Knots, i.e. embeddings of spheres
(→Proof of classification of (4k-1)-knots in 6k-space) |
(→Proof of classification of (4k-1)-knots in 6k-space) |
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{{beginthm|Lemma|[Whitney lemma]}}\label{whitney} | {{beginthm|Lemma|[Whitney lemma]}}\label{whitney} | ||
− | Let $ | + | Let $u: P \rightarrow W$ be a map from a connected oriented $p$-manifold $P$ to a simply connected oriented $(p+q)$-manifold $W$. If $p, q \geq 3$, then |
− | # If $q \geq p$, there is a homotopy $ | + | # If $q \geq p$, there is a homotopy $u_t$ such that $u_0 = u$ and $u_1(P)$ is an embedding. |
− | # Suppose in addition that $ | + | # 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. |
{{endthm}} | {{endthm}} | ||
Line 134: | Line 134: | ||
{{beginproof}} 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 $\widetilde{x_i}: S^{2k} \rightarrow V$ represent the homotopy class $h^{-1}(x_i)$. | {{beginproof}} 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 $\widetilde{x_i}: S^{2k} \rightarrow V$ represent the homotopy class $h^{-1}(x_i)$. | ||
Now we perform the following inductive procedure. | Now we perform the following inductive procedure. | ||
− | At the $i$-th step of the procedure assume that the maps $ | + | At the $i$-th step of the procedure assume that the maps $g_1, \ldots, g_{i-1}$ are already constructed, and we construct $g_i$. |
First, applying item 1 of Lemma \ref{whitney} to $\widetilde{x_i}$, | First, applying item 1 of Lemma \ref{whitney} to $\widetilde{x_i}$, | ||
we may suppose that $\widetilde{x_i}$ itself is an embedding. As $2k \geq 3$ and $V$ is simply connected, $V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$ 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 $j$. | we may suppose that $\widetilde{x_i}$ itself is an embedding. As $2k \geq 3$ and $V$ is simply connected, $V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$ 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 $j$. | ||
− | Further, we apply item 2 of Lemma \ref{whitney} to $ | + | Further, we apply item 2 of Lemma \ref{whitney} to $u$ equal to $g_j$, $v$ equal to $\widetilde{x_i}$, and $W$ equal to $V \setminus \bigcup\limits_{l<j}g_l(S^{2k})$, for any $j < i$. |
<!-- Observe that Lemma \ref{whitney} 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. | <!-- Observe that Lemma \ref{whitney} 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 $i$, $j$, --> | Also, as $x_i \cap_V x_j = 0$ for any $i$, $j$, --> | ||
− | As the result, $\widetilde{x_i}$ is replaced with a homotopic embedding $g_i$, and the images of $g_1, \ldots, g_i$ are disjoint. | + | As the result, $\widetilde{x_i}$ is replaced with a homotopic embedding $g_i$, and the images of $g_1, \ldots, g_i$ are pairwise disjoint. |
After the step $m$ we obtain the required set of embeddings. | After the step $m$ we obtain the required set of embeddings. | ||
{{endproof}} | {{endproof}} | ||
Line 150: | Line 150: | ||
{{endthm}} | {{endthm}} | ||
− | {{beginproof}} | + | {{beginproof}} |
− | + | 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 $ | + | Hence $g(S^{2k})$ has a framing in $V$. |
− | Hence $ | + | |
<!-- and $f_1$ is tangent to $G(D^{2k+1})$.--> | <!-- and $f_1$ is tangent to $G(D^{2k+1})$.--> | ||
− | Identify all the normal spaces | + | Identify all the normal spaces of $G(D^{2k+1})$ with the normal space at $G(0)$. |
− | The normal framing $a$ of $ | + | The normal framing $a$ of $g(S^{2k})$ in $V$ is orthogonal to $G(D^{2k+1})$. |
So $a$ defines a map $S^{2k}\to V_{4k,2k}$. | 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. | 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 $ | + | 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 $\zeta=0$. | It suffices to prove that $\zeta=0$. | ||
