High codimension links

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1 Introduction

This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.

See general introduction on embeddings, notation and conventions in [Skopenkov2016c, ${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings. See [[High_codimension_embeddings#Introduction|general introduction on embeddings]], [[High_codimension_embeddings#Notation and conventions|notation and conventions]] in \cite[$\S\S$ 1, $\S$ 2].

Componentwise version of embedded connected sum [Skopenkov2016c, $\S$ 5] defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$ . See [Skopenkov2006, Figure 3.3], [Haefliger1966], [Haefliger1966a], [Skopenkov2015, Group Structure Lemma 2.2 and Remark 2.3.a].

2 The Hopf linking

Recall that for each $n$ -manifold $N$ and $m\ge2n+2$ , every two embeddings $N\to\Rr^m$ are isotopic [Skopenkov2016c, General Position Theorem 3.1]. Here the restriction $m\ge2n+2$ is sharp for non-connected manifolds.

For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding.

For $n=1$ the Hopf linking is shown in [Skopenkov2006, Figure 2.1.a]. For arbitrary $n$ (including $n=1$ ) the image of the Hopf Linking is the union of two $n$ -spheres

$\displaystyle \partial D^{n+1}\times0\quad\text{and}\quad 0\times\partial D^{n+1}\quad\text{in}\quad\partial(D^{n+1}\times D^{n+1}).$ $\displaystyle \partial D^{n+1}\times0\quad\text{and}\quad 0\times\partial D^{n+1}\quad\text{in}\quad\partial(D^{n+1}\times D^{n+1}).$

Alternatively, these spheres are given by equations:

$\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..$ $\displaystyle \left\{\begin{array}{c} x_1=\dots=x_n=0\\ x_{n+1}^2+\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..$

This embedding is distinguished from the standard embedding by the linking coefficient.

3 The Zeeman construction and linking coefficient

The following table was obtained by Zeeman around 1960:

$\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 \end{array}$ $\displaystyle \begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \\ \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 \end{array}$

3.1 Definitions of the Zeeman map and linking coefficient

Take $x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$ . Take any map $\varphi:S^p\to\partial D^{m-q}$ . Define embedding $\tau(x)$ on $S^p$ to be the composition

$\displaystyle S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$ $\displaystyle S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$

where $i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard [Skopenkov2006, Figure 3.2].

Clearly, $\tau$ is well-defined and is a homomorphism.

Fix orientations of $S^p$ , $S^q$ , $S^m$ and $D^{m-p}$ . Take an embedding $f:S^p\sqcup S^q\to S^m$ . Take an embedding $g:D^{m-q}\to S^m$ such that $gD^{m-q}$ intersects $fS^q$ transversally at exactly one point with positive sign [Skopenkov2006, Figure 3.1]. Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence.

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

Let $h$ be a homotopy inverse of $h'$ . Define

$\displaystyle \lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$ $\displaystyle \lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$

(a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$ . Clearly, $\lambda$ is a homomorphism.

(b) For $m=p+q+1$ there is a simpler alternative `homological' definition. That definition works for $m=q+2$ as well.

(c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$ .

(d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$ , if the restriction of $f$ to $S^q$ is unknotted).

(e) Clearly, $\lambda\tau=\id\pi_p(S^{m-q-1})$ , even for $m=q+2$ . So $\lambda$ is surjective and $\tau$ is injective.

3.2 Classification in the `metastable' range

If $1\le p\le q$ , then both $\lambda$ and $\tau$ are isomorphisms for $m\ge\frac p2+q+2$ and for $m\ge\frac{3q}2+2$ , in the PL and smooth cases, respectively.

The surjectivity of $\lambda$ (=the injectivity of $\tau$ ) follows from $\lambda\tau=\id$ . The injectivity of $\lambda$ (=the surjectivity of $\tau$ ) is proved in [Haefliger1962t], [Zeeman1962] (or follows from [Skopenkov2006, the Haefliger-Weber Theorem 5.4 and the Deleted Product Lemma 5.3.a]).

An analogue of this result holds for links with many components [Haefliger1966a]: the collection of pairwise linking coefficients is bijective for $2m\ge3n+4$ and $n$ -dimensional links in $\R^m$ .

