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, 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
.
![f:S^{4k-1}\to S^{6k}](/images/math/b/5/a/b5aaa31fc93cad500eeb3be985b335e4.png)
![\overline f:V\to D^{6k+1}](/images/math/9/f/5/9f501444fdb7f4d4ae1df68fcb525c1e.png)
![4k](/images/math/0/d/5/0d5777efe31988f31ab28b4fe0b65ac9.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![S^{4k-1}](/images/math/4/7/a/47a1015fcab977d6665ea137faa42d19.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda^*(c)\in\Z](/images/math/c/0/7/c07b4397d9e29bf012c6c84e5d8899f7.png)
![f(V)](/images/math/9/6/8/968250213d93a9a7bd801905302f3e71.png)
![\overline f(c)](/images/math/2/e/f/2ef1fcc94a63c02c6d8b3439db0778b6.png)
![\lambda^*:H_{2k}(V;\Z)\to\Z](/images/math/9/0/f/90fa622df53a3032c56ca8596ddde93a.png)
![\lambda\in H_{2k}(V,\partial;\Z)](/images/math/3/d/4/3d4ab3607763fca104c99b898a160a8f.png)
![\lambda^*[c]=\lambda\cap_V[c]](/images/math/b/2/8/b2878e69f979f9a0ca580faf4d104941.png)
![[c]\in H_{2k}(V;\Z)](/images/math/8/9/f/89f1bbfc9e8e5259c8e842991bb8018b.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\cap_V](/images/math/a/0/2/a02908ed6080691f0efe0a4168262421.png)
![H_{2k}(V;\Z)](/images/math/8/8/b/88b72e33307b3cefbf640d44bb973c7f.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![\rho_2[c]\cap_V[c]=\overline{w_{2k}}(c\subset V)=\rho_2[c]\cap_VPDw_{2k}(V)=0](/images/math/7/8/8/788a4e3686035885f1e93cfe29a761c1.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda\cap_V\lambda](/images/math/f/9/4/f949e7b5a359f80ce0ed5d166ae6957c.png)
![\displaystyle \varkappa(f):=\lambda\cap_V\lambda/2.](/images/math/3/d/c/3dc13f01f2d003319abab12c4e44c28d.png)
Since the signature of is zero, there is a symplectic basis
in
.
Then clearly
![\displaystyle \varkappa(f) = \sum\limits_{j=1}^s \lambda^*(\beta_j)\lambda^*(\alpha_j).](/images/math/5/d/e/5de79b31dfd0d4d96dc24a5bb861d954.png)
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
:
![\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}](/images/math/b/5/d/b5db6794a965eaffb93a0f335929215a.png)
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:
![\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~.](/images/math/8/3/b/83bb39e23e367d1f73a0d9f448fd6357.png)
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 a submanifold
such that
and
.
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 a submanifold
such that
and
.
Recall that isotopy classes of embeddings
are in 1--1 correspondence with
-cobordism classes of oriented submanifolds of
diffeomorphic to
for
, ???
, cf. [Haefliger1966, 1.8], [Kervaire1965].
Hence
is isotopic to standart embedding.
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 [???]].
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
.
Make the following inductive procedure.
At the
-th step!!!,
, assume that the maps
are already constructed, and we construct
.
Applying item 1 of Lemma 6.3 to , we may suppose that
is an embedding.
As and
is simply connected,
is simply connected for any
.
The algebraic intersection number of
and
is zero for any
.
Hence we can !!!apply item 2 of Lemma 6.3 to !!!
, \
, and
, for any
.
As the result,
is replaced by a homotopic embedding
, and the images of
are pairwise disjoint.
After the step we obtain a required set of embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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.
Since over
the manifold
has a framing, we obtain that
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
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 6.6.
Let be a
-submanifold of
and let
be an embedding such that
.
Then there is a smooth submanifold
homeomorphic to
and 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
for every
.
For denote by
the homotopy class of the shift of
by the first vector of the framing of
on
.
Since
, we have
.
