Embeddings just below the stable range: classification
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Contents |
1 Introduction
Most of 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], [Skopenkov2006, Theorem 2.5]. In this page we summarize the situation for
and
is a connected, as well as in some more general situations.
For the classification of embeddings of some disconnected manifolds see [Skopenkov2016h].
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. Denote
.
2 Classification
For the next theorem, the Whitney invariant is defined in
5 below.
Theorem 2.1. Assume that is a closed connected
-manifold, and either
or
and we are in the PL category.
(a) If is oriented, the Whitney invariant,
![\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),](/images/math/0/c/0/0c01fbf8f4f9771a7e251feb59559526.png)
is a 1-1 correspondence.
(b) If is non-orientable, then there is a 1-1 correspondence
![\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.](/images/math/2/4/e/24e59ff646a210203da373f9a602a0bf.png)
Remark 2.2 (Comments on the proof).
Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in 2], [Hudson1969,
11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].
Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in 2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.
In Part (b) we replaced the kernel from [Bausum1975, Theorem 43] by
. This is possible because, as a specialist could see,
is given by multiplication with the first Stefel-Whitney class
(which equals to the first Wu class
[Milnor&Stasheff1974, Theorem 11.4]). Since
is non-orientable,
. So by Poincaré duality,
.
The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, 5].
It would be interesting to check if part (b) is equivalent to different forms of description of
[Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
The classification of smooth embeddings of 3-manifolds in is more complicated, see Theorem 6.3 below for
or [Skopenkov2016t].
Concerning embeddings of connected -manifolds in
see [Yasui1984] for
, [Skopenkov2016f] for
, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.
Theorem 2.1 is generalized to a description of for closed
-connected
-manifolds
, see Theorem 6.2.
3 Hudson tori
Tex syntax errorto just
Tex syntax error.
Example 3.1. Let us construct, for any and
, a smooth embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
We start with the cases .
Take the standard inclusion .
The 'standard embedding'
is given by the standard inclusions
Tex syntax error
Tex syntax erroranalogously to
Tex syntax error, where
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
Take the embedding given by
Tex syntax error
Tex syntax errorjoins the images of
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\Hud_n(1)](/images/math/5/2/d/52d8f018b81e36930436db3c1454a796.png)
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For we repeat the above construction of
replacing
by
copies
of
,
.
The copies are outside
and are `parallel' to
.
The copies have the standard orientation for
or the opposite orientation for
.
Then we make embedded connected sum along natural segments joining every
-th copy to the
-th
copy.
We obtain an embedding
which has disjoint images with
.
Let
be the linked embedded connected sum of
and
.
The original motivation for Hudson was that is not isotopic to
for any
(this is a particular case of Proposition 3.2 below). One might guess that
is not isotopic
to
for
and that a
-valued invariant of
can be defined by the homotopy class of the map
Tex syntax error
However, this is only true for odd.
Proposition 3.2.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for even, by Theorem 2.1.
This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of
for the PL category).
It would be interesting to find an explicit construction of an isotopy between
and
, cf. [Vrabec1977,
5].
Analogously,
is not isotopic to
if
.
It would be interesting to know if the converse holds, e.g. is
(PS or smoothly) isotopic to
.
Example 3.3.
Take any .
Take a map
of degree
(so we can take
).
Recall that
.
Define the smooth embedding
to be the composition
Tex syntax error
Let us present a geometric description of this embedding.
Define a map by
.
This map gives an embedding
Tex syntax error
Tex syntax erroris the union of the graphs of the maps
![\overline a](/images/math/4/f/f/4ff0b2c40da836a1b7c801879c156a0d.png)
![-\overline a](/images/math/8/e/5/8e5eb3db0a5bbf6fec8a239782cdefa0.png)
![t\in S^{n-1}](/images/math/d/0/b/d0bb33dc2f67081c20c164b1a684501f.png)
![D^{n+1}\times t](/images/math/c/a/7/ca745a1e30fa17ec700f9d04c9fe8744.png)
![D^n\times t](/images/math/5/b/f/5bf99f848e37cae0bce4fe823e689da7.png)
![S^0\times t\to D^n\times t](/images/math/0/8/6/086108c6e6d192ebde0c3f5406917ebd.png)
![\Hud_n'(a)](/images/math/b/1/c/b1c5f108b78c001bf133a56e2ddc8c83.png)
![S^1\times t\to D^{n+1}\times t](/images/math/d/a/d/dad76014c549bc406ac1a4c3a7304320.png)
![t](/images/math/3/0/b/30b2ab8dc1496d06b230a71d8962af5d.png)
Remark 3.4. (a) The analogue of Proposition 3.2 for replaced to
holds, with an analogous proof.
(b) The embeddings and
are smoothly isotopic for
and are PS isotopic for
[Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]).
This follows by calculation of the Whitney invariant (Remark 5.3.d below).
It would be interesting to know if they are smoothly isotopic for
.
It would be interesting to know if they are piecewise smoothly isotopic for
.
(c) For Example 3.3 gives 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.d below.
(d) Analogously one constructs the Hudson torus for
and
or, more generally,
for
and
.
There are versions
of these constructions corresponding to Definition 3.3.
For
this corresponds to the Zeeman map
[Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a,
2.3].
It would be interesting to know if the links
are isotopic, cf. [Skopenkov2015a, Remark 2.7.b].
These constructions can be further generalized [Skopenkov2016k].
4 Action by linked embedded connected sum
In this section we generalize the construction of the Hudson torus .
Let
be a closed connected oriented
-manifold.
We work in the smooth category which we omit.
Apparently analogous results hold for
in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).
Example 4.1.
For any , an embedding
and
, we shall construct an embedding
.
This embedding is said to be obtained by linked embedded connected sum of
with an
-sphere representing the `homology Alexander dual'
of
(defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).
Represent by an embedding
.
By definition, the class
is represented by properly oriented
.
Since any orientable bundle over
is trivial,
.
Take an embedding
whose image is
and which represents
.
By embedded surgery on
we obtain an embedding
representing
(see details in Proposition 4.2 below).
Define
to be the linked embedded connected sum of
and
, along some arc joining their images.
Proposition 4.2 (Embedded surgery).
For any , a neighborhood
of a codimension at least 3 subpolyhedron in
and an embedding
there is an embedding
homologous to
.
Proof. Take a vector field on normal to
.
Extend
along this vector field to a map
.
![2n>4](/images/math/3/9/f/39f044554ddcec9904d56e948f6ae327.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![\overline b](/images/math/6/9/3/6931221e5bb1220fb689387c4a965a3e.png)
Tex syntax errormisses
![U\cup g(S^1\times S^{n-1})](/images/math/8/5/e/85e5e4507d7c16a18c918dadc48f9292.png)
Since , we have
.
Hence the standard
-framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).](/images/math/a/a/3/aa3cea2126097ada817f3b858e41c7c0.png)
Take an embedding such that
Tex syntax error
with proper orientation so that is homologous to
. QED
The isotopy class of the embedding is independent of the choises in the construction.
The independence of the arc and of the maps
follows by
and by Proposition 4.3 below, respectively.
By Definition 5.1 of the Whitney invariant, is
for
odd and
for
even.
Thus by Theorem 2.1.a for
all isotopy classes of embeddings
can be obtained from any chosen embedding
by the above construction.
Proposition 4.3. For any both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of
on
.
This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].
5 The Whitney invariant
Let be a closed
-manifold.
Take an embedding
.
Fix an orientation on
.
For any other embedding
we define the Whitney invariant
![\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).](/images/math/5/2/4/5241adb18efe69df96fbf1725c71c755.png)
Here the coefficients are
if
is oriented and
is odd, and are
otherwise.
Tex syntax erroris defined as the homology class of the closure of the self-intersection set of a general position homotopy
![H](/images/math/2/f/b/2fbada10033dab2ef3330c6cb17a3a0c.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see
also [HaefligerHirsch1963].
The definition in the PL category is analogous [Hudson1969, 11], [Vrabec1977, p. 145],
[Skopenkov2006,
2.4 `The Whitney invariant'].
We begin by presenting a simpler definition, Definition 5.1, for a particular case.
For Theorem 2.1 only the case is required.
Definition 5.1.
Assume that is
-connected and
.
Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of
and
to
are isotopic, cf. [Takase2006, Lemma 2.2].
(Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of
and
to
are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since
is
-connected,
retracts to an
-dimensional polyhedron.
Therefore these restrictions are isotopic.)
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
').
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
Tex syntax erroris a compact
![(2n+1-m)](/images/math/0/6/3/063e854d764c6b9c09f9df1bcf76a85b.png)
![\partial N_0](/images/math/f/9/6/f96c15bf3a49cdf2051f311be0c68a80.png)
So carries a homology class with
coefficients.
If
is odd and
is oriented, then
has a natural orientation defined below,
and so carries a homology class with
coefficients.
Define
to be the homology class:
Tex syntax error
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error) is defined (for
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![x_f\in f\cap F](/images/math/7/f/b/7fb6f0e49619c2ef0b26f48589ee7685.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![(\xi_f,\eta_f)](/images/math/6/9/9/6994fd91df6289bee65aefab6bc3b565.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
![x_F\in B^n\times I](/images/math/3/f/6/3f69b52e956c6a6ca178831a6fe42489.png)
![Fx_F=fx_f](/images/math/c/e/2/ce2e87a9d982adc19aabb66241f97573.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![x_F](/images/math/c/2/9/c2952c4745794ea5ae3e306b271fdf36.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![df(x_f)\xi_f=dF(x_F)\xi_F](/images/math/1/b/a/1bab6ec8f7be5007b81716de60cd2716.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![(\xi_F,\eta_F)](/images/math/c/8/a/c8aa786734dd3b5a2d8c5c97b416f234.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)](/images/math/2/1/d/21d94058634f7edd21782b7de8c7c1c9.png)
![fx_f=Fx_F](/images/math/f/2/1/f216388cdabee8cb9a80855668711cfa.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Definition 5.2.
Assume that .
Take a general position homotopy
between
and
.
Tex syntax errorof the self-intersection set carries a cycle mod 2. If
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![2m\ge3n+2](/images/math/a/c/4/ac42ffc32fee13f8930d2fe0bef58252.png)
Tex syntax errorcan be assumed to be a submanifold. In general, since
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
Define the Whitney invariant to be the homology class:
Tex syntax error
Clearly, if
is isotopic to
.
Hence the Whitney invariant defines a map
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).](/images/math/c/3/0/c30f51328f41351aac3861e4f870de3e.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. is independent of the choice of a general position homotopy
from
to
.
Tex syntax errorfor a general position homotopy
![H_{01}:N\times I\times I\to\Rr^m\times I\times I](/images/math/7/d/9/7d974d2c2ed915a105323b69b287d03e.png)
![H_0,H_1:N\times I\to\Rr^m\times I](/images/math/0/8/7/0874f958e49857acc9c56909b71752e6.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
(b) Definition 5.1 is a particular case of Definition 5.2.
(Indeed, if on
, we can take
to be fixed on
. See details in [Skopenkov2010, Difference
Lemma 2.4].)
Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice
of
and of the isotopy making
outside
.
