Embeddings just below the stable range: classification
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Contents |
1 Introduction
Recall the unknotting theorem that if is a connected manifold of dimension
, then there is just one isotopy class of embedding
if
. In this page we summarise the situation for
, and give references to the case
.
For notation and conventions see high codimension embeddings.
2 Classification
Classification Theorem 2.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s,\end{array}\right.](/images/math/9/7/7/977eb87ba150391f256831cd135a87cc.png)
is bijective if either or
and CAT=PL [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]
The Whitney invariant is defined below.
The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of see Embeddings of highly-connected manifolds. Some estimations of
for a closed
-connected
-manifold
and
(including
) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.
3 Examples
Together with the Haefliger knotted sphere , Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).
For define the
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection, we recall for and
. Hudson's construction of embedings
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
Take the standard embeddings (where
means homothety with coefficient 2) and
.
Fix a point
.
The Hudson torus
is the embedded connected sum of
![\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.](/images/math/5/8/2/582d83ee56b6551b010d068fd3f28ee3.png)
(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)
For instead of an embedded
-sphere
we can take
copies
(
) of
-sphere outside
`parallel' to
.
Then we join these spheres by tubes so that the homotopy class of the resulting embedding
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/5/e/4/5e4f894b5aa579c4e813954dd0f3454f.png)
Let be the connected sum of this embedding with the above standard
embedding
.
Clearly, is isotopic to the standard embedding.
Proposition 3.1.
For odd
is isotopic to
if and only
if
.
For even
is isotopic to
if and only if
.
In particular, is not isotopic to
for each
(this was the original motivation for Hudson).
Proposition 3.1 follows by Remark 5.e and, for even, by Theorem 4, both results from classification just below the stable range.
It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of
Proposition 3.1 for
.
3.2 Hudson tori 2
In this subsection we give, for and
another construction of embeddings
![\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/f/0/3/f03bdba139558d9b127d28e84c53d4b5.png)
Define a map to be the constant
on one
component
and the standard embedding
on the other component.
This map gives an
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset \Rr^{2n}.](/images/math/7/a/7/7a7bb377fc8f6be96b696d21d08e3952.png)
(See Figure 2.2 of [Skopenkov2006].)
Each disk intersects the image of this embedding at two
points lying in
.
Extend this embedding
for each
to an
embedding
.
(See Figure 2.3 of [Skopenkov2006].)
Thus we obtain the Hudson torus
![\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/4/4/d/44d0d089254c0e5a3b52719760ea1909.png)
The embedding is obtained in the same way starting from a map
of degree
.
The same proposition as above holds with replaced to
.
3.3 Remarks
We have is PL isotopic to
[Skopenkov2006a].
It would be interesting to prove the smooth analogue of this result.
For these construction give what we call the
Hudson torus.
The
Hudson torus is constructed analogously and is the
composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the Hudson torus
for
or, more generally,
for
and
for
.
3.4 An action of the first homology group on embeddings
In this subsection, for and for orientable
with
, we construct an embedding
from an embedding
.
For , represent
by an embedding
. Since any orientable bundle over
is trivial,
. Identify
with
. It remains to make an embedded surgery of
to obtain an
-sphere
, and then we set
.
Take a vector field on normal to
. Extend
along this vector field to a smooth map
. Since
and
, by general position we may assume that
is an embedding and
misses
. Since
, we have
.
Hence the standard framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}](/images/math/4/0/f/40f08ff9a14fa1038d146656cb8fbf18.png)
![\displaystyle \mbox{Let}\qquad \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.](/images/math/9/8/5/985515b95fa1549044a22b99def3232e.png)
This construction generalizes the construction of (from
).
Clearly, is
or
. Thus unless
and CAT=DIFF
- all isotopy classes of embedings
can be obtained (from a certain given embedding
) by the above construction;
- the above construction defines an action
.
4 The Whitney invariant (for either n odd or N orientable)
Fix orientations on and, if
is even, on
. Fix an embedding
. For an embedding
the restrictions of
and
to
are regular homotopic [Hirsch1959]. Since
has an
-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of
and assume that
on
. Take a general position homotopy
relative to
between the restrictions of
and
to
. Then
(i.e. `the intersection of this homotopy with
') is a 1-manifold (possibly non-compact) without boundary. Define
to be the homology class of the closure of this 1-manifold:
![\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).](/images/math/2/b/b/2bbc331d0f1495918467d3ff52161652.png)
The orientation on is defined for
orientable as follows. (This orientation is defined for each
but used only for odd
.) For each point
take a vector at
tangent to
. Complete this vector to a positive base tangent to
. Since
, by general position there is a unique point
such that
. The tangent vector at
thus gives a tangent vector at
to
. Complete this vector to a positive base tangent to
, where the orientation on
comes from
. The union of the images of the constructed two bases is a base at
of
. If this base is positive, then call the initial vector of
positive. Since a change of the orientation on
forces a change of the orientation of the latter base of
, it follows that this condition indeed defines an orientation on
.
Remark 4.1.
- The Whitney invariant is well-defined, i.e. independent of the choice of
and of the isotopy making
outside
. This is so because the above definition is clearly equivalent to the following:
is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set
of a general position homotopy
between
and
. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need
-coefficients when
is even.