Since
, we have
for
.
Hence by Hurewicz Theorem!!!
implies
.
Therefore there are extensions of
to maps
such that
.
Take such that
for any
.
Take tubular neighborhoods
of
such that
for any
.
Since
, we have that algebraic intersection number of
and
is zero.
We have
.
Apply item 2 of Lemma 6.3 to
, \
the inclusion of
into
, and
.
So we may assume that
does not intersect
.
Hence we may suppose that
.
Apply Lemma 6.5 to so on
one by one, and to the manifolds
.
Denote by
the resulting maps.
Define manifolds
for
inductively.
!!!Let
, and let
be a manifold
obtained using Lemma 6.6 for
and
.
By Lemma 6.6,
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
and inclusion
.
References
- [???] Template:???
- [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.
- [Kervaire1965] M. A. Kervaire, Les noeuds de dimensions sup'erieures, Bull. Soc. Math. France 93 (1965), 225-271.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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
.
![f:S^{4k-1}\to S^{6k}](/images/math/b/5/a/b5aaa31fc93cad500eeb3be985b335e4.png)
![\overline f:V\to D^{6k+1}](/images/math/9/f/5/9f501444fdb7f4d4ae1df68fcb525c1e.png)
![4k](/images/math/0/d/5/0d5777efe31988f31ab28b4fe0b65ac9.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![S^{4k-1}](/images/math/4/7/a/47a1015fcab977d6665ea137faa42d19.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda^*(c)\in\Z](/images/math/c/0/7/c07b4397d9e29bf012c6c84e5d8899f7.png)
![f(V)](/images/math/9/6/8/968250213d93a9a7bd801905302f3e71.png)
![\overline f(c)](/images/math/2/e/f/2ef1fcc94a63c02c6d8b3439db0778b6.png)
![\lambda^*:H_{2k}(V;\Z)\to\Z](/images/math/9/0/f/90fa622df53a3032c56ca8596ddde93a.png)
![\lambda\in H_{2k}(V,\partial;\Z)](/images/math/3/d/4/3d4ab3607763fca104c99b898a160a8f.png)
![\lambda^*[c]=\lambda\cap_V[c]](/images/math/b/2/8/b2878e69f979f9a0ca580faf4d104941.png)
![[c]\in H_{2k}(V;\Z)](/images/math/8/9/f/89f1bbfc9e8e5259c8e842991bb8018b.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\cap_V](/images/math/a/0/2/a02908ed6080691f0efe0a4168262421.png)
![H_{2k}(V;\Z)](/images/math/8/8/b/88b72e33307b3cefbf640d44bb973c7f.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![\rho_2[c]\cap_V[c]=\overline{w_{2k}}(c\subset V)=\rho_2[c]\cap_VPDw_{2k}(V)=0](/images/math/7/8/8/788a4e3686035885f1e93cfe29a761c1.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda\cap_V\lambda](/images/math/f/9/4/f949e7b5a359f80ce0ed5d166ae6957c.png)
![\displaystyle \varkappa(f):=\lambda\cap_V\lambda/2.](/images/math/3/d/c/3dc13f01f2d003319abab12c4e44c28d.png)
Since the signature of is zero, there is a symplectic basis
in
.
Then clearly
![\displaystyle \varkappa(f) = \sum\limits_{j=1}^s \lambda^*(\beta_j)\lambda^*(\alpha_j).](/images/math/5/d/e/5de79b31dfd0d4d96dc24a5bb861d954.png)
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
:
![\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}](/images/math/b/5/d/b5db6794a965eaffb93a0f335929215a.png)
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:
![\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~.](/images/math/8/3/b/83bb39e23e367d1f73a0d9f448fd6357.png)
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 a submanifold
such that
and
.
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 a submanifold
such that
and
.
Recall that isotopy classes of embeddings
are in 1--1 correspondence with
-cobordism classes of oriented submanifolds of
diffeomorphic to
for
, ???
, cf. [Haefliger1966, 1.8], [Kervaire1965].
