Embeddings just below the stable range: classification
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Contents |
1 Introduction
This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn the theory of embeddings.
Recall the Whitney-Wu Unknotting Theorem: if is a connected manifold of dimension
, and
, then every two embeddings
are isotopic [Skopenkov2016c, Theorem 3.2]. In this page we summarize the situation for
and some more general situations.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
2].
2 Classification
For the next theorem, the Whitney invariant is defined Section 5 below.
Theorem 2.1.
Let be a closed connected
-manifold.
The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.](/images/math/8/0/9/809207b464d4e42615836c778b7a5d17.png)
is bijective if either or
and the PL category.
This is proved in [Haefliger1962b], [Haefliger&Hirsch1963] in the smooth category, and in [Weber1967] in the PL category, cf. [Bausum1975]. A minor miscalculation for the non-orientable case was corrected in [Vrabec1977].
The classification of smooth embeddings of 3-manifolds in is more complicated [Skopenkov2016t].
For embeddings of -manifolds in
see the case of 4-manifolds [Skopenkov2016f], [Yasui1984] for
and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 2.1 is generalized in 5 to a description of
for closed
-connected
-manifolds
.
3 Hudson tori
Together with the Haefliger knotted sphere [Skopenkov2016t], the examples of Hudson tori presented below were the first examples of embeddings in co-dimension greater than 2 which are not isotopic to the standard embedding defined below. (Hudson's construction [Hudson1963] was not as explicit as those below.)
For we define the standard embedding
as the composition the standard embeddings
.
Let .
Let us construct, for each and
, an embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
The reader might first consider the case .
Definition 3.1.
(This construction, as opposed to Definition 3.2, works for .)
Take the standard embeddings
(where
means homothety with coefficient 2) and
.
Take embedded sphere and embedded torus
![\displaystyle 2\partial D^{n+1}\times *\subset 2D^{n+1}\times S^{n-1}\subset\Rr^{2n}\quad\text{and}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/0/a/a/0aac68f6ce851fcfedef491078d78627.png)
Join them by an arc whose interior misses the two embedded manifolds.
The Hudson torus is the embedded connected sum of the two embeddings along this arc, compatible with the orientation.
(Unlike the unlinked embedded connected sum [Skopenkov2016c] this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)
Definition 3.2.
For instead of
we take
copies
(
) of
-sphere outside
`parallel' to
, with standard orientation for
or the opposite orientation for
. Then we make embedded connected sum by tubes joining each
-th copy to
-th copy.
We obtain an embedding
.
Let
be the linked embedded connected sum of
with the embedding
from Definition 3.1.
Clearly, is isotopic to the standard embedding.
The original motivation for Hudson was that is not isotopic to
for each
(this is a particular case of Proposition 3.3 below).
One guesses that is not isotopic to
for
.
And that a
-valued invariant exists and is `realized' by the homotopy class of the map
![\displaystyle S^n\overset g\to S^{2n}-D^{n+1}\times S^{n-1}\sim S^{2n}-S^{n-1}\sim S^n \quad\text{which is}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/6/0/0/600b6c1dbe3e66e9e32f403f4060383c.png)
However, this is only true for odd.
Proposition 3.3.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 3.3 follows by calculation of the Whitney invariant (Remark 5.3.e below) and, for even, by Theorem 2.1.
Analogously,
is not isotopic to
if
.
It would be interesting to know if the converse holds.
E.g. is
isotopic to
?
It would also be interesting to find an explicit construction of an isotopy between
and
, cf. [Vrabec1977,
5].
Definition 3.4.
Let us give, for and
, another construction of embeddings
![\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/f/0/3/f03bdba139558d9b127d28e84c53d4b5.png)
Define a map to be the constant
on one component
and the `standard inclusion'
on the other component.
This map gives an embedding
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/b/4/4/b44063272e7093e36017894c775f40ff.png)
(See [Skopenkov2006, Figure 2.2].
The image of this embedding is the union of the standard and the graph of the identity map in
.)
Take any .