Line 170: | Line 169: | ||
Since $\pi_{2k}(V_{4k,2k})\cong\Z$, we obtain that $j=0$ for $a=b=2k$. | Since $\pi_{2k}(V_{4k,2k})\cong\Z$, we obtain that $j=0$ for $a=b=2k$. | ||
This and $\partial\zeta=0$ imply that $\zeta=0$. | This and $\partial\zeta=0$ imply that $\zeta=0$. | ||
− | |||
<!-- (In \cite[the last but one paragraph of 3.5]{Haefliger1962} perhaps one has to replace `$\partial\zeta$ is the obstruction to trivializing the normal bundle of $g(S^{2k})$ in $V$' by `$\partial\zeta$ is the obstruction to trivializing te orthogonal complement to the $f_1$-direction of the normal bundle of $V$ restricted to $g(S^{2k})$'?) | <!-- (In \cite[the last but one paragraph of 3.5]{Haefliger1962} perhaps one has to replace `$\partial\zeta$ is the obstruction to trivializing the normal bundle of $g(S^{2k})$ in $V$' by `$\partial\zeta$ is the obstruction to trivializing te orthogonal complement to the $f_1$-direction of the normal bundle of $V$ restricted to $g(S^{2k})$'?) | ||
− | |||
(No, as we use different obstruction; our is orthogonal to Haefliger's) | (No, as we use different obstruction; our is orthogonal to Haefliger's) | ||
− | |||
Alternatively: | Alternatively: | ||
Consider a map of the long exact sequences associated to the inclusion $S^{2k} = SO_{2k+1}/SO_{2k} \rightarrow SO_{4k}/SO_{2k} = V_{4k, 2k}$: | Consider a map of the long exact sequences associated to the inclusion $S^{2k} = SO_{2k+1}/SO_{2k} \rightarrow SO_{4k}/SO_{2k} = V_{4k, 2k}$: | ||
− | |||
$\xymatrix{\pi_{2k}(SO_{2k+1}) \ar[r]^{p = 0} \ar[d]^{s} & \pi_{2k}(S^{2k}) \ar[d] \\ | $\xymatrix{\pi_{2k}(SO_{2k+1}) \ar[r]^{p = 0} \ar[d]^{s} & \pi_{2k}(S^{2k}) \ar[d] \\ | ||
\pi_{2k}(SO_{4k})\ar[r]^{p'=0} & \pi_{2k}(V_{4k, 2k})\ar[r]^{\partial '} & \pi_{2k-1}(SO_{2k})}$ | \pi_{2k}(SO_{4k})\ar[r]^{p'=0} & \pi_{2k}(V_{4k, 2k})\ar[r]^{\partial '} & \pi_{2k-1}(SO_{2k})}$ | ||
− | |||
Since $s$ is surjective, we obtain that $p'=0$. | Since $s$ is surjective, we obtain that $p'=0$. | ||
− | |||
Hence $\partial'$ is injective. | Hence $\partial'$ is injective. | ||
The class $\partial'\zeta$ is??? the obstruction to trivializing the orthogonal complement of $g(S^{2k})$, | The class $\partial'\zeta$ is??? the obstruction to trivializing the orthogonal complement of $g(S^{2k})$, | ||
cf. \cite[3.4, Lemma]{Haefliger1962}. | cf. \cite[3.4, Lemma]{Haefliger1962}. | ||
− | |||
− | |||
Suggestion: interpreting $\partial'\zeta$ as an obstruction, we derive that $\partial'\zeta$ is zero, as $V$ is framed and the first vector of the framing is tangent to $G_1$. --> | Suggestion: interpreting $\partial'\zeta$ as an obstruction, we derive that $\partial'\zeta$ is zero, as $V$ is framed and the first vector of the framing is tangent to $G_1$. --> | ||
− | + | <!-- Therefore, $\partial ' \zeta =0$. Since $\partial '$ is injective, we obtain $\zeta = 0$. -->{{endproof}} | |
− | <!-- Therefore, $\partial ' \zeta =0$. Since $\partial '$ is injective, we obtain $\zeta = 0$. --> | + | |
− | {{endproof}} | + | |
− | + | ||
{{beginthm|Lemma}} \label{l:smoothen} | {{beginthm|Lemma}} \label{l:smoothen} | ||
Denote by $G$ and $g$ the embeddings from Lemma \ref{l:multi_spherical_modification}. | Denote by $G$ and $g$ the embeddings from Lemma \ref{l:multi_spherical_modification}. |
Revision as of 16:38, 18 February 2024
This page has not been refereed. The information given here might be incomplete or provisional. |
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, 1, 3].
2 Examples
There are smooth embeddings 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 one constructs a smooth embedding , see [Skopenkov2016h, 5]. For even is not smoothly isotopic to the standard embedding; represents a generator of [Haefliger1962].
