Knots, i.e. embeddings of spheres

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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, $\S$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\S$1, $\S$$\S$3].

2 Examples

There are smooth embeddings $S^{2l-1}\to\Rr^{3l}$$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]).

Example 2.1. (a) Analogously to the Haefliger trefoil knot for any $l>1$$l>1$ one constructs a smooth embedding $t:S^{2l-1}\to\Rr^{3l}$$t:S^{2l-1}\to\Rr^{3l}$, see [Skopenkov2016h, $\S$$\S$5]. For $l$$l$ even $t$$t$ is not smoothly isotopic to the standard embedding; $t$$t$ represents a generator of $E_D^{3l}(S^{2l-1})\cong\Zz$$E_D^{3l}(S^{2l-1})\cong\Zz$ [Haefliger1962].

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

(b) For any $k=1,3,7$$k=1,3,7$ let $\eta\in\pi_{4k-1}(S^{2k})$$\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})$$\zeta:\pi_{4k-1}(S^{2k})\to E_D^{6k}(S^{4k-1}\sqcup S^{4k-1})$ the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum $\#\zeta\eta$$\#\zeta\eta$ of the components of (a representative of) $\zeta\eta$$\zeta\eta$ is not smoothly isotopic to the standard embedding; $\#\zeta\eta$$\#\zeta\eta$ is a generator of $E_D^{6k}(S^{4k-1})\cong\Z$$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$$\varkappa:E^{6k}_D(S^{4k-1})\to\Z$. The definition is motivated by Haefliger's proof that any embedding $S^n\to S^m$$S^n\to S^m$ is isotopic to the standard embedding for $2m\ge3n+4$$2m\ge3n+4$, and by analyzing what obstructs carrying this proof for $2m=3n+3$$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$$V$ is zero, there is a symplectic basis $\alpha_1,\ldots,\alpha_s,\beta_1,\ldots,\beta_s$$\alpha_1,\ldots,\alpha_s,\beta_1,\ldots,\beta_s$ in $H_{2k}(V;\Z)$$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$$6k$-space, discovered in [Guillou&Marin1986], [Takase2004], see [Skopenkov2016t, the Kreck Invariant Lemma 4.5]. For a definition by Kreck, and for a generalization to 3-manifolds see [Skopenkov2016t, $\S$$\S$4].

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

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

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

4 Classification

Theorem 4.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For $m-n\ge3$$m-n\ge3$ the group $E^m_D(S^n)$$E^m_D(S^n)$ is finite unless $n=4k-1$$n=4k-1$ and $m\le6k$$m\le6k$, when $E^m_D(S^n)$$E^m_D(S^n)$ is the sum of $\Z$$\Z$ and a finite group.

Theorem 4.2 (Haefliger-Milgram). We have the following table for the group $E^m_D(S^n)$$E^m_D(S^n)$; in the whole table $k\ge1$$k\ge1$; in the fifth column $k\ne2$$k\ne2$; in the last two columns $k\ge2$$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$$k$ is odd, are presented in [Haefliger1966, 8.15] (see also $\S$$\S$6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, $\S$$\S$3]; there is a typo in [Haefliger1966, 8.15]: $C^{3k}_{4k−2}=0$$C^{3k}_{4k−2}=0$ should be $C^{4k}_{8k−2}=0$$C^{4k}_{8k−2}=0$). The remaining results follow from [Haefliger1966, 8.15] and [Milgram1972, Theorem F]. Alternative proofs for the cases $(m,n)=(7,4),(6,3)$$(m,n)=(7,4),(6,3)$ are given in [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].

Theorem 4.3 [Milgram1972, Corollary G]. We have $E^m_D(S^n)=0$$E^m_D(S^n)=0$ if and only if either $2m\ge3n+4$$2m\ge3n+4$, or $(m,n)=(6k+4,4k+2)$$(m,n)=(6k+4,4k+2)$, or $(m,n)=(3k,2k)$$(m,n)=(3k,2k)$ and $k\equiv3,11\mod12$$k\equiv3,11\mod12$, or $(m,n)=(3k+2,2k+2)$$(m,n)=(3k+2,2k+2)$ and $k\equiv14,22\mod24$$k\equiv14,22\mod24$.