![W(f)](/images/math/2/2/1/221f32285325c78c47de94fd7c7f75ab.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error). For the reflection
![\sigma:\Rr^m\to\Rr^m](/images/math/6/1/9/6193d98c9ab0422c651027e2264dd25a.png)
![W(\sigma\circ f)=-W(f)](/images/math/1/9/2/1924babe23e26ae0ce7bf8327aff3f63.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error; for Definition 5.1 also observe that we may assume that
![f=f_0=\sigma\circ f](/images/math/c/6/c/c6cf229f35059deca4f1195f751730d9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
(d) For the Hudson tori is
or
for
,
and
.
For this is clear by Definition 5.1. For
and
this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For
the proof is analogous.
![W(f\#g)=W(f)](/images/math/7/6/e/76eb58f9e926b43c618caded88d8bf11.png)
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![g:S^n\to\Rr^m](/images/math/f/b/3/fb3a60c7de4f595f5e0bfa65d1fa30c7.png)
![W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0](/images/math/d/b/d/dbdd409aa4c1f3c7c9265288b0c274ba.png)
![H_f](/images/math/0/4/b/04bcba99b75cca1cc5abd4e5fddb95d8.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![H_g](/images/math/f/7/b/f7b9ec248e54e11f3b2ed76020fd35a5.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![H_f\sharp H_g](/images/math/2/8/e/28e29f776c97e9d5df24c5584b361f57.png)
![f\#g](/images/math/0/a/a/0aabaf11c5162ade8ba1ee6647598003.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax erroris null-homologous in
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![N\cong N\#S^n](/images/math/2/f/1/2f18f52244a27abfdb750b1a4c8861ba.png)
(f) For and
the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h,
3].
(g) The Whitney invariant need not be a bijection for . This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking
even,
non-orientable,
and applying by Theorem 2.1.b.
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically
-connected
-manifold,
. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings
are isotopic when
and
. In this section we generalize Theorem 2.1 to a description of
and further to
for
.
6.1 Examples
Some simple examples are the Hudson tori .
Example 6.1 (cf. [Skopenkov2010, Definition 1.4]).
Assume that is
-connected and
.
Then for an embedding
and a class
one can construct an embedding
by linked embedded connected sum analogously to the case
presented in Example 4.1.
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.
The embedding has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].
6.2 Classification
Theorem 6.2. Let be a closed oriented homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})](/images/math/1/f/9/1f9ca0866bacc6e8147523f65e9df3c8.png)
is a bijection, provided in the smooth category or
in the PL category.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in
[Weber1967], [Hudson1969,
11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993,
7]. The proof actually used the homological
-connectedness assumption (basically because the
-connectedness was used to ensure high enough connectedness of the complement in
to the image of
, by Alexander duality and simple connectedness of the complement, so homological
-connectedness of
is sufficient).
For Theorem 6.2 is covered by Theorem 2.1; for
it is not.
For
the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3].
For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].
An inverse to the map of Theorem 6.2 is given by Example 6.1.
Because of the existence of knotted spheres 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 spheres
. E.g.
for any
[Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2].
The following result for
was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1].
Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008,
4].
Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. 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 any 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
.
How does one describe when
is not
-connected?
For general
see the sentence on
at the end of
2.
We can say more as the connectivity
of
increases.
Some estimations of
for a closed
-connected
-manifold
are presented in [Skopenkov2010]. For
one can go even further:
Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. 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/6/4/3/643341df90bc99621c0740a72f5ab4ef.png)
The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5].
For
Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because
is
-connected).
For
Theorem 6.4 is covered by Theorem 6.2; for
it is not.
For application to knotted tori see [Skopenkov2016k, Theorem 5.1].
For generalization to arbitrary manifolds see survey [Skopenkov2006,
5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4
can be replaced by
for any
.
7 An orientation on the self-intersection set
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
Tex syntax errorof the self-intersection set of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax errorsuch that
- both
and
Tex syntax error
are subpolyhedra of some triangulation of,
- we have
and
-
is an open manifold consisting of self-transverse double points of
.
Definition 7.1 (A canonical orientation on ).
Take points
away from
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is oriented, 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
.
Remark 7.2 (Properties of the orientation just defined on )..
- A change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.
- The orientation on
need not extend to
Tex syntax error
: take the smooth coneover a general position map
having only two transverse self-intersection points, where the smooth cone is defined by
, for
and
.
- The orientation on
extends to
Tex syntax error
ifis odd [Hudson1969, Lemma 11.4].
Remark 7.3 (A canonical orientation on for
even).
This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.
Take a -base
at a point
. Since
is oriented, 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
at
.
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
.
We remark that a change of the orientation of forces changes of the signs of
and so does not change the orientation of
.
8 References
- [Adachi1993] M. Adachi, Embeddings and immersions, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension
dans
, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension
dans
, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Haefliger1966] A. Haefliger, Differential embeddings of
in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [HaefligerHirsch1963] Template:HaefligerHirsch1963
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1967] J. Hudson, Piecewise linear embeddings, Ann. of Math. (2) 85 (1967) 1–31. MR0215308 (35 #6149) Zbl 0153.25601
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Koschorke2013] U. Koschorke, Immersion, http://www.map.mpim-bonn.mpg.de/Immersion
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in
, Comment. Math. Helv. 72 (1997), 543-555.
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [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.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [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
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1977] J. Vrabec, Knotting a
-connected closed
-manifold in
, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Yasui1984] T. Yasui, Enumerating embeddings of
-manifolds in Euclidean
-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019
![n>1](/images/math/f/3/2/f32ba7f7c2722280ef2c82ff05d61be1.png)
![m \ge2n+1](/images/math/f/3/1/f312fc5cb48b012a9b7b515034a931eb.png)
![N \to\Rr^m](/images/math/e/1/d/e1d83d016b26cc4d71e3d872594d0c42.png)
![m=2n\ge6](/images/math/9/9/0/990e2dcb0be4d51d59ba1986b7a1257c.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. Denote
.
2 Classification
For the next theorem, the Whitney invariant is defined in
5 below.
Theorem 2.1. Assume that is a closed connected
-manifold, and either
or
and we are in the PL category.
(a) If is oriented, the Whitney invariant,
![\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),](/images/math/0/c/0/0c01fbf8f4f9771a7e251feb59559526.png)
is a 1-1 correspondence.
(b) If is non-orientable, then there is a 1-1 correspondence
![\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.](/images/math/2/4/e/24e59ff646a210203da373f9a602a0bf.png)
Remark 2.2 (Comments on the proof).
Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in 2], [Hudson1969,
11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].
Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in 2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.
In Part (b) we replaced the kernel from [Bausum1975, Theorem 43] by
. This is possible because, as a specialist could see,
is given by multiplication with the first Stefel-Whitney class
(which equals to the first Wu class
[Milnor&Stasheff1974, Theorem 11.4]). Since
is non-orientable,
. So by Poincaré duality,
.
The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, 5].
It would be interesting to check if part (b) is equivalent to different forms of description of
[Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
The classification of smooth embeddings of 3-manifolds in is more complicated, see Theorem 6.3 below for
or [Skopenkov2016t].
Concerning embeddings of connected -manifolds in
see [Yasui1984] for
, [Skopenkov2016f] for
, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.
Theorem 2.1 is generalized to a description of for closed
-connected
-manifolds
, see Theorem 6.2.
3 Hudson tori
Tex syntax errorto just
Tex syntax error.
Example 3.1. Let us construct, for any and
, a smooth embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
We start with the cases .
Take the standard inclusion .
The 'standard embedding'
is given by the standard inclusions
Tex syntax error
Tex syntax erroranalogously to
Tex syntax error, where
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
Take the embedding given by
Tex syntax error
Tex syntax errorjoins the images of
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\Hud_n(1)](/images/math/5/2/d/52d8f018b81e36930436db3c1454a796.png)
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For we repeat the above construction of
replacing
by
copies
of
,
.
The copies are outside
and are `parallel' to
.
The copies have the standard orientation for
or the opposite orientation for
.
Then we make embedded connected sum along natural segments joining every
-th copy to the
-th
copy.
We obtain an embedding
which has disjoint images with
.
Let
be the linked embedded connected sum of
and
.
The original motivation for Hudson was that is not isotopic to
for any
(this is a particular case of Proposition 3.2 below). One might guess that
is not isotopic
to
for
and that a
-valued invariant of
can be defined by the homotopy class of the map
Tex syntax error
However, this is only true for odd.
Proposition 3.2.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for even, by Theorem 2.1.
This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of
for the PL category).
It would be interesting to find an explicit construction of an isotopy between
and
, cf. [Vrabec1977,
5].
Analogously,
is not isotopic to
if
.
It would be interesting to know if the converse holds, e.g. is
(PS or smoothly) isotopic to
.
Example 3.3.
Take any .
Take a map
of degree
(so we can take
).
Recall that
.
Define the smooth embedding
to be the composition
Tex syntax error
Let us present a geometric description of this embedding.
Define a map by
.
This map gives an embedding
Tex syntax error
Tex syntax erroris the union of the graphs of the maps
![\overline a](/images/math/4/f/f/4ff0b2c40da836a1b7c801879c156a0d.png)
![-\overline a](/images/math/8/e/5/8e5eb3db0a5bbf6fec8a239782cdefa0.png)
![t\in S^{n-1}](/images/math/d/0/b/d0bb33dc2f67081c20c164b1a684501f.png)
![D^{n+1}\times t](/images/math/c/a/7/ca745a1e30fa17ec700f9d04c9fe8744.png)
![D^n\times t](/images/math/5/b/f/5bf99f848e37cae0bce4fe823e689da7.png)
![S^0\times t\to D^n\times t](/images/math/0/8/6/086108c6e6d192ebde0c3f5406917ebd.png)
![\Hud_n'(a)](/images/math/b/1/c/b1c5f108b78c001bf133a56e2ddc8c83.png)
![S^1\times t\to D^{n+1}\times t](/images/math/d/a/d/dad76014c549bc406ac1a4c3a7304320.png)
![t](/images/math/3/0/b/30b2ab8dc1496d06b230a71d8962af5d.png)
Remark 3.4. (a) The analogue of Proposition 3.2 for replaced to
holds, with an analogous proof.
(b) The embeddings and
are smoothly isotopic for
and are PS isotopic for
[Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]).
This follows by calculation of the Whitney invariant (Remark 5.3.d below).
It would be interesting to know if they are smoothly isotopic for
.
It would be interesting to know if they are piecewise smoothly isotopic for
.
(c) For Example 3.3 gives 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.d below.
(d) Analogously one constructs the Hudson torus for
and
or, more generally,
for
and
.
There are versions
of these constructions corresponding to Definition 3.3.
For
this corresponds to the Zeeman map
[Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a,
2.3].
It would be interesting to know if the links
are isotopic, cf. [Skopenkov2015a, Remark 2.7.b].
These constructions can be further generalized [Skopenkov2016k].
4 Action by linked embedded connected sum
In this section we generalize the construction of the Hudson torus .
Let
be a closed connected oriented
-manifold.
We work in the smooth category which we omit.
Apparently analogous results hold for
in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).
Example 4.1.
For any , an embedding
and
, we shall construct an embedding
.
This embedding is said to be obtained by linked embedded connected sum of
with an
-sphere representing the `homology Alexander dual'
of
(defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).
Represent by an embedding
.
By definition, the class
is represented by properly oriented
.
Since any orientable bundle over
is trivial,
.