It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
(b) For any let be the homotopy class of the Hopf map. Denote by the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum of the components of (a representative of) is not smoothly isotopic to the standard embedding; is a generator of [Skopenkov2015a, Corollary 2.13].
3 Invariants
Let us define the Haefliger invariant . The definition is motivated by Haefliger's proof that any embedding is isotopic to the standard embedding for , and by analyzing what obstructs carrying this proof for .
By [Haefliger1962, 2.1, 2.2] any embedding has a framing extendable to a framed embedding of a -manifold whose boundary is , and whose signature is zero. For an integer -cycle in let be the linking number of with a slight shift of along the first vector of the framing. This defines a map . 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 such that for any . Since has a normal framing, its intersection form is even. (Indeed, represent a class in by a closed oriented -submanifold . Then because has a normal framing.) Hence is an even integer. DefineSince the signature of is zero, there is a symplectic basis in . Then clearly
For an alternative definition via Seifert surfaces in -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, 4].
Sketch of a proof that is well-defined (i.e. is independent of , , and the framings), and is invariant under isotopy of . [Haefliger1962, Theorem 2.6] Analogously one defines and for a framed -submanifold of . Since is a characteristic number, it is independent of framed cobordism. So defines a homomorphism . The latter group is finite by the Serre theorem. Hence the homomorphism is trivial.
Since is a characteristic number, it is independent of framed cobordism of a framed (and hence of the isotopy of a framed ).
Therefore is a well-defined invariant of a framed cobordism class of a framed . By [Haefliger1962, 2.9] (cf. [Haefliger1962, 2.2 and 2.3]) is also independent of the framing of extendable to a framing of some -manifold having trivial signature. QED
For definition of the attaching invariant see [Haefliger1966], [Skopenkov2005, 3].
4 Classification
Theorem 4.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 4.2 (Haefliger-Milgram). We have the following table for the group ; in the whole table ; in the fifth column ; in the last two columns :
Proof for the first four columns, and for the fifth column when is odd, are presented in [Haefliger1966, 8.15] (see also 6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]; there is a typo in [Haefliger1966, 8.15]: should be ). The remaining results follow from [Haefliger1966, 8.15] and [Milgram1972, Theorem F]. Alternative proofs for the cases are given in [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
Theorem 4.3 [Milgram1972, Corollary G]. We have if and only if either , or , or and , or and .
For a description of 2-components of 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 are and . Hopefully application of Kreck surgery could be useful to find these groups, cf. [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 4.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here is the space of maps of degree . Restricting a map from to identifies as a subspace of . Define . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 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 ), 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 is injective for .
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 be a framed -connected -submanifold of such that , signature of is zero, and . Then there is an embedding such that .
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 such that there is a framed -connected -submanifold of with zero signature such that and . Then by Lemma 6.2 there is an extension such that . From Smale Theorem it follows that is isotopic to the standard embedding.
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 be a map from a connected oriented -manifold to a simply connected oriented -manifold . If , then
- If , there is a homotopy such that and is an embedding.
- Suppose in addition that and there is a map with from a connected oriented -manifold such that the algebraic intersection number of and is zero. Then there is a homotopy relative to the boundary such that and does not intersect . If is an embedding, the homotopy can be chosen so that is an embedding.
Lemma 6.4. Let be a -connected -manifold, and are such that for every . Then there are embeddings with pairwise disjoint images representing , respectively.
Proof. As is -connected, the Hurewicz map is an isomorphism. For an element , let represent the homotopy class . Now we perform the following inductive procedure. At the -th step of the procedure assume that the maps are already constructed, and we construct . First, applying item 1 of Lemma 6.3 to , we may suppose that itself is an embedding. As and is simply connected, is simply connected for any . The algebraic intersection number of and is zero for any . Further, we apply item 2 of Lemma 6.3 to equal to , equal to , and equal to , for any . As the result, is replaced with a homotopic embedding , and the images of are pairwise disjoint. After the step we obtain the required set of embeddings.
Lemma 6.5.[cf. Proposition 3.3 in [Haefliger1962]] Let be an orientable -submanifold of , and be an embedding such that , orthogonal to , and over the manifold has a framing whose first vector is tangent to . Assume that has zero algebraic self-intersection in . Then there is an embedding extending such that .
Proof. Since has zero algebraic self-intersection in , the Euler class of the normal bundle of in is zero. Hence has a framing in .