For a description of 2-components of $E^m_D(S^n)$$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)$$E^m_D(S^n)$ are $E^8_D(S^5)$$E^8_D(S^5)$ and $E^{11}_D(S^7)$$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$$m\ge n+3$ the group $E^m_D(S^n)$$E^m_D(S^n)$ has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].

Theorem 4.4 [Haefliger1966]. For $q\ge3$$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$$SG_q$ is the space of maps $S^{q-1} \to S^{q-1}$$S^{q-1} \to S^{q-1}$ of degree $1$$1$. Restricting a map from $SO_q$$SO_q$ to $S^{q-1} \subset \Rr^q$$S^{q-1} \subset \Rr^q$ identifies $SO_q$$SO_q$ as a subspace of $SG_q$$SG_q$. Define $SG:=SG_1\cup\ldots\cup SG_q\cup\ldots$$SG:=SG_1\cup\ldots\cup SG_q\cup\ldots$. Analogously define $SO$$SO$. Let $s$$s$ be the stabilization homomorphism. The attaching invariant $a$$a$ and the map $u$$u$ are defined in [Haefliger1966], see also [Skopenkov2005, $\S$$\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$$\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

Theorem 6.1. The Haefliger invariant $\varkappa:E_D^{6k}(S^{4k-1})\to\Zz$$\varkappa:E_D^{6k}(S^{4k-1})\to\Zz$ is injective for $k>1$$k>1$.

The proof is a certain simplification of [Haefliger1962]. We present an exposition structured to make it more accessible to non-specialists.

Lemma 6.2. Let $V$$V$ be a framed $(2k-1)$$(2k-1)$-connected $4k$$4k$-submanifold of $D^{6k+1}$$D^{6k+1}$ such that $S^{4k-1}\cong \partial V \subset \partial D^{6k+1}$$S^{4k-1}\cong \partial V \subset \partial D^{6k+1}$, signature of $V$$V$ is zero, and $\varkappa(V) = 0$$\varkappa(V) = 0$. Then there is an embedding $G:D^{4k}\to D^{6k+1}$$G:D^{4k}\to D^{6k+1}$ such that $G(\partial D^{4k})=\partial V$$G(\partial D^{4k})=\partial V$.

Proof of Theorem 6.1 using Lemma 6.2. 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}$$g:S^{4k-1}\to \partial D^{6k+1}$ such that $\varkappa(g)=0$$\varkappa(g)=0$ there is a framed $(2k-1)$$(2k-1)$-connected $4k$$4k$-submanifold $V$$V$ of $D^{6k+1}$$D^{6k+1}$ with zero signature such that $S^{4k-1}\cong \partial V \subset \partial D^{6k+1}$$S^{4k-1}\cong \partial V \subset \partial D^{6k+1}$ and $\varkappa(V) = 0$$\varkappa(V) = 0$. Then from Lemma 6.2 it follows that there is an extension $G:D^{4k}\to D^{6k+1}$$G:D^{4k}\to D^{6k+1}$ such that $G|_{S^{4k-1}} = g$$G|_{S^{4k-1}} = g$. From Smale Theorem it follows that $g$$g$ is isotopic to the standard embedding. $\Box$$\Box$

To prove Lemma 6.2 we need Lemma 6.3, Lemma 6.4 and Lemma 6.5.

All the manifolds below can have non-empty boundaries.