The disk
intersects the image of this embedding by two points lying in
, i.e., by the image of an embedding
. Extend the latter embedding to an embedding
. See [Skopenkov2006, Figure 2.3].) Thus we obtain the Hudson torus
![\displaystyle \Hud_n'(1):S^1\times S^{n-1}\overset h\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/9/e/6/9e6b6ae1d9510c53b83d92f275fbe3a8.png)
Here , where
is identified with
.
The embedding is obtained in the same way starting from a map
of degree
instead of the `standard inclusion'.
Remark 3.5.
(a) The analogue of Proposition 3.3 for replaced to
holds, with analogous proof.
(b) Embeddings and
are smoothly isotopic for
and are PL isotopic for
[Skopenkov2006a]. It would be interesting to know if they are isotopic for
, or are smoothly isotopic for
.
(c) For these construction give what we call the left Hudson torus.
The right Hudson torus is constructed analogously.
It is the composition of the left Hudson torus and the exchanging factors autodiffeomorphism of
.
The right and the left Hudson tori are not isotopic by Remark 5.3.e below.
(d) Analogously one constructs the Hudson torus for
or, more generally,
for
.
There are versions of these constructions corresponding to Definition 3.4.
For
this corresponds to the Zeeman construction
[Skopenkov2016h] and its composition with the second unframed Kirby move.
It would be interesting to know if links
are isotopic, cf. [Skopenkov2015a, Remark 2.9.b].
These constructions could be further generalized [Skopenkov2016k].
4 Action by linked embedded connected sum
In this subsection we generalize the construction of Hudson torus .
For
, a closed connected orientable
-manifold
, an embedding
and
, we construct an embedding
.
This embedding is obtained by linked embedded connected sum of
with an
-sphere representing homology Alexander dual of
.
More precisely, represent by an embedding
.
Since any orientable bundle over
is trivial,
.
Identify
with
.
In the next paragraph we recall definition of embedded surgery of
which yields an embedding
.
Then we define
to be the (linked) embedded connected sum of
and
(along certain arc joining their images).
Take a vector field on normal to
.
Extend
along this vector field to a map
.
Since
and
, by general position we may assume that
is an embedding and
misses
.
Since
, we have
.
Hence the standard framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}.](/images/math/7/e/a/7ea4f359e00a962162a9fa1e8ebde8c9.png)
Define embedding by setting
![\displaystyle g(S^n):\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n](/images/math/0/8/d/08d183411d367189cf8992d220a5ebde.png)
with natural orientation.
By Definition 5.1 of the Whitney invariant, . Thus by Theorem 2.1
- all isotopy classes of embeddings
can be obtained from any chosen embedding
by the above construction;
- linked embedded connected sum defines a free transitive action of
on
, unless
in the smooth category.
Parametric connected sum also defines a free transitive action of on
for
[Skopenkov2014, Remark 18.a].
5 The Whitney invariant
Let be a closed
-manifold and
embeddings.
Roughly speaking,
is defined as the homology class of the self-intersection set
of a general position homotopy
between
and
.
This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010].
The definition in the PL category is analogous [Hudson1969,
12], [Vrabec1977, p. 145],
[Skopenkov2006,
2.4].
Before we present a simpler Definition 5.1 for a particular case.
For Theorem 2.1 only the case
is required.
Fix an orientation on . Assume that either
is even or
is oriented.
Definition 5.1.
Assume that is
-connected and
.
Then restrictions of
and
to
are regular homotopic [Hirsch1959].
Since
is
-connected,
has an
-dimensional spine.
Therefore these restrictions are isotopic, cf. [Haefliger&Hirsch1963, 3.1.b], [Takase2006, Lemma 2.2].
So we can make an isotopy of
and assume that
on
.
Take a general position homotopy
relative to
between the restrictions of
and
to
.
Let
(`the intersection of this homotopy with
').
Since
, by general position
is a compact
-manifold whose boundary is contained in
.
So
carries a homology class with
coefficients.