Hence
is isotopic to standart embedding.
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 [???]].
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
.
Make the following inductive procedure.
At the
-th step!!!,
, assume that the maps
are already constructed, and we construct
.
Applying item 1 of Lemma 6.3 to , we may suppose that
is an embedding.
As and
is simply connected,
is simply connected for any
.
The algebraic intersection number of
and
is zero for any
.
Hence we can !!!apply item 2 of Lemma 6.3 to !!!
, \
, and
, for any
.
As the result,
is replaced by a homotopic embedding
, and the images of
are pairwise disjoint.
After the step we obtain a required set of embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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.
Since over
the manifold
has a framing, we obtain that
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
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 6.6.
Let be a
-submanifold of
and let
be an embedding such that
.
Then there is a smooth submanifold
homeomorphic to
and 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
for every
.
For denote by
the homotopy class of the shift of
by the first vector of the framing of
on
.
Since
, we have
.
Since
, we have
for
.
Hence by Hurewicz Theorem!!!
implies
.
Therefore there are extensions of
to maps
such that
.
Take such that
for any
.
Take tubular neighborhoods
of
such that
for any
.
Since
, we have that algebraic intersection number of
and
is zero.
We have
.
Apply item 2 of Lemma 6.3 to
, \
the inclusion of
into
, and
.
So we may assume that
does not intersect
.
Hence we may suppose that
.
Apply Lemma 6.5 to so on
one by one, and to the manifolds
.
Denote by
the resulting maps.
Define manifolds
for
inductively.
!!!Let
, and let
be a manifold
obtained using Lemma 6.6 for
and
.
By Lemma 6.6,
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
and inclusion
.
References
- [???] Template:???
- [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.
- [Kervaire1965] M. A. Kervaire, Les noeuds de dimensions sup'erieures, Bull. Soc. Math. France 93 (1965), 225-271.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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
.
![f:S^{4k-1}\to S^{6k}](/images/math/b/5/a/b5aaa31fc93cad500eeb3be985b335e4.png)
![\overline f:V\to D^{6k+1}](/images/math/9/f/5/9f501444fdb7f4d4ae1df68fcb525c1e.png)
![4k](/images/math/0/d/5/0d5777efe31988f31ab28b4fe0b65ac9.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![S^{4k-1}](/images/math/4/7/a/47a1015fcab977d6665ea137faa42d19.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda^*(c)\in\Z](/images/math/c/0/7/c07b4397d9e29bf012c6c84e5d8899f7.png)
![f(V)](/images/math/9/6/8/968250213d93a9a7bd801905302f3e71.png)
![\overline f(c)](/images/math/2/e/f/2ef1fcc94a63c02c6d8b3439db0778b6.png)
![\lambda^*:H_{2k}(V;\Z)\to\Z](/images/math/9/0/f/90fa622df53a3032c56ca8596ddde93a.png)
![\lambda\in H_{2k}(V,\partial;\Z)](/images/math/3/d/4/3d4ab3607763fca104c99b898a160a8f.png)
![\lambda^*[c]=\lambda\cap_V[c]](/images/math/b/2/8/b2878e69f979f9a0ca580faf4d104941.png)
![[c]\in H_{2k}(V;\Z)](/images/math/8/9/f/89f1bbfc9e8e5259c8e842991bb8018b.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\cap_V](/images/math/a/0/2/a02908ed6080691f0efe0a4168262421.png)
![H_{2k}(V;\Z)](/images/math/8/8/b/88b72e33307b3cefbf640d44bb973c7f.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![\rho_2[c]\cap_V[c]=\overline{w_{2k}}(c\subset V)=\rho_2[c]\cap_VPDw_{2k}(V)=0](/images/math/7/8/8/788a4e3686035885f1e93cfe29a761c1.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda\cap_V\lambda](/images/math/f/9/4/f949e7b5a359f80ce0ed5d166ae6957c.png)
![\displaystyle \varkappa(f):=\lambda\cap_V\lambda/2.](/images/math/3/d/c/3dc13f01f2d003319abab12c4e44c28d.png)
Since the signature of is zero, there is a symplectic basis
in
.