Lemma 6.3 [Whitney lemma]. Let $k: P \rightarrow W$$k: P \rightarrow W$ be a map from a connected $p$$p$-manifold $P$$P$ to a simply connected $(p+q)$$(p+q)$-manifold $W$$W$. If $p, q \geq 3$$p, q \geq 3$, then

1. There is a homotopy $k_t$$k_t$ such that $k_0 = k$$k_0 = k$ and $k_1(P)$$k_1(P)$ is an embedding.???
2. Suppose in addition that there is a map $y???: Q \rightarrow W$$y???: Q \rightarrow W$ from a connected $q$$q$-manifold $Q$$Q$ such that the algebraic intersection number??? of $k(P)$$k(P)$ and $y(Q)$$y(Q)$ is zero. Then there is a homotopy $y_t$$y_t$ relative to the boundary such that $y_0 = y$$y_0 = y$ and $y_1(Q)$$y_1(Q)$ does not intersect $k(P)$$k(P)$. If $y$$y$ is an embedding, the homotopy $y_t$$y_t$ can be chosen so that $y_1$$y_1$ is an embedding.

Lemma 6.4. Let $V$$V$ be a $(2k-1)$$(2k-1)$-connected $4k$$4k$-manifold, and $x_1, \ldots, x_m \in H_{2k}(V)$$x_1, \ldots, x_m \in H_{2k}(V)$ are such that $x_i \cap_V x_j = 0$$x_i \cap_V x_j = 0$ for every $i,j$$i,j$. Then there are embeddings $g_1, \ldots, g_m: \, S^{2k} \rightarrow V$$g_1, \ldots, g_m: \, S^{2k} \rightarrow V$ with pairwise disjoint images representing $x_1, \ldots, x_m$$x_1, \ldots, x_m$, respectively.

Proof. As $V$$V$ is $(2k-1)$$(2k-1)$-connected, the Hurewicz map $h: \pi_{2k}(V) \rightarrow H_{2k}(V)$$h: \pi_{2k}(V) \rightarrow H_{2k}(V)$ is an isomorphism. For an element $x_i \in H_{2k}(V)$$x_i \in H_{2k}(V)$, let $v_i: S^{2k} \rightarrow V$$v_i: S^{2k} \rightarrow V$ represent the homotopy class $h^{-1}(x_i)$$h^{-1}(x_i)$. Now we perform the following inductive procedure. At the $i$$i$-th step of the procedure assume that the maps $g_j$$g_j$ from the assertion are already constructed for any $j < i$$j < i$, and we construct $g_i$$g_i$. First, applying item 1 of Lemma 6.3 to $v_i$$v_i$, we may suppose that $v_i$$v_i$ itself is an embedding. Further, we apply item 2 of Lemma 6.3, ???setting $g_j$$g_j$ as $k$$k$, $v_i$$v_i$ as $y$$y$, and $V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$$V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$ as $W$$W$, for any $j < i$$j < i$. ???Observe that Lemma 6.3 is applicable as $V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$$V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$ is simply connected provided $2k \geq 3$$2k \geq 3$ and $V$$V$ is simply connected. Also, as $x_i \cap_V x_j = 0$$x_i \cap_V x_j = 0$ for any $i$$i$, $j$$j$, the algebraic intersection number for all the pairs of maps above??? is zero. As the result, $v_i$$v_i$ is replaced with a homotopic embedding $g_i$$g_i$, and the images of $g_1, \ldots, g_i$$g_1, \ldots, g_i$ are disjoint. After the step $m$$m$ we obtain the ???desired set of embeddings.

$\square$$\square$

Lemma 6.5.[cf. Proposition 3.3 in [Ha62]] Let $V$$V$ be an orientable $4k$$4k$-submanifold of $B^{6k+1}$$B^{6k+1}$, and $???g_1,\ldots,g_s:D^{2k+1}\to B^{6k+1}$$???g_1,\ldots,g_s:D^{2k+1}\to B^{6k+1}$ be embeddings with pairwise disjoint images such that $g_i(D^{2k+1})\cap V = g_i(S^{2k})$$g_i(D^{2k+1})\cap V = g_i(S^{2k})$, $g_i(D^{2k+1})$$g_i(D^{2k+1})$ are orthogonal to $V$$V$, and over every $g_i(S^{2k})$$g_i(S^{2k})$ the manifold $V$$V$ has a framing whose first vector is tangent to $g_i(D^{2k+1})$$g_i(D^{2k+1})$. Assume that $g_i(S^{2k})$$g_i(S^{2k})$ has zero algebraic self-intersection in $V$$V$ for every $i$$i$. Then there are embeddings $G_1,\ldots,G_s:D^{2k+1}\times D^{2k}\to B^{6k+1}$$G_1,\ldots,G_s:D^{2k+1}\times D^{2k}\to B^{6k+1}$ extending $g_1,\ldots,g_s$$g_1,\ldots,g_s$ such that $G_i(S^{2k}\times D^{2k})\subset V$$G_i(S^{2k}\times D^{2k})\subset V$ for every $i$$i$.

