Knots, i.e. embeddings of spheres

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[edit] 1 Introduction

We work in a smooth category. In particular, terms embedding and smooth embedding or map and smooth map are used interchangeably. For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, ${{Stub}} == Introduction == <wikitex>; See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction on embeddings]], [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] in \cite[$\S\S$ 1, $\S$ 3].

[edit] 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].

[edit] 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].

[edit] 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].

[edit] 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].

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

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 $B^{6k+1}$ such that $S^{4k-1}\cong \partial V \subset \partial B^{6k+1}$ , signature of $V$ is zero, and $\varkappa(V) = 0$ . Then there is a submanifold $V'\subset B^{6k+1}$ such that $V'\cong D^{4k}$ and $\partial V'=\partial V$ .

Proof of Theorem 6.1 using Lemma 6.2. By the first three paragraphs of the proof of Theorem 3.1 in [Haefliger1962], for any embedding $g:S^{4k-1}\to \partial B^{6k+1}$ such that $\varkappa(g)=0$ there is a framed $(2k-1)$ -connected $4k$ -submanifold $V$ of $B^{6k+1}$ with zero signature such that $g(S^{4k-1}) = \partial V \subset \partial B^{6k+1}$ and $\varkappa(V) = 0$ . Then by Lemma 6.2 there is a submanifold $V'\subset B^{6k+1}$ such that $D^{4k} \cong V'$ and $\partial V'=\partial V$ . Recall that isotopy classes of embeddings $S^q\to S^n$ are in 1--1 correspondence with $h$ -cobordism classes of oriented submanifolds of $S^n$ diffeomorphic to $S^q$ for $n\ge5$ , $n\ge q+3$ , cf. [Haefliger1966, 1.8], [Kervaire1965]. Hence $g$ is isotopic to standart embedding. $\Box$

To prove Lemma 6.2 we need Lemmas 6.3, 6.4 and 6.5. Below manifolds can have non-empty boundaries.

Let $u: P \rightarrow W$ be a map from a connected oriented $p$ -manifold $P$ to a simply connected oriented $(p+q)$ -manifold $W$ . If $p, q \geq 3$ , then

If $q \geq p$ , there is a homotopy $u_t$ such that $u_0 = u$ and $u_1(P)$ is an embedding.
Suppose in addition that $u(\partial P) \subseteq \partial W$ and there is a map $v: Q \rightarrow W$ with $v(\partial Q) \subseteq \partial W$ from a connected oriented $q$ -manifold $Q$ such that the algebraic intersection number of $u(P)$ and $v(Q)$ is zero. Then there is a homotopy $v_t$ relative to the boundary such that $v_0 = v$ and $v_1(Q)$ does not intersect $u(P)$ . If $v$ is an embedding, the homotopy $v_t$ can be chosen so that $v_1$ is an embedding.

Below we denote by $h: \pi_{m}(\cdot) \rightarrow H_{m}(\cdot)$ the Hurewicz map.

Let $V$ be a $(2k-1)$ -connected $4k$ -manifold, and let $x_1, \ldots, x_{s} \in H_{2k}(V)$ be homology classes such that $x_i \cap_V x_j = 0$ for every $i,j$ . Then there are embeddings $g_1, \ldots, g_s: \, S^{2k} \rightarrow V$ with pairwise disjoint images representing $x_1, \ldots, x_s$ , respectively.

As $V$ is $(2k-1)$ -connected, $h: \pi_{2k}(V) \rightarrow H_{2k}(V)$ is an isomorphism. For an element $x_i \in H_{2k}(V)$ , let $\widetilde{x_i}: S^{2k} \rightarrow V$ be a representative of the homotopy class $h^{-1}(x_i)$ . Applying item 1 of Lemma 6.3 to $\widetilde{x_i}$ , we may assume that $\widetilde{x_i}$ is an embedding.

Make the following inductive procedure. At the $i$ -th step, $i=1,\ldots,s$ , assume that the embeddings $g_1, \ldots, g_{i-1}$ are already constructed, and we construct $g_i$ . Since $2k \geq 3$ and $V$ is simply connected, $W:=V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$ is simply connected for any $j < i$ . The algebraic intersection number of $g_j(S^{2k})$ and $\widetilde{x_i}(S^{2k})$ is zero for any $j$ . Hence we can apply item 2 of Lemma 6.3 to $u=g_j$ and $v=\widetilde{x_i}$ and $W$ as above for any $j < i$ . So $\widetilde{x_i}$ is replaced by a homotopic embedding $g_i$ , and the images of $g_1, \ldots, g_i$ are pairwise disjoint. After $s$ -th step we obtain a required set of embeddings.

