})\cong S^{2q+1}$; | + | * 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 given | + | * 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_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.$$ | \left\{\begin{array}{c} x_{q+2}=\dots=x_{2q+1}=0\\ x_1^2+\dots+x_q^2+(x_{q+1}-1)^2=1.\end{array}\right.$$ | ||
− | This embedding is distinguished from the standard embedding by [[#The linking coefficient|the linking | + | This embedding is distinguished from the standard embedding by [[#The linking coefficient|the linking number]] (cf. $\S$\ref{s:inv}). |
− | + | (b) For any $p,q$ there is an embedding $S^p\sqcup S^q\to\Rr^{p+q+1}$ which is not isotopic to the standard embedding. | |
− | + | Analogously to (a), the image is the union of two spheres which can be described as follows: | |
− | * or given | + | * 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: | ||
$$\left\{\begin{array}{c} x_1=\dots=x_p=0\\ x_{p+1}^2+\dots+x_{p+q+1}^2=1,\end{array}\right. \qquad\text{and}\qquad | $$\left\{\begin{array}{c} x_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.$$ | \left\{\begin{array}{c} x_{p+2}=\dots=x_{p+q+1}=0\\ x_1^2+\dots+x_p^2+(x_{p+1}-1)^2=1.\end{array}\right.$$ | ||
− | This embedding is distinguished from the standard embedding | + | This embedding is also distinguished from the standard embedding by [[#The linking coefficient|the linking number]] (cf. $\S$\ref{s:inv}). |
+ | {{endthm}} | ||
− | {{beginthm|Definition|(The | + | {{beginthm|Definition|(A link with prescribed [[#The linking coefficient|linking coefficient]])}}\label{dz} |
− | We define | + | We define the `Zeeman' map |
+ | [[Image:linking_coefficient.jpg|thumb|350px|Figure 3: A link with prescribed linking coefficient]] | ||
$$\zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.$$ | $$\zeta=\zeta_{m,p,q}:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)\quad\text{for}\quad p\le q.$$ | ||
− | |||
For a map $x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$ let | For a map $x:S^p\to S^{m-q-1}$ representing an element of $\pi_p(S^{m-q-1})$ let | ||
$$\overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad | $$\overline\zeta\phantom{}_x:S^p\to S^m\quad\text{be the composition}\quad | ||
− | S^p\overset{x\times | + | S^p\overset{x\times{\rm i}}\to S^{m-q-1}\times S^q\overset{{\rm i}_{m,q}}\to S^m,$$ |
− | + | see Figure 3. We have $\overline\zeta_x(S^p)\cap{\rm i}(S^q)\subset{\rm i}_{m,q}(S^{m-q-1}\times S^p)\cap{\rm i}_{m,q}(0\times S^q) =\emptyset$. Let $\zeta[x]:=[\overline\zeta_x\sqcup{\rm i}]$. | |
− | We have $\overline\zeta_x(S^p)\cap | + | <!--where $i$ is the natural `standard embedding' defined in \cite[$\S$2.1]{Skopenkov2015a}--> |
− | Let $\zeta[x]:=[\overline\zeta_x\sqcup | + | |
{{endthm}} | {{endthm}} | ||
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<wikitex>;\label{s:inv} | <wikitex>;\label{s:inv} | ||
− | Here we define the linking coefficient and discuss | + | Here we define the linking coefficient and discuss its properties. |
Fix orientations of the standard spheres and balls. | Fix orientations of the standard spheres and balls. | ||
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$$\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.$$ | $$\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 $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 $ | + | 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. | Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. | ||
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{{beginthm|Remark}}\label{lkrem} | {{beginthm|Remark}}\label{lkrem} | ||
− | (a) Clearly, $\lambda[f]$ is | + | (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. | One can check that $\lambda$ is a homomorphism. | ||
− | (b) | + | (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) | + | (c) Clearly, $\lambda\zeta=\id\pi_p(S^{m-q-1})$ for the [[#Examples|Zeeman map]] $\zeta$. So $\lambda$ is surjective and $\zeta$ is injective. |
− | (d) | + | (d) For $m=p+q+1$ there is a simpler alternative definition 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}} | {{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. | |
− | $m\ge\frac p2+q+2$. | + | |
− | The | + | |
− | {{beginthm|Definition|(The $\alpha$-invariant)}} We define a map | + | {{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$. | $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)$. | 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 | + | 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|}.$$ | $$\widetilde f:S^p\times S^q\to S^{m-1}\quad\text{by}\quad\widetilde f(x,y)=\frac{fx-fy}{|fx-fy|}.$$ | ||
For $p,q\le m-2$ define the $\alpha$-invariant by | For $p,q\le m-2$ define the $\alpha$-invariant by | ||
− | $$\alpha | + | $$\alpha[f]=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$$ |
− | The second isomorphism in this formula is | + | The 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. | + | 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 | + | 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 | + | (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 $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$.) |
− | $(S^p\times S^q,S^p\vee S^q)$ 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]] --> |
{{endthm}} | {{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}}} | {{beginthm|Lemma|\cite[Lemma 5.1]{Kervaire1959a}}} | ||
− | We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$. | + | We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ for $p,q\le m-3$. |
