Knots, i.e. embeddings of spheres

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1 Introduction

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, ${{Stub}} == Introduction == <wikitex>; For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$ 1, $\S$ 3].

2 Examples

There are smooth embeddings $S^{2l-1}\to\Rr^{3l}$ which are not smoothly isotopic to the standard embedding. They are PS (piecewise smoothly) isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).

(a) Analogously to the Haefliger trefoil knot for any $l>1$ one constructs a smooth embedding $t:S^{2l-1}\to\Rr^{3l}$ , see [Skopenkov2016h, $\S$ 5]. For $l$ even $t$ is not smoothly isotopic to the standard embedding; $t$ represents a generator of $E_D^{3l}(S^{2l-1})\cong\Zz$ [Haefliger1962].

It would be interesting to know if for $l>1$ odd this embedding is a generator of $E_D^{3l}(S^{2l-1})\cong\Zz_2$ . The last phrase of [Haefliger1962t] suggests that this is true for $l=3$ .

(b) For any $k=1,3,7$ let $\eta\in\pi_{4k-1}(S^{2k})$ be the homotopy class of the Hopf map. Denote by $\zeta:\pi_{4k-1}(S^{2k})\to E_D^{6k}(S^{4k-1}\sqcup S^{4k-1})$ the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum $\#\zeta\eta$ of the components of (a representative of) $\zeta\eta$ is not smoothly isotopic to the standard embedding; $\#\zeta\eta$ is a generator of $E_D^{6k}(S^{4k-1})\cong\Z$ [Skopenkov2015a, Corollary 2.13].

3 Invariants

Let us define the Haefliger invariant $\varkappa:E^{6k}_D(S^{4k-1})\to\Z$ . The definition is motivated by Haefliger's proof that any embedding $S^n\to S^m$ is isotopic to the standard embedding for $2m\ge3n+4$ , and by analyzing what obstructs carrying this proof for $2m=3n+3$ .

By [Haefliger1962, 2.1, 2.2] any embedding $f:S^{4k-1}\to S^{6k}$ $f:S^{4k-1}\to S^{6k}$ has a framing extendable to a framed embedding $\overline f:V\to D^{6k+1}$ $\overline f:V\to D^{6k+1}$ of a $4k$ $4k$ -manifold $V$ $V$ whose boundary is $S^{4k-1}$ $S^{4k-1}$ , and whose signature is zero. For an integer $2k$ $2k$ -cycle $c$ $c$ in $V$ $V$ let $\lambda^*(c)\in\Z$ $\lambda^*(c)\in\Z$ be the linking number of $f(V)$ $f(V)$ with a slight shift of $\overline f(c)$ $\overline f(c)$ along the first vector of the framing. This defines a map $\lambda^*:H_{2k}(V;\Z)\to\Z$ $\lambda^*:H_{2k}(V;\Z)\to\Z$ . This map is a homomorphism (as opposed to the Arf map defined in a similar way [Pontryagin1959]). Then by Lefschetz duality there is a unique $\lambda\in H_{2k}(V,\partial;\Z)$ $\lambda\in H_{2k}(V,\partial;\Z)$ such that $\lambda^*[c]=\lambda\cap_V[c]$ $\lambda^*[c]=\lambda\cap_V[c]$ for any $[c]\in H_{2k}(V;\Z)$ $[c]\in H_{2k}(V;\Z)$ . Since $V$ $V$ has a normal framing, its intersection form $\cap_V$ $\cap_V$ is even. (Indeed, represent a class in $H_{2k}(V;\Z)$ $H_{2k}(V;\Z)$ by a closed oriented $2k$ $2k$ -submanifold $c$ $c$ . Then $\rho_2[c]\cap_V[c]=\overline{w_{2k}}(c\subset V)=\rho_2[c]\cap_VPDw_{2k}(V)=0$ $\rho_2[c]\cap_V[c]=\overline{w_{2k}}(c\subset V)=\rho_2[c]\cap_VPDw_{2k}(V)=0$ because $V$ $V$ has a normal framing.) Hence $\lambda\cap_V\lambda$ $\lambda\cap_V\lambda$ is an even integer. Define

$\displaystyle \varkappa(f):=\lambda\cap_V\lambda/2.$ $\displaystyle \varkappa(f):=\lambda\cap_V\lambda/2.$