Proof. By general position, it suffices to prove the existence of $G_1,\ldots,G_s$$G_1,\ldots,G_s$ for $s=1$$s=1$. So assume that $s=1$$s=1$. Since $g_1(S^{2k})$$g_1(S^{2k})$ has zero algebraic self-intersection in $V$$V$, the Euler class of the normal bundle of $g_1(S^{2k})$$g_1(S^{2k})$ in $V$$V$ is zero. Hence $g_1(S^{2k})$$g_1(S^{2k})$ has a framing in $V$$V$.

Identify all the normal spaces!!! of $G_1(D^{2k+1})$$G_1(D^{2k+1})$ with the normal space at $G_1(0)$$G_1(0)$. The normal framing $a$$a$ of $g_1(S^{2k})$$g_1(S^{2k})$ in $V$$V$ is orthogonal to $G_1(D^{2k+1})$$G_1(D^{2k+1})$. So $a$$a$ defines a map $S^{2k}\to V_{4k,2k}$$S^{2k}\to V_{4k,2k}$. Let $\zeta=\in\pi_{2k}(V_{4k,2k})$$\zeta=\in\pi_{2k}(V_{4k,2k})$ be the homotopy class of this map. This is the obstruction to extending $a$$a$ to a normal $2k$$2k$-framing of $G_1$$G_1$ in $B^{6k+1}$$B^{6k+1}$ (so apriori $\zeta=\zeta(a)$$\zeta=\zeta(a)$). It suffices to prove that $\zeta=0$$\zeta=0$.

Consider the exact sequence of the bundle $SO_{a+b}/SO_b = V_{a+b,a}$$SO_{a+b}/SO_b = V_{a+b,a}$: $\pi_q(SO_{a+b}) \overset{j}\to \pi_q(V_{a+b,a}) \overset{\partial}\to \pi_{q-1}(SO_b)$$\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$$\partial\zeta=0$: if $\alpha:S^q\to V_{a,b}$$\alpha:S^q\to V_{a,b}$ is a map, then $\partial[\alpha]$$\partial[\alpha]$ is the obstruction to trivialization of the orthogonal complement to the field of $b$$b$-frames in $S^q\times\R^a$$S^q\times\R^a$ corresponding to $\alpha$$\alpha$. By [Fomenko&Fuchs2016, Corollary in $\S$$\S$25.4] $\pi_{2k}(SO_{4k})$$\pi_{2k}(SO_{4k})$ is a finite group. Since $\pi_{2k}(V_{4k,2k})\cong\Z$$\pi_{2k}(V_{4k,2k})\cong\Z$, we obtain that $j=0$$j=0$ for $a=b=2k$$a=b=2k$. This and $\partial\zeta=0$$\partial\zeta=0$ imply that $\zeta=0$$\zeta=0$.

(In [Haefliger1962, the last but one paragraph of 3.5] perhaps one has to replace `$\partial\zeta$$\partial\zeta$ is the obstruction to trivializing the normal bundle of $g(S^{2k})$$g(S^{2k})$ in $V$$V$' by `$\partial\zeta$$\partial\zeta$ is the obstruction to trivializing te orthogonal complement to the $f_1$$f_1$-direction of the normal bundle of $V$$V$ restricted to $g(S^{2k})$$g(S^{2k})$'?)