3.3 Alpha-invariant

By Freudenthal Suspension Theorem $\Sigma^{\infty}:\pi_p(S^{m-q-1})\to\pi^S_{p+q+1-m}$ is an isomorphism for $m\ge\frac p2+q+2$ . The stable suspension of the linking coefficient can be described alternatively as follows. For an embedding $f:S^p\sqcup S^q\to S^m$ define a map [Skopenkov2006, Figure 3.1]

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

For $p\le q\le m-2$ define the $\alpha$ -invariant by

$\displaystyle \alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$ $\displaystyle \alpha(f)=[\widetilde f]\in[S^p\times S^q,S^{m-1}]\overset{v^*}\cong\pi_{p+q}(S^{m-1})\cong\pi^S_{p+q+1-m}.$

The second isomorphism in this formula is given by the Freudenthal Suspension Theorem. The map $v:S^p\times S^q\to\frac{S^p\times S^q}{S^p\vee S^q}\cong S^{p+q}$ is the quotient map. See [Skopenkov2006, Figure 3.4]. The map $v^*$ is an isomorphism for $m\ge q+2$ .

(For $m\ge q+3$ this follows by general position and for $m=q+2$ by the cofibration Barratt-Puppe exact sequence of pair $(S^p\times S^q,S^p\vee S^q)$ and by the existence of a retraction $\Sigma(S^p\times S^q)\to\Sigma(S^p\vee S^q)$ .)

We have $\alpha=\pm\Sigma^{\infty}\lambda_{12}$ by [Kervaire1959a, Lemma 5.1]. Hence $\Sigma^{\infty}\lambda_{12}=\pm\Sigma^{\infty}\lambda_{21}$ .

Note that $\alpha$ -invariant can be defined in more general situations [Koschorke1988], [Skopenkov2006, $\S$ 5].

4 Classification below the metastable range

4.1 Higher-dimensional Borromean rings

Let us present an example of non-injectivity of the collection of pairwise linking coefficients.

The Borromean rings $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose restrictions to 2-componented sublinks are trivial [Haefliger1962, 4.1], [Haefliger1962t].

Denote coordinates in $\Rr^{3l}$ by $(x,y,z)=(x_1,\dots,x_l,y_1,\dots,y_l,z_1,\dots,z_l)$ . The Borromean rings [Skopenkov2006, Figures 3.5 and 3.6] are the three spheres given by the following three systems of equations:

$\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..$ $\displaystyle \left\{\begin{array}{c} x=0\\ |y|^2+2|z|^2=1\end{array}\right., \qquad \left\{\begin{array}{c} y=0\\ |z|^2+2|x|^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} z=0\\ |x|^2+2|y|^2=1 \end{array}\right..$

This embedding is distinguished from the standard embedding by the well-known Massey invariant [Skopenkov2017] (or because the connected sum of the three components yields a non-trivial knot [Haefliger1962]).

4.2 Higher-dimensional Whitehead link

Let us present an example of non-injectivity of the linking coefficient.

The Whitehead link $w:S^{2l-1}\sqcup S^{2l-1}\to\R^{3l}$ is a non-trivial embedding whose linking coefficient $\lambda_{12}(w)$ is trivial.

The Whitehead link is obtained from Borromean rings by joining two components with a tube. We have $\lambda_{12}(w)=0$ because by moving two of the three Borromean rings and self-intersecting them, we can drag the third ring apart (see details in [Skopenkov2006a]). For $l\ne1,3,7$ the Whitehead link is distinguished from the standard embedding by $\lambda_{21}(w)=[\iota_l,\iota_l]\ne0$ . This fact should be well-known, but I do not know a published proof except [Skopenkov2015a, Lemma 2.18]. For $l=1,3,7$ the Whitehead link is is distinguished from the standard embedding by more complicated invariants [Skopenkov2006a], [Haefliger1962t, \S3].

This example shows that the dimension restriction is sharp in Theorem 3.4.

This example seems to be discovered by Whitehead, in connection with Whitehead product.

Cf. the Haefliger Trefoil knot [Skopenkov2016t].

4.3 Classification

Let $C_q^{m-q}:=E^m_D(S^q)$ . For some information on this group see [Skopenkov2006, $\S$ 3.3], [Skopenkov2016k].