Since the signature of $V$ is zero, there is a symplectic basis $\alpha_1,\ldots,\alpha_s,\beta_1,\ldots,\beta_s$ in $H_{2k}(V;\Z)$ . Then clearly

$\displaystyle \varkappa(f) = \sum\limits_{j=1}^s \lambda^*(\beta_j)\lambda^*(\alpha_j).$ $\displaystyle \varkappa(f) = \sum\limits_{j=1}^s \lambda^*(\beta_j)\lambda^*(\alpha_j).$

For an alternative definition via Seifert surfaces in $6k$ -space, discovered in [Guillou&Marin1986], [Takase2004], see [Skopenkov2016t, the Kreck Invariant Lemma 4.5]. For a definition by Kreck, and for a generalization to 3-manifolds see [Skopenkov2016t, $\S$ 4].

Sketch of a proof that $\varkappa(f)$ is well-defined (i.e. is independent of $V$ , $\overline f$ , and the framings), and is invariant under isotopy of $f$ . [Haefliger1962, Theorem 2.6] Analogously one defines $\lambda(V)$ and $\varkappa(V):=\lambda(V)\cap_V\lambda(V)/2$ for a framed $4k$ -submanifold $V$ of $S^{6k+1}$ . Since $\varkappa(V)$ is a characteristic number, it is independent of framed cobordism. So $\varkappa(V)$ defines a homomorphism $\Omega_{fr}^{4k}(6k+1)=\pi_{6k+1}(S^{2k+1})\to\Z$ . The latter group is finite by the Serre theorem. Hence the homomorphism is trivial.

Since $\varkappa(f)$ is a characteristic number, it is independent of framed cobordism of a framed $f$ (and hence of the isotopy of a framed $f$ ).

Therefore $\varkappa(f)$ is a well-defined invariant of a framed cobordism class of a framed $f$ . By [Haefliger1962, 2.9] (cf. [Haefliger1962, 2.2 and 2.3]) $\varkappa(f)$ is also independent of the framing of $f$ extendable to a framing of some $4k$ -manifold $V$ having trivial signature. QED

For definition of the attaching invariant $E^{n+q}_D(S^n)\to\pi_n(G_q,SO_q)$ see [Haefliger1966], [Skopenkov2005, $\S$ 3].

4 Classification

For $m-n\ge3$ the group $E^m_D(S^n)$ is finite unless $n=4k-1$ and $m\le6k$ , when $E^m_D(S^n)$ is the sum of $\Z$ and a finite group.

We have the following table for the group $E^m_D(S^n)$ ; in the whole table $k\ge1$ ; in the fifth column $k\ne2$ ; in the last two columns $k\ge2$ :

$\displaystyle \begin{array}{c|c|c|c|c|c|c|c} (m,n) &2m\ge3n+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k+4,4k+2) &(12k+7,8k+4) & (12k+1,8k)\\ \hline E^m_D(S^n)&0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 \end{array}$ $\displaystyle \begin{array}{c|c|c|c|c|c|c|c} (m,n) &2m\ge3n+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k+4,4k+2) &(12k+7,8k+4) & (12k+1,8k)\\ \hline E^m_D(S^n)&0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 \end{array}$

Proof for the first four columns, and for the fifth column when $k$ is odd, are presented in [Haefliger1966, 8.15] (see also $\S$ 6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, $\S$ 3]; there is a typo in [Haefliger1966, 8.15]: $C^{3k}_{4k−2}=0$ should be $C^{4k}_{8k−2}=0$ ). The remaining results follow from [Haefliger1966, 8.15] and [Milgram1972, Theorem F]. Alternative proofs for the cases $(m,n)=(7,4),(6,3)$ are given in [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].

We have $E^m_D(S^n)=0$ if and only if either $2m\ge3n+4$ , or $(m,n)=(6k+4,4k+2)$ , or $(m,n)=(3k,2k)$ and $k\equiv3,11\mod12$ , or $(m,n)=(3k+2,2k+2)$ and $k\equiv14,22\mod24$ .

For a description of 2-components of $E^m_D(S^n)$ see [Milgram1972, Theorem F]. Observe that no reliable reference (containing complete proofs) of results announced in [Milgram1972] appeared. Thus, strictly speaking, the corresponding results are conjectures.