Take an embedding
whose image is
and which represents
.
By embedded surgery on
we obtain an embedding
representing
(see details in Proposition 4.2 below).
Define
to be the linked embedded connected sum of
and
, along some arc joining their images.
Proposition 4.2 (Embedded surgery).
For any , a neighborhood
of a codimension at least 3 subpolyhedron in
and an embedding
there is an embedding
homologous to
.
Proof. Take a vector field on normal to
.
Extend
along this vector field to a map
.
![2n>4](/images/math/3/9/f/39f044554ddcec9904d56e948f6ae327.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![\overline b](/images/math/6/9/3/6931221e5bb1220fb689387c4a965a3e.png)
Tex syntax errormisses
![U\cup g(S^1\times S^{n-1})](/images/math/8/5/e/85e5e4507d7c16a18c918dadc48f9292.png)
Since , we have
.
Hence the standard
-framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).](/images/math/a/a/3/aa3cea2126097ada817f3b858e41c7c0.png)
Take an embedding such that
Tex syntax error
with proper orientation so that is homologous to
. QED
The isotopy class of the embedding is independent of the choises in the construction.
The independence of the arc and of the maps
follows by
and by Proposition 4.3 below, respectively.
By Definition 5.1 of the Whitney invariant, is
for
odd and
for
even.
Thus by Theorem 2.1.a for
all isotopy classes of embeddings
can be obtained from any chosen embedding
by the above construction.
Proposition 4.3. For any both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of
on
.
This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].
5 The Whitney invariant
Let be a closed
-manifold.
Take an embedding
.
Fix an orientation on
.
For any other embedding
we define the Whitney invariant
![\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).](/images/math/5/2/4/5241adb18efe69df96fbf1725c71c755.png)
Here the coefficients are
if
is oriented and
is odd, and are
otherwise.
Tex syntax erroris defined as the homology class of the closure of the self-intersection set of a general position homotopy
![H](/images/math/2/f/b/2fbada10033dab2ef3330c6cb17a3a0c.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see
also [HaefligerHirsch1963].
The definition in the PL category is analogous [Hudson1969, 11], [Vrabec1977, p. 145],
[Skopenkov2006,
2.4 `The Whitney invariant'].
We begin by presenting a simpler definition, Definition 5.1, for a particular case.
For Theorem 2.1 only the case is required.
Definition 5.1.
Assume that is
-connected and
.
Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of
and
to
are isotopic, cf. [Takase2006, Lemma 2.2].
(Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of
and
to
are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since
is
-connected,
retracts to an
-dimensional polyhedron.
Therefore these restrictions are isotopic.)
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
').
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
Tex syntax erroris a compact
![(2n+1-m)](/images/math/0/6/3/063e854d764c6b9c09f9df1bcf76a85b.png)
![\partial N_0](/images/math/f/9/6/f96c15bf3a49cdf2051f311be0c68a80.png)
So carries a homology class with
coefficients.
If
is odd and
is oriented, then
has a natural orientation defined below,
and so carries a homology class with
coefficients.
Define
to be the homology class:
Tex syntax error
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error) is defined (for
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![x_f\in f\cap F](/images/math/7/f/b/7fb6f0e49619c2ef0b26f48589ee7685.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![(\xi_f,\eta_f)](/images/math/6/9/9/6994fd91df6289bee65aefab6bc3b565.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
![x_F\in B^n\times I](/images/math/3/f/6/3f69b52e956c6a6ca178831a6fe42489.png)
![Fx_F=fx_f](/images/math/c/e/2/ce2e87a9d982adc19aabb66241f97573.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![x_F](/images/math/c/2/9/c2952c4745794ea5ae3e306b271fdf36.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![df(x_f)\xi_f=dF(x_F)\xi_F](/images/math/1/b/a/1bab6ec8f7be5007b81716de60cd2716.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![(\xi_F,\eta_F)](/images/math/c/8/a/c8aa786734dd3b5a2d8c5c97b416f234.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)](/images/math/2/1/d/21d94058634f7edd21782b7de8c7c1c9.png)
![fx_f=Fx_F](/images/math/f/2/1/f216388cdabee8cb9a80855668711cfa.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Definition 5.2.
Assume that .
Take a general position homotopy
between
and
.
Tex syntax errorof the self-intersection set carries a cycle mod 2. If
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![2m\ge3n+2](/images/math/a/c/4/ac42ffc32fee13f8930d2fe0bef58252.png)
Tex syntax errorcan be assumed to be a submanifold. In general, since
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
Define the Whitney invariant to be the homology class:
Tex syntax error
Clearly, if
is isotopic to
.
Hence the Whitney invariant defines a map
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).](/images/math/c/3/0/c30f51328f41351aac3861e4f870de3e.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. is independent of the choice of a general position homotopy
from
to
.
Tex syntax errorfor a general position homotopy
![H_{01}:N\times I\times I\to\Rr^m\times I\times I](/images/math/7/d/9/7d974d2c2ed915a105323b69b287d03e.png)
![H_0,H_1:N\times I\to\Rr^m\times I](/images/math/0/8/7/0874f958e49857acc9c56909b71752e6.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
(b) Definition 5.1 is a particular case of Definition 5.2.
(Indeed, if on
, we can take
to be fixed on
. See details in [Skopenkov2010, Difference
Lemma 2.4].)
Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice
of
and of the isotopy making
outside
.
![W(f)](/images/math/2/2/1/221f32285325c78c47de94fd7c7f75ab.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error). For the reflection
![\sigma:\Rr^m\to\Rr^m](/images/math/6/1/9/6193d98c9ab0422c651027e2264dd25a.png)
![W(\sigma\circ f)=-W(f)](/images/math/1/9/2/1924babe23e26ae0ce7bf8327aff3f63.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error; for Definition 5.1 also observe that we may assume that
![f=f_0=\sigma\circ f](/images/math/c/6/c/c6cf229f35059deca4f1195f751730d9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
(d) For the Hudson tori is
or
for
,
and
.
For this is clear by Definition 5.1. For
and
this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For
the proof is analogous.
![W(f\#g)=W(f)](/images/math/7/6/e/76eb58f9e926b43c618caded88d8bf11.png)
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![g:S^n\to\Rr^m](/images/math/f/b/3/fb3a60c7de4f595f5e0bfa65d1fa30c7.png)
![W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0](/images/math/d/b/d/dbdd409aa4c1f3c7c9265288b0c274ba.png)
![H_f](/images/math/0/4/b/04bcba99b75cca1cc5abd4e5fddb95d8.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![H_g](/images/math/f/7/b/f7b9ec248e54e11f3b2ed76020fd35a5.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![H_f\sharp H_g](/images/math/2/8/e/28e29f776c97e9d5df24c5584b361f57.png)
![f\#g](/images/math/0/a/a/0aabaf11c5162ade8ba1ee6647598003.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax erroris null-homologous in
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![N\cong N\#S^n](/images/math/2/f/1/2f18f52244a27abfdb750b1a4c8861ba.png)
(f) For and
the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h,
3].
(g) The Whitney invariant need not be a bijection for . This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking
even,
non-orientable,
and applying by Theorem 2.1.b.
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically
-connected
-manifold,
. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings
are isotopic when
and
. In this section we generalize Theorem 2.1 to a description of
and further to
for
.
6.1 Examples
Some simple examples are the Hudson tori .
Example 6.1 (cf. [Skopenkov2010, Definition 1.4]).
Assume that is
-connected and
.
Then for an embedding
and a class
one can construct an embedding
by linked embedded connected sum analogously to the case
presented in Example 4.1.
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.
The embedding has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].
6.2 Classification
Theorem 6.2. Let be a closed oriented homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})](/images/math/1/f/9/1f9ca0866bacc6e8147523f65e9df3c8.png)
is a bijection, provided in the smooth category or
in the PL category.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in
[Weber1967], [Hudson1969,
11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993,
7]. The proof actually used the homological
-connectedness assumption (basically because the
-connectedness was used to ensure high enough connectedness of the complement in
to the image of
, by Alexander duality and simple connectedness of the complement, so homological
-connectedness of
is sufficient).
For Theorem 6.2 is covered by Theorem 2.1; for
it is not.
For
the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3].
For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].
An inverse to the map of Theorem 6.2 is given by Example 6.1.
Because of the existence of knotted spheres 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 spheres
. E.g.
for any
[Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2].
The following result for
was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1].
Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008,
4].
Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. 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 any 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
.
How does one describe when
is not
-connected?
For general
see the sentence on
at the end of
2.
We can say more as the connectivity
of
increases.
Some estimations of
for a closed
-connected
-manifold
are presented in [Skopenkov2010]. For
one can go even further:
Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. 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/6/4/3/643341df90bc99621c0740a72f5ab4ef.png)
The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5].
For
Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because
is
-connected).
For
Theorem 6.4 is covered by Theorem 6.2; for
it is not.
For application to knotted tori see [Skopenkov2016k, Theorem 5.1].
For generalization to arbitrary manifolds see survey [Skopenkov2006,
5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4
can be replaced by
for any
.
7 An orientation on the self-intersection set
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
Tex syntax errorof the self-intersection set of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax errorsuch that
- both
and
Tex syntax error
are subpolyhedra of some triangulation of,
- we have
and
-
is an open manifold consisting of self-transverse double points of
.
Definition 7.1 (A canonical orientation on ).
Take points
away from
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is oriented, 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
.
Remark 7.2 (Properties of the orientation just defined on )..
- A change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.
- The orientation on
need not extend to
Tex syntax error
: take the smooth coneover a general position map
having only two transverse self-intersection points, where the smooth cone is defined by
, for
and
.
- The orientation on
extends to
Tex syntax error
ifis odd [Hudson1969, Lemma 11.4].
Remark 7.3 (A canonical orientation on for
even).
This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.
Take a -base
at a point
. Since
is oriented, 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
at
.
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
.
We remark 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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, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1977] J. Vrabec, Knotting a
-connected closed
-manifold in
, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Yasui1984] T. Yasui, Enumerating embeddings of
-manifolds in Euclidean
-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019
![n>1](/images/math/f/3/2/f32ba7f7c2722280ef2c82ff05d61be1.png)
![m \ge2n+1](/images/math/f/3/1/f312fc5cb48b012a9b7b515034a931eb.png)
![N \to\Rr^m](/images/math/e/1/d/e1d83d016b26cc4d71e3d872594d0c42.png)
![m=2n\ge6](/images/math/9/9/0/990e2dcb0be4d51d59ba1986b7a1257c.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. Denote
.
2 Classification
For the next theorem, the Whitney invariant is defined in
5 below.
Theorem 2.1. Assume that is a closed connected
-manifold, and either
or
and we are in the PL category.
(a) If is oriented, the Whitney invariant,
![\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),](/images/math/0/c/0/0c01fbf8f4f9771a7e251feb59559526.png)
is a 1-1 correspondence.
(b) If is non-orientable, then there is a 1-1 correspondence
![\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.](/images/math/2/4/e/24e59ff646a210203da373f9a602a0bf.png)
Remark 2.2 (Comments on the proof).
Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in 2], [Hudson1969,
11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].
Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in 2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.
In Part (b) we replaced the kernel from [Bausum1975, Theorem 43] by
. This is possible because, as a specialist could see,
is given by multiplication with the first Stefel-Whitney class
(which equals to the first Wu class
[Milnor&Stasheff1974, Theorem 11.4]). Since
is non-orientable,
. So by Poincaré duality,
.