{{endthm}} | {{endthm}} | ||
− | Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$. | + | Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$, even though in general $\lambda_{12} \neq \pm \lambda_{21}$ as we explain in Example \ref{belmetwhi}.a,c. |
− | Note that $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite[$\S$5]{Skopenkov2006}. | + | Note that the $\alpha$-invariant can be defined in more general situations \cite{Koschorke1988}, \cite[$\S$5]{Skopenkov2006}. |
</wikitex> | </wikitex> | ||
− | ==Classification in the metastable range== | + | == Classification in the metastable range == |
<wikitex>; | <wikitex>; | ||
{{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} | {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} | ||
− | + | If $1\le p\le q$, then both $\lambda$ and $\zeta$ are isomorphisms for $m\ge\frac{3q}2+2$ in the smooth category, and for $m\ge\frac p2+q+2$ in the PL category. | |
− | + | ||
− | + | ||
{{endthm}} | {{endthm}} | ||
The surjectivity of $\lambda$ (or the injectivity of $\zeta$) follows from $\lambda\zeta=\id$. | 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} | + | The injectivity of $\lambda$ (or the surjectivity of $\zeta$) is proved in \cite[Theorem in $\S$5]{Haefliger1962t}, \cite{Zeeman1962} |
(or follows from \cite[the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]{Skopenkov2006}). | (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 \cite{ | + | An analogue of this result holds for links with many components, each of the same dimension \cite[Theorem in $\S$5]{Haefliger1962t}. Let $(q) = (q, \dots, q)$ be the $s$-tuple consisting entirely of some positive integer $q$. |
− | {{beginthm|Theorem}}\label{t:lkmany} | + | {{beginthm|Theorem}}\label{t:lkmany} The collection of pairwise linking coefficients |
− | The collection of pairwise linking coefficients | + | |
$$\bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$ | $$\bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : E^m(S^{(q)}) \to (\pi_{2q+1-m}^S)^{s(s-1)/2}$$ | ||
− | is | + | is a 1-1 correspondence for $m\ge\frac{3q}2+2$. |
− | + | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
==Examples beyond the metastable range== | ==Examples beyond the metastable range== | ||
− | <wikitex>; | + | <wikitex>;\label{s:ebmr} |
− | + | For $l=1$ the results of this section are parts of low-dimensional link theory, so they were known well before given references. | |
− | + | First 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 (higher-dimensional) | + | (a) There is a non-trivial embedding $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ whose restriction to each 2-component sublink is trivial \cite[4.1]{Haefliger1962}, \cite[$\S$6]{Haefliger1962t}. |
− | are | + | |
+ | In order to construct such an embedding, 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 spheres']] are 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} 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.$$ | \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1. \end{array}\right.$$ | ||
− | The required embedding is any embedding whose image | + | [[Image:borromean_hidim_new.jpg|thumb|400px|Figure 6: The Borromean rings]] |
− | { | + | See Figure 6. |
+ | The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres. | ||
+ | <!--See [[Media:35.pdf|the figure]] and [[Media:36.pdf|another figure]]. \cite[Figures 3.5 and 3.6]{Skopenkov2006} --> | ||
− | + | (b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. | |
− | + | More precisely, each two of the Borromean spheres span two (intersecting) $2l$-disks disjoint from the remaining sphere. | |
− | + | (c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint $2l$-disks (intersecting the remaining sphere). | |
+ | Moreover, we can take these $2l$-disks so that | ||
+ | |||
+ | * each one of them intersects the remaining sphere transversely by an $(l-1)$-sphere; | ||
+ | |||
+ | * the obtained two disjoint $(l-1)$-spheres in the remaining sphere have linking number $\pm1$, i.e. one of them spans an $l$-disk (in the remaining sphere) itersecting the other transversely at exacly one point. | ||
+ | |||
+ | (d) The Borromean link is distinguished from the standard embedding by ''triple linking number'' of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). | ||
+ | Take a 3-component link, i.e. an embedding $g:S^{2l-1}_1\sqcup S^{2l-1}_2\sqcup S^{2l-1}_3\to S^{3l}$. | ||
+ | Assume that $g$ is ''pairwise unlinked'', i.e. every two components are contained in disjoint smooth balls. | ||
+ | Let $D_2,D_3\subset S^{3l}$ be disjoint oriented embedded $2l$-disks in general position to $g_1:=g|_{S^{2l-1}_1}$, and such that $g(S^{2l-1}_i)=\partial D_i$ for $i=2,3$. | ||
+ | Then for $j=2,3$ the preimage $g_1^{-1}D_j$ is an oriented $(l-1)$-submanifold of $S^{2l-1}_1$ missing $g_1^{-1}D_{5-j}$. | ||
+ | Let $\mu(g)$ be the linking number of $g_1^{-1}D_2$ and $g_1^{-1}D_3$ in $S^{2l-1}_1$. | ||
+ | |||
+ | (e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of $D_2,D_3$, and of the isotopy of $g$. | ||
+ | The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number \cite[$\S$4]{Haefliger1962t}, \cite{HaefligerSteer1965}, \cite[proof of Theorem 9.4]{Haefliger1966a}, \cite{Massey1968}, \cite[$\S$7]{Massey1990} by the well-known `linking number' definition of the Whitehead invariant $\pi_{2l-1}(S^l\vee S^l)\to\Z$ \cite[$\S$2, Sketch of a proof of (b1)]{Skopenkov2020e}. Cf. \cite[\S9.1]{Moriyama2008}. If $g$ is pairwise unlinked, then the number $\mu$ is independent of permutation of the components, up to multiplication by $\pm1$ \cite{HaefligerSteer1965} (this can be easily proved directly). | ||