The lowest-dimensional unknown groups $E^m_D(S^n)$ are $E^8_D(S^5)$ and $E^{11}_D(S^7)$ . Hopefully application of Kreck surgery could be useful to find these groups, cf. [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].

For $m\ge n+3$ the group $E^m_D(S^n)$ has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].

For $q\ge3$ there is the following exact sequence of abelian groups:

$\displaystyle \ldots \to \pi_{n+1}(SG,SO) \xrightarrow{~u~} E^{n+q}_D(S^n) \xrightarrow{~a~} \pi_n(SG_q,SO_q) \xrightarrow{~s~} \pi_n(SG,SO) \xrightarrow{~u~} E^{n+q-1}_D(S^{n-1})\to \ldots~.$ $\displaystyle \ldots \to \pi_{n+1}(SG,SO) \xrightarrow{~u~} E^{n+q}_D(S^n) \xrightarrow{~a~} \pi_n(SG_q,SO_q) \xrightarrow{~s~} \pi_n(SG,SO) \xrightarrow{~u~} E^{n+q-1}_D(S^{n-1})\to \ldots~.$

Here $SG_q$ is the space of maps $S^{q-1} \to S^{q-1}$ of degree $1$ . Restricting a map from $SO_q$ to $S^{q-1} \subset \Rr^q$ identifies $SO_q$ as a subspace of $SG_q$ . Define $SG:=SG_1\cup\ldots\cup SG_q\cup\ldots$ . Analogously define $SO$ . Let $s$ be the stabilization homomorphism. The attaching invariant $a$ and the map $u$ are defined in [Haefliger1966], see also [Skopenkov2005, $\S$ 3].

5 Some remarks on codimension 2 knots

For the best known specific case, i.e. for codimension 2 embeddings of spheres (in particular, for the classical theory of knots in $\Rr^3$ ), a complete readily calculable classification (in the sense of Remark 1.2 of [Skopenkov2016c]) is neither known nor expected at the time of writing. However, there is a vast literature on codimension 2 knots. See e.g. the interesting papers [Farber1981], [Farber1983], [Kearton1983], [Farber1984].

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a] and [Ranicki1998].

6 Proof of classification of (4k-1)-knots in 6k-space

Here we prove that the Haefliger invariant $\varkappa:E_D^{6k}(S^{4k-1})\to\Zz$ is injective for $k>1$ . The proof is a certain simplification of [Haefliger1962]. We present an exposition structured to make it more accessible to non-specialists.

Let $V$ be a framed $(2k-1)$ -connected $4k$ -submanifold of $D^{6k+1}$ such that $S^{4k-1}\cong \partial V \subset \partial D^{6k+1}$ , signature of $V$ is zero, and $\varkappa(V) = 0$ . Then there is an embedding $G:D^{4k}\to D^{6k+1}$ such that $G(\partial D^{4k})=\partial V$ .

The Haefliger invariant $\varkappa:E_D^{6k}(S^{4k-1})\to\Zz$ is injective for $k>1$ .

Proof using Lemma 6.1. By the first three paragraphs of the proof of Theorem 3.1 in [Ha62], for any embedding $g:S^{4k-1}\to \partial D^{6k+1}$ such that $\varkappa(g)=0$ there is a framed $(2k-1)$ -connected $4k$ -submanifold $V$ of $D^{6k+1}$ with zero signature, such that $S^{4k-1}\cong \partial V \subset \partial D^{6k+1}$ and $\varkappa(V) = 0$ . Then from Lemma 6.1 it follows that there is an extension $G:D^{4k}\to D^{6k+1}$ such that $G|_{S^{4k-1}} = g$ . From Smale Theorem it follows that $g$ is isotopic to the standard embedding. $\Box$

!!!Below we assume that all maps are smooth and in general position. For a manifold $M$ denote by $\cap_M$ the intersection form, by $\partial M$ the boundary of $M$ , and by $\mathrm{Int} M$ the interior of $M$ . The symbol $[ \cdot ]$ denotes the integral fundamental class of a manifold, the homotopy class of a map or a homology class of a cycle, depending on the context.

To prove Lemma 6.1 we need Lemma 6.4 and Lemma 6.5.

We also??? formulate below a suitable version of the Whitney Theorem.