$\square$ $\square$

Let $V$ be an orientable $4k$ -submanifold of $B^{6k+1}$ , and $g:D^{2k+1}\to B^{6k+1}$ be an embedding such that $g(D^{2k+1})\cap V = g(S^{2k})$ and over $g(S^{2k})$ the manifold $V$ has a framing whose first vector is tangent to $g(D^{2k+1})$ . Assume that $g(S^{2k})$ has zero algebraic self-intersection in $V$ . Then $g$ extends to an embedding $G: D^{2k+1}\times D^{2k}\to B^{6k+1}$ such that $G(S^{2k}\times D^{2k})\subset V$ .

(A slightly different proof is presented in the proof of Proposition 3.3 in [Haefliger1962].) Since $g(S^{2k})$ has zero algebraic self-intersection in $V$ , the Euler class of the normal bundle of $g(S^{2k})$ in $V$ is zero. Since over $g(S^{2k})$ the manifold $V$ has a framing, we obtain that $g(S^{2k})$ has a framing in $V$ .

Identify all the normal spaces of $G(D^{2k+1})$ with the normal space at $G(0)$ . The normal framing $a$ of $g(S^{2k})$ in $V$ is orthogonal to $G(D^{2k+1})$ . So $a$ defines a map $S^{2k}\to V_{4k,2k}$ . Let $\zeta \in\pi_{2k}(V_{4k,2k})$ be the homotopy class of this map. This is the obstruction to extending $a$ to a normal $2k$ -framing of $G$ in $B^{6k+1}$ (so apriori $\zeta=\zeta(a)$ ). It suffices to prove that $\zeta=0$ .

Consider the exact sequence of the bundle $SO_{a+b}/SO_b = V_{a+b,a}$ : $\pi_q(SO_{a+b}) \overset{j}\to \pi_q(V_{a+b,a}) \overset{\partial}\to \pi_{q-1}(SO_b)$ . By the following well-known assertion, $\partial\zeta=0$ : if $\alpha\in\pi_q(V_{a,b})$ , then $\partial\alpha$ is the obstruction to trivialization of the orthogonal complement to the field of $b$ -frames in $S^q\times\R^a$ corresponding to a representative of $\alpha$ .

Consider a map of the exact sequences associated to the inclusion $S^{2k} = SO_{2k+1}/SO_{2k} \to V_{4k,2k} = SO_{4k}/SO_{2k}$ . The composition $\pi_{2k}(S^{2k})\overset{i}\to\pi_{2k}(V_{4k,2k})\overset{\partial}\to\pi_{2k-1}(SO_{2k})$ is the boundary map $\partial'$ . The group $\pi_{2k-1}(SO_{2k})$ is in natural 1--1 correspondence with the group of $2k$ -bundles over $S^{2k}$ . The image $\partial'\iota_{2k}$ is the tangent bundle $\tau$ of $S^{2k}$ . Since the Euler class of $n\tau$ is $2n\ne0$ , the map $\partial'$ is injective. Since $i$ is an isomorphism, the map $\partial$ is injective. This and $\partial\zeta=0$ imply that $\zeta=0$ .

Alternatively, by [Fomenko&Fuchs2016, Corollary in $\S$ 25.4] $\pi_{2k}(SO_{4k})$ is a finite group (in [Fomenko&Fuchs2016, Corollary in $\S$ 25.4] the formula for $\pi_q(SO_{2m})$ is correct, although the formula for $\pi_q(SO_{2m+1})$ is incorrect because $\pi_3(SO_3)\cong\Z\oplus\Z$ ). Since $\pi_{2k}(V_{4k,2k})\cong\Z$ , we obtain that $j=0$ for $a=b=2k$ . This and $\partial\zeta=0$ imply that $\zeta=0$ .

$\square$ $\square$

Let $V$ be a $4k$ -submanifold of $B^{6k+1}$ and let $G:D^{2k+1}\times D^{2k}\to B^{6k+1}$ be an embedding such that $G(S^{2k}\times D^{2k})\subset V$ . Then there is a smooth submanifold $V'\subset B^{6k+1}$ homeomorphic to $V\backslash ( G(S^{2k}\times D^{2k})) \bigcup G(D^{2k+1}\times S^{2k-1})$ and such that $V\backslash G(D^{2k+1}\times D^{2k})=V'\backslash G(D^{2k+1}\times D^{2k})$ .

Lemma 6.6 is essentialy proved in [Haefliger1962, $\S$ 3.3].