(a) [Haefliger1966a] If $p,q\le m-3$ , then

$\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$ $\displaystyle E^m_D(S^p\sqcup S^q)\cong E^m_{PL}(S^p\sqcup S^q)\oplus C^{m-q}_q\oplus C^{m-p}_p.$

(b) [Haefliger1966a, Theorem 10.7], [Skopenkov2009] If $p\le q\le m-3$ and $3m\ge2p+2q+6$ , then there is a map $\varkappa$ for which the following map is an isomorphism

$\displaystyle \lambda_{12}\oplus\varkappa:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).$ $\displaystyle \lambda_{12}\oplus\varkappa:E^m_{PL}(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})\oplus \pi_{p+q+2-m}(SO, SO_{m-p-1}).$

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

Part (b) implies that

$\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}.$ $\displaystyle E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\cong\pi_{2l-1}(S^l)\oplus\Z_{(l)}.$

This isomorphism is defined for $l\ge2$ , $l\ne3,7$ by map

$\displaystyle \lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l).$ $\displaystyle \lambda_{12}\oplus\lambda_{21}:E^{3l}_{PL}(S^{2l-1}\sqcup S^{2l-1})\to\pi_{2l-1}(S^l)\oplus\pi_{2l-1}(S^l).$

This map is injective for $l\ge2$ , $l\ne3,7$ ; the image of this map is $\{(a,b)\ :\ \Sigma a=\Sigma b\}$ [Haefliger1962t]. Thus part (b) shows that $\lambda_{12}\oplus\lambda_{21}$ is in general not injective.

5 Further discussion

The set $E^m(S^{n_1}\sqcup\dots\sqcup S^{n_s})$ for $m\ge n_i+3$ has been described in terms of exact sequences involving homotopy groups of spheres [Haefliger1966], [Haefliger1966a], cf. [Levine1965], [Habegger1986].

See [Skopenkov2009], [Crowley&Ferry&Skopenkov2011], [Avvakumov2016].

6 References

[Avvakumov2016] S. Avvakumov, The classification of certain linked 3-manifolds in 6-space, Moscow Mathematical Journal, 16:1 (2016) 1-25. http://arxiv.org/abs/1408.3918.
[Crowley&Ferry&Skopenkov2011] D. Crowley, S.C. Ferry, M. Skopenkov, The rational classification of links of codimension >2, Forum Math. 26 (2014), 239-269. https://arxiv.org/abs/1106.1455
[Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
[Haefliger1962] A. Haefliger, Knotted $(4k-1)$ -spheres in $6k$ -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 $S^n$ in $S^{n+q}$ for $q>2$ , Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
[Haefliger1966a] A. Haefliger, Enlacements de sphères en co-dimension supérieure à 2, Comment. Math. Helv.41 (1966), 51-72. MR0212818 (35 #3683) Zbl 0149.20801
[Kervaire1959a] M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopf's invariant, Ann. of Math. 62 (1959) 345--362.
[Koschorke1988] U. Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61:4 (1988) 383--415.
[Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
[Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
[Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
[Skopenkov2009] M. Skopenkov, Suspension theorems for links and link maps. Proc. AMS 137 (2009) 359--369. arxiv:math/0610320, version 2 or higher
[Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
[Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

[Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
[Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
[Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
[Skopenkov2017] A. Skopenkov, Algebraic Topology From Algorithmic Viewpoint, draft of a book.
[Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069