The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, 5].
It would be interesting to check if part (b) is equivalent to different forms of description of
[Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
The classification of smooth embeddings of 3-manifolds in is more complicated, see Theorem 6.3 below for
or [Skopenkov2016t].
Concerning embeddings of connected -manifolds in
see [Yasui1984] for
, [Skopenkov2016f] for
, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.
Theorem 2.1 is generalized to a description of for closed
-connected
-manifolds
, see Theorem 6.2.
3 Hudson tori
Tex syntax errorto just
Tex syntax error.
Example 3.1. Let us construct, for any and
, a smooth embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
We start with the cases .
Take the standard inclusion .
The 'standard embedding'
is given by the standard inclusions
Tex syntax error
Tex syntax erroranalogously to
Tex syntax error, where
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
Take the embedding given by
Tex syntax error
Tex syntax errorjoins the images of
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\Hud_n(1)](/images/math/5/2/d/52d8f018b81e36930436db3c1454a796.png)
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For we repeat the above construction of
replacing
by
copies
of
,
.
The copies are outside
and are `parallel' to
.
The copies have the standard orientation for
or the opposite orientation for
.
Then we make embedded connected sum along natural segments joining every
-th copy to the
-th
copy.
We obtain an embedding
which has disjoint images with
.
Let
be the linked embedded connected sum of
and
.
The original motivation for Hudson was that is not isotopic to
for any
(this is a particular case of Proposition 3.2 below). One might guess that
is not isotopic
to
for
and that a
-valued invariant of
can be defined by the homotopy class of the map
Tex syntax error
However, this is only true for odd.
Proposition 3.2.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for even, by Theorem 2.1.
This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of
for the PL category).
It would be interesting to find an explicit construction of an isotopy between
and
, cf. [Vrabec1977,
5].
Analogously,
is not isotopic to
if
.
It would be interesting to know if the converse holds, e.g. is
(PS or smoothly) isotopic to
.
Example 3.3.
Take any .
Take a map
of degree
(so we can take
).
Recall that
.
Define the smooth embedding
to be the composition
Tex syntax error
Let us present a geometric description of this embedding.
Define a map by
.
This map gives an embedding
Tex syntax error
Tex syntax erroris the union of the graphs of the maps
![\overline a](/images/math/4/f/f/4ff0b2c40da836a1b7c801879c156a0d.png)
![-\overline a](/images/math/8/e/5/8e5eb3db0a5bbf6fec8a239782cdefa0.png)
![t\in S^{n-1}](/images/math/d/0/b/d0bb33dc2f67081c20c164b1a684501f.png)
![D^{n+1}\times t](/images/math/c/a/7/ca745a1e30fa17ec700f9d04c9fe8744.png)
![D^n\times t](/images/math/5/b/f/5bf99f848e37cae0bce4fe823e689da7.png)
![S^0\times t\to D^n\times t](/images/math/0/8/6/086108c6e6d192ebde0c3f5406917ebd.png)
![\Hud_n'(a)](/images/math/b/1/c/b1c5f108b78c001bf133a56e2ddc8c83.png)
![S^1\times t\to D^{n+1}\times t](/images/math/d/a/d/dad76014c549bc406ac1a4c3a7304320.png)
![t](/images/math/3/0/b/30b2ab8dc1496d06b230a71d8962af5d.png)
Remark 3.4. (a) The analogue of Proposition 3.2 for replaced to
holds, with an analogous proof.
(b) The embeddings and
are smoothly isotopic for
and are PS isotopic for
[Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]).
This follows by calculation of the Whitney invariant (Remark 5.3.d below).
It would be interesting to know if they are smoothly isotopic for
.
It would be interesting to know if they are piecewise smoothly isotopic for
.
(c) For Example 3.3 gives 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.d below.
(d) Analogously one constructs the Hudson torus for
and
or, more generally,
for
and
.
There are versions
of these constructions corresponding to Definition 3.3.
For
this corresponds to the Zeeman map
[Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a,
2.3].
It would be interesting to know if the links
are isotopic, cf. [Skopenkov2015a, Remark 2.7.b].
These constructions can be further generalized [Skopenkov2016k].
4 Action by linked embedded connected sum
In this section we generalize the construction of the Hudson torus .
Let
be a closed connected oriented
-manifold.
We work in the smooth category which we omit.
Apparently analogous results hold for
in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).
Example 4.1.
For any , an embedding
and
, we shall construct an embedding
.
This embedding is said to be obtained by linked embedded connected sum of
with an
-sphere representing the `homology Alexander dual'
of
(defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).
Represent by an embedding
.
By definition, the class
is represented by properly oriented
.
Since any orientable bundle over
is trivial,
.
Take an embedding
whose image is
and which represents
.
By embedded surgery on
we obtain an embedding
representing
(see details in Proposition 4.2 below).
Define
to be the linked embedded connected sum of
and
, along some arc joining their images.
Proposition 4.2 (Embedded surgery).
For any , a neighborhood
of a codimension at least 3 subpolyhedron in
and an embedding
there is an embedding
homologous to
.
Proof. Take a vector field on normal to
.
Extend
along this vector field to a map
.
![2n>4](/images/math/3/9/f/39f044554ddcec9904d56e948f6ae327.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![\overline b](/images/math/6/9/3/6931221e5bb1220fb689387c4a965a3e.png)
Tex syntax errormisses
![U\cup g(S^1\times S^{n-1})](/images/math/8/5/e/85e5e4507d7c16a18c918dadc48f9292.png)
Since , we have
.
Hence the standard
-framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).](/images/math/a/a/3/aa3cea2126097ada817f3b858e41c7c0.png)
Take an embedding such that
Tex syntax error
with proper orientation so that is homologous to
. QED
The isotopy class of the embedding is independent of the choises in the construction.
The independence of the arc and of the maps
follows by
and by Proposition 4.3 below, respectively.
By Definition 5.1 of the Whitney invariant, is
for
odd and
for
even.
Thus by Theorem 2.1.a for
all isotopy classes of embeddings
can be obtained from any chosen embedding
by the above construction.
Proposition 4.3. For any both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of
on
.
This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].
5 The Whitney invariant
Let be a closed
-manifold.
Take an embedding
.
Fix an orientation on
.
For any other embedding
we define the Whitney invariant
![\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).](/images/math/5/2/4/5241adb18efe69df96fbf1725c71c755.png)
Here the coefficients are
if
is oriented and
is odd, and are
otherwise.
Tex syntax erroris defined as the homology class of the closure of the self-intersection set of a general position homotopy
![H](/images/math/2/f/b/2fbada10033dab2ef3330c6cb17a3a0c.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see
also [HaefligerHirsch1963].
The definition in the PL category is analogous [Hudson1969, 11], [Vrabec1977, p. 145],
[Skopenkov2006,
2.4 `The Whitney invariant'].
We begin by presenting a simpler definition, Definition 5.1, for a particular case.
For Theorem 2.1 only the case is required.
Definition 5.1.
Assume that is
-connected and
.
Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of
and
to
are isotopic, cf. [Takase2006, Lemma 2.2].
(Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of
and
to
are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since
is
-connected,
retracts to an
-dimensional polyhedron.
Therefore these restrictions are isotopic.)
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
').
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
Tex syntax erroris a compact
![(2n+1-m)](/images/math/0/6/3/063e854d764c6b9c09f9df1bcf76a85b.png)
![\partial N_0](/images/math/f/9/6/f96c15bf3a49cdf2051f311be0c68a80.png)
So carries a homology class with
coefficients.
If
is odd and
is oriented, then
has a natural orientation defined below,
and so carries a homology class with
coefficients.
Define
to be the homology class:
Tex syntax error
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error) is defined (for
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![x_f\in f\cap F](/images/math/7/f/b/7fb6f0e49619c2ef0b26f48589ee7685.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![(\xi_f,\eta_f)](/images/math/6/9/9/6994fd91df6289bee65aefab6bc3b565.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
![x_F\in B^n\times I](/images/math/3/f/6/3f69b52e956c6a6ca178831a6fe42489.png)
![Fx_F=fx_f](/images/math/c/e/2/ce2e87a9d982adc19aabb66241f97573.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![x_F](/images/math/c/2/9/c2952c4745794ea5ae3e306b271fdf36.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![df(x_f)\xi_f=dF(x_F)\xi_F](/images/math/1/b/a/1bab6ec8f7be5007b81716de60cd2716.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![(\xi_F,\eta_F)](/images/math/c/8/a/c8aa786734dd3b5a2d8c5c97b416f234.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)](/images/math/2/1/d/21d94058634f7edd21782b7de8c7c1c9.png)
![fx_f=Fx_F](/images/math/f/2/1/f216388cdabee8cb9a80855668711cfa.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Definition 5.2.
Assume that .
Take a general position homotopy
between
and
.
Tex syntax errorof the self-intersection set carries a cycle mod 2. If
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![2m\ge3n+2](/images/math/a/c/4/ac42ffc32fee13f8930d2fe0bef58252.png)
Tex syntax errorcan be assumed to be a submanifold. In general, since
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
Define the Whitney invariant to be the homology class:
Tex syntax error
Clearly, if
is isotopic to
.
Hence the Whitney invariant defines a map
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).](/images/math/c/3/0/c30f51328f41351aac3861e4f870de3e.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. is independent of the choice of a general position homotopy
from
to
.
Tex syntax errorfor a general position homotopy
![H_{01}:N\times I\times I\to\Rr^m\times I\times I](/images/math/7/d/9/7d974d2c2ed915a105323b69b287d03e.png)
![H_0,H_1:N\times I\to\Rr^m\times I](/images/math/0/8/7/0874f958e49857acc9c56909b71752e6.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
(b) Definition 5.1 is a particular case of Definition 5.2.
(Indeed, if on
, we can take
to be fixed on
. See details in [Skopenkov2010, Difference
Lemma 2.4].)
Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice
of
and of the isotopy making
outside
.
![W(f)](/images/math/2/2/1/221f32285325c78c47de94fd7c7f75ab.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error). For the reflection
![\sigma:\Rr^m\to\Rr^m](/images/math/6/1/9/6193d98c9ab0422c651027e2264dd25a.png)
![W(\sigma\circ f)=-W(f)](/images/math/1/9/2/1924babe23e26ae0ce7bf8327aff3f63.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error; for Definition 5.1 also observe that we may assume that
![f=f_0=\sigma\circ f](/images/math/c/6/c/c6cf229f35059deca4f1195f751730d9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
(d) For the Hudson tori is
or
for
,
and
.
For this is clear by Definition 5.1. For
and
this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For
the proof is analogous.
![W(f\#g)=W(f)](/images/math/7/6/e/76eb58f9e926b43c618caded88d8bf11.png)
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![g:S^n\to\Rr^m](/images/math/f/b/3/fb3a60c7de4f595f5e0bfa65d1fa30c7.png)
![W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0](/images/math/d/b/d/dbdd409aa4c1f3c7c9265288b0c274ba.png)
![H_f](/images/math/0/4/b/04bcba99b75cca1cc5abd4e5fddb95d8.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![H_g](/images/math/f/7/b/f7b9ec248e54e11f3b2ed76020fd35a5.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![H_f\sharp H_g](/images/math/2/8/e/28e29f776c97e9d5df24c5584b361f57.png)
![f\#g](/images/math/0/a/a/0aabaf11c5162ade8ba1ee6647598003.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax erroris null-homologous in
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![N\cong N\#S^n](/images/math/2/f/1/2f18f52244a27abfdb750b1a4c8861ba.png)
(f) For and
the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h,
3].