+ | |||
+ | (f) The Borromean link is non-trivial also 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[$\S$4]{Skopenkov2016c}) yields a non-trivial knot \cite[Theorem 4.3]{Haefliger1962}, cf. [[3-manifolds_in_6-space#Examples|the Haefliger Trefoil knot]] \cite[Example 2.1]{Skopenkov2016t}. | ||
+ | {{endthm}} | ||
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}. | 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|Whitehead link | + | {{beginthm|Example|(Whitehead link)}}\label{belmetwhi} (a) For every positive integer $l$ there is a non-trivial embedding $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient $\lambda_{21}(w)$ 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 the second and the third 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$3, $\S$4]{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.) | |
+ | |||
+ | (b) The second component is null-homotopic in the complement of the first component by Example \ref{belmetbor}.b. | ||
+ | |||
+ | (c) For $l\ne1,3,7$ the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. | ||
+ | More precisely, $\lambda_{12}(w)$ equals to the Whitehead square $[\iota_l,\iota_l]\ne0$ of the generator $\iota_l\in\pi_l(S^l)$ \cite[end of $\S$6]{Haefliger1962t} (the proof is not hard put presumably was not published before \cite[the Whitehead Link Lemma 2.14]{Skopenkov2015a}, \cite[the Whitehead Link Lemma 3.1]{Skopenkov2024} for $l$ even). | ||
+ | For $l=1,3,7$ the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant \cite{Skopenkov2006a}. | ||
+ | |||
+ | (d) This example (in higher dimensions, i.e. for $l>1$) seems to have been discovered by Whitehead, in connection with Whitehead product. <!--It would be interesting to find a publication where it first appeared.--> | ||
+ | |||
+ | (e) For some results on links $S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ related to the Whitehead link see \cite[$\S$2.5]{Skopenkov2015a}. | ||
{{endthm}} | {{endthm}} | ||
+ | </wikitex> | ||
− | We have $\lambda_{ | + | == Linked manifolds == |
− | + | <wikitex>; | |
− | + | 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 high-connectivity assumptions are made on the source manifolds. | |
− | + | ||
− | \cite{ | + | {{beginthm|Theorem}}\label{t:man} Assume that $N_1,\ldots,N_s$ are closed connected manifolds and $\frac{m-1}2=\dim N_i\ge2$ 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 |
+ | $$\bigoplus\limits_{1\le i<j\le s}\lambda_{ij} : | ||
+ | E^m(N_1\sqcup\ldots\sqcup N_s) \to \Z^{\frac{t(t-1)}2} \oplus \Z_2^{\frac{s(s-1)}2-\frac{t(t-1)}2}$$ | ||
+ | is well-defined and is a 1-1 correspondence, where $t$ of $N_1,\ldots, N_s$ are orientable and $s-t$ | ||
+ | are not. | ||
+ | {{endthm}} | ||
+ | |||
+ | We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension. | ||
+ | |||
+ | {{beginthm|Theorem}}\label{t:mang} Let $N_1$ and $N_2$ be closed $n$-dimensional homologically $(2n-m+1)$-connected orientable manifolds. 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 | ||
+ | $$\alpha:E^m(N_1\sqcup N_2)\to\pi_{2n-m+1}^S$$ | ||
+ | is well-defined and is a 1-1 correspondence, provided $2m\ge3n+4$. | ||
+ | {{endthm}} | ||
+ | |||
+ | Theorems \ref{t:man} and \ref{t:mang} are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category \cite{Ivansic&Horvatic1974}. They are also corollaries of \cite[the Haefliger-Weber Theorem 5.4]{Skopenkov2006} (in both categories); 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}. | ||
+ | See \cite[Proposition 1.2]{Skopenkov2000} for the ''link map'' analogue. | ||
+ | |||
+ | 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 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[$\S$1]{Avvakumov2016}. | ||
+ | |||
+ | {{beginthm|Theorem}}\label{t:avv} \cite[Theorem 1]{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 $2k$. | ||
+ | {{endthm}} | ||
+ | |||
+ | 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{c:avv} \cite[Corollary 1]{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 $g'$. | ||
+ | {{endthm}} | ||
− | + | For an unpublished generalization of Theorem \ref{t:avv} and Corollary \ref{c:avv} see \cite{Avvakumov2017}. | |
</wikitex> | </wikitex> | ||
− | == | + | == Reduction to the case with unknotted components == |
<wikitex>; | <wikitex>; | ||
+ | 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$1.5]{Crowley&Ferry&Skopenkov2011}. | ||
− | {{beginthm|Theorem}}\label{dpl} | + | Define the restriction homomorphism by mapping the isotopy class of a link to the ordered $s$-tuple of the isotopy classes of its components: |
− | \cite{Haefliger1966a} | + | $$ r \colon E^m_D(S^{(n)}) \to \bigoplus_{i=1}^s E^m_D(S^{n_i}), \quad r[f]:=\oplus_{i=1}^s [f|_{S^{n_i}} \colon S^{n_i} \to S^m].$$ |
− | $$E^m_D(S^{(n)})\ | + | Take $s$ pairwise disjoint $m$-discs in $S^m$, i.e. take an embedding $g:D^m_1\sqcup\ldots\sqcup D^m_s\to S^m$. Define |
+ | $$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)}).$$ | ||
+ | Informally, the homomorphism $u$ is obtained by taking embedded connected sums of components with knots $h_i:S^{n_i}\to S^m$ representing the elements of $E^m_D(S^{n_i})$ inverse to the components, whose images $h_i(S^{n_i})$ are small and are close to the components. | ||
+ | |||
+ | {{beginthm|Theorem}}\label{dpl} \cite[Theorem 2.4]{Haefliger1966a} For $n_1,\ldots,n_s\le m-3$, the homomorphism | ||
+ | $$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})$$ | ||
+ | is an isomorphism. | ||