Identify all the normal spaces of with the normal space at . The normal framing of in is orthogonal to . So defines a map . Let be the homotopy class of this map. This is the obstruction to extending to a normal -framing of in (so apriori ). It suffices to prove that .
Consider the exact sequence of the bundle : . By the following well-known assertion, : if is a map, then is the obstruction to trivialization of the orthogonal complement to the field of -frames in corresponding to . By [Fomenko&Fuchs2016, Corollary in 25.4] is a finite group (in [Fomenko&Fuchs2016, Corollary in 25.4] the formula for is correct, although the formula for is incorrect because ). Since , we obtain that for . This and imply that .
Lemma 6.6. Denote by and the embeddings from Lemma 6.5. There is smooth manifold such that is homeomorphic to and . Additionaly,
- ;
- if then for ;
- can be chosen such that .
Below the symbol 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 in such that , and for any . From Lemma 6.4 it follows that there are embeddings with pairwise disjoint images such that .
Denote by for the result of shifting of by the first vector of the framing of . Since , we have . Since , we have for . Since , we have . Therefore there are extensions of to maps such that .
Take such that for any and take neighborhoods of such that for any . Since , we have that algebraic intersection number of and is zero. Since and we have . Applying item 2 of Lemma 6.3 to as , embedding of into as and as we may suppose that does not intersect . Hence we may suppose that does not intersect .
Apply Lemma 6.5 one by one to maps as for and to the manifolds . Denote by the resulting mappings. Define inductively manifolds for such that and is a manifold as in statement of Lemma 6.6 for manifold as and map as . By items 1, 2 of Lemma 6.6, it follows that and for . Since is a symplectic basis in , it follows that . Then from Generalized Poincare conjecture proved by Smale it follows that . Hence . Then take by the composition of a diffeomorphism such that and inclusion .
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
- [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -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 in for , 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
2 Examples
There are smooth embeddings 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 one constructs a smooth embedding , see [Skopenkov2016h, 5]. For even is not smoothly isotopic to the standard embedding; represents a generator of [Haefliger1962].
It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
(b) For any let be the homotopy class of the Hopf map. Denote by the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum of the components of (a representative of) is not smoothly isotopic to the standard embedding; is a generator of [Skopenkov2015a, Corollary 2.13].
3 Invariants
Let us define the Haefliger invariant . The definition is motivated by Haefliger's proof that any embedding is isotopic to the standard embedding for , and by analyzing what obstructs carrying this proof for .
By [Haefliger1962, 2.1, 2.2] any embedding has a framing extendable to a framed embedding of a -manifold whose boundary is , and whose signature is zero. For an integer -cycle in let be the linking number of with a slight shift of along the first vector of the framing. This defines a map . 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 such that for any . Since has a normal framing, its intersection form is even. (Indeed, represent a class in by a closed oriented -submanifold . Then because has a normal framing.) Hence is an even integer. DefineSince the signature of is zero, there is a symplectic basis in . Then clearly
For an alternative definition via Seifert surfaces in -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, 4].
Sketch of a proof that is well-defined (i.e. is independent of , , and the framings), and is invariant under isotopy of . [Haefliger1962, Theorem 2.6] Analogously one defines and for a framed -submanifold of . Since is a characteristic number, it is independent of framed cobordism. So defines a homomorphism . The latter group is finite by the Serre theorem. Hence the homomorphism is trivial.
Since is a characteristic number, it is independent of framed cobordism of a framed (and hence of the isotopy of a framed ).
Therefore is a well-defined invariant of a framed cobordism class of a framed . By [Haefliger1962, 2.9] (cf. [Haefliger1962, 2.2 and 2.3]) is also independent of the framing of extendable to a framing of some -manifold having trivial signature. QED
For definition of the attaching invariant see [Haefliger1966], [Skopenkov2005, 3].
4 Classification
Theorem 4.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 4.2 (Haefliger-Milgram). We have the following table for the group ; in the whole table ; in the fifth column ; in the last two columns :
Proof for the first four columns, and for the fifth column when is odd, are presented in [Haefliger1966, 8.15] (see also 6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]; there is a typo in [Haefliger1966, 8.15]: should be ). The remaining results follow from [Haefliger1966, 8.15] and [Milgram1972, Theorem F]. Alternative proofs for the cases are given in [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
Theorem 4.3 [Milgram1972, Corollary G]. We have if and only if either , or , or and , or and .
