Knots, i.e. embeddings of spheres

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Contents

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, \S1, \S3].

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]).

Example 2.1. (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, \S5]. For l even t is not smoothly isotopic to the standard embedding; t represents a generator of E_D^{3l}(S^{2l-1})\cong\Zz [Haefliger1962].

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

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

3 Invariants

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

By [Haefliger1962, 2.1, 2.2] any embedding f:S^{4k-1}\to S^{6k} has a framing extendable to a framed embedding \overline f:V\to D^{6k+1} of a 4k-manifold V whose boundary is S^{4k-1}, and whose signature is zero. For an integer 2k-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 [Pontryagin1959]). Then by Lefschetz duality there is a unique \lambda\in H_{2k}(V,\partial;\Z) such that \lambda^*[c]=\lambda\cap_V[c] for any [c]\in H_{2k}(V;\Z). Since V has a normal framing, its intersection form \cap_V is even. (Indeed, represent a class in H_{2k}(V;\Z) by a closed oriented 2k-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
\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).

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, \S4].

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, \S3].

4 Classification

Theorem 4.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. 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.

Theorem 4.2 (Haefliger-Milgram). 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}

Proof for the first four columns, and for the fifth column when k is odd, are presented in [Haefliger1966, 8.15] (see also \S6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, \S3]; 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].

Theorem 4.3 [Milgram1972, Corollary G]. 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].

Theorem 4.4 [Haefliger1966]. 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~.

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, \S3].

5 Some remarks on codimension 2 knots

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

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

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

Theorem 6.1. 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.

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

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

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

We formulate below a suitable version of the Whitney Lemma. Note that all the manifolds below can have non-empty boundaries.

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

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


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

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

\square

Lemma 6.5.[cf. Proposition 3.3 in [Ha62]] Let V be an orientable 4k-submanifold of B^{6k+1}, and 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}). Assume that either g_i(S^{2k}) has zero algebraic self-intersection in V for every i, or g_i(S^{2k}) has a framing in V for every i, or V has a framing in B^{6k+1}. Then there are embeddings G_1,\ldots,G_s:D^{2k+1}\times D^{2k}\to B^{6k+1} extending g_1,\ldots,g_s such that G_i(S^{2k}\times D^{2k})\subset V for every i.

Moreover, one can smoothen V':=V\backslash ???(\bigcup G_i(S^{2k}\times D^{2k-1})) \bigcup G_i(D^{2k+1}\times S^{2k-2}) to obtain a smooth submanifold of B^{6k+1}.

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

Denote by \zeta\in\pi_{2k}(V_{4k,2k}) the obstruction to extending a normal framing of g_i(S^{2k}) in V to a normal 2k-framing of g_i in B^{6k+1}. Consider the exact sequence of the bundle SO_{2k}\to SO_{4k}\to V_{4k,2k}. Since g_i(S^{2k}) has a framing in V, we have \partial\zeta_i=0\in\pi_{2k-1}(SO_{2k}), cf. [Haefliger1962, 3.4, Lemma]. For k odd from the exact sequence of the bundle we obtain \zeta_i=0.

Identify all fibers of the normal bundle of G(D^{2k+1}) with the fiber over G(0) via some trivialization T. As the normal bundle to g(S^{2k}) is trivial, there is a framing of g(S^{2k}) in V. This framing is orthogonal to G(D^{2k+1}). The image of this framing under T defines an element \zeta \in \pi_{2k}(V_{4k, 2k}).