{{endthm}} | {{endthm}} | ||
− | + | For [[Knots,_i.e._embeddings_of_spheres|some information on the groups $E^m_D(S^q)$]] see \cite{Skopenkov2016s}, \cite[$\S$3.3]{Skopenkov2006}. | |
− | + | </wikitex> | |
− | + | ==Classification beyond the metastable range== | |
+ | <wikitex>;\label{s:cbmr} | ||
− | {{beginthm| | + | {{beginthm|Theorem}}\label{belmethae} |
− | \cite[Theorem 10.7]{Haefliger1966a}, \cite{Skopenkov2009} | + | \cite[Theorem 10.7]{Haefliger1966a}, \cite[Theorem 1.1]{Skopenkov2009}, \cite[Theorem 1.1]{Skopenkov2006b} |
If $p\le q\le m-3$ and $3m\ge2p+2q+6$, then there is a homomorphism $\beta$ for which the following map is an isomorphism | If $p\le q\le m-3$ and $3m\ge2p+2q+6$, then there is a homomorphism $\beta$ for which the following map is an isomorphism | ||
$$\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}).$$ | $$\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}).$$ | ||
{{endthm}} | {{endthm}} | ||
− | The map $\beta$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in | + | The map $\beta$ and its right inverse $\pi_{p+q+2-m}(SO, SO_{m-p-1})\to\ker\lambda_{12}$ are constructed in \cite[$\S$10]{Haefliger1966a} and in \cite[8.13]{Haefliger1966}, respectively. For alternative geometric (and presumably equivalent) definitions of $\beta$ see \cite[$\S$3]{Skopenkov2009}, \cite[$\S$5]{Skopenkov2006b}, cf. \cite[$\S$2]{Skopenkov2007} and \cite[$\S$2.3]{Crowley&Skopenkov2016}. For a historical remark see \cite[the second paragraph in p. 2]{Skopenkov2009}. |
<!--The case $3m\ge4q+6$ (when by general position no quadruple linking appears) is called a `2-metastable range' for embeddings of $q$-polyhdera in $\R^m$. --> | <!--The case $3m\ge4q+6$ (when by general position no quadruple linking appears) is called a `2-metastable range' for embeddings of $q$-polyhdera in $\R^m$. --> | ||
{{beginthm|Remark}}\label{r:bel} | {{beginthm|Remark}}\label{r:bel} | ||
− | (a) | + | (a) Theorem \ref{belmethae} implies that for any $l\ge2$ we have an isomorphism |
$$\lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{\varepsilon(l)}.$$ | $$\lambda_{12}\oplus\beta:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\Z_{\varepsilon(l)}.$$ | ||
− | (b) | + | (b) For any $l\not\in\{1,3,7\}$, the map |
$$\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)$$ | $$\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)$$ | ||
− | is injective and its image is $\{(a,b)\ :\ \Sigma a= | + | is injective and its image is $\{(a,b)\ :\ \Sigma(a+(-1)^lb)=0\}$. |
− | + | ||
− | + | For $l\ge4$ see \cite[$\S$6]{Haefliger1962t}. The following proof for $l=2$ and general remark are intended for specialists. | |
− | + | For $l=2$ there is an exact sequence $\pi_5(S^3) \to \ker \lambda_{12} \xrightarrow{\lambda_{21}|_{ \ker \lambda_{12} } } \pi_3(S^2) \xrightarrow{\Sigma} \pi_4(S^3)$, | |
− | $ | + | where $\lambda_{12}$ is $\lambda_{12,PL}$ <!-- not $\lambda_{12,D}$ --> \cite[Corollary 10.3]{Haefliger1966a}. |
− | + | We have $\pi_3(S^2)\cong\Zz$, $\pi_4(S^3)\cong\Zz_2$ and $\Sigma$ is the reduction modulo 2. | |
+ | By the exactness of the previous sequence, <!--$\text{im} \, \lambda_{21}= 2\Z$. --> $ \lambda_{21}(\ker \lambda_{12}) = 2\Z$. | ||
+ | By (a) $\ker \lambda_{12} \cong \Z$. | ||
+ | Hence $\lambda_{21}|_{\ker\lambda_{12}}$ is injective. | ||
+ | We have $\Sigma\lambda_{21}=\Sigma\lambda_{12}$ by \cite[Proposition 10.2]{Haefliger1966a}. | ||
+ | So the formula of (b) follows. | ||
+ | |||
+ | Analogously to \cite[Theorem 3.5]{Skopenkov2009} using geometric definitons of $\beta$ \cite[$\S$3]{Skopenkov2009}, \cite[$\S$5]{Skopenkov2006b} and geomeric interpretation of the EHP sequence $\ldots\overset H\to\pi_l(SO,SO_l)\overset P\to\pi_{2l-1}(S^l)\overset\Sigma\to\pi_{l-1}^S\overset H\to\ldots$ \cite[Main Theorem in $\S$1]{Koschorke&Sanderson1977} one can possibly prove that $\pm P\beta=\lambda_{12}+(-1)^l\lambda_{21}$. Then (b) would follow. | ||
+ | |||
+ | (c) For any $l\in\{1,3,7\}$ the map $\lambda_{12}\oplus\lambda_{21}$ in (b) above is not injective \cite[$\S$6]{Haefliger1962t}. | ||
{{endthm}} | {{endthm}} | ||
+ | |||
+ | {{beginthm|Theorem}}\label{three} | ||
+ | For any $l>2$ there an isomorphism | ||
+ | $$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)}^3\oplus\Z$$ | ||
+ | which is the sum of 3 pairwise invariants of Remark \ref{r:bel}.a, and the triple linking number ($\S$\ref{s:ebmr}). | ||
+ | {{endthm}} | ||
+ | |||
+ | This follows from \cite[Theorem 9.4]{Haefliger1966a}, see also \cite[$\S$6]{Haefliger1962t}. | ||
</wikitex> | </wikitex> | ||
− | == Classification in codimension 3 == | + | == Classification in codimension at least 3 == |
− | <wikitex>; | + | <wikitex>;\label{s:cl3} |
− | In this | + | In this section we assume that $(n)$ is an $s$-tuple such that $m\ge n_i+3$. |
− | + | For this case a readily calculable classification of $E^m_{PL}(S^{(n)})\otimes\Qq$ was obtained in | |
− | + | \cite[Theorem 1.9]{Crowley&Ferry&Skopenkov2011}. In particular, \cite[$\S$1.2]{Crowley&Ferry&Skopenkov2011} contains necessary and sufficient conditions on $(n)$ which determine when $E^m_{PL}(S^{(n)})$ is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of \cite[Theorem 1.9 and $\S$1.2]{Crowley&Ferry&Skopenkov2011} and describe the methods of its proof in Definition \ref{d:hclink} and Theorem \ref{thm:hclink} below. | |
− | + | <!--, as well as an effective procedure for computing the rank of the group $E^m_{PL}(S^{(n)})$.--> | |
− | $ | + | |
− | + | ||
− | + | {{beginthm|Theorem}}\label{t:cfs} There are algorithms which for integers $m,n_1,\ldots,n_s>0$ | |