For
odd it has a natural orientation defined below, and so carries a homology class with
coefficients.
Define
to be the homology class:
![\displaystyle W(f):=[Cl(f\cap F)]\in H_{2n-m+1}(N_0,\partial N_0;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)}).](/images/math/6/2/3/62386be7f5df7183aaf4ec148a9b7a61.png)
The orientation on (extendable to
) is defined for
odd as follows.
For each point
take a vector at
tangent to
.
Complete this vector to a positive base tangent to
.
Since
, by general position there is a unique point
such that
.
The tangent vector at
thus gives a tangent vector at
to
.
Complete this vector to a positive base tangent to
, where the orientation on
comes from
.
The union of the images of the constructed two bases is a base at
of
.
If the latter base is positive, then call the initial vector of
positive.
Since a change of the orientation on
forces a change of the orientation of the latter base of
, this condition indeed defines an orientation on
.
Definition 5.2.
Assume that .
Take a general position homotopy
between
and
.
Since
, by general position the closure
of the self-intersection set has codimension 2 singularities.
So the closure carries a homology class with
coefficients.
(Note that
can be assumed to be a submanifold for
.)
For
odd it has a natural orientation (
7) and so carries a homology class with
coefficients.
Define the Whitney invariant to be the homology class:
![\displaystyle W(f):=[Cl\Sigma(H)]\in H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)}).](/images/math/0/1/c/01c3c7c72bc5b0f3d57f8c8fad32aa8b.png)
Clearly, (for both definitions).
The definition of depends on the choice of
, but we write
not
for brevity.
Remark 5.3.
(a) The Whitney invariant is well-defined by Definition 5.2, i.e. independent of the choice of , analogously to [Skopenkov2006,
2.4].
(The orientation is defined for each
but used only for odd
.
When
is even, for
being well-defined we need
-coefficients.)
(b) Definition 5.1 is equivalent to Definition 5.2.
(Indeed, if on
, we can take
to be fixed on
.)
Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice of
and of the isotopy making
outside
.
(c) Clearly, is not changed throughout isotopy of
.
Hence it gives a map
.
(d) Since a change of the orientation on forces a change of the orientation on
, the class
is independent of the choice of the orientation on
. For the reflection
with respect to a hyperplane we have
(because we may assume that
on
and because a change of the orientation of
forces a change of the orientation of
).
(e) For the Hudson tori is
or
for
, is
for
.
(f) for each embeddings
and
.
(e) For the Whitney invariant is the collection of pairwise linking coefficients of the components of
, cf. Remark 3.2.b of [Skopenkov2016h].
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically
-connected
-manifold,
. Recall the unknotting theorem [Skopenkov2016c] that every two embeddings
are isotopic when
and
. In this section we present description of
generalizing Theorem 2.1, and its generalization to
for
.
6.1 Examples
Simplest examples are Hudson tori .
Example 6.1 (linked embedded connected sum, cf. [Skopenkov2010, Definition 1.4]).
If is
-connected, then for an embedding
and a class
one can construct an embedding
by linked connected sum analogously to the case
.
We have for the Whitney invariant. Hence by Theorem 6.2 below this construction gives a free transitive action of
on
(provided
or
in the PL or smooth categories, respectively).
If
, then this construction gives only a construction of embeddings
for each
but not a well-defined action of
on
.
Embedding can alternatively be constructed using parametric connected sum [Skopenkov2014, Remark 18.a].
6.2 Classification just below the stable range
Theorem 6.2. Let be a closed orientable homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{(n-k-1)})](/images/math/9/f/d/9fd4546ac04cbcb280f3b8d4e00c9a98.png)
is a bijection, provided or
in the PL or smooth categories, respectively.
This was proved homotopically -connected manifolds in [Haefliger&Hirsch1963] in the smooth category, and in [Hudson1969,
11], [Boechat&Haefliger1970], [Boechat1971] in the PL category, cf. [Vrabec1977]. The proof works for homologically
-connected manifolds.
For Theorem 6.2 is covered by Theorem 2.1; for
it is not. The PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c] for
.