(No, as we use different obstruction; our is orthogonal to Haefliger's)

Alternatively: Consider a map of the long exact sequences associated to the inclusion $S^{2k} = SO_{2k+1}/SO_{2k} \rightarrow SO_{4k}/SO_{2k} = V_{4k, 2k}$$S^{2k} = SO_{2k+1}/SO_{2k} \rightarrow SO_{4k}/SO_{2k} = V_{4k, 2k}$:

$\xymatrix{\pi_{2k}(SO_{2k+1}) \ar[r]^{p = 0} \ar[d]^{s} & \pi_{2k}(S^{2k}) \ar[d] \\ \pi_{2k}(SO_{4k})\ar[r]^{p'=0} & \pi_{2k}(V_{4k, 2k})\ar[r]^{\partial '} & \pi_{2k-1}(SO_{2k})}$

By [Fomenko&Fuchs2016, Corollary in $\S$$\S$25.4] $\pi_{2k}(SO_{2k+1})$$\pi_{2k}(SO_{2k+1})$ is a finite group. Since $\pi_{2k}(S^{2k})\cong\Z$$\pi_{2k}(S^{2k})\cong\Z$, we obtain that $p=0$$p=0$. Since $s$$s$ is surjective, we obtain that $p'=0$$p'=0$.

Hence $\partial'$$\partial'$ is injective. The class $\partial'\zeta$$\partial'\zeta$ is??? the obstruction to trivializing the orthogonal complement of $g(S^{2k})$$g(S^{2k})$, cf. [Haefliger1962, 3.4, Lemma].

Suggestion: interpreting $\partial'\zeta$$\partial'\zeta$ as an obstruction, we derive that $\partial'\zeta$$\partial'\zeta$ is zero, as $V$$V$ is framed and the first vector of the framing is tangent to $G_1$$G_1$.

Therefore, $\partial ' \zeta =0$$\partial ' \zeta =0$. Since $\partial '$$\partial '$ is injective, we obtain $\zeta = 0$$\zeta = 0$.

$\square$$\square$

Lemma 6.6. Denote by $G_1,\ldots, G_s$$G_1,\ldots, G_s$ the embeddings from Lemma 6.5. There is smooth manifold $V'\subset D^{6k+1}$$V'\subset D^{6k+1}$ such that $V'$$V'$ is homeomorphic to the $V\backslash (\bigcup G_i(???S^{2k}\times D^{2k-1})) \bigcup G_i(D^{2k+1}\times S^{2k-2})$$V\backslash (\bigcup G_i(???S^{2k}\times D^{2k-1})) \bigcup G_i(D^{2k+1}\times S^{2k-2})$ and $\partial V'=\partial V$$\partial V'=\partial V$.

Proof. For any $G_i$$G_i$ choose smole??? neighborhood $U_i$$U_i$ of $G_i(D^{2k+1})$$G_i(D^{2k+1})$ and smooth diffeomorphism $\phi_i: \mathbb{R}^{6k+1}\to U_i$$\phi_i: \mathbb{R}^{6k+1}\to U_i$ such that $\phi_i^{-1}\circ G_i(D^{2k+1}\times D^{2k})$$\phi_i^{-1}\circ G_i(D^{2k+1}\times D^{2k})$ is defined by $x^2\leq 1, y^2\leq 1, z=0$$x^2\leq 1, y^2\leq 1, z=0$ in some coordinates $(x, y, z)=(x_1,\ldots, x_{2k+1}, y_1, \ldots, y_{2k}, z_1, \ldots, z_{2k})$$(x, y, z)=(x_1,\ldots, x_{2k+1}, y_1, \ldots, y_{2k}, z_1, \ldots, z_{2k})$ of $\mathbb{R}^{6k+1}$$\mathbb{R}^{6k+1}$.

