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

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

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

Lemma 6.3 [Whitney lemma; [Prasolov2007], \S22]. Let u: P \rightarrow W be a map from a connected oriented p-manifold P to a simply connected oriented (p+q)-manifold W. If p, q \geq 3, then

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

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

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

Proof. As V is (2k-1)-connected, h: \pi_{2k}(V) \rightarrow H_{2k}(V) is an isomorphism. For an element x_i \in H_{2k}(V), let \widetilde{x_i}: S^{2k} \rightarrow V be a representative of the homotopy class h^{-1}(x_i). Make the following inductive procedure. At the i-th step, i=1,\ldots,s, assume that the maps g_1, \ldots, g_{i-1} are already constructed, and we construct g_i.

Applying item 1 of Lemma 6.3 to \widetilde{x_i}, we may assume that \widetilde{x_i} is an embedding.

Since 2k \geq 3 and V is simply connected, W:=V \setminus \bigcup\limits_{l<j}g_l(S^{2k}) is simply connected for any j. The algebraic intersection number of g_j(S^{2k}) and \widetilde{x_i}(S^{2k}) is zero for any j. Hence we can apply item 2 of Lemma 6.3 to u=g_j and v=\widetilde{x_i} and W as above for any j < i. As the result, \widetilde{x_i} is replaced by a homotopic embedding g_i, and the images of g_1, \ldots, g_i are pairwise disjoint.

After the step s we obtain a required set of embeddings.

\square

Lemma 6.5. Let V be an orientable 4k-submanifold of B^{6k+1}, and g:D^{2k+1}\to B^{6k+1} be an embedding such that g(D^{2k+1})\cap V = g(S^{2k}) and over g(S^{2k}) the manifold V has a framing whose first vector is tangent to g(D^{2k+1}). Assume that g(S^{2k}) has zero algebraic self-intersection in V. Then g extends to an embedding G: D^{2k+1}\times D^{2k}\to B^{6k+1} such that G(S^{2k}\times D^{2k})\subset V.

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

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

Consider the exact sequence of the bundle SO_{a+b}/SO_b = V_{a+b,a}: \pi_q(SO_{a+b}) \overset{j}\to \pi_q(V_{a+b,a}) \overset{\partial}\to \pi_{q-1}(SO_b). By the following well-known assertion, \partial\zeta=0: if \alpha:S^q\to V_{a,b} is a map, then \partial[\alpha] is the obstruction to trivialization of the orthogonal complement to the field of b-frames in S^q\times\R^a corresponding to \alpha. By [Fomenko&Fuchs2016, Corollary in \S25.4] \pi_{2k}(SO_{4k}) is a finite group (in [Fomenko&Fuchs2016, Corollary in \S25.4] the formula for \pi_q(SO_{2m}) is correct, although the formula for \pi_q(SO_{2m+1}) is incorrect because \pi_3(SO_3)\cong\Z\oplus\Z). Since \pi_{2k}(V_{4k,2k})\cong\Z, we obtain that j=0 for a=b=2k. This and \partial\zeta=0 imply that \zeta=0.

\square

Lemma 6.6. Let V be a 4k-submanifold of B^{6k+1} and let G:D^{2k+1}\times D^{2k}\to B^{6k+1} be an embedding such that G(S^{2k}\times D^{2k})\subset V. Then there is a smooth submanifold V'\subset B^{6k+1} homeomorphic to V\backslash ( G(S^{2k}\times D^{2k})) \bigcup G(D^{2k+1}\times S^{2k-1}) and such that V\backslash G(D^{2k+1}\times D^{2k})=V'\backslash G(D^{2k+1}\times D^{2k}).

Lemma 6.6 is essentialy proved in [Haefliger1962, \S3.3].

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 [Haefliger1962], there is a basis \alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_s in H_{2k}(V) such that \alpha_i\cap \alpha_j=\beta_i\cap \beta_j = 0, \alpha_i\cap \beta_j=\delta_{i, j} and \lambda^*(\alpha_i)=0 for any i, j. From Lemma 6.4 it follows that there are embeddings f_1, \ldots, f_s:S^{2k}\to V with pairwise disjoint images such that f_{i*}[S^{2k}]=\alpha_i for every i=1,\ldots,s.

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

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

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

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.

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