An inverse to the map of Theorem 6.2 is given be Example 6.1.
By Theorem 6.2 the Whitney invariant is bijective for
.
It is in fact a group isomorphism (for the group structure introduced in [Skopenkov2006a], [Skopenkov2015], [Skopenkov2015a]).
The generator is Hudson torus
.
Also, for
by Theorem 6.2 the Whitney invariants
![\displaystyle W^{3q}_{q-1,q}:E^{3q}_{PL}(S^{q-1}\times S^q)\to\Zz_{(q)} \quad\text{and}\quad W^{3q+1}_{q,q}:E^{3q+1}_{PL}(S^q\times S^q)\to\Zz_{(q)}\oplus \Zz_{(q)}](/images/math/6/a/9/6a9b6f86dfc99684b89237328c77e5cc.png)
are bijective.
In the smooth category for even
is not injective (see the next subsection), and
is not surjective [Boechat1971], [Skopenkov2016f], and
is not injective [Skopenkov2016f].
6.3 Classification in the presence of smoothly knotted spheres
Because of the existence of knots the analogues of Theorem 6.2 for in the PL case, and for
in the smooth case are false.
So for the smooth category,
and
closed connected, a classification of
is much harder: for 40 years the only known complete readily calculable classification results were for homology sphere
.
The following result for
was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].
Theorem 6.3 [Skopenkov2008]. Let be a closed orientable homologically
-connected
-manifold. Then the Whitney invariant
![\displaystyle W:E^{6l}_D(N)\to H_{2l-1}(N)](/images/math/b/7/6/b760d141e7e83636033b5274cd238183.png)
is surjective and for each the analogue of the Kreck invariant
![\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}](/images/math/7/f/f/7ff2f8eade12bc02562772d968d484ce.png)
is a 1-1 correspondence, where is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
.
E.g. by Theorem 6.3 the Whitney invariant is surjective and for each
there is a 1-1 correspondence
.
6.4 Classification further below the stable range
How to describe for
?
See remarks on
in
.
Some estimations of
for a closed
-connected
-manifold
are presented in [Skopenkov2010].
For
one can go even further:
Theorem 6.4 [Becker&Glover1971]. Let be a closed
-connected
-manifold embeddable into
,
and
. Then there is a 1-1 correspondence
![\displaystyle E^m(N)\to [N_0;V_{m,n+1}].](/images/math/d/9/d/d9d98867f20932b861cd8be7892e278f.png)
For this is the same as General Position Theorem 2.1 [Skopenkov2016c] (because
is
-connected).
For
this is covered by Theorem 6.2; for
it is not.
E.g. by Theorem 6.4 there is a 1-1 correspondence for
and
.
For a generalization see [Skopenkov2016k] (to knotted tori) and [Skopenkov2002].
Observe that in Theorem 6.4 can be replaced by
for each
.
7 An orientation on the self-intersection set
Let be a general position smooth map of an oriented
-manifold
. Assume that
so that the closure
of the self-intersection set of
has codimension 2 singularities. Then
- (1)
has a natural orientation.
- (2) the natural orientation on
need not extend to
: take the cone
over a general position map
having only one self-intersection point.
- (3) the natural orientation on
extend to
if
is odd [Hudson1969, Lemma 11.4].
- (4)
has a natural orientation if
is even.
Let us prove (1). Take points outside singularities of
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is orientable, we can take positive
-bases
and
at
and
normal to
and to
. If the base
of
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and
.
(Note that a change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.)
Let us prove (4). Take a -base
at a point
outside singularities of
. Since
is orientable, we can take a positive
-base
normal to
in one sheet of
. Analogously construct an
-base
for the other sheet of
.
Since
is even, the orientation of the base
of
does not depend on choosing the first and the other sheet of
.
If the base
is positive, then call the base
positive.
This is well-defined because a change of the sign of
forces changes of the signs of
and so of
.
(Note that a change of the orientation of
forces changes of the signs of
and so does not change the orientation of
.)
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