, $\S]{Skopenkov2016c}. Componentwise version of [[High_codimension_embeddings#Embedded_connected_sum|embedded connected sum]] \cite[$\S]{Skopenkov2016c} defines a commutative group structure on $E^m(S^p\sqcup S^q)$ for $m-3\ge p,q$. See \cite[Figure 3.3]{Skopenkov2006}, \cite{Haefliger1966}, \cite{Haefliger1966a}, \cite[Group Structure Lemma 2.2 and Remark 2.3.a]{Skopenkov2015}. == The Hopf linking == ; [[High_codimension_embeddings:_classification#Unknotting_theorems|Recall that]] for each $n$-manifold $N$ and $m\ge2n+2$, every two embeddings $N\to\Rr^m$ are isotopic \cite[General Position Theorem 3.1]{Skopenkov2016c}. Here the restriction $m\ge2n+2$ is sharp for non-connected manifolds. {{beginthm|Example|the Hopf linking}}\label{hopf} For each $n$ there is an embedding $S^n\sqcup S^n\to\Rr^{2n+1}$ which is not isotopic to the standard embedding. {{endthm}} For $n=1$ the Hopf linking is shown in \cite[Figure 2.1.a]{Skopenkov2006}. For arbitrary $n$ (including $n=1$) the image of the Hopf Linking is the union of two $n$-spheres $$\partial D^{n+1}\times0\quad\text{and}\quad 0\times\partial D^{n+1}\quad\text{in}\quad\partial(D^{n+1}\times D^{n+1}).$$ Alternatively, these spheres are given by equations: $$\left\{\begin{array}{c} x_1=\dots=x_n=0\ x_{n+1}^2+\dots+x_{2n+1}^2=1\end{array}\right. \qquad\text{and}\qquad \left\{\begin{array}{c} x_{n+2}=\dots=x_{2n+1}=0\ x_1^2+\dots+x_n^2+(x_{n+1}-1)^2=1\end{array}\right..$$ This embedding is distinguished from the standard embedding by [[#The Zeeman construction and linking coefficient|the linking coefficient]]. == The Zeeman construction and linking coefficient == ; The following table was obtained by Zeeman around 1960: $$\begin{array}{c|c|c|c|c|c|c|c} m &\ge2q+2 &2q+1 &2q\ge8 &2q-1\ge11 &2q-2\ge14 &2q-3\ge17 &2q-4\ge20 \ \#E^m(S^q\sqcup S^q) &1 &\infty &2 &2 &24 &1 &1 \end{array}$$ ====Definitions of the Zeeman map and linking coefficient==== ; {{beginthm|Definition|of the Zeeman map $\tau:\pi_p(S^{m-q-1})\to E^m(S^p\sqcup S^q)$ for $p\le q$}}\label{dz} Take $x\in\pi_p(S^{m-q-1})$ Define embedding $\tau(x)$ on $S^q$ to be the standard embedding into $\R^m$. Take any map $\varphi:S^p\to\partial D^{m-q}$. Define embedding $\tau(x)$ on $S^p$ to be the composition $$S^p\overset{x\times i}\to\partial D^{m-q}\times S^q \subset D^{m-q}\times S^q\subset\R^m,$$ where $i:S^p\to S^q$ is the equatorial inclusion and the latter inclusion is the standard \cite[Figure 3.2]{Skopenkov2006}. {{endthm}} Clearly, $\tau$ is well-defined and is a homomorphism. {{beginthm|Definition|of linking coefficient $\lambda=\lambda_{12}:E^m(S^p\sqcup S^q)\to\pi_p(S^{m-q-1})$ for $m\ge q+3$}}\label{dl} Fix orientations of $S^p$, $S^q$, $S^m$ and $D^{m-p}$. Take an embedding $f:S^p\sqcup S^q\to S^m$. Take an embedding $g:D^{m-q}\to S^m$ such that $gD^{m-q}$ intersects $fS^q$ transversally at exactly one point with positive sign \cite[Figure 3.1]{Skopenkov2006}. Then the restriction $h':S^{m-q-1}\to S^m-fS^q$ of $g$ to $\partial D^{m-q}$ is a homotopy equivalence. (Indeed, since $m\ge q+3$, the complement $S^m-fS^q$ is simply-connected. By Alexander duality $h'$ induces isomorphism in homology. Hence by Hurewicz and Whitehead theorems $h'$ is a homotopy equivalence.) Let $h$ be a homotopy inverse of $h'$. Define $$\lambda(f)=\lambda_{12}(f):=[S^p\overset{f|_{S^p}}\to S^m-fS^q\overset h\to S^{m-q-1}]\in\pi_p(S^{m-q-1}).$$ {{endthm}} {{beginthm|Remark}}\label{lkrem} (a) Clearly, $\lambda(f)$ is indeed independent of $g,h',h$. Clearly, $\lambda$ is a homomorphism. (b) For $m=p+q+1$ there is a simpler alternative `homological' definition. That definition works for $m=q+2$ as well. (c) Analogously one can define $\lambda_{21}(f)\in\pi_q(S^{m-p-1})$ for $m\ge p+3$. (d) This definition works for $m=q+2$ if $S^m-fS^q$ is simply-connected (or, equivalently for $q>4$, if the restriction of $f$ to $S^q$ is unknotted). (e) Clearly, $\lambda\tau=\id\pi_p(S^{m-q-1})$, even for $m=q+2$. So $\lambda$ is surjective and $\tau$ is injective. {{endthm}} ====Classification in the `metastable' range==== ; {{beginthm|The Haefliger-Zeeman Theorem}}\label{lkhaze} If \S1, $\S$ $\S$ 2].