− | + | ||
− | + | (a) calculate ${\rm rk}E^m_{PL}(S^{(n)})$. | |
− | + | (b) determine whether $E^m_{PL}(S^{(n)})$ is finite. | |
{{endthm}} | {{endthm}} | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
+ | {{beginthm|Definition|(The Haefliger link sequence)}}\label{d:hclink} | ||
+ | In \cite[1.2-1.5]{Haefliger1966a} Haefliger defined a long exact sequence of abelian groups | ||
+ | $$ \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. | ||
+ | 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}$. | ||
+ | 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})\subset\pi_{m-1}(W)$. | ||
+ | |||
+ | It can be shown that each component of a link $f:S^{(n)} \to S^m$ has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement $C_f$. Analogously to Definition \ref{dl} there is a canonical homotopy equivalence $C_f \sim W$. It can also be shown that the homotopy class in $C_f$ of the push off of the $j$th component 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 generalization of the linking coefficient. Finally, define $\lambda:= \oplus_{j=1}^s \lambda_j$. | ||
+ | |||
+ | Taking the Whitehead product with the class of the inclusion of $S^{m-n_j-1}$ into $W$ 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} | {{beginthm|Theorem}}\label{thm:hclink} | ||
(a) \cite[Theorem 1.3]{Haefliger1966a} The Haefliger link sequence is exact. | (a) \cite[Theorem 1.3]{Haefliger1966a} The Haefliger link sequence is exact. | ||
− | (b) | + | (b) The map $\lambda\otimes\Qq \colon E^m_{PL}(S^{(n)})\otimes\Qq \to \ker(w\otimes\Qq)$ is an isomorphism. |
{{endthm}} | {{endthm}} | ||
− | Part (b) follows because $\mu \otimes \Qq = 0$ \cite[Lemma 1.3]{Crowley&Ferry&Skopenkov2011}, and so the link | + | 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 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> | ||
Latest revision as of 09:56, 26 April 2024
This page has been accepted for publication in the Bulletin of the Manifold Atlas. |
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1 Introduction
Most of this page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.
On this page we describe readily calculable classifications of embeddings of closed disconnected manifolds into up to isotopy, and more generally of spaces which are a disjoint union of manifolds of varying dimensions. The presently known cases of such classifications are embeddings , where are spheres (or even closed manifolds) and for every , under some further restrictions. For a related classification of knotted tori see [Skopenkov2016k].
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
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One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
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For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2020e] A. Skopenkov, Extendability of simplicial maps is undecidable, Discr. Comp. Geom., 69:1 (2023), 250–259. arXiv:2008.00492
- [Skopenkov2024] A. Skopenkov, The band connected sum and the second Kirby move for higher-dimensional links, submitted to arXiv on April 7, 2024.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2020e] A. Skopenkov, Extendability of simplicial maps is undecidable, Discr. Comp. Geom., 69:1 (2023), 250–259. arXiv:2008.00492
- [Skopenkov2024] A. Skopenkov, The band connected sum and the second Kirby move for higher-dimensional links, submitted to arXiv on April 7, 2024.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2020e] A. Skopenkov, Extendability of simplicial maps is undecidable, Discr. Comp. Geom., 69:1 (2023), 250–259. arXiv:2008.00492
- [Skopenkov2024] A. Skopenkov, The band connected sum and the second Kirby move for higher-dimensional links, submitted to arXiv on April 7, 2024.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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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- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
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For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
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One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2020e] A. Skopenkov, Extendability of simplicial maps is undecidable, Discr. Comp. Geom., 69:1 (2023), 250–259. arXiv:2008.00492
- [Skopenkov2024] A. Skopenkov, The band connected sum and the second Kirby move for higher-dimensional links, submitted to arXiv on April 7, 2024.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2020e] A. Skopenkov, Extendability of simplicial maps is undecidable, Discr. Comp. Geom., 69:1 (2023), 250–259. arXiv:2008.00492
- [Skopenkov2024] A. Skopenkov, The band connected sum and the second Kirby move for higher-dimensional links, submitted to arXiv on April 7, 2024.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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
- [Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -space, Ann. of Math. (2) 75 (1962), 452–466. MR0145539 (26 #3070) Zbl 0105.17407
- [Haefliger1962t] A. Haefliger, Differentiable links, Topology, 1 (1962) 241--244
- [Haefliger1966] A. Haefliger, Differential embeddings of in for , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
- [HaefligerSteer1965] A. Haefliger and B. Steer, Symmetry of linking coefficients, Comment. Math. Helv., 39 (1965), 259--270.pdf
- [Ivansic&Horvatic1974] I. Ivan\v{s}i\'c and K. Horvati\'c, On unlinking of polyhedra, Glasnik Mat. 9(29) (1974) 147-153.