(g) The Whitney invariant need not be a bijection for . This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking
even,
non-orientable,
and applying by Theorem 2.1.b.
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically
-connected
-manifold,
. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings
are isotopic when
and
. In this section we generalize Theorem 2.1 to a description of
and further to
for
.
6.1 Examples
Some simple examples are the Hudson tori .
Example 6.1 (cf. [Skopenkov2010, Definition 1.4]).
Assume that is
-connected and
.
Then for an embedding
and a class
one can construct an embedding
by linked embedded connected sum analogously to the case
presented in Example 4.1.
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.
The embedding has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].
6.2 Classification
Theorem 6.2. Let be a closed oriented homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})](/images/math/1/f/9/1f9ca0866bacc6e8147523f65e9df3c8.png)
is a bijection, provided in the smooth category or
in the PL category.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in
[Weber1967], [Hudson1969,
11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993,
7]. The proof actually used the homological
-connectedness assumption (basically because the
-connectedness was used to ensure high enough connectedness of the complement in
to the image of
, by Alexander duality and simple connectedness of the complement, so homological
-connectedness of
is sufficient).
For Theorem 6.2 is covered by Theorem 2.1; for
it is not.
For
the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3].
For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].
An inverse to the map of Theorem 6.2 is given by Example 6.1.
Because of the existence of knotted spheres 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 spheres
. E.g.
for any
[Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2].
The following result for
was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1].
Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008,
4].
Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. 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 any 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
.
How does one describe when
is not
-connected?
For general
see the sentence on
at the end of
2.
We can say more as the connectivity
of
increases.
Some estimations of
for a closed
-connected
-manifold
are presented in [Skopenkov2010]. For
one can go even further:
Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. 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/6/4/3/643341df90bc99621c0740a72f5ab4ef.png)
The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5].
For
Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because
is
-connected).
For
Theorem 6.4 is covered by Theorem 6.2; for
it is not.
For application to knotted tori see [Skopenkov2016k, Theorem 5.1].
For generalization to arbitrary manifolds see survey [Skopenkov2006,
5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4
can be replaced by
for any
.
7 An orientation on the self-intersection set
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
Tex syntax errorof the self-intersection set of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax errorsuch that
- both
and
Tex syntax error
are subpolyhedra of some triangulation of,
- we have
and
-
is an open manifold consisting of self-transverse double points of
.
Definition 7.1 (A canonical orientation on ).
Take points
away from
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is oriented, 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
.
Remark 7.2 (Properties of the orientation just defined on )..
- A change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.
- The orientation on
need not extend to
Tex syntax error
: take the smooth coneover a general position map
having only two transverse self-intersection points, where the smooth cone is defined by
, for
and
.
- The orientation on
extends to
Tex syntax error
ifis odd [Hudson1969, Lemma 11.4].
Remark 7.3 (A canonical orientation on for
even).
This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.
Take a -base
at a point
. Since
is oriented, 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
at
.
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
.
We remark that a change of the orientation of forces changes of the signs of
and so does not change the orientation of
.
8 References
- [Adachi1993] M. Adachi, Embeddings and immersions, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension
dans
, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension
dans
, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Haefliger1966] A. Haefliger, Differential embeddings of
in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [HaefligerHirsch1963] Template:HaefligerHirsch1963
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1967] J. Hudson, Piecewise linear embeddings, Ann. of Math. (2) 85 (1967) 1–31. MR0215308 (35 #6149) Zbl 0153.25601
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Koschorke2013] U. Koschorke, Immersion, http://www.map.mpim-bonn.mpg.de/Immersion
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in
, Comment. Math. Helv. 72 (1997), 543-555.
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [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.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [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
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1977] J. Vrabec, Knotting a
-connected closed
-manifold in
, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Yasui1984] T. Yasui, Enumerating embeddings of
-manifolds in Euclidean
-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019
![n>1](/images/math/f/3/2/f32ba7f7c2722280ef2c82ff05d61be1.png)
![m \ge2n+1](/images/math/f/3/1/f312fc5cb48b012a9b7b515034a931eb.png)
![N \to\Rr^m](/images/math/e/1/d/e1d83d016b26cc4d71e3d872594d0c42.png)
![m=2n\ge6](/images/math/9/9/0/990e2dcb0be4d51d59ba1986b7a1257c.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. Denote
.
2 Classification
For the next theorem, the Whitney invariant is defined in
5 below.
Theorem 2.1. Assume that is a closed connected
-manifold, and either
or
and we are in the PL category.
(a) If is oriented, the Whitney invariant,
![\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),](/images/math/0/c/0/0c01fbf8f4f9771a7e251feb59559526.png)
is a 1-1 correspondence.
(b) If is non-orientable, then there is a 1-1 correspondence
![\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.](/images/math/2/4/e/24e59ff646a210203da373f9a602a0bf.png)
Remark 2.2 (Comments on the proof).
Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in 2], [Hudson1969,
11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].
Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in 2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.
In Part (b) we replaced the kernel from [Bausum1975, Theorem 43] by
. This is possible because, as a specialist could see,
is given by multiplication with the first Stefel-Whitney class
(which equals to the first Wu class
[Milnor&Stasheff1974, Theorem 11.4]). Since
is non-orientable,
. So by Poincaré duality,
.
The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, 5].
It would be interesting to check if part (b) is equivalent to different forms of description of
[Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
The classification of smooth embeddings of 3-manifolds in is more complicated, see Theorem 6.3 below for
or [Skopenkov2016t].
Concerning embeddings of connected -manifolds in
see [Yasui1984] for
, [Skopenkov2016f] for
, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.
Theorem 2.1 is generalized to a description of for closed
-connected
-manifolds
, see Theorem 6.2.
3 Hudson tori
Tex syntax errorto just
Tex syntax error.
Example 3.1. Let us construct, for any and
, a smooth embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
We start with the cases .
Take the standard inclusion .
The 'standard embedding'
is given by the standard inclusions
Tex syntax error
Tex syntax erroranalogously to
Tex syntax error, where
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
Take the embedding given by
Tex syntax error
Tex syntax errorjoins the images of
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\Hud_n(1)](/images/math/5/2/d/52d8f018b81e36930436db3c1454a796.png)
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For we repeat the above construction of
replacing
by
copies
of
,
.
The copies are outside
and are `parallel' to
.
The copies have the standard orientation for
or the opposite orientation for
.
Then we make embedded connected sum along natural segments joining every
-th copy to the
-th
copy.
We obtain an embedding
which has disjoint images with
.
Let
be the linked embedded connected sum of
and
.
The original motivation for Hudson was that is not isotopic to
for any
(this is a particular case of Proposition 3.2 below). One might guess that
is not isotopic
to
for
and that a
-valued invariant of
can be defined by the homotopy class of the map
Tex syntax error
However, this is only true for odd.
Proposition 3.2.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for even, by Theorem 2.1.
This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of
for the PL category).
It would be interesting to find an explicit construction of an isotopy between
and
, cf. [Vrabec1977,
5].
Analogously,
is not isotopic to
if
.
It would be interesting to know if the converse holds, e.g. is
(PS or smoothly) isotopic to
.
Example 3.3.
Take any .
Take a map
of degree
(so we can take
).
Recall that
.
Define the smooth embedding
to be the composition
Tex syntax error
Let us present a geometric description of this embedding.
Define a map by
.
This map gives an embedding
Tex syntax error
Tex syntax erroris the union of the graphs of the maps
![\overline a](/images/math/4/f/f/4ff0b2c40da836a1b7c801879c156a0d.png)
![-\overline a](/images/math/8/e/5/8e5eb3db0a5bbf6fec8a239782cdefa0.png)
![t\in S^{n-1}](/images/math/d/0/b/d0bb33dc2f67081c20c164b1a684501f.png)
![D^{n+1}\times t](/images/math/c/a/7/ca745a1e30fa17ec700f9d04c9fe8744.png)
![D^n\times t](/images/math/5/b/f/5bf99f848e37cae0bce4fe823e689da7.png)
![S^0\times t\to D^n\times t](/images/math/0/8/6/086108c6e6d192ebde0c3f5406917ebd.png)
![\Hud_n'(a)](/images/math/b/1/c/b1c5f108b78c001bf133a56e2ddc8c83.png)
![S^1\times t\to D^{n+1}\times t](/images/math/d/a/d/dad76014c549bc406ac1a4c3a7304320.png)
![t](/images/math/3/0/b/30b2ab8dc1496d06b230a71d8962af5d.png)
Remark 3.4. (a) The analogue of Proposition 3.2 for replaced to
holds, with an analogous proof.
(b) The embeddings and
are smoothly isotopic for
and are PS isotopic for
[Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]).
This follows by calculation of the Whitney invariant (Remark 5.3.d below).
It would be interesting to know if they are smoothly isotopic for
.
It would be interesting to know if they are piecewise smoothly isotopic for
.
(c) For Example 3.3 gives 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.d below.
(d) Analogously one constructs the Hudson torus for
and
or, more generally,
for
and
.
There are versions
of these constructions corresponding to Definition 3.3.
For
this corresponds to the Zeeman map
[Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a,
2.3].
It would be interesting to know if the links
are isotopic, cf. [Skopenkov2015a, Remark 2.7.b].
These constructions can be further generalized [Skopenkov2016k].
4 Action by linked embedded connected sum
In this section we generalize the construction of the Hudson torus .
Let
be a closed connected oriented
-manifold.
We work in the smooth category which we omit.
Apparently analogous results hold for
in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).
Example 4.1.
For any , an embedding
and
, we shall construct an embedding
.
This embedding is said to be obtained by linked embedded connected sum of
with an
-sphere representing the `homology Alexander dual'
of
(defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).
Represent by an embedding
.
By definition, the class
is represented by properly oriented
.
Since any orientable bundle over
is trivial,
.
Take an embedding
whose image is
and which represents
.
By embedded surgery on
we obtain an embedding
representing
(see details in Proposition 4.2 below).
Define
to be the linked embedded connected sum of
and
, along some arc joining their images.
Proposition 4.2 (Embedded surgery).
For any , a neighborhood
of a codimension at least 3 subpolyhedron in
and an embedding
there is an embedding
homologous to
.
Proof. Take a vector field on normal to
.
Extend
along this vector field to a map
.
![2n>4](/images/math/3/9/f/39f044554ddcec9904d56e948f6ae327.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![\overline b](/images/math/6/9/3/6931221e5bb1220fb689387c4a965a3e.png)
Tex syntax errormisses
![U\cup g(S^1\times S^{n-1})](/images/math/8/5/e/85e5e4507d7c16a18c918dadc48f9292.png)
Since , we have
.
Hence the standard
-framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).](/images/math/a/a/3/aa3cea2126097ada817f3b858e41c7c0.png)
Take an embedding such that
Tex syntax error
with proper orientation so that is homologous to
. QED
The isotopy class of the embedding is independent of the choises in the construction.
