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, $\S$${{Stub}} == Introduction == <wikitex>; For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$1, $\S$$\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). 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^m_D(S^q)\to\pi_q(G_{m-q},SO_{m-q})$$E^m_D(S^q)\to\pi_q(G_{m-q},SO_{m-q})$ see [Haefliger1966], [Skopenkov2005, $\S$$\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$$k$ is odd, are presented in [Haefliger1966, 8.15] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, $\S$$\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$$SG_q$ be the space of maps $S^{q-1} \to S^{q-1}$$S^{q-1} \to S^{q-1}$ of degree $1$$1$. Restricting an element of $SO_q$$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$$SG_q$. Let $SG:=SG_1\cup\ldots\cup SG_q\cup\ldots$$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 References

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• [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
• [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
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• [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

, $\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). 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'' $E^m_D(S^q)\to\pi_q(G_{m-q},SO_{m-q})$ 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} (or 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 $SG_q$ be 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}$$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). 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^m_D(S^q)\to\pi_q(G_{m-q},SO_{m-q})$$E^m_D(S^q)\to\pi_q(G_{m-q},SO_{m-q})$ see [Haefliger1966], [Skopenkov2005, $\S$$\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$$k$ is odd, are presented in [Haefliger1966, 8.15] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, $\S$$\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$$SG_q$ be the space of maps $S^{q-1} \to S^{q-1}$$S^{q-1} \to S^{q-1}$ of degree $1$$1$. Restricting an element of $SO_q$$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$$SG_q$. Let $SG:=SG_1\cup\ldots\cup SG_q\cup\ldots$$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 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
• [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.
• [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

$. Restricting an element of $SO_q$ to $S^{q-1} \subset \Rr^q$ identifies $SO_q$ as a subspace of $SG_q$. Let $SG:=SG_1\cup\ldots\cup SG_q\cup\ldots$. Analogously define $SO$. Let $s$ be the stabilization homomorphism. The attaching invariant $a$ and the map $u$ are defined in \cite{Haefliger1966}, see also \cite[$\S]{Skopenkov2005}. {{endthm}} == 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 [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|Remark 1.2]] of \cite{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 \cite{Farber1981}, \cite{Farber1983}, \cite{Kearton1983}, \cite{Farber1984}. On the other hand, if one studies embeddings up to the weaker relation of [[Isotopy|''concordance'']], then much is known. See e.g. \cite{Levine1969a} and \cite{Ranicki1998}. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S1, $\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). 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^m_D(S^q)\to\pi_q(G_{m-q},SO_{m-q})$$E^m_D(S^q)\to\pi_q(G_{m-q},SO_{m-q})$ see [Haefliger1966], [Skopenkov2005, $\S$$\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$$k$ is odd, are presented in [Haefliger1966, 8.15] (or are deduced from that paper using simple calculations, cf. [Skopenkov2005, $\S$$\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$$SG_q$ be the space of maps $S^{q-1} \to S^{q-1}$$S^{q-1} \to S^{q-1}$ of degree $1$$1$. Restricting an element of $SO_q$$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$$SG_q$. Let $SG:=SG_1\cup\ldots\cup SG_q\cup\ldots$$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].

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