Then clearly
![\displaystyle \varkappa(f) = \sum\limits_{j=1}^s \lambda^*(\beta_j)\lambda^*(\alpha_j).](/images/math/5/d/e/5de79b31dfd0d4d96dc24a5bb861d954.png)
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
:
![\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}](/images/math/b/5/d/b5db6794a965eaffb93a0f335929215a.png)
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:
![\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~.](/images/math/8/3/b/83bb39e23e367d1f73a0d9f448fd6357.png)
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 a submanifold
such that
and
.
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 a submanifold
such that
and
.
Recall that isotopy classes of embeddings
are in 1--1 correspondence with
-cobordism classes of oriented submanifolds of
diffeomorphic to
for
, ???
, cf. [Haefliger1966, 1.8], [Kervaire1965].
Hence
is isotopic to standart embedding.
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 [???]].
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
.
Make the following inductive procedure.
At the
-th step!!!,
, assume that the maps
are already constructed, and we construct
.
Applying item 1 of Lemma 6.3 to , we may suppose that
is an embedding.
As and
is simply connected,
is simply connected for any
.
The algebraic intersection number of
and
is zero for any
.
Hence we can !!!apply item 2 of Lemma 6.3 to !!!
, \
, and
, for any
.
As the result,
is replaced by a homotopic embedding
, and the images of
are pairwise disjoint.
After the step we obtain a required set of embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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.
Since over
the manifold
has a framing, we obtain that
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
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 6.6.
Let be a
-submanifold of
and let
be an embedding such that
.
Then there is a smooth submanifold
homeomorphic to
and 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
for every
.
For denote by
the homotopy class of the shift of
by the first vector of the framing of
on
.
Since
, we have
.
Since
, we have
for
.
Hence by Hurewicz Theorem!!!
implies
.
Therefore there are extensions of
to maps
such that
.
Take such that
for any
.
Take tubular neighborhoods
of
such that
for any
.
Since
, we have that algebraic intersection number of
and
is zero.
We have
.
Apply item 2 of Lemma 6.3 to
, \
the inclusion of
into
, and
.
So we may assume that
does not intersect
.
Hence we may suppose that
.
Apply Lemma 6.5 to so on
one by one, and to the manifolds
.
Denote by
the resulting maps.
Define manifolds
for
inductively.
!!!Let
, and let
be a manifold
obtained using Lemma 6.6 for
and
.
By Lemma 6.6,
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
and inclusion
.
References
- [???] Template:???
- [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.
- [Kervaire1965] M. A. Kervaire, Les noeuds de dimensions sup'erieures, Bull. Soc. Math. France 93 (1965), 225-271.
- [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
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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
.