Let $\pi: P^p \rightarrow W^{p+q}$ be a map from a connected $p$ -manifold $P???$ to a simply connected $(p+q)$ -manifold $W^{p+q}$ . !!!If $p, q \geq 3$ , then then

1. There is a homotopy $\pi_t$ such that $\pi_0 = \pi$ and $\pi_1???(P^p)$ has no ???self-intersections.
2. Suppose in addition that there is another??? map $\tau: Q^q \rightarrow W^{p+q}$ from a connected manifold $Q^q$ ???, such that the algebraic intersection number of $\pi(P^p)$ and $\tau(Q^q)$ is zero.

Then there is a homotopy $\tau_t$ ???relative to the boundary such that $\tau_0 = \tau$ and $\tau_1(Q^q)$ does not intersect $\pi(P^p)$ . If $\tau$ is an embedding, the homotopy $\tau_t$ can be chosen to be ???in the class of embeddings. ???In addition, the homotopy can be chosen so that $\tau_t|_{\partial Q^q} \equiv \tau|_{\partial Q^q}$ .

???All the manifolds above can have non-empty boundaries.

Let $V$ be a $(2k-1)$ -connected $4k$ -manifold. Then for any $x_1, \ldots, x_m \in H_{2k}(V)$ such that $x_i \cap_V x_j = 0$ for any $i,j$ , there are embeddings $g_1, \ldots, g_m: \, S^{2k} \rightarrow V$ with pairwise disjoint images and such that $g_{i*}[S^{2k}] = x_i$ .

As $V$ is $(2k-1)$ -connected, the Hurewicz map $h_*: \pi_{2k}(V) \rightarrow H_{2k}(V)$ is an isomorphism. So every class in $H_{2k}(V)$ can be realised by a ???sphere. Indeed???, given an element $x_i \in H_{2k}(V)$ , let $w_i: S^{2k} \rightarrow V$ represent the homotopy class $h_*^{-1}(x_i)$ . Then the Hurewicz theorem??? states that $w_{i*}[S^{2k}] = x_i$ . Given such a map $w_i$ , we say that $x_i$ is realised by [??? $S^{2k}$ via the map] $w_i$ .

Now we perform the following inductive procedure. At the $i$ -th step of the procedure, the maps $g_j$ from the assertion are already constructed [for $x_j$ ] for any $j < i$ and??? we construct $g_i$ . First, we realise $x_i$ by [a $2k$ -sphere via???] some map $w_i$ and??? we apply item 1 of Theorem 6.3 to $w_i$ to replace it??? with some homotopic embedding. Therefore, we may suppose that $w_i$ itself is an embedding. Further, we apply item 2 of Theorem 6.3 to each triple $(g_j, w_i, V \setminus (\bigcup\limits_{l<j}g_l(S^{2k}))$ , where we set $g_j$ as $\pi$ , $w_i$ as $\tau$ , and $V \setminus (\bigcup\limits_{l<j}g_l(S^{2k}))$ as $W^{p+q}$ , for any $j < i$ . As the result, $w_i$ is replaced with a homotopic embedding $g_i$ , and the images of $g_1, \ldots, g_i$ are disjoint. After the step $m$ we obtain the desired set of embeddings.

???Observe that Theorem 6.3 is applicable as all the maps above are from $S^{2k}$ to $V$ , where $V$ is simply connected, and $2k \geq ???4$ . Also, as $x_i \cap_V x_j = 0$ for any $i$ , $j$ , the algebraic intersection number for all the pairs of maps above is zero.

$\square$ $\square$

[cf. Proposition 3.3 in [Ha62]] Let $k>1$ and $V$ be a framed $4k$ -submanifold of $D^{6k+1}$ such that $H_m(V)=0$ for every $0< m< 2k$ ???. Let $g_1, \ldots, g_s:S^{2k}\to V$ be embeddings with pairwise disjoint images extendable to the embeddings $g_i???:D^{2k+1}\to D^{6k+1}$ with pairwise disjoint images such that $g_i(D^{2k+1})\cap V= g_i(S^{2k})$ ???. Then there are extensions of $g_i$ to embeddings $G_i:D^{2k+1}\times D^{2k-1}$ such that

1. $G_i(S^{2k}\times D^{2k-1})\subset V$ ;
2. $V':=V\backslash (\bigcup G_i(S^{2k}\times D^{2k-1})) \bigcup G_i(D^{2k+1}\times S^{2k-2})$ is a ???smooth submanifold of $D^{6k+1}$ .