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

\xymatrix{\pi_{2k}(SO_{2k})\ar[r]^{i} \ar[d]^{\equiv} & \pi_{2k}(SO_{2k+1}) \ar[r]^{p = 0} \ar[d]^{s} & \pi_{2k}(S^{2k}) \ar[r]^{\partial}\ar[d] & \pi_{2k-1}(SO_{2k})\ar[d]^{\equiv} \\ \pi_{2k}(SO_{2k})\ar[r]^{i'} & \pi_{2k}(SO_{6k+1})\ar[r]^{p'=0} & \pi_{2k}(V_{4k, 2k})\ar[r]^{\partial '} & \pi_{2k-1}(SO_{2k})}

From Milnor [??, proof of 5.11] it follows that \partial is injective, therefore i is surjective. As s is surjective, from commutativity of the diagram i' is also surjective. Therefore, \partial ' is injective. By Lemma 3.4 from [Ha62], \partial' \zeta is the obstruction to trivializing the normal bundle of g(S^{2k}), which is trivial. Therefore, \partial ' \zeta =0 and \zeta = 0 as \partial ' is injective. Thus, the obstruction is always zero.

\square

Proof. Let us smoothen V'.

Fix some i. From \zeta_i=0 it follows that there are orthonormal vector fields v_1, \ldots, v_{2k} on D{'}^{2k} such that \phi_i^{-1}(V) is tangent to v_j along \phi_i^{-1}(g_i(S^{2k})) for any j\leq 2k. Since every fiber of D{'}^{2k} is trivial, there are orthonormal vector fields v_{2k+1}, \ldots, v_{4k} on D{'}^{2k} such that v_j\perp v_m for any j\leq 2k and 2k<m\leq 4k. Let us choose an extensions of v_1,\ldots, v_{4k} to smole neighborhood of D' such that

  1. \phi_i^{-1}(V) is tangent to vector fields v_1, \ldots, v_{2k};
  2. the map \psi: D{'}^{2k}\times \mathbb{R}^{4k}\to\mathbb{R}^{6k+1}, defined by formula \psi(x, y)=\gamma_y(||y||), where x\in D{'}^{2k}, y=(y_1, \ldots, y_{4k})\in \mathbb{R}^{4k} and where \gamma_y:\mathbb{R}\to\mathbb{R}^{6k+1} is a solution of Cauchy problem

\begin{cases}  \gamma_y'(t)=\frac{y_1}{||y||}v_1(t)+\ldots +\frac{y_{4k}}{||y||}v_{4k}(t) \\  \gamma_y(0)=x,  \end{cases} is the diffeomorphism between D{'}^{2k}\times \mathbb{R}^{4k} and image of \psi.

  1. \psi(D{'}^{2k}\times \mathbb{R}^{4k})\cap \phi^{-1}(V)\cong S^{2k}\times \mbox{Int} D^{2k}.

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

\square

Below the symbol [ \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\} in H_{2k}(V) such that \alpha_i\cap \alpha_j=\beta_i\cap \beta_j = 0, \alpha_i\cap \beta_j=\delta_{i, j} and \lambda^*(\alpha_i)=0 for any i, j. From Lemma 6.4 it follows that there are embeddings f_1, \ldots, f_s:S^{2k}\to V with pairwise disjoint images such that f_{1*}[S^{2k}]=\alpha_1, \ldots, f_{s*}[S^{2k}]=\alpha_s.

Denote by f_{i}':S^{2k}\to D^{6k+1}\backslash V for i\leq s the result of shifting of f^i by the first vector of the framing of V. Since \lambda(f_{i*}[S^{2k}])=\lambda(\alpha_i)=0, we have f_{i*}'([S^{2k}])=0 \in H_{2k}(D^{6k+1}\backslash V). Since \mbox{dim} D^{6k+1}-\mbox{dim} V=6k+1-4k=2k+1, we have \pi_m(D^{6k+1}\backslash \mbox{Int}{V}) = 0 for m<2k. Since 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). Therefore there are extensions of f_1, \ldots, f_s to maps!!! 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}).

Denote by V' the manifold as in Lemma 6.5 for V and g_1, \ldots, g_s as described above UNAPPLICABLE. Since \left\{\alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s\right\} is a basis in H_{2k}(V), it follows that H_{2k}(V')\simeq 0. Hence \pi_1(V')\simeq 0 and H_m(V')\simeq 0 for 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}. Hence V'\cong D^{4k}. Then we set by G a diffeomorphism between D^{4k} and V'\subset D^{6k+1} such that G|_{S^{4k-1}}=g. \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).

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