Denote by $a:\mathbb{R}\to \mathbb{R}$$a:\mathbb{R}\to \mathbb{R}$ a smooth monotonous map such that $a(x)=0$$a(x)=0$ for $x\leq \frac{1}{2}$$x\leq \frac{1}{2}$ and $a(x)=1$$a(x)=1$ for $x\geq 1$$x\geq 1$. Denote by $g:D^{2k+1}\times S^{2k-1}\to \mathbb{R}^{6k+1}$$g:D^{2k+1}\times S^{2k-1}\to \mathbb{R}^{6k+1}$ the smooth map, defined by the formula $g(x, y):=(x,y\cdot a(x^2), 0)$$g(x, y):=(x,y\cdot a(x^2), 0)$. Take by $V'$$V'$ the manifold $V\backslash (\bigcup G_i(S^{2k}\times D^{2k-1})) \bigcup \phi_i\circ g(D^{2k+1}\times S^{2k-2})$$V\backslash (\bigcup G_i(S^{2k}\times D^{2k-1})) \bigcup \phi_i\circ g(D^{2k+1}\times S^{2k-2})$. $V'$$V'$ is homeomorphic to $V$$V$. Therefore, map $f:V\to V'$$f:V\to V'$ defined by equation $f(x)=\begin{cases} x\text{, if }x\in V\backslash (\bigcup G_i(S^{2k}\times D^{2k-1})) \\g\circ G_i^{-1}\circ \phi_i^{-1}(x)\text{, if }x\in G_i(S^{2k}\times D^{2k-1}) \end{cases}$$f(x)=\begin{cases} x\text{, if }x\in V\backslash (\bigcup G_i(S^{2k}\times D^{2k-1})) \\g\circ G_i^{-1}\circ \phi_i^{-1}(x)\text{, if }x\in G_i(S^{2k}\times D^{2k-1}) \end{cases}$ is the homeomorphism. Since $V'$$V'$ coincide with $V$$V$ outside the $\bigcup \phi_i\circ g(D^{2k+1}\times S^{2k-2})$$\bigcup \phi_i\circ g(D^{2k+1}\times S^{2k-2})$ and $V'$$V'$ is smooth in $\bigcup U_i$$\bigcup U_i$, we have that $V'$$V'$ is smooth.

$\square$$\square$

Below the symbol $[ \cdot ]$$[ \cdot ]$ denotes the integral fundamental class of a manifold or the homotopy class of a map, depending on the context.

Proof of Lemma 6.2 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\}$$\left\{\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s\right\}$ in $H_{2k}(V)$$H_{2k}(V)$ such that $\alpha_i\cap \alpha_j=\beta_i\cap \beta_j = 0$$\alpha_i\cap \alpha_j=\beta_i\cap \beta_j = 0$, $\alpha_i\cap \beta_j=\delta_{i, j}$$\alpha_i\cap \beta_j=\delta_{i, j}$ and $\lambda^*(\alpha_i)=0$$\lambda^*(\alpha_i)=0$ for any $i, j$$i, j$. From Lemma 6.4 it follows that there are embeddings $f_1, \ldots, f_s:S^{2k}\to V$$f_1, \ldots, f_s:S^{2k}\to V$ with pairwise disjoint images such that $f_{1*}[S^{2k}]=\alpha_1, \ldots, f_{s*}[S^{2k}]=\alpha_s$$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$$f_{i}':S^{2k}\to D^{6k+1}\backslash V$ for $i\leq s$$i\leq s$ the result of shifting of $f_i$$f_i$ by the first vector of the framing of $V$$V$. Since $\lambda(f_{i*}[S^{2k}])=\lambda(\alpha_i)=0$$\lambda(f_{i*}[S^{2k}])=\lambda(\alpha_i)=0$, we have $f_{i*}'([S^{2k}])=0 \in H_{2k}(D^{6k+1}\backslash V)$$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$$\mbox{dim} D^{6k+1}-\mbox{dim} V=6k+1-4k=2k+1$, we have $\pi_m(D^{6k+1}\backslash \mbox{Int}{V}) = 0$$\pi_m(D^{6k+1}\backslash \mbox{Int}{V}) = 0$ for $m<2k$$m<2k$. Since $f_{i*}'([S^{2k}])=0\in H_{2k}(D^{6k+1}\backslash V)$$f_{i*}'([S^{2k}])=0\in H_{2k}(D^{6k+1}\backslash V)$, we have $[f_i']=0\in \pi_{2k}(D^{6k+1}\backslash V)$$[f_i']=0\in \pi_{2k}(D^{6k+1}\backslash V)$. Therefore there are extensions of $f_1, \ldots, f_s$$f_1, \ldots, f_s$ to maps $g_1, \ldots, g_s:D^{2k+1}\to B^{6k+1}$$g_1, \ldots, g_s:D^{2k+1}\to B^{6k+1}$ such that $g_i(D^{2k+1})\cap V = g_i(S^{2k})$$g_i(D^{2k+1})\cap V = g_i(S^{2k})$.