- [Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
- [Koschorke&Sanderson1977] U. Koschorke and B. Sanderson, Geometric interpretation of the generalized Hopf invariant, Math. Scand. 41 (1977) 199--217.
- [Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
- [Massey1968] W. S. Massey, Higher order linking numbers, Proc. Conf. on Algebraic Topology, Univ. Illinois, Chicago Circle, Chicago, Ill., (1968) pp. 174--205. MR0254832 (40 #8039), see also MR1625365 (99e:57016) massey.pdf.
- [Massey1990] W. S. Massey, Homotopy classification of 3-component links of codimension greater than 2, Topol. Appl. 34 (1990) 269--300.
- [Moriyama2008] T. Moriyama, An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number, J. Math. Sci. Univ. Tokyo, 18 (2011), 193--237. arXiv:0806.3733.
- [Skopenkov2000] A. Skopenkov, On the generalized Massey-Rolfsen invariant for link maps, Fund. Math. 165 (2000), 1-15.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [Skopenkov2006b] M. Skopenkov, A formula for the group of links in the 2-metastable dimension, arxiv:math/0610320v1
- [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
- [Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
- [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Skopenkov2020e] A. Skopenkov, Extendability of simplicial maps is undecidable, Discr. Comp. Geom., 69:1 (2023), 250–259. arXiv:2008.00492
- [Skopenkov2024] A. Skopenkov, The band connected sum and the second Kirby move for higher-dimensional links, submitted to arXiv on April 7, 2024.
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
For an -tuple denote . Although is not a manifold when are not all equal, embeddings and isotopy between such embeddings are defined analogously to the case of manifolds [Skopenkov2016i]. Denote by the set of embeddings up to isotopy.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3].
The following table was obtained by Zeeman around 1960 (see the Haefliger-Zeeman Theorem 4.1 below):
A component-wise version of embedded connected sum [Skopenkov2016c, 5] defines a commutative group structure on the set for [Haefliger1966a, 2.5], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a], [Avvakumov2016, 1], [Avvakumov2017, 1.4], see Figure 1.
The standard embedding is defined by . Fix pairwise disjoint -discs . The standard embedding is defined by taking the union of the compositions of the standard embeddings with the fixed inclusions .
2 Examples
Recall that for any -manifold and , every two embeddings are isotopic [Skopenkov2016c, General Position Theorem 2.1], [Skopenkov2006, General Position Theorem 2.1]. The following example shows that the restriction is sharp for non-connected manifolds.
Example 2.1 (The Hopf Link). (a) For every positive integer there is an embedding , which is not isotopic to the standard embedding.
For the Hopf link is shown in Figure 2. For all the image of the Hopf link is the union of two -spheres which can be described as follows:
- either the spheres are and in ;
- or they are given as the sets of points in satisfying the following equations:
This embedding is distinguished from the standard embedding by the linking number (cf. 3).
(b) For any there is an embedding which is not isotopic to the standard embedding.
Analogously to (a), the image is the union of two spheres which can be described as follows:
- either the spheres are and in .
- or they are given as the points in satisfying the following equations:
This embedding is also distinguished from the standard embedding by the linking number (cf. 3).
Definition 2.2 (A link with prescribed linking coefficient). We define the `Zeeman' map
For a map representing an element of let
Tex syntax error
Tex syntax error. Let
Tex syntax error.
One can easily check that is well-defined and is a homomorphism.
3 The linking coefficient
Here we define the linking coefficient and discuss its properties. Fix orientations of the standard spheres and balls.
Definition 3.1 (The linking coefficient). We define a map
Take an embedding representing an element . Take an embedding such that intersects transversely at exactly one point with positive sign; see Figure 4.
Then the restriction of to is a homotopy equivalence.
(Indeed, since , by general position the complement is simply-connected. By Alexander duality, induces isomorphism in homology. Hence by the Hurewicz and Whitehead theorems is a homotopy equivalence.)
Let be a homotopy inverse of . Define
Remark 3.2. (a) Clearly, is well-defined, i.e. is independent of the choices of and of the representative of . One can check that is a homomorphism.
(b) Analogously one can define for , by exchanging and in the above definition.
(c) Clearly, for the Zeeman map . So is surjective and is injective.
(d) For there is a simpler alternative definition using homological ideas. That definition can be generalized to the case where the components are closed orientable manifolds. Cf. Remark 5.3.f of [Skopenkov2016e].
Recall that by the Freudenthal Suspension Theorem the stable suspension homomorphism is an isomorphism for . The stable suspension of the linking coefficient can be described as follows.
Definition 3.3 (The -invariant). We define a map for . Take an embedding representing an element . Define the Gauss map (see Figure 4)
For define the -invariant by
The second isomorphism in this formula is the suspension isomorphism. The map is the quotient map, see Figure 5.