As $V$ is a framed manifold, the normal bundle of $g_i(S^{2k})\subset V$ is trivial. Hence for $i\leq s$ there are??? disjoint neighborhoods $U_i$ of $g_i(D^{2k+1})$ and homeomorphisms $\phi_i:\mathbb{R}^{6k+1}\to U_i$ such that:

in coordinates $(x, y, z)=(x_1, \ldots, x_{2k+1}, y_1, \ldots, y_{2k}, z_1, \ldots, z_{2k})$ of $\mathbb{R}^{6k+1}$ we have $\phi_i^{-1}(g_i(D^{2k+1}))$ is defined by $y=z=0$ , $x^2=\Sigma x_i^2\leq 1$ ;

in coordinates $(x, y, z)$ of $\mathbb{R}^{6k+1}$ we have that $\pi_i^{-1}(V)$ is defined by $-x^2+y^2=-a(x^2+y^2), z=0$ , where $a(x^2+y^2)$ is a decreasing function of $x^2+y^2$ , equal to $1$ for $x^2+y^2\leq 1$ and $0$ for $x^2+y^2\geq 2$

We take by??? the exten???tion of $g_i$ $g_i$ an embedding $G_i:D^{2k+1}\times D^{2k-1}\to D^{6k+1}$ $G_i:D^{2k+1}\times D^{2k-1}\to D^{6k+1}$ such that $\phi^{-1}(G_i(D^{2k+1}\times D^{2k-1}))$ $\phi^{-1}(G_i(D^{2k+1}\times D^{2k-1}))$ is defined by $-a(x^2+y^2)\leq -x^2+y^2\leq a(x^2+y^2)$ $-a(x^2+y^2)\leq -x^2+y^2\leq a(x^2+y^2)$ .

$\square$ $\square$

Proof of Lemma 6.1 using Lemmas 6.4, 6.5.. By the fourth??? paragraph of the proof of Theorem 3.1 in [Ha62], there is a basis $\left\{\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s\right\}$ in $H_{2k}(V)$ such that $\alpha_i\cap \alpha_j=\beta_i\cap \beta_j = 0$ , $\alpha_i\cap \beta_j=\delta_{i, j}$ and $\lambda???(\alpha_i)=0$ for any $i, j$ . From Lemma 6.4 it follows that there are embeddings $f_1, \ldots, f_s:S^{2k}\to V$ with pairwise disjoint images, such that $f_{1*}[S^{2k}]=\alpha_1, \ldots, f_{s*}[S^{2k}]=\alpha_s$ . Denote by $f_{i}':S^{2k}\to D^{6k+1}\backslash V$ for $i\leq s$ the result of shifting of $f^i$ by the first vector of the framing of $V$ . The condition $\lambda(f_{i*}[S^{2k}])=\lambda(\alpha_i)=0$ means that $f_{i*}'([S^{2k}])=0 \in H_{2k}(D^{6k+1}\backslash V)$ . Since $\mbox{dim} D^{6k+1}-\mbox{dim} V=6k+1-4k=2k+1$ , we have $\pi_m(D^{6k+1}\backslash \mbox{Int}{V}) = 0$ for $m<2k$ . Since the cycle $f_{i*}'([S^{2k}])$ is homologous to zero we have $[f_{i}']=0\in \pi_{2k}(D^{6k+1}\backslash V)$ . Therefore there are extensions of $f_1, \ldots, f_s$ to $g_1, \ldots, g_s:D^{2k+1}\to D^{6k+1}$ such that $g_i(D^{2k+1})\cap V = g_i(S^{2k})$ .

Denote by $V'$ the manifold as in item 2 of Lemma 6.5 for $V$ and $g_1, \ldots, g_s$ as described above. Since $\left\{\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s\right\}$ is a basis in $H_{2k}(V)$ it follows that $H_{2k}(V')\simeq 0$ . Hence $H_m(V')\simeq 0$ for $0< m\leq 2k$ and $V'\cong D^{4k}$ . Then we set $G=id:V'\cong D^{4k}\to D^{6k+1}$ . $\Box$

References

[Crowley&Skopenkov2008] D. Crowley and A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, II, Intern. J. Math., 22:6 (2011) 731-757. Available at the arXiv:0808.1795.
[Farber1981] M. Sh. Farber, Classification of stable fibered knots, Mat. Sb. (N.S.), 115(157):2(6) (1981) 223–262.
[Farber1983] M. Sh. Farber, The classification of simple knots, Russian Math. Surveys, 38:5 (1983).