???Set $\varepsilon>0$$\varepsilon>0$ such that $g_i(\varepsilon D^{2k+1})\cap V= g_i(\mbox{Int} D^{2k+1})\cap V$$g_i(\varepsilon D^{2k+1})\cap V= g_i(\mbox{Int} D^{2k+1})\cap V$. As $\lambda^*(g_{i*}[S^{2k}])=0$$\lambda^*(g_{i*}[S^{2k}])=0$ for any $i$$i$, the algebraic intersection number of $g_i(\varepsilon D^{2k+1})$$g_i(\varepsilon D^{2k+1})$ and $V$$V$ is zero. Applying item 2 of Lemma6.3 to $g_i|_{\varepsilon D^{2k+1}}$$g_i|_{\varepsilon D^{2k+1}}$ as $y$$y$, embedding of $V$$V$ into $D^{6k+1}$$D^{6k+1}$ as $k$$k$ and $D^{6k+1}$$D^{6k+1}$ as $W$$W$ we may suppose that $g_i(\varepsilon D^{2k+1})$$g_i(\varepsilon D^{2k+1})$ does not intersect $V$$V$. Hance for any $i$$i$ renamed by $g_i:D^{2k+1}\to D^{6k+1}\backslash V$$g_i:D^{2k+1}\to D^{6k+1}\backslash V$ a map which coincides with $g_i$$g_i$ on $D^{2k+1}\backslash \varepsilon D^{2k+1}$$D^{2k+1}\backslash \varepsilon D^{2k+1}$ and $g_i(\varepsilon D^{2k+1})\cap V=\emptyset$$g_i(\varepsilon D^{2k+1})\cap V=\emptyset$.

???Denote by $G_1, \ldots, G_s$$G_1, \ldots, G_s$ the maps as in Lemma 6.5 for $V$$V$ and $g_1, \ldots, g_s$$g_1, \ldots, g_s$ as described above. Denote by $V'$$V'$ the manifold as in Lemma 6.6. Since $\left\{\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s\right\}$$\left\{\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s\right\}$ is a basis in $H_{2k}(V)$$H_{2k}(V)$, ???it follows that $H_{2k}(V')= 0$$H_{2k}(V')= 0$. Since $\pi_1(V)= 0$$\pi_1(V)= 0$ we have $\pi_1(V')= 0$$\pi_1(V')= 0$. Hence??? $H_m(V')= 0$$H_m(V')= 0$ for $0< m\leq 2k$$0< m\leq 2k$. From Generalized Poincare conjecture proved by Smale it follows that $V'\cup_{\partial V'=\partial D^{4k}} D^{4k}\cong S^{4k}$$V'\cup_{\partial V'=\partial D^{4k}} D^{4k}\cong S^{4k}$. Hence $V'\cong D^{4k}$$V'\cong D^{4k}$. Then we set by??? $G$$G$ a ???diffeomorphism between $D^{4k}$$D^{4k}$ and $V'\subset D^{6k+1}$$V'\subset D^{6k+1}$ such that $G|_{S^{4k-1}}=g$$G|_{S^{4k-1}}=g$. $\Box$$\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.
• [Fomenko&Fuchs2016] A. T. Fomenko and D. B. Fuks, Homotopical Topology. Translated from the Russian. Graduate Texts in Mathematics, 273. Springer-Verlag, Berlin, 2016. DOI 10.1007/978-3-319-23488-5.

• [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)$$(4k-1)$-spheres in $6k$$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$$S^n$ in $S^{n+q}$$S^{n+q}$ for $q>2$$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:0603429MR2434474 (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