The independence of the arc and of the maps
follows by
and by Proposition 4.3 below, respectively.
By Definition 5.1 of the Whitney invariant, is
for
odd and
for
even.
Thus by Theorem 2.1.a for
all isotopy classes of embeddings
can be obtained from any chosen embedding
by the above construction.
Proposition 4.3. For any both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of
on
.
This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].
5 The Whitney invariant
Let be a closed
-manifold.
Take an embedding
.
Fix an orientation on
.
For any other embedding
we define the Whitney invariant
![\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).](/images/math/5/2/4/5241adb18efe69df96fbf1725c71c755.png)
Here the coefficients are
if
is oriented and
is odd, and are
otherwise.
Tex syntax erroris defined as the homology class of the closure of the self-intersection set of a general position homotopy
![H](/images/math/2/f/b/2fbada10033dab2ef3330c6cb17a3a0c.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see
also [HaefligerHirsch1963].
The definition in the PL category is analogous [Hudson1969, 11], [Vrabec1977, p. 145],
[Skopenkov2006,
2.4 `The Whitney invariant'].
We begin by presenting a simpler definition, Definition 5.1, for a particular case.
For Theorem 2.1 only the case is required.
Definition 5.1.
Assume that is
-connected and
.
Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of
and
to
are isotopic, cf. [Takase2006, Lemma 2.2].
(Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of
and
to
are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since
is
-connected,
retracts to an
-dimensional polyhedron.
Therefore these restrictions are isotopic.)
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
').
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
Tex syntax erroris a compact
![(2n+1-m)](/images/math/0/6/3/063e854d764c6b9c09f9df1bcf76a85b.png)
![\partial N_0](/images/math/f/9/6/f96c15bf3a49cdf2051f311be0c68a80.png)
So carries a homology class with
coefficients.
If
is odd and
is oriented, then
has a natural orientation defined below,
and so carries a homology class with
coefficients.
Define
to be the homology class:
Tex syntax error
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error) is defined (for
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![x_f\in f\cap F](/images/math/7/f/b/7fb6f0e49619c2ef0b26f48589ee7685.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![(\xi_f,\eta_f)](/images/math/6/9/9/6994fd91df6289bee65aefab6bc3b565.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
![x_F\in B^n\times I](/images/math/3/f/6/3f69b52e956c6a6ca178831a6fe42489.png)
![Fx_F=fx_f](/images/math/c/e/2/ce2e87a9d982adc19aabb66241f97573.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![x_F](/images/math/c/2/9/c2952c4745794ea5ae3e306b271fdf36.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![df(x_f)\xi_f=dF(x_F)\xi_F](/images/math/1/b/a/1bab6ec8f7be5007b81716de60cd2716.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![(\xi_F,\eta_F)](/images/math/c/8/a/c8aa786734dd3b5a2d8c5c97b416f234.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)](/images/math/2/1/d/21d94058634f7edd21782b7de8c7c1c9.png)
![fx_f=Fx_F](/images/math/f/2/1/f216388cdabee8cb9a80855668711cfa.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Definition 5.2.
Assume that .
Take a general position homotopy
between
and
.
Tex syntax errorof the self-intersection set carries a cycle mod 2. If
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![2m\ge3n+2](/images/math/a/c/4/ac42ffc32fee13f8930d2fe0bef58252.png)
Tex syntax errorcan be assumed to be a submanifold. In general, since
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
Define the Whitney invariant to be the homology class:
Tex syntax error
Clearly, if
is isotopic to
.
Hence the Whitney invariant defines a map
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).](/images/math/c/3/0/c30f51328f41351aac3861e4f870de3e.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. is independent of the choice of a general position homotopy
from
to
.
Tex syntax errorfor a general position homotopy
![H_{01}:N\times I\times I\to\Rr^m\times I\times I](/images/math/7/d/9/7d974d2c2ed915a105323b69b287d03e.png)
![H_0,H_1:N\times I\to\Rr^m\times I](/images/math/0/8/7/0874f958e49857acc9c56909b71752e6.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
(b) Definition 5.1 is a particular case of Definition 5.2.
(Indeed, if on
, we can take
to be fixed on
. See details in [Skopenkov2010, Difference
Lemma 2.4].)
Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice
of
and of the isotopy making
outside
.
![W(f)](/images/math/2/2/1/221f32285325c78c47de94fd7c7f75ab.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error). For the reflection
![\sigma:\Rr^m\to\Rr^m](/images/math/6/1/9/6193d98c9ab0422c651027e2264dd25a.png)
![W(\sigma\circ f)=-W(f)](/images/math/1/9/2/1924babe23e26ae0ce7bf8327aff3f63.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error; for Definition 5.1 also observe that we may assume that
![f=f_0=\sigma\circ f](/images/math/c/6/c/c6cf229f35059deca4f1195f751730d9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
(d) For the Hudson tori is
or
for
,
and
.
For this is clear by Definition 5.1. For
and
this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For
the proof is analogous.
![W(f\#g)=W(f)](/images/math/7/6/e/76eb58f9e926b43c618caded88d8bf11.png)
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![g:S^n\to\Rr^m](/images/math/f/b/3/fb3a60c7de4f595f5e0bfa65d1fa30c7.png)
![W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0](/images/math/d/b/d/dbdd409aa4c1f3c7c9265288b0c274ba.png)
![H_f](/images/math/0/4/b/04bcba99b75cca1cc5abd4e5fddb95d8.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![H_g](/images/math/f/7/b/f7b9ec248e54e11f3b2ed76020fd35a5.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![H_f\sharp H_g](/images/math/2/8/e/28e29f776c97e9d5df24c5584b361f57.png)
![f\#g](/images/math/0/a/a/0aabaf11c5162ade8ba1ee6647598003.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax erroris null-homologous in
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![N\cong N\#S^n](/images/math/2/f/1/2f18f52244a27abfdb750b1a4c8861ba.png)
(f) For and
the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h,
3].
(g) The Whitney invariant need not be a bijection for . This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking
even,
non-orientable,
and applying by Theorem 2.1.b.
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically
-connected
-manifold,
. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings
are isotopic when
and
. In this section we generalize Theorem 2.1 to a description of
and further to
for
.
6.1 Examples
Some simple examples are the Hudson tori .
Example 6.1 (cf. [Skopenkov2010, Definition 1.4]).
Assume that is
-connected and
.
Then for an embedding
and a class
one can construct an embedding
by linked embedded connected sum analogously to the case
presented in Example 4.1.
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.
The embedding has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].
6.2 Classification
Theorem 6.2. Let be a closed oriented homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})](/images/math/1/f/9/1f9ca0866bacc6e8147523f65e9df3c8.png)
is a bijection, provided in the smooth category or
in the PL category.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in
[Weber1967], [Hudson1969,
11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993,
7]. The proof actually used the homological
-connectedness assumption (basically because the
-connectedness was used to ensure high enough connectedness of the complement in
to the image of
, by Alexander duality and simple connectedness of the complement, so homological
-connectedness of
is sufficient).
For Theorem 6.2 is covered by Theorem 2.1; for
it is not.
For
the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3].
For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].
An inverse to the map of Theorem 6.2 is given by Example 6.1.
Because of the existence of knotted spheres 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 spheres
. E.g.
for any
[Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2].
The following result for
was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1].
Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008,
4].
Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. 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 any 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
.
How does one describe when
is not
-connected?
For general
see the sentence on
at the end of
2.
We can say more as the connectivity
of
increases.
Some estimations of
for a closed
-connected
-manifold
are presented in [Skopenkov2010]. For
one can go even further:
Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. 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/6/4/3/643341df90bc99621c0740a72f5ab4ef.png)
The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5].
For
Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because
is
-connected).
For
Theorem 6.4 is covered by Theorem 6.2; for
it is not.
For application to knotted tori see [Skopenkov2016k, Theorem 5.1].
For generalization to arbitrary manifolds see survey [Skopenkov2006,
5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4
can be replaced by
for any
.
7 An orientation on the self-intersection set
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
Tex syntax errorof the self-intersection set of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax errorsuch that
- both
and
Tex syntax error
are subpolyhedra of some triangulation of,
- we have
and
-
is an open manifold consisting of self-transverse double points of
.
Definition 7.1 (A canonical orientation on ).
Take points
away from
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is oriented, 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
.
Remark 7.2 (Properties of the orientation just defined on )..
- A change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.
- The orientation on
need not extend to
Tex syntax error
: take the smooth coneover a general position map
having only two transverse self-intersection points, where the smooth cone is defined by
, for
and
.
- The orientation on
extends to
Tex syntax error
ifis odd [Hudson1969, Lemma 11.4].
Remark 7.3 (A canonical orientation on for
even).
This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.
Take a -base
at a point
. Since
is oriented, 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
at
.
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
.
We remark that a change of the orientation of forces changes of the signs of
and so does not change the orientation of
.
8 References
- [Adachi1993] M. Adachi, Embeddings and immersions, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001
- [Avvakumov2017] S. Avvakumov, The classification of linked 3-manifolds in 6-space, Algebraic & Geometric Topology, to appear. arxiv preprint.
- [Bausum1975] D. R. Bausum, Embeddings and immersions of manifolds in Euclidean space, Trans. Amer. Math. Soc. 213 (1975), 263–303. MR0474330 (57 #13976) Zbl 0323.57017
- [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
- [Boechat&Haefliger1970] J. Boéchat and A. Haefliger, Plongements différentiables des variétés orientées de dimension
dans
, (French) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 156–166. MR0270384 (42 #5273) Zbl 0199.27003
- [Boechat1971] J. Boéchat, Plongements de variétées différentiables orientées de dimension
dans
, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
- [Haefliger1962b] A. Haefliger, Plongements de variétés dans le domain stable, Séminaire Bourbaki, 245 (1962).
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Haefliger1966] A. Haefliger, Differential embeddings of
in
for
, Ann. of Math. (2) 83 (1966), 402–436. MR0202151 (34 #2024) Zbl 0151.32502
- [HaefligerHirsch1963] Template:HaefligerHirsch1963
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1967] J. Hudson, Piecewise linear embeddings, Ann. of Math. (2) 85 (1967) 1–31. MR0215308 (35 #6149) Zbl 0153.25601
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Koschorke2013] U. Koschorke, Immersion, http://www.map.mpim-bonn.mpg.de/Immersion
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Skopenkov1997] A. Skopenkov, On the deleted product criterion for embeddability of manifolds in
, Comment. Math. Helv. 72 (1997), 543-555.
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [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.
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
- [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
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
- [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.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
- [Skopenkov2016s] A. Skopenkov, Knots, i.e. embeddings of spheres, preprint.
- [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.
- [Takase2006] M. Takase, Homology 3-spheres in codimension three, Internat. J. Math. 17 (2006), no.8, 869–885.
arxiv:math/0506464 MR2261638 (2007g:57049) Zbl 1113.57013
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1977] J. Vrabec, Knotting a
-connected closed
-manifold in
, Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
- [Weber1967] C. Weber, Plongements de polyedres dans le domaine metastable, Comment. Math. Helv. 42 (1967), 1-27. MR0238330 (38 #6606) Zbl 0152.22402
- [Yasui1984] T. Yasui, Enumerating embeddings of
-manifolds in Euclidean
-space, J. Math. Soc. Japan 36 (1984), no.4, 555–576. MR759414 Zbl 0557.57019
![n>1](/images/math/f/3/2/f32ba7f7c2722280ef2c82ff05d61be1.png)
![m \ge2n+1](/images/math/f/3/1/f312fc5cb48b012a9b7b515034a931eb.png)
![N \to\Rr^m](/images/math/e/1/d/e1d83d016b26cc4d71e3d872594d0c42.png)
![m=2n\ge6](/images/math/9/9/0/990e2dcb0be4d51d59ba1986b7a1257c.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. Denote
.