Proof of Lemma 6.2 using Lemmas 6.4, 6.5. By the fourth paragraph of the proof of Theorem 3.1 in [Haefliger1962], there is a basis $\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s$ 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 representing $x_1, \ldots, x_s$ , respectively. [!!!such that $f_{i*}[S^{2k}]=\alpha_i$ for every $i=1,\ldots,s$ ]

For $i=1,\ldots,s$ denote by $\alpha_i'\in\pi_{2k}(B^{6k+1}\backslash V)$ the homotopy class of the shift of $f_i$ by the first vector of the framing of $V$ on $f_i(S^{2k})$ . Since $\lambda^*(\alpha_i)=0$ , we have $h\alpha_i' =0 \in H_{2k}(B^{6k+1}\backslash V)$ . Since $\mbox{dim} B^{6k+1}-\mbox{dim} V=6k+1-4k=2k+1$ , the complement $B^{6k+1}\backslash V$ is $(2k-1)$ -connected. Hence by Hurewicz Theorem $h\alpha_i'=0$ implies $\alpha_i'=0$ . Therefore there are extensions $g_1, \ldots, g_s:D^{2k+1}\to B^{6k+1}$ of $f_1, \ldots, f_s$ such that $g_i(D^{2k+1})\cap V = g_i(S^{2k})$ .

Take $\varepsilon>0$ such that $g_i(\varepsilon D^{2k+1})\cap V= g_i(\mbox{Int} D^{2k+1})\cap V$ for any $i\leq s$ . Take a tubular neighborhoods $U_i$ of $g_i(S^{2k})$ such that $g_i(D^{2k+1})\backslash U_i=g_i(\varepsilon D^{2k+1})$ for any $i\leq s$ . The algebraic intersection number of $g_i(\varepsilon D^{2k+1})$ and $V\backslash U_i$ equals $\lambda^*(\alpha_i)=\lambda^*(g_{i*}[S^{2k}])=0$ . We have $\pi_1(B^{6k+1}\backslash U_i)=\pi_1(B^{6k+1}\backslash g_i(S^{2k}))=0$ . So we can apply item 2 of Lemma 6.3 to $v=g_i|_{\varepsilon D^{2k+1}}$ , $k:V\backslash U_i\to B^{6k+1}\backslash U_i$ the inclusion, and $W=B^{6k+1}\backslash U_i$ . So we may assume that $g_i(\varepsilon D^{2k+1})$ does not intersect $V \backslash U_i$ . Hence we may assume that $g_i(D^{2k+1})\cap V=g_i(S^{2k})$ .

Apply Lemma 6.5 to $g=g_1, \ldots, g_s$ one by one, and to the manifold $V$ . Denote by $G_1, \ldots, G_s$ the resulting maps. Define manifolds $V^{i}$ for $0\leq i\leq s$ inductively. Let $V^0:=V$ , and let $V^{i}$ be a manifold $V'$ obtained applying Lemma 6.6 for $V=V^{i-1}$ and $G=G_i$ . By Lemma 6.6, $\pi_1(V)=\pi_1(V^s)= 0$ and $H_j(V)=H_j(V^s)= 0$ for $j<2k$ . Since $\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s$ is a symplectic basis in $H_{2k}(V)$ , it follows that $H_{2k}(V^s)= 0$ . Then from Generalized Poincare conjecture proved by Smale it follows that $V^s\cup_{\partial V^s=\partial D^{4k}} D^{4k}\cong S^{4k}$ . Hence $D^{4k} \cong V^s$ . Then take $V':=V^s$ . $\Box$

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, $\S]{Skopenkov2016c}. == Examples == ; Analogously to [[3-manifolds_in_6-space#Examples|the Haefliger trefoil knot]] for $k>1$ one constructs a smooth embedding $t:S^{2k-1}\to\Rr^{3k}$. For $k$ even this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz$; it is not ''smoothly'' isotopic to the standard embedding, but is ''piecewise smoothly'' isotopic to it \cite{Haefliger1962}. It would be interesting to know if for $k$ odd this embedding is a generator of $E_D^{3k}(S^{2k-1})\cong\Zz_2$. The last phrase of \cite{Haefliger1962t} suggests that this is true for $k=3$. == Readily calculable classification == ; For $m\ge n+3$ the group $E^m(S^n)$ has been described in terms of exact sequences involving the homotopy groups of spheres and the homotopy groups of the pairs $(SG_n,SO_n)$ for $n = n_i+1$ and $n_i$ \cite{Haefliger1966a}, cf. \cite{Levine1965}, \cite{Milgram1972}, \cite{Habegger1986}. Here $SG_n$ is the space of maps $f \colon S^{n-1} \to S^{n-1}$ of degree \S1, $\S$ $\S$ 3].

[edit] 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].

[edit] 3 Invariants