For a description of 2-components of 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 are and . Hopefully application of Kreck surgery could be useful to find these groups, cf. [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 4.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here is the space of maps of degree . Restricting a map from to identifies as a subspace of . Define . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 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 ), 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 is injective for .
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 be a framed -connected -submanifold of such that , signature of is zero, and . Then there is an embedding such that .
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 such that there is a framed -connected -submanifold of with zero signature such that and . Then by Lemma 6.2 there is an extension such that . From Smale Theorem it follows that is isotopic to the standard embedding.
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 be a map from a connected oriented -manifold to a simply connected oriented -manifold . If , then
- If , there is a homotopy such that and is an embedding.
- Suppose in addition that and there is a map with from a connected oriented -manifold such that the algebraic intersection number of and is zero. Then there is a homotopy relative to the boundary such that and does not intersect . If is an embedding, the homotopy can be chosen so that is an embedding.
Lemma 6.4. Let be a -connected -manifold, and are such that for every . Then there are embeddings with pairwise disjoint images representing , respectively.
Proof. As is -connected, the Hurewicz map is an isomorphism. For an element , let represent the homotopy class . Now we perform the following inductive procedure. At the -th step of the procedure assume that the maps are already constructed, and we construct . First, applying item 1 of Lemma 6.3 to , we may suppose that itself is an embedding. As and is simply connected, is simply connected for any . The algebraic intersection number of and is zero for any . Further, we apply item 2 of Lemma 6.3 to equal to , equal to , and equal to , for any . As the result, is replaced with a homotopic embedding , and the images of are pairwise disjoint. After the step we obtain the required set of embeddings.
Lemma 6.5.[cf. Proposition 3.3 in [Haefliger1962]] Let be an orientable -submanifold of , and be an embedding such that , orthogonal to , and over the manifold has a framing whose first vector is tangent to . Assume that has zero algebraic self-intersection in . Then there is an embedding extending such that .
Proof. Since has zero algebraic self-intersection in , the Euler class of the normal bundle of in is zero. Hence has a framing in .
Identify all the normal spaces of with the normal space at . The normal framing of in is orthogonal to . So defines a map . Let be the homotopy class of this map. This is the obstruction to extending to a normal -framing of in (so apriori ). It suffices to prove that .
Consider the exact sequence of the bundle : . By the following well-known assertion, : if is a map, then is the obstruction to trivialization of the orthogonal complement to the field of -frames in corresponding to . By [Fomenko&Fuchs2016, Corollary in 25.4] is a finite group (in [Fomenko&Fuchs2016, Corollary in 25.4] the formula for is correct, although the formula for is incorrect because ). Since , we obtain that for . This and imply that .
Lemma 6.6. Denote by and the embeddings from Lemma 6.5. There is smooth manifold such that is homeomorphic to and . Additionaly,
- ;
- if then for ;
- can be chosen such that .
Below the symbol 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 in such that , and for any . From Lemma 6.4 it follows that there are embeddings with pairwise disjoint images such that .
Denote by for the result of shifting of by the first vector of the framing of . Since , we have . Since , we have for . Since , we have . Therefore there are extensions of to maps such that .
Take such that for any and take neighborhoods of such that for any . Since , we have that algebraic intersection number of and is zero. Since and we have . Applying item 2 of Lemma 6.3 to as , embedding of into as and as we may suppose that does not intersect . Hence we may suppose that does not intersect .
Apply Lemma 6.5 one by one to maps as for and to the manifolds . Denote by the resulting mappings. Define inductively manifolds for such that and is a manifold as in statement of Lemma 6.6 for manifold as and map as . By items 1, 2 of Lemma 6.6, it follows that and for . Since is a symplectic basis in , it follows that . Then from Generalized Poincare conjecture proved by Smale it follows that . Hence . Then take by the composition of a diffeomorphism such that and inclusion .
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
- [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -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 in for , 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
2 Examples
There are smooth embeddings 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 one constructs a smooth embedding , see [Skopenkov2016h, 5]. For even is not smoothly isotopic to the standard embedding; represents a generator of [Haefliger1962].
It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
(b) For any let be the homotopy class of the Hopf map. Denote by the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum of the components of (a representative of) is not smoothly isotopic to the standard embedding; is a generator of [Skopenkov2015a, Corollary 2.13].
3 Invariants
Let us define the Haefliger invariant . The definition is motivated by Haefliger's proof that any embedding is isotopic to the standard embedding for , and by analyzing what obstructs carrying this proof for .