2 Classification
For the next theorem, the Whitney invariant is defined in
5 below.
Theorem 2.1. Assume that is a closed connected
-manifold, and either
or
and we are in the PL category.
(a) If is oriented, the Whitney invariant,
![\displaystyle W:E^{2n}(N)\to H_1(N;\Zz_{\varepsilon(n-1)}),](/images/math/0/c/0/0c01fbf8f4f9771a7e251feb59559526.png)
is a 1-1 correspondence.
(b) If is non-orientable, then there is a 1-1 correspondence
![\displaystyle E^{2n}(N)\to \begin{cases} H_1(N;\Zz_2) & n\text{ is odd}\\ \Zz\oplus\Zz_2^{s-1}\quad\mbox{where}\quad H_1(N;\Zz_2)\cong\Zz_2^s & n\text{ is even}\end{cases}.](/images/math/2/4/e/24e59ff646a210203da373f9a602a0bf.png)
Remark 2.2 (Comments on the proof).
Part (a) is proved in [Haefliger&Hirsch1963, Theorem 2.4] in the smooth category, and in [Weber1967, Theorem 4' in 2], [Hudson1969,
11], [Vrabec1977, Theorems 1.1 and 1.2] in the PL category, see also [Haefliger1962b, 1.3.e], [Haefliger1963], [Bausum1975, Theorem 43].
Part (b) is proved in [Bausum1975, Theorem 43] in the smooth category. By [Weber1967, Theorems 1 and 1' in 2], [Skopenkov1997, Theorem 1.1.c] the proof works also in the PL category.
In Part (b) we replaced the kernel from [Bausum1975, Theorem 43] by
. This is possible because, as a specialist could see,
is given by multiplication with the first Stefel-Whitney class
(which equals to the first Wu class
[Milnor&Stasheff1974, Theorem 11.4]). Since
is non-orientable,
. So by Poincaré duality,
.
The 1-1 correspondence from (b) can presumably be defined as a (generalized) Whitney invariant, see [Vrabec1977], but the proof used 'the Haefliger-Wu invariant' whose definition can be found e.g. in [Skopenkov2006, 5].
It would be interesting to check if part (b) is equivalent to different forms of description of
[Haefliger1962b, 1.3.e], [Haefliger1963], [Vrabec1977, Theorem 1.1].
The classification of smooth embeddings of 3-manifolds in is more complicated, see Theorem 6.3 below for
or [Skopenkov2016t].
Concerning embeddings of connected -manifolds in
see [Yasui1984] for
, [Skopenkov2016f] for
, and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for manifolds with boundary.
Theorem 2.1 is generalized to a description of for closed
-connected
-manifolds
, see Theorem 6.2.
3 Hudson tori
Tex syntax errorto just
Tex syntax error.
Example 3.1. Let us construct, for any and
, a smooth embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
We start with the cases .
Take the standard inclusion .
The 'standard embedding'
is given by the standard inclusions
Tex syntax error
Tex syntax erroranalogously to
Tex syntax error, where
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
Take the embedding given by
Tex syntax error
Tex syntax errorjoins the images of
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\Hud_n(1)](/images/math/5/2/d/52d8f018b81e36930436db3c1454a796.png)
![\Hud_n(0)](/images/math/4/a/f/4af6f5ce195d8f168b5b889e4f281f1c.png)
![g_1](/images/math/4/4/b/44be6ace5668ee8cd97c7accc925aab4.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For we repeat the above construction of
replacing
by
copies
of
,
.
The copies are outside
and are `parallel' to
.
The copies have the standard orientation for
or the opposite orientation for
.
Then we make embedded connected sum along natural segments joining every
-th copy to the
-th
copy.
We obtain an embedding
which has disjoint images with
.
Let
be the linked embedded connected sum of
and
.
The original motivation for Hudson was that is not isotopic to
for any
(this is a particular case of Proposition 3.2 below). One might guess that
is not isotopic
to
for
and that a
-valued invariant of
can be defined by the homotopy class of the map
Tex syntax error
However, this is only true for odd.
Proposition 3.2.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 3.2 follows by calculation of the Whitney invariant (Remark 5.3.d below) and, for even, by Theorem 2.1.
This proposition holds with the same proof in the piecewise smooth category, whose definition is recalled in [Skopenkov2016f, Remark 1.1]). Proposition 3.2 also holds in the PL category (with an analogous construction of
for the PL category).
It would be interesting to find an explicit construction of an isotopy between
and
, cf. [Vrabec1977,
5].
Analogously,
is not isotopic to
if
.
It would be interesting to know if the converse holds, e.g. is
(PS or smoothly) isotopic to
.
Example 3.3.
Take any .
Take a map
of degree
(so we can take
).
Recall that
.
Define the smooth embedding
to be the composition
Tex syntax error
Let us present a geometric description of this embedding.
Define a map by
.
This map gives an embedding
Tex syntax error
Tex syntax erroris the union of the graphs of the maps
![\overline a](/images/math/4/f/f/4ff0b2c40da836a1b7c801879c156a0d.png)
![-\overline a](/images/math/8/e/5/8e5eb3db0a5bbf6fec8a239782cdefa0.png)
![t\in S^{n-1}](/images/math/d/0/b/d0bb33dc2f67081c20c164b1a684501f.png)
![D^{n+1}\times t](/images/math/c/a/7/ca745a1e30fa17ec700f9d04c9fe8744.png)
![D^n\times t](/images/math/5/b/f/5bf99f848e37cae0bce4fe823e689da7.png)
![S^0\times t\to D^n\times t](/images/math/0/8/6/086108c6e6d192ebde0c3f5406917ebd.png)
![\Hud_n'(a)](/images/math/b/1/c/b1c5f108b78c001bf133a56e2ddc8c83.png)
![S^1\times t\to D^{n+1}\times t](/images/math/d/a/d/dad76014c549bc406ac1a4c3a7304320.png)
![t](/images/math/3/0/b/30b2ab8dc1496d06b230a71d8962af5d.png)
Remark 3.4. (a) The analogue of Proposition 3.2 for replaced to
holds, with an analogous proof.
(b) The embeddings and
are smoothly isotopic for
and are PS isotopic for
[Skopenkov2006a, commutativity of the left upper square in the Restriction Lemma 5.2], [Skopenkov2015a, Lemma 2.15.c] (see [Skopenkov2016f, Remark 1.2]).
This follows by calculation of the Whitney invariant (Remark 5.3.d below).
It would be interesting to know if they are smoothly isotopic for
.
It would be interesting to know if they are piecewise smoothly isotopic for
.
(c) For Example 3.3 gives 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.d below.
(d) Analogously one constructs the Hudson torus for
and
or, more generally,
for
and
.
There are versions
of these constructions corresponding to Definition 3.3.
For
this corresponds to the Zeeman map
[Skopenkov2016h, Definition 2.2] and its composition with 'the unframed second Kirby move' [Skopenkov2015a,
2.3].
It would be interesting to know if the links
are isotopic, cf. [Skopenkov2015a, Remark 2.7.b].
These constructions can be further generalized [Skopenkov2016k].
4 Action by linked embedded connected sum
In this section we generalize the construction of the Hudson torus .
Let
be a closed connected oriented
-manifold.
We work in the smooth category which we omit.
Apparently analogous results hold for
in the PL and PS categories (see [Skopenkov2016f, Remark 1.2]).
Example 4.1.
For any , an embedding
and
, we shall construct an embedding
.
This embedding is said to be obtained by linked embedded connected sum of
with an
-sphere representing the `homology Alexander dual'
of
(defined in [Skopenkov2005, Alexander Duality Lemma 4.6]).
Represent by an embedding
.
By definition, the class
is represented by properly oriented
.
Since any orientable bundle over
is trivial,
.
Take an embedding
whose image is
and which represents
.
By embedded surgery on
we obtain an embedding
representing
(see details in Proposition 4.2 below).
Define
to be the linked embedded connected sum of
and
, along some arc joining their images.
Proposition 4.2 (Embedded surgery).
For any , a neighborhood
of a codimension at least 3 subpolyhedron in
and an embedding
there is an embedding
homologous to
.
Proof. Take a vector field on normal to
.
Extend
along this vector field to a map
.
![2n>4](/images/math/3/9/f/39f044554ddcec9904d56e948f6ae327.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![U](/images/math/d/3/8/d38dd561606a88c6780fa908c4968cff.png)
![\overline b](/images/math/6/9/3/6931221e5bb1220fb689387c4a965a3e.png)
Tex syntax errormisses
![U\cup g(S^1\times S^{n-1})](/images/math/8/5/e/85e5e4507d7c16a18c918dadc48f9292.png)
Since , we have
.
Hence the standard
-framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat b:D^2\times D^{n-1}\to\R^{2n}-U\quad\text{such that}\quad \widehat b(\partial D^2\times D^{n-1})\subset g(S^1\times S^{n-1}).](/images/math/a/a/3/aa3cea2126097ada817f3b858e41c7c0.png)
Take an embedding such that
Tex syntax error
with proper orientation so that is homologous to
. QED
The isotopy class of the embedding is independent of the choises in the construction.
The independence of the arc and of the maps
follows by
and by Proposition 4.3 below, respectively.
By Definition 5.1 of the Whitney invariant, is
for
odd and
for
even.
Thus by Theorem 2.1.a for
all isotopy classes of embeddings
can be obtained from any chosen embedding
by the above construction.
Proposition 4.3. For any both the linked embedded connected sum and parametric connected sum (introduced in [Skopenkov2006a], [Skopenkov2015a]) define free transitive actions of
on
.
This follows by Theorem 2.1.a and by [Skopenkov2014, Remark 18.a].
5 The Whitney invariant
Let be a closed
-manifold.
Take an embedding
.
Fix an orientation on
.
For any other embedding
we define the Whitney invariant
![\displaystyle W(f, f_0)=W_{f_0}(f)=W(f)\in H_{2n-m+1}(N;\Zz_N).](/images/math/5/2/4/5241adb18efe69df96fbf1725c71c755.png)
Here the coefficients are
if
is oriented and
is odd, and are
otherwise.
Tex syntax erroris defined as the homology class of the closure of the self-intersection set of a general position homotopy
![H](/images/math/2/f/b/2fbada10033dab2ef3330c6cb17a3a0c.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
This is formalized in Definition 5.2 in the smooth category, following [Skopenkov2010], see
also [HaefligerHirsch1963].
The definition in the PL category is analogous [Hudson1969, 11], [Vrabec1977, p. 145],
[Skopenkov2006,
2.4 `The Whitney invariant'].
We begin by presenting a simpler definition, Definition 5.1, for a particular case.
For Theorem 2.1 only the case is required.
Definition 5.1.
Assume that is
-connected and
.