The map is a 1--1 correspondence for . (For this follows by general position and for by the cofibration Barratt-Puppe exact sequence of pair and by the existence of a retraction .)
One can easily check that is well-defined and for is a homomorphism.
Lemma 3.4 [Kervaire1959a, Lemma 5.1]. We have for .
Hence , even though in general as we explain in Example 5.2.a,c.
Note that the -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, 5].
4 Classification in the metastable range
The Haefliger-Zeeman Theorem 4.1. If , then both and are isomorphisms for in the smooth category, and for in the PL category.
The surjectivity of (or the injectivity of ) follows from . The injectivity of (or the surjectivity of ) is proved in [Haefliger1962t, Theorem in 5], [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 in 5]. Let be the -tuple consisting entirely of some positive integer .
Theorem 4.2. The collection of pairwise linking coefficients
is a 1-1 correspondence for .
5 Examples beyond the metastable range
For the results of this section are parts of low-dimensional link theory, so they were known well before given references.
First 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). (a) There is a non-trivial embedding whose restriction to each 2-component sublink is trivial [Haefliger1962, 4.1], [Haefliger1962t, 6].
In order to construct such an embedding, denote coordinates in by . The (higher-dimensional) `Borromean spheres' are given by the following three systems of equations:
See Figure 6. The required embedding, called Borromean link, is any embedding whose image is the union of the Borromean spheres.
(b) By moving two of the Borromean spheres and self-intersecting them, we can drag the remaining sphere far away from the two. More precisely, each two of the Borromean spheres span two (intersecting) -disks disjoint from the remaining sphere.
(c) Restriction of the Borromean link to each 2-component sublink is trivial, because each two of the Borromean spheres span two disjoint -disks (intersecting the remaining sphere). Moreover, we can take these -disks so that
- each one of them intersects the remaining sphere transversely by an -sphere;
- the obtained two disjoint -spheres in the remaining sphere have linking number , i.e. one of them spans an -disk (in the remaining sphere) itersecting the other transversely at exacly one point.
(d) The Borromean link is distinguished from the standard embedding by triple linking number of Milnor-Haefliger-Steer-Massey defined as follows (cf. (c)). Take a 3-component link, i.e. an embedding . Assume that is pairwise unlinked, i.e. every two components are contained in disjoint smooth balls. Let be disjoint oriented embedded -disks in general position to , and such that for . Then for the preimage is an oriented -submanifold of missing . Let be the linking number of and in .
(e) In a standard way one checks that the triple linking number is well-defined, i.e. is independent of the choice of , and of the isotopy of . The above definition is equivalent to the standard definition of Milnor-Haefliger-Steer-Massey number [Haefliger1962t, 4], [HaefligerSteer1965], [Haefliger1966a, proof of Theorem 9.4], [Massey1968], [Massey1990, 7] by the well-known `linking number' definition of the Whitehead invariant [Skopenkov2020e, 2, Sketch of a proof of (b1)]. Cf. [Moriyama2008, \S9.1]. If is pairwise unlinked, then the number is independent of permutation of the components, up to multiplication by [HaefligerSteer1965] (this can be easily proved directly).
(f) The Borromean link is non-trivial also because joining the three components with two tubes (i.e. `linked analogue' of embedded connected sum of the components [Skopenkov2016c, 4]) yields a non-trivial knot [Haefliger1962, Theorem 4.3], cf. the Haefliger Trefoil knot [Skopenkov2016t, Example 2.1].
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). (a) For every positive integer there is a non-trivial embedding such that the second component is null-homotopic in the complement of the first component (i.e. the linking coefficient is trivial).
Such an embedding is obtained from the Borromean rings embedding by joining the second and the third components with a tube, i.e. by the `linked analogue' of the embedded connected sum of the components [Skopenkov2016c, 3, 4]; see also the Wikipedia article on the Whitehead link. (For the connected sum is not well-defined, so we take the specific connected sum (w) from Figure 7.)
(b) The second component is null-homotopic in the complement of the first component by Example 5.1.b.
(c) For the Whitehead link is non-trivial because the first component is not null-homotopic in the complement of the second component. More precisely, equals to the Whitehead square of the generator [Haefliger1962t, end of 6] (the proof is not hard put presumably was not published before [Skopenkov2015a, the Whitehead Link Lemma 2.14], [Skopenkov2024, the Whitehead Link Lemma 3.1] for even). For the Whitehead link is distinguished from the standard embedding by more complicated Sato-Levine invariant [Skopenkov2006a].
(d) This example (in higher dimensions, i.e. for ) seems to have been discovered by Whitehead, in connection with Whitehead product.
(e) For some results on links related to the Whitehead link see [Skopenkov2015a, 2.5].
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 high-connectivity assumptions are made on the source manifolds.
Theorem 6.1. Assume that are closed connected manifolds and for every . Then for an embedding and every one defines the linking coefficient , see Remark 3.2.e. We have if both and are orientable, and otherwise. Then the collection of pairwise linking coefficients
is well-defined and is a 1-1 correspondence, where of are orientable and are not.
We present the general case of the classification of embeddings of disjoint highly-connected manifolds only for the case of two oriented components of the same dimension.
Theorem 6.2. Let and be closed -dimensional homologically -connected orientable manifolds. For an embedding one can define the invariant analogously to Definition 3.3. Then
is well-defined and is a 1-1 correspondence, provided .