By [Haefliger1962, 2.1, 2.2] any embedding has a framing extendable to a framed embedding of a -manifold whose boundary is , and whose signature is zero. For an integer -cycle in let be the linking number of with a slight shift of along the first vector of the framing. This defines a map . 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 such that for any . Since has a normal framing, its intersection form is even. (Indeed, represent a class in by a closed oriented -submanifold . Then because has a normal framing.) Hence is an even integer. DefineSince the signature of is zero, there is a symplectic basis in . Then clearly
For an alternative definition via Seifert surfaces in -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, 4].
Sketch of a proof that is well-defined (i.e. is independent of , , and the framings), and is invariant under isotopy of . [Haefliger1962, Theorem 2.6] Analogously one defines and for a framed -submanifold of . Since is a characteristic number, it is independent of framed cobordism. So defines a homomorphism . The latter group is finite by the Serre theorem. Hence the homomorphism is trivial.
Since is a characteristic number, it is independent of framed cobordism of a framed (and hence of the isotopy of a framed ).
Therefore is a well-defined invariant of a framed cobordism class of a framed . By [Haefliger1962, 2.9] (cf. [Haefliger1962, 2.2 and 2.3]) is also independent of the framing of extendable to a framing of some -manifold having trivial signature. QED
For definition of the attaching invariant see [Haefliger1966], [Skopenkov2005, 3].
4 Classification
Theorem 4.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 4.2 (Haefliger-Milgram). We have the following table for the group ; in the whole table ; in the fifth column ; in the last two columns :
Proof for the first four columns, and for the fifth column when is odd, are presented in [Haefliger1966, 8.15] (see also 6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]; there is a typo in [Haefliger1966, 8.15]: should be ). The remaining results follow from [Haefliger1966, 8.15] and [Milgram1972, Theorem F]. Alternative proofs for the cases are given in [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
Theorem 4.3 [Milgram1972, Corollary G]. We have if and only if either , or , or and , or and .
For a description of 2-components of 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 are and . Hopefully application of Kreck surgery could be useful to find these groups, cf. [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 4.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here is the space of maps of degree . Restricting a map from to identifies as a subspace of . Define . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 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 ), 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 is injective for .
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 be a framed -connected -submanifold of such that , signature of is zero, and . Then there is an embedding such that .
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 such that there is a framed -connected -submanifold of with zero signature such that and . Then by Lemma 6.2 there is an extension such that . From Smale Theorem it follows that is isotopic to the standard embedding.
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 be a map from a connected oriented -manifold to a simply connected oriented -manifold . If , then
- If , there is a homotopy such that and is an embedding.
- Suppose in addition that and there is a map with from a connected oriented -manifold such that the algebraic intersection number of and is zero. Then there is a homotopy relative to the boundary such that and does not intersect . If is an embedding, the homotopy can be chosen so that is an embedding.
Lemma 6.4. Let be a -connected -manifold, and are such that for every . Then there are embeddings with pairwise disjoint images representing , respectively.
Proof. As is -connected, the Hurewicz map is an isomorphism. For an element , let represent the homotopy class . Now we perform the following inductive procedure. At the -th step of the procedure assume that the maps are already constructed, and we construct . First, applying item 1 of Lemma 6.3 to , we may suppose that itself is an embedding. As and is simply connected, is simply connected for any . The algebraic intersection number of and is zero for any . Further, we apply item 2 of Lemma 6.3 to equal to , equal to , and equal to , for any . As the result, is replaced with a homotopic embedding , and the images of are pairwise disjoint. After the step we obtain the required set of embeddings.
Lemma 6.5.[cf. Proposition 3.3 in [Haefliger1962]] Let be an orientable -submanifold of , and be an embedding such that , orthogonal to , and over the manifold has a framing whose first vector is tangent to . Assume that has zero algebraic self-intersection in . Then there is an embedding extending such that .
Proof. Since has zero algebraic self-intersection in , the Euler class of the normal bundle of in is zero. Hence has a framing in .
Identify all the normal spaces of with the normal space at . The normal framing of in is orthogonal to . So defines a map . Let be the homotopy class of this map. This is the obstruction to extending to a normal -framing of in (so apriori ). It suffices to prove that .
Consider the exact sequence of the bundle : . By the following well-known assertion, : if is a map, then is the obstruction to trivialization of the orthogonal complement to the field of -frames in corresponding to . By [Fomenko&Fuchs2016, Corollary in 25.4] is a finite group (in [Fomenko&Fuchs2016, Corollary in 25.4] the formula for is correct, although the formula for is incorrect because ). Since , we obtain that for . This and imply that .