[Farber1984] M. Sh. Farber, An algebraic classification of some even-dimensional spherical knots I, II, Trans. Amer. Math. Soc. 281 (1984), 507-528; 529-570.
[Guillou&Marin1986] L. Guillou and A.Marin, Eds., A la r\'echerche de la topologie perdue, 1986, Progress in Math., 62, Birkhauser, Basel
[Ha62] Template:Ha62
[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
[Kearton1983] C. Kearton, An algebraic classification of certain simple even-dimensional knots, Trans. Amer. Math. Soc. 176 (1983), 1–53.
[Levine1965] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR0180981 (31 #5211) Zbl 0136.21102
[Levine1969a] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR0246314 (39 #7618) Zbl 0176.22101
[Milgram1972] R. J. Milgram, On the Haefliger knot groups, Bull. of the Amer. Math. Soc., 78:5 (1972) 861--865.
[Pontryagin1959] L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, Amer. Math. Soc. Translations, Ser. 2, Vol. 11, Providence, R.I. (1959), 1–114. MR0115178 (22 #5980) Zbl 0084.19002
[Ranicki1998] A. Ranicki, High-dimensional knot theory, Springer-Verlag, 1998. MR1713074 (2000i:57044) Zbl 1059.19003
[Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
[Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429 MR2434474 (2010e:57028) Zbl 1167.57013

[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.
[Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
[Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
[Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
[Takase2004] M. Takase, A geometric formula for Haefliger knots, Topology 43 (2004), no.6, 1425–1447. MR2081431 (2005e:57032) Zbl 1060.57021

