Knots, i.e. embeddings of spheres
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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; [Prasolov2007], 22].
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.
Below we denote by the Hurewicz map.
Lemma 6.4. Let be a
-connected
-manifold, and let
be homology classes such that
for every
.
Then there are embeddings
with pairwise disjoint images representing
, respectively.
Proof. As is
-connected,
is an isomorphism.
For an element
, let
be a representative of the homotopy class
.
!!!Applying item 1 of Lemma 6.3 to
, we may assume that
is an embedding.
Make the following inductive procedure.
At the -th step,
, assume that the embeddings
are already constructed, and we construct
.
Since
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
and
as above for any
.
So
is replaced by a homotopic embedding
, and the images of
are pairwise disjoint.
After
-th step we obtain a required set of embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 6.5.
Let be an orientable
-submanifold of
, and
be an embedding such that
and over
the manifold
has a framing whose first vector is tangent to
.
Assume that
has zero algebraic self-intersection in
.
Then
extends to an embedding
such that
.
Proof. (A slightly different proof is presented in the proof of Proposition 3.3 in [Haefliger1962].)
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
, then
is the obstruction to trivialization of the orthogonal complement to the field of
-frames in
corresponding to a representative of
.
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
.
Lemma 6.6 is essentialy proved in [Haefliger1962, 3.3].
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 representing
, respectively.
[!!!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
, the complement
is
-connected.
Hence by Hurewicz Theorem
implies
.
Therefore there are extensions
of
such that
.
Take such that
for any
.
Take tubular neighborhoods
of
such that
for any
.
The algebraic intersection number of
and
equals
.
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 assume that
.
Apply Lemma 6.5 to one by one, and to the manifold
.
Denote by
the resulting maps.
Define manifolds
for
inductively.
Let
, and let
be a manifold
obtained applying 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
.
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