Lemma 6.6. Denote by and the embeddings from Lemma 6.5. There is smooth manifold such that is homeomorphic to and . Additionaly,
- ;
- if then for ;
- can be chosen such that .
Below the symbol 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 in such that , and for any . From Lemma 6.4 it follows that there are embeddings with pairwise disjoint images such that .
Denote by for the result of shifting of by the first vector of the framing of . Since , we have . Since , we have for . Since , we have . Therefore there are extensions of to maps such that .
Take such that for any and take neighborhoods of such that for any . Since , we have that algebraic intersection number of and is zero. Since and we have . Applying item 2 of Lemma 6.3 to as , embedding of into as and as we may suppose that does not intersect . Hence we may suppose that does not intersect .
Apply Lemma 6.5 one by one to maps as for and to the manifolds . Denote by the resulting mappings. Define inductively manifolds for such that and is a manifold as in statement of Lemma 6.6 for manifold as and map as . By items 1, 2 of Lemma 6.6, it follows that and for . Since is a symplectic basis in , it follows that . Then from Generalized Poincare conjecture proved by Smale it follows that . Hence . Then take by the composition of a diffeomorphism such that and inclusion .
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
- [Habegger1986] N. Habegger, Knots and links in codimension greater than 2, Topology, 25:3 (1986) 253--260.
- [Haefliger1962] A. Haefliger, Knotted -spheres in -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 in for , 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
2 Examples
There are smooth embeddings 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 one constructs a smooth embedding , see [Skopenkov2016h, 5]. For even is not smoothly isotopic to the standard embedding; represents a generator of [Haefliger1962].
It would be interesting to know if for odd this embedding is a generator of . The last phrase of [Haefliger1962t] suggests that this is true for .
(b) For any let be the homotopy class of the Hopf map. Denote by the Zeeman map, see [Skopenkov2016h, Definition 2.2]. The embedded connected sum of the components of (a representative of) is not smoothly isotopic to the standard embedding; is a generator of [Skopenkov2015a, Corollary 2.13].
3 Invariants
Let us define the Haefliger invariant . The definition is motivated by Haefliger's proof that any embedding is isotopic to the standard embedding for , and by analyzing what obstructs carrying this proof for .
By [Haefliger1962, 2.1, 2.2] any embedding has a framing extendable to a framed embedding of a -manifold whose boundary is , and whose signature is zero. For an integer -cycle in let be the linking number of with a slight shift of along the first vector of the framing. This defines a map . 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 such that for any . Since has a normal framing, its intersection form is even. (Indeed, represent a class in by a closed oriented -submanifold . Then because has a normal framing.) Hence is an even integer. DefineSince the signature of is zero, there is a symplectic basis in . Then clearly
For an alternative definition via Seifert surfaces in -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, 4].
Sketch of a proof that is well-defined (i.e. is independent of , , and the framings), and is invariant under isotopy of . [Haefliger1962, Theorem 2.6] Analogously one defines and for a framed -submanifold of . Since is a characteristic number, it is independent of framed cobordism. So defines a homomorphism . The latter group is finite by the Serre theorem. Hence the homomorphism is trivial.
Since is a characteristic number, it is independent of framed cobordism of a framed (and hence of the isotopy of a framed ).
Therefore is a well-defined invariant of a framed cobordism class of a framed . By [Haefliger1962, 2.9] (cf. [Haefliger1962, 2.2 and 2.3]) is also independent of the framing of extendable to a framing of some -manifold having trivial signature. QED
For definition of the attaching invariant see [Haefliger1966], [Skopenkov2005, 3].
4 Classification
Theorem 4.1 [Levine1965, Corollary in p. 44], [Haefliger1966]. For the group is finite unless and , when is the sum of and a finite group.
Theorem 4.2 (Haefliger-Milgram). We have the following table for the group ; in the whole table ; in the fifth column ; in the last two columns :
Proof for the first four columns, and for the fifth column when is odd, are presented in [Haefliger1966, 8.15] (see also 6; some proofs are deduced from that paper using simple calculations, cf. [Skopenkov2005, 3]; there is a typo in [Haefliger1966, 8.15]: should be ). The remaining results follow from [Haefliger1966, 8.15] and [Milgram1972, Theorem F]. Alternative proofs for the cases are given in [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
Theorem 4.3 [Milgram1972, Corollary G]. We have if and only if either , or , or and , or and .