, $\S]{Skopenkov2016c}. == Examples == ; There are smooth embeddings $S^{2l-1}\to\Rr^{3l}$ which are not smoothly isotopic to the standard embedding. They are PS (piecewise smoothly) isotopic to the standard embedding (by the Zeeman [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem 2.3]] of \cite{Skopenkov2016c} and \cite[Remark 1.1]{Skopenkov2016f}). {{beginthm|Example}}\label{e:gen} (a) Analogously to [[3-manifolds_in_6-space#Examples|the Haefliger trefoil knot]] for any $l>1$ one constructs a smooth embedding $t:S^{2l-1}\to\Rr^{3l}$, see \cite[$\S]{Skopenkov2016h}. For $l$ even $t$ is not smoothly isotopic to the standard embedding; $t$ represents a generator of $E_D^{3l}(S^{2l-1})\cong\Zz$ \cite{Haefliger1962}. It would be interesting to know if for $l>1$ odd this embedding is a generator of $E_D^{3l}(S^{2l-1})\cong\Zz_2$. The last phrase of \cite{Haefliger1962t} suggests that this is true for $l=3$. (b) For any $k=1,3,7$ let $\eta\in\pi_{4k-1}(S^{2k})$ be the homotopy class of the Hopf map. Denote by $\zeta:\pi_{4k-1}(S^{2k})\to E_D^{6k}(S^{4k-1}\sqcup S^{4k-1})$ [[High_codimension_links#Examples|the Zeeman map]], see \cite[Definition 2.2]{Skopenkov2016h}. The embedded connected sum $\#\zeta\eta$ of the components of (a representative of) $\zeta\eta$ is not smoothly isotopic to the standard embedding; $\#\zeta\eta$ is a generator of $E_D^{6k}(S^{4k-1})\cong\Z$ \cite[Corollary 2.13]{Skopenkov2015a}. {{endthm}} == Invariants == ; Let us define the ''Haefliger invariant'' $\varkappa:E^{6k}_D(S^{4k-1})\to\Z$. The definition is motivated by Haefliger's proof that any embedding $S^n\to S^m$ is isotopic to the standard embedding for m\ge3n+4$, and by analyzing what obstructs carrying this proof for m=3n+3$. By \cite[2.1, 2.2]{Haefliger1962} any embedding $f:S^{4k-1}\to S^{6k}$ has a framing extendable to a framed embedding $\overline f:V\to D^{6k+1}$ of a k$-manifold $V$ whose boundary is $S^{4k-1}$, and whose signature is zero. For an integer k$-cycle $c$ in $V$ let $\lambda^*(c)\in\Z$ be the linking number of $f(V)$ with a slight shift of $\overline f(c)$ along the first vector of the framing. This defines a map $\lambda^*:H_{2k}(V;\Z)\to\Z$. This map is a homomorphism (as opposed to the Arf map defined in a similar way \cite{Pontryagin1959}). Then by Lefschetz duality there is a unique $\lambda\in H_{2k}(V,\partial;\Z)$ such that $\lambda^*[c]=\lambda\cap_V[c]$ for any $[c]\in H_{2k}(V;\Z)$. Since $V$ has a normal framing, its intersection form $\cap_V$ is even. (Indeed, represent a class in $H_{2k}(V;\Z)$ by a closed oriented k$-submanifold $c$. Then $\rho_2[c]\cap_V[c]=\overline{w_{2k}}(c\subset V)=\rho_2[c]\cap_VPDw_{2k}(V)=0$ because $V$ has a normal framing.) Hence $\lambda\cap_V\lambda$ is an even integer. Define $$\varkappa(f):=\lambda\cap_V\lambda/2.$$ Since the signature of $V$ is zero, there is a symplectic basis $\alpha_1,\ldots,\alpha_s,\beta_1,\ldots,\beta_s$ in $H_{2k}(V;\Z)$. Then clearly $$\varkappa(f) = \sum\limits_{j=1}^s \lambda^*(\beta_j)\lambda^*(\alpha_j).$$ For an alternative definition via Seifert surfaces in k$-space, discovered in \cite{Guillou&Marin1986}, \cite{Takase2004}, see \cite[the Kreck Invariant Lemma 4.5]{Skopenkov2016t}. For a definition by Kreck, and for a generalization to 3-manifolds see \cite[$\S]{Skopenkov2016t}. ''Sketch of a proof that $\varkappa(f)$ is well-defined (i.e. is independent of $V$, $\overline f$, and the framings), and is invariant under isotopy of $f$.'' \cite[Theorem 2.6]{Haefliger1962} Analogously one defines $\lambda(V)$ and $\varkappa(V):=\lambda(V)\cap_V\lambda(V)/2$ for a framed k$-submanifold $V$ of $S^{6k+1}$. Since $\varkappa(V)$ is a characteristic number, it is independent of framed cobordism. So $\varkappa(V)$ defines a homomorphism $\Omega_{fr}^{4k}(6k+1)=\pi_{6k+1}(S^{2k+1})\to\Z$. The latter group is finite by the Serre theorem. Hence the homomorphism is trivial. Since $\varkappa(f)$ is a characteristic number, it is independent of framed cobordism of a framed $f$ (and hence of the isotopy of a framed $f$). Therefore $\varkappa(f)$ is a well-defined invariant of a framed cobordism class of a framed $f$. By \cite[2.9]{Haefliger1962} (cf. \cite[2.2 and 2.3]{Haefliger1962}) $\varkappa(f)$ is also independent of the framing of $f$ extendable to a framing of some k$-manifold $V$ having trivial signature. QED For definition of the ''attaching invariant'' $E^{n+q}_D(S^n)\to\pi_n(G_q,SO_q)$ see \cite{Haefliger1966}, \cite[$\S]{Skopenkov2005}. == Classification == ; {{beginthm|Theorem|\cite[Corollary in p. 44]{Levine1965}, \cite{Haefliger1966}}}\label{t:leha} For $m-n\ge3$ the group $E^m_D(S^n)$ is finite unless $n=4k-1$ and $m\le6k$, when $E^m_D(S^n)$ is the sum of $\Z$ and a finite group. {{endthm}} {{beginthm|Theorem|(Haefliger-Milgram)}}\label{t:hami} We have the following table for the group $E^m_D(S^n)$; in the whole table $k\ge1$; in the fifth column $k\ne2$; in the last two columns $k\ge2$: $$\begin{array}{c|c|c|c|c|c|c|c} (m,n) &2m\ge3n+4 &(6k,4k-1) &(6k+3,4k+1) &(7,4) &(6k+4,4k+2) &(12k+7,8k+4) & (12k+1,8k)\ \hline E^m_D(S^n)&0 &\Z &\Z_2 &\Z_{12} &0 &\Z_4 &\Z_2\oplus\Z_2 \end{array}$$ {{endthm}} Proof for the first four columns, and for the fifth column when $k$ is odd, are presented in \cite[8.15]{Haefliger1966} (see also $\S; some proofs are deduced from that paper using simple calculations, cf. \cite[$\S]{Skopenkov2005}; there is a typo in \cite[8.15]{Haefliger1966}: $C^{3k}_{4k−2}=0$ should be $C^{4k}_{8k−2}=0$). The remaining results follow from \cite[8.15]{Haefliger1966} and \cite[Theorem F]{Milgram1972}. Alternative proofs for the cases $(m,n)=(7,4),(6,3)$ are given in \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008}, \cite{Skopenkov2008}. {{beginthm|Theorem|\cite[Corollary G]{Milgram1972}}}\label{t:mi} We have $E^m_D(S^n)=0$ if and only if either m\ge3n+4$, or $(m,n)=(6k+4,4k+2)$, or $(m,n)=(3k,2k)$ and $k\equiv3,11\mod12$, or $(m,n)=(3k+2,2k+2)$ and $k\equiv14,22\mod24$. {{endthm}} For a description of 2-components of $E^m_D(S^n)$ see \cite[Theorem F]{Milgram1972}. Observe that no reliable reference (containing complete proofs) of results announced in \cite{Milgram1972} appeared. Thus, strictly speaking, the corresponding results are conjectures. The lowest-dimensional unknown groups $E^m_D(S^n)$ are $E^8_D(S^5)$ and $E^{11}_D(S^7)$. Hopefully application of Kreck surgery could be useful to find these groups, cf. \cite{Skopenkov2005}, \cite{Crowley&Skopenkov2008}, \cite{Skopenkov2008}. For $m\ge n+3$ the group $E^m_D(S^n)$ has been described as follows, in terms of exact sequences \cite{Haefliger1966}, cf. \cite{Levine1965}, \cite{Haefliger1966a}, \cite{Milgram1972}, \cite{Habegger1986}. {{beginthm|Theorem|\cite{Haefliger1966}}}\label{t:knots} For $q\ge3$ there is the following exact sequence of abelian groups: $$ \ldots \to \pi_{n+1}(SG,SO) \xrightarrow{~u~} E^{n+q}_D(S^n) \xrightarrow{~a~} \pi_n(SG_q,SO_q) \xrightarrow{~s~} \pi_n(SG,SO) \xrightarrow{~u~} E^{n+q-1}_D(S^{n-1})\to \ldots~.$$ Here $SG_q$ is the space of maps $S^{q-1} \to S^{q-1}$ of degree \S1, $\S$ $\S$ 3].