Then by [Haefliger&Hirsch1963, Theorem 3.1.b] restrictions of
and
to
are isotopic, cf. [Takase2006, Lemma 2.2].
(Here is sketch of an argument. Using the Smale-Hirsch classification of immersions we obtain that restrictions of
and
to
are `regular homotopic', see [Koschorke2013, Definition 2.7]. Since
is
-connected,
retracts to an
-dimensional polyhedron.
Therefore these restrictions are isotopic.)
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
').
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
Tex syntax erroris a compact
![(2n+1-m)](/images/math/0/6/3/063e854d764c6b9c09f9df1bcf76a85b.png)
![\partial N_0](/images/math/f/9/6/f96c15bf3a49cdf2051f311be0c68a80.png)
So carries a homology class with
coefficients.
If
is odd and
is oriented, then
has a natural orientation defined below,
and so carries a homology class with
coefficients.
Define
to be the homology class:
Tex syntax error
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error) is defined (for
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![x_f\in f\cap F](/images/math/7/f/b/7fb6f0e49619c2ef0b26f48589ee7685.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![(\xi_f,\eta_f)](/images/math/6/9/9/6994fd91df6289bee65aefab6bc3b565.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![n+2(n+1)<2m](/images/math/4/9/2/492957e7b3a24f7196e54cb98a8a21b0.png)
![x_F\in B^n\times I](/images/math/3/f/6/3f69b52e956c6a6ca178831a6fe42489.png)
![Fx_F=fx_f](/images/math/c/e/2/ce2e87a9d982adc19aabb66241f97573.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![x_f](/images/math/2/5/0/250a6d662fd43f608d2b574ce8359a4d.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![x_F](/images/math/c/2/9/c2952c4745794ea5ae3e306b271fdf36.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![df(x_f)\xi_f=dF(x_F)\xi_F](/images/math/1/b/a/1bab6ec8f7be5007b81716de60cd2716.png)
![\xi_F](/images/math/c/a/c/cacbcda5312fdc1a88e434b80e637b6a.png)
![(\xi_F,\eta_F)](/images/math/c/8/a/c8aa786734dd3b5a2d8c5c97b416f234.png)
![B^n\times I](/images/math/2/0/0/200a06208941104fdb81594ea819ffda.png)
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\zeta=(df(x_f)\xi_f,df(x_f)\eta_f,dF(x_F)\eta_F)](/images/math/2/1/d/21d94058634f7edd21782b7de8c7c1c9.png)
![fx_f=Fx_F](/images/math/f/2/1/f216388cdabee8cb9a80855668711cfa.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![\xi_f](/images/math/8/d/2/8d28b94db8e9d6ece3c3b912fac89c7b.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
![\zeta](/images/math/4/d/9/4d93e5b9f68d271cbf8df4a50bdbfefd.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Definition 5.2.
Assume that .
Take a general position homotopy
between
and
.
Tex syntax errorof the self-intersection set carries a cycle mod 2. If
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![2m\ge3n+2](/images/math/a/c/4/ac42ffc32fee13f8930d2fe0bef58252.png)
Tex syntax errorcan be assumed to be a submanifold. In general, since
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![m-n](/images/math/b/6/e/b6e54703653c23e61afe266a3bdbdd5f.png)
Define the Whitney invariant to be the homology class:
Tex syntax error
Clearly, if
is isotopic to
.
Hence the Whitney invariant defines a map
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N;\Zz_N),\quad [f] \mapsto W(f).](/images/math/c/3/0/c30f51328f41351aac3861e4f870de3e.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. is independent of the choice of a general position homotopy
from
to
.
Tex syntax errorfor a general position homotopy
![H_{01}:N\times I\times I\to\Rr^m\times I\times I](/images/math/7/d/9/7d974d2c2ed915a105323b69b287d03e.png)
![H_0,H_1:N\times I\to\Rr^m\times I](/images/math/0/8/7/0874f958e49857acc9c56909b71752e6.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
(b) Definition 5.1 is a particular case of Definition 5.2.
(Indeed, if on
, we can take
to be fixed on
. See details in [Skopenkov2010, Difference
Lemma 2.4].)
Hence the Whitney invariant is well-defined by Definition 5.1, i.e. independent of the choice
of
and of the isotopy making
outside
.
![W(f)](/images/math/2/2/1/221f32285325c78c47de94fd7c7f75ab.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error). For the reflection
![\sigma:\Rr^m\to\Rr^m](/images/math/6/1/9/6193d98c9ab0422c651027e2264dd25a.png)
![W(\sigma\circ f)=-W(f)](/images/math/1/9/2/1924babe23e26ae0ce7bf8327aff3f63.png)
![\Rr^m](/images/math/7/f/2/7f261d3a1d5315d963aab7981fccbd80.png)
![f\cap F](/images/math/6/a/3/6a3602509e8d9d182f7d67881eff9fc0.png)
Tex syntax error; for Definition 5.1 also observe that we may assume that
![f=f_0=\sigma\circ f](/images/math/c/6/c/c6cf229f35059deca4f1195f751730d9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
(d) For the Hudson tori is
or
for
,
and
.
For this is clear by Definition 5.1. For
and
this was proved in [Hudson1963] (with a different but equivalent definition of the Whitney invariant; using and proving a particular case of Remark 5.3.f). For
the proof is analogous.
![W(f\#g)=W(f)](/images/math/7/6/e/76eb58f9e926b43c618caded88d8bf11.png)
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![g:S^n\to\Rr^m](/images/math/f/b/3/fb3a60c7de4f595f5e0bfa65d1fa30c7.png)
![W(f\#g)-W(f)=W(f\#g,f_0)-W(f,f_0)=W(f\#g,f)=0](/images/math/d/b/d/dbdd409aa4c1f3c7c9265288b0c274ba.png)
![H_f](/images/math/0/4/b/04bcba99b75cca1cc5abd4e5fddb95d8.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![H_g](/images/math/f/7/b/f7b9ec248e54e11f3b2ed76020fd35a5.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![H_f\sharp H_g](/images/math/2/8/e/28e29f776c97e9d5df24c5584b361f57.png)
![f\#g](/images/math/0/a/a/0aabaf11c5162ade8ba1ee6647598003.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax erroris null-homologous in
![S^n](/images/math/a/f/d/afd38444b95e8b5abbf51c458ea39cbc.png)
![N\cong N\#S^n](/images/math/2/f/1/2f18f52244a27abfdb750b1a4c8861ba.png)
(f) For and
the Whitney invariant equals to the pair of linking coefficients [Skopenkov2016h,
3].
(g) The Whitney invariant need not be a bijection for . This is seen, for example, by applying Theorem 6.4 below in case of knotted tori [Skopenkov2016k, Theorem 5.1]) or by taking
even,
non-orientable,
and applying by Theorem 2.1.b.
6 A generalization to highly-connected manifolds
In this section let be a closed orientable homologically
-connected
-manifold,
. Recall the unknotting theorem [Skopenkov2016c, Theorem 2.4] that all embeddings
are isotopic when
and
. In this section we generalize Theorem 2.1 to a description of
and further to
for
.
6.1 Examples
Some simple examples are the Hudson tori .
Example 6.1 (cf. [Skopenkov2010, Definition 1.4]).
Assume that is
-connected and
.
Then for an embedding
and a class
one can construct an embedding
by linked embedded connected sum analogously to the case
presented in Example 4.1.
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.
The embedding has an alternative construction using parametric connected sum [Skopenkov2014, Remark 18.a].
6.2 Classification
Theorem 6.2. Let be a closed oriented homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N;\Zz_{\varepsilon(n-k-1)})](/images/math/1/f/9/1f9ca0866bacc6e8147523f65e9df3c8.png)
is a bijection, provided in the smooth category or
in the PL category.
This was proved for -connected manifolds in the smooth category [Haefliger&Hirsch1963, Theorem 2.4], and in the PL category in
[Weber1967], [Hudson1969,
11], cf. [Boechat&Haefliger1970, Theorem 1.6], [Boechat1971, Theorem 4.2], [Vrabec1977, Theorems 1.1 and 1.2], [Adachi1993,
7]. The proof actually used the homological
-connectedness assumption (basically because the
-connectedness was used to ensure high enough connectedness of the complement in
to the image of
, by Alexander duality and simple connectedness of the complement, so homological
-connectedness of
is sufficient).
For Theorem 6.2 is covered by Theorem 2.1; for
it is not.
For
the PL case of Theorem 6.2 gives nothing but the Zeeman Unknotting Spheres Theorem [Skopenkov2016c, Theorem 2.3].
For the case of knotted tori see [Skopenkov2016k, Theorem 3.1].
An inverse to the map of Theorem 6.2 is given by Example 6.1.
Because of the existence of knotted spheres 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 spheres
. E.g.
for any
[Haefliger1966, Corollary 8.14], [Skopenkov2016s, Theorem 3.2].
The following result for
was obtained using the Boéchat-Haefliger formula for the smoothing obstruction [Boechat&Haefliger1970, Theorem 2.1], [Boechat1971, Theorem 5.1].
Using that formula one can define the higher-dimensional Kreck invariant [Skopenkov2008,
4].
Theorem 6.3 [Skopenkov2008, Higher-dimensional Classification Theorem]. 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 any 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
.
How does one describe when
is not
-connected?
For general
see the sentence on
at the end of
2.
We can say more as the connectivity
of
increases.
Some estimations of
for a closed
-connected
-manifold
are presented in [Skopenkov2010]. For
one can go even further:
Theorem 6.4 [Becker&Glover1971, Corollary 1.3]. 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/6/4/3/643341df90bc99621c0740a72f5ab4ef.png)
The 1-1 correspondence is constructed using `the Haefliger-Wu invariant' involving the configuration space of distinct pairs [Skopenkov2006, 5].
For
Theorem 6.4 is the same as General Position Theorem [Skopenkov2016c, Theorem 2.1] (because
is
-connected).
For
Theorem 6.4 is covered by Theorem 6.2; for
it is not.
For application to knotted tori see [Skopenkov2016k, Theorem 5.1].
For generalization to arbitrary manifolds see survey [Skopenkov2006,
5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Observe that in Theorem 6.4
can be replaced by
for any
.
7 An orientation on the self-intersection set
![f:N\to\Rr^m](/images/math/e/3/d/e3d1bfaded40e9891440dbe20c1baeeb.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![m\ge n+2](/images/math/1/f/2/1f2b3a0c2d4e3f11b684651d72a3e81d.png)
Tex syntax errorof the self-intersection set of
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
Tex syntax errorsuch that
- both
and
Tex syntax error
are subpolyhedra of some triangulation of,
- we have
and
-
is an open manifold consisting of self-transverse double points of
.
Definition 7.1 (A canonical orientation on ).
Take points
away from
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is oriented, 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
.
Remark 7.2 (Properties of the orientation just defined on )..
- A change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.
- The orientation on
need not extend to
Tex syntax error
: take the smooth coneover a general position map
having only two transverse self-intersection points, where the smooth cone is defined by
, for
and
.
- The orientation on
extends to
Tex syntax error
ifis odd [Hudson1969, Lemma 11.4].
Remark 7.3 (A canonical orientation on for
even).
This remark is added as a complement for Definition 7.1 but is not needed for the definition of the Whitney invariant.
Take a -base
at a point
. Since
is oriented, 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
at
.
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
.
We remark 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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