Theorems 6.1 and 6.2 are easily deduced from the Zeeman/Haefliger Unknotting Theorem in the PL/smooth category [Ivansic&Horvatic1974]. They are also corollaries of [Skopenkov2006, the Haefliger-Weber Theorem 5.4] (in both categories); the calculations are analogous to the construction of the 1-1 correspondence in Definition 3.3. See [Skopenkov2000, Proposition 1.2] for the link map analogue.
Now we present an extension of Theorems 6.1 and 6.2 to a case where and . In particular, for the results below , and the manifolds are only -connected. An embedding is Brunnian if its restriction to each component is isotopic to the standard embedding. For each triple of integers such that is even, Avvakumov has constructed a Brunnian embedding , which appears in the next result [Avvakumov2016, 1].
Theorem 6.3. [Avvakumov2016, Theorem 1] Any Brunnian embedding is isotopic to for some integers such that is even. Two embeddings and are isotopic if and only if and both and are divisible by .
The proof uses M. Skopenkov's classification of embeddings (Theorem 8.1 for ). The following corollary shows that the relation between the embeddings and is not trivial.
Corollary 6.4. [Avvakumov2016, Corollary 1], cf. [Skopenkov2016t, Corollary 3.5.b] There exist embeddings and such that the componentwise embedded connected sum is isotopic to but is not isotopic to .
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 is an -tuple such that .
Define to be the subgroup of links all whose components are unknotted, i.e. isotopic to the standard embedding . We remark that [Haefliger1966a, 2.4, 2.6 and 9.3], [Crowley&Ferry&Skopenkov2011, 1.5].
Define the restriction homomorphism by mapping the isotopy class of a link to the ordered -tuple of the isotopy classes of its components:
Take pairwise disjoint -discs in , i.e. take an embedding . Define
Then is a right inverse of the restriction homomorphism , i.e. . The unknotting homomorphism is defined to be the homomorphism
Informally, the homomorphism is obtained by taking embedded connected sums of components with knots representing the elements of inverse to the components, whose images are small and are close to the components.
For some information on the groups see [Skopenkov2016s], [Skopenkov2006, 3.3].
8 Classification beyond the metastable range
Theorem 8.1. [Haefliger1966a, Theorem 10.7], [Skopenkov2009, Theorem 1.1], [Skopenkov2006b, Theorem 1.1] If and , then there is a homomorphism for which the following map is an isomorphism
The map and its right inverse are constructed in [Haefliger1966a, 10] and in [Haefliger1966, 8.13], respectively. For alternative geometric (and presumably equivalent) definitions of see [Skopenkov2009, 3], [Skopenkov2006b, 5], cf. [Skopenkov2007, 2] and [Crowley&Skopenkov2016, 2.3]. For a historical remark see [Skopenkov2009, the second paragraph in p. 2].
Remark 8.2. (a) Theorem 8.1 implies that for any we have an isomorphism
(b) For any , the map
is injective and its image is .
For see [Haefliger1962t, 6]. The following proof for and general remark are intended for specialists.
For there is an exact sequence , where is [Haefliger1966a, Corollary 10.3]. We have , and is the reduction modulo 2. By the exactness of the previous sequence, . By (a) . Hence is injective. We have by [Haefliger1966a, Proposition 10.2]. So the formula of (b) follows.
Analogously to [Skopenkov2009, Theorem 3.5] using geometric definitons of [Skopenkov2009, 3], [Skopenkov2006b, 5] and geomeric interpretation of the EHP sequence [Koschorke&Sanderson1977, Main Theorem in 1] one can possibly prove that . Then (b) would follow.
(c) For any the map in (b) above is not injective [Haefliger1962t, 6].
Theorem 8.3. For any there an isomorphism
which is the sum of 3 pairwise invariants of Remark 8.2.a, and the triple linking number (5).
This follows from [Haefliger1966a, Theorem 9.4], see also [Haefliger1962t, 6].
9 Classification in codimension at least 3
In this section we assume that is an -tuple such that . For this case a readily calculable classification of was obtained in [Crowley&Ferry&Skopenkov2011, Theorem 1.9]. In particular, [Crowley&Ferry&Skopenkov2011, 1.2] contains necessary and sufficient conditions on which determine when is finite. These results are elementary, but the precise formulations are technical. Hence, we only state the following corollary of [Crowley&Ferry&Skopenkov2011, Theorem 1.9 and 1.2] and describe the methods of its proof in Definition 9.2 and Theorem 9.3 below.
Theorem 9.1. There are algorithms which for integers
(a) calculateTex syntax error.
(b) determine whether is finite.
Definition 9.2 (The Haefliger link sequence). In [Haefliger1966a, 1.2-1.5] Haefliger defined a long exact sequence of abelian groups
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 -tuples are the same for different terms. Denote . For any and positive integer denote by the homomorphism induced by the collapse map onto to the -component of the wedge. Denote and .
It can be shown that each component of a link has a non-zero normal vector field. So each component of a link can be pushed off along such a vector field into the complement . Analogously to Definition 3.1 there is a canonical homotopy equivalence . It can also be shown that the homotopy class in of the push off of the th component gives a well-defined map . In fact, the map is a generalization of the linking coefficient. Finally, define .
Taking the Whitehead product with the class of the inclusion of into defines a homomorphism . Define .
The definition of the homomorphism is given in [Haefliger1966a, 1.5].
Theorem 9.3. (a) [Haefliger1966a, Theorem 1.3] The Haefliger link sequence is exact.
(b) The map is an isomorphism.
Part (b) follows because [Crowley&Ferry&Skopenkov2011, Lemma 1.3], and so the Haefliger link sequence after tensoring with the rational numbers splits into short exact sequences.
In general, the computation of the 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 (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.
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