2 Examples

There are smooth embeddings $S^{2l-1}\to\Rr^{3l}$ which are not smoothly isotopic to the standard embedding. They are PS (piecewise smoothly) isotopic to the standard embedding (by the Zeeman Unknotting Spheres Theorem 2.3 of [Skopenkov2016c] and [Skopenkov2016f, Remark 1.1]).

(a) Analogously to the Haefliger trefoil knot for any $l>1$ one constructs a smooth embedding $t:S^{2l-1}\to\Rr^{3l}$ , see [Skopenkov2016h, $\S$ 5]. For $l$ even $t$ is not smoothly isotopic to the standard embedding; $t$ represents a generator of $E_D^{3l}(S^{2l-1})\cong\Zz$ [Haefliger1962].

It would be interesting to know if for $l>1$ odd this embedding is a generator of $E_D^{3l}(S^{2l-1})\cong\Zz_2$ . The last phrase of [Haefliger1962t] suggests that this is true for $l=3$ .

(b) For any $k=1,3,7$ let $\eta\in\pi_{4k-1}(S^{2k})$ be the homotopy class of the Hopf map. Denote by $\zeta:\pi_{4k-1}(S^{2k})\to E_D^{6k}(S^{4k-1}\sqcup S^{4k-1})$ the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum $\#\zeta\eta$ of the components of (a representative of) $\zeta\eta$ is not smoothly isotopic to the standard embedding; $\#\zeta\eta$ is a generator of $E_D^{6k}(S^{4k-1})\cong\Z$ [Skopenkov2015a, Corollary 2.13].

3 Invariants