For a description of 2-components of 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 are and . Hopefully application of Kreck surgery could be useful to find these groups, cf. [Skopenkov2005], [Crowley&Skopenkov2008], [Skopenkov2008].
For the group has been described as follows, in terms of exact sequences [Haefliger1966], cf. [Levine1965], [Haefliger1966a], [Milgram1972], [Habegger1986].
Theorem 4.4 [Haefliger1966]. For there is the following exact sequence of abelian groups:
Here is the space of maps of degree . Restricting a map from to identifies as a subspace of . Define . Analogously define . Let be the stabilization homomorphism. The attaching invariant and the map are defined in [Haefliger1966], see also [Skopenkov2005, 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 ), 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 is injective for .
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 be a framed -connected -submanifold of such that , signature of is zero, and . Then there is an embedding such that .
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 such that there is a framed -connected -submanifold of with zero signature such that and . Then by Lemma 6.2 there is an extension such that . From Smale Theorem it follows that is isotopic to the standard embedding.
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 be a map from a connected oriented -manifold to a simply connected oriented -manifold . If , then
- If , there is a homotopy such that and is an embedding.
- Suppose in addition that and there is a map with from a connected oriented -manifold such that the algebraic intersection number of and is zero. Then there is a homotopy relative to the boundary such that and does not intersect . If is an embedding, the homotopy can be chosen so that is an embedding.
Lemma 6.4. Let be a -connected -manifold, and are such that for every . Then there are embeddings with pairwise disjoint images representing , respectively.
Proof. As is -connected, the Hurewicz map is an isomorphism. For an element , let represent the homotopy class . Now we perform the following inductive procedure. At the -th step of the procedure assume that the maps are already constructed, and we construct . First, applying item 1 of Lemma 6.3 to , we may suppose that itself is an embedding. As and is simply connected, is simply connected for any . The algebraic intersection number of and is zero for any . Further, we apply item 2 of Lemma 6.3 to equal to , equal to , and equal to , for any . As the result, is replaced with a homotopic embedding , and the images of are pairwise disjoint. After the step we obtain the required set of embeddings.
Lemma 6.5.[cf. Proposition 3.3 in [Haefliger1962]] Let be an orientable -submanifold of , and be an embedding such that , orthogonal to , and over the manifold has a framing whose first vector is tangent to . Assume that has zero algebraic self-intersection in . Then there is an embedding extending such that .
Proof. Since has zero algebraic self-intersection in , the Euler class of the normal bundle of in is zero. Hence has a framing in .
Identify all the normal spaces of with the normal space at . The normal framing of in is orthogonal to . So defines a map . Let be the homotopy class of this map. This is the obstruction to extending to a normal -framing of in (so apriori ). It suffices to prove that .
Consider the exact sequence of the bundle : . By the following well-known assertion, : if is a map, then is the obstruction to trivialization of the orthogonal complement to the field of -frames in corresponding to . By [Fomenko&Fuchs2016, Corollary in 25.4] is a finite group (in [Fomenko&Fuchs2016, Corollary in 25.4] the formula for is correct, although the formula for is incorrect because ). Since , we obtain that for . This and imply that .
Lemma 6.6. Denote by and the embeddings from Lemma 6.5. There is smooth manifold such that is homeomorphic to and . Additionaly,
- ;
- if then for ;
- can be chosen such that .
Below the symbol 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 in such that , and for any . From Lemma 6.4 it follows that there are embeddings with pairwise disjoint images such that .
Denote by for the result of shifting of by the first vector of the framing of . Since , we have . Since , we have for . Since , we have . Therefore there are extensions of to maps such that .
Take such that for any and take neighborhoods of such that for any . Since , we have that algebraic intersection number of and is zero. Since and we have . Applying item 2 of Lemma 6.3 to as , embedding of into as and as we may suppose that does not intersect . Hence we may suppose that does not intersect .
Apply Lemma 6.5 one by one to maps as for and to the manifolds . Denote by the resulting mappings. Define inductively manifolds for such that and is a manifold as in statement of Lemma 6.6 for manifold as and map as . By items 1, 2 of Lemma 6.6, it follows that and for . Since is a symplectic basis in , it follows that . Then from Generalized Poincare conjecture proved by Smale it follows that . Hence . Then take by the composition of a diffeomorphism such that and inclusion .
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