![f:S^{4k-1}\to S^{6k}](/images/math/b/5/a/b5aaa31fc93cad500eeb3be985b335e4.png)
![\overline f:V\to D^{6k+1}](/images/math/9/f/5/9f501444fdb7f4d4ae1df68fcb525c1e.png)
![4k](/images/math/0/d/5/0d5777efe31988f31ab28b4fe0b65ac9.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![S^{4k-1}](/images/math/4/7/a/47a1015fcab977d6665ea137faa42d19.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda^*(c)\in\Z](/images/math/c/0/7/c07b4397d9e29bf012c6c84e5d8899f7.png)
![f(V)](/images/math/9/6/8/968250213d93a9a7bd801905302f3e71.png)
![\overline f(c)](/images/math/2/e/f/2ef1fcc94a63c02c6d8b3439db0778b6.png)
![\lambda^*:H_{2k}(V;\Z)\to\Z](/images/math/9/0/f/90fa622df53a3032c56ca8596ddde93a.png)
![\lambda\in H_{2k}(V,\partial;\Z)](/images/math/3/d/4/3d4ab3607763fca104c99b898a160a8f.png)
![\lambda^*[c]=\lambda\cap_V[c]](/images/math/b/2/8/b2878e69f979f9a0ca580faf4d104941.png)
![[c]\in H_{2k}(V;\Z)](/images/math/8/9/f/89f1bbfc9e8e5259c8e842991bb8018b.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\cap_V](/images/math/a/0/2/a02908ed6080691f0efe0a4168262421.png)
![H_{2k}(V;\Z)](/images/math/8/8/b/88b72e33307b3cefbf640d44bb973c7f.png)
![2k](/images/math/d/e/b/deb18f521fe11b601162415def052d2e.png)
![c](/images/math/3/e/3/3e33e017cd76b9b7e6c7364fb91e2e90.png)
![\rho_2[c]\cap_V[c]=\overline{w_{2k}}(c\subset V)=\rho_2[c]\cap_VPDw_{2k}(V)=0](/images/math/7/8/8/788a4e3686035885f1e93cfe29a761c1.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\lambda\cap_V\lambda](/images/math/f/9/4/f949e7b5a359f80ce0ed5d166ae6957c.png)
![\displaystyle \varkappa(f):=\lambda\cap_V\lambda/2.](/images/math/3/d/c/3dc13f01f2d003319abab12c4e44c28d.png)
Since the signature of is zero, there is a symplectic basis
in
.
Then clearly
![\displaystyle \varkappa(f) = \sum\limits_{j=1}^s \lambda^*(\beta_j)\lambda^*(\alpha_j).](/images/math/5/d/e/5de79b31dfd0d4d96dc24a5bb861d954.png)
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
:
![\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}](/images/math/b/5/d/b5db6794a965eaffb93a0f335929215a.png)
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:
![\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~.](/images/math/8/3/b/83bb39e23e367d1f73a0d9f448fd6357.png)
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 a submanifold
such that
and
.
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 a submanifold
such that
and
.
Recall that isotopy classes of embeddings
are in 1--1 correspondence with
-cobordism classes of oriented submanifolds of
diffeomorphic to
for
, ???
, cf. [Haefliger1966, 1.8], [Kervaire1965].
Hence
is isotopic to standart embedding.
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 [???]].
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
.
Make the following inductive procedure.
At the
-th step!!!,
, assume that the maps
are already constructed, and we construct
.
Applying item 1 of Lemma 6.3 to , we may suppose that
is an embedding.
As and
is simply connected,
is simply connected for any
.
The algebraic intersection number of
and
is zero for any
.
Hence we can !!!apply item 2 of Lemma 6.3 to !!!
, \
, and
, for any
.
As the result,
is replaced by a homotopic embedding
, and the images of
are pairwise disjoint.
After the step we obtain a required set of embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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.
Since over
the manifold
has a framing, we obtain that
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
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 6.6.
Let be a
-submanifold of
and let
be an embedding such that
.
Then there is a smooth submanifold
homeomorphic to
and 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
for every
.
For denote by
the homotopy class of the shift of
by the first vector of the framing of
on
.
Since
, we have
.
Since
, we have
for
.
Hence by Hurewicz Theorem!!!
implies
.
Therefore there are extensions of
to maps
such that
.
Take such that
for any
.
Take tubular neighborhoods
of
such that
for any
.
Since
, we have that algebraic intersection number of
and
is zero.
We have
.
Apply item 2 of Lemma 6.3 to
, \
the inclusion of
into
, and
.
So we may assume that
does not intersect
.
Hence we may suppose that
.
Apply Lemma 6.5 to so on
one by one, and to the manifolds
.
Denote by
the resulting maps.
Define manifolds
for
inductively.
!!!Let
, and let
be a manifold
obtained using Lemma 6.6 for
and
.
By Lemma 6.6,
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
and inclusion
.
References
- [???] Template:???
- [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.
- [Kervaire1965] M. A. Kervaire, Les noeuds de dimensions sup'erieures, Bull. Soc. Math. France 93 (1965), 225-271.
- [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