- Clearly,
. The definition of
depends on the choice of
, but we write
not
for brevity.
- Since a change of the orientation on
forces a change of the orientation on
, the class
is independent of the choice of the orientation on
. For the reflection
with respect to a hyperplane we have
(because we may assume that
on
and because a change of the orientation of
forces a change of the orientation of
).
- The above definition makes sense for each
, not only for
.
- Clearly,
is
or
for
for the Hudson tori.
for each embeddings
and
.
5 A generalization to highly-connected manifolds
Examples are the above Hudson tori . See also [Milgram&Rees1971].
5.1 Classification
Theorem 5.1. Let be a closed orientable homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})](/images/math/6/e/f/6ef58c45d6f2c416944ab02e70082041.png)
is a bijection, provided or
in the PL or DIFF categories, respectively.
Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977]
-connected manifolds. The proof works for
-connected manifolds.
E.g. by Theorem 5.1 we obtain that the Whitney invariant is bijective for
.
It is in fact a group isomorphism; the generator is the Hudson torus.
The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for .
Analogously to Theorem 5.1 it may be proved that
- if
is a closed connected non-orientable
-manifold, then
![\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},](/images/math/9/9/f/99f142acee2cec8c57f33cae640c57d3.png)
provided or
in the PL or DIFF categories, respectively.
This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).
Because of the existence of knots the analogues of Theorem 5.1 for in the PL case, and for
in the smooth case are false. So for the smooth category and
a classification is much harder: for 40 years the
known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.
Theorem 5.2. [Skopenkov2006] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
, where
is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
. E.g. by Theorem 5.2 we obtain that
- the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
.
5.2 The Whitney invariant
6 References
- [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
- [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).
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
- [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.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [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
- [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
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![n>1](/images/math/f/3/2/f32ba7f7c2722280ef2c82ff05d61be1.png)
![N \to \Rr^m](/images/math/5/7/8/5783af70788b1a317af63b4be92832ae.png)
![m \geq 2n + 1](/images/math/3/3/8/338ad0c0ef7cedda6ae1e913ae3b5973.png)
![m = 2n\ge8](/images/math/7/b/5/7b5f4bcb390563f9db21e9fec11cdf91.png)
![m=2n-1\ge9](/images/math/3/e/c/3ecf9f1d79d81bb67c392fc47332094c.png)
For notation and conventions see high codimension embeddings.
2 Classification
Classification Theorem 2.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s,\end{array}\right.](/images/math/9/7/7/977eb87ba150391f256831cd135a87cc.png)
is bijective if either or
and CAT=PL [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]
The Whitney invariant is defined below.
The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of see Embeddings of highly-connected manifolds. Some estimations of
for a closed
-connected
-manifold
and
(including
) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.
3 Examples
Together with the Haefliger knotted sphere , Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).
For define the
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection, we recall for and
. Hudson's construction of embedings
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
Take the standard embeddings (where
means homothety with coefficient 2) and
.
Fix a point
.
The Hudson torus
is the embedded connected sum of
![\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.](/images/math/5/8/2/582d83ee56b6551b010d068fd3f28ee3.png)
(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)
For instead of an embedded
-sphere
we can take
copies
(
) of
-sphere outside
`parallel' to
.
Then we join these spheres by tubes so that the homotopy class of the resulting embedding
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/5/e/4/5e4f894b5aa579c4e813954dd0f3454f.png)
Let be the connected sum of this embedding with the above standard
embedding
.
Clearly, is isotopic to the standard embedding.
Proposition 3.1.
For odd
is isotopic to
if and only
if
.
For even
is isotopic to
if and only if
.
In particular, is not isotopic to
for each
(this was the original motivation for Hudson).
Proposition 3.1 follows by Remark 5.e and, for even, by Theorem 4, both results from classification just below the stable range.
It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of
Proposition 3.1 for
.
3.2 Hudson tori 2
In this subsection we give, for and
another construction of embeddings
![\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/f/0/3/f03bdba139558d9b127d28e84c53d4b5.png)
Define a map to be the constant
on one
component
and the standard embedding
on the other component.
This map gives an
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset \Rr^{2n}.](/images/math/7/a/7/7a7bb377fc8f6be96b696d21d08e3952.png)
(See Figure 2.2 of [Skopenkov2006].)
Each disk intersects the image of this embedding at two
points lying in
.
Extend this embedding
for each
to an
embedding
.
(See Figure 2.3 of [Skopenkov2006].)
Thus we obtain the Hudson torus
![\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/4/4/d/44d0d089254c0e5a3b52719760ea1909.png)
The embedding is obtained in the same way starting from a map
of degree
.
The same proposition as above holds with replaced to
.
3.3 Remarks
We have is PL isotopic to
[Skopenkov2006a].
It would be interesting to prove the smooth analogue of this result.
For these construction give what we call the
Hudson torus.
The
Hudson torus is constructed analogously and is the
composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the Hudson torus
for
or, more generally,
for
and
for
.
3.4 An action of the first homology group on embeddings
In this subsection, for and for orientable
with
, we construct an embedding
from an embedding
.
For , represent
by an embedding
. Since any orientable bundle over
is trivial,
. Identify
with
. It remains to make an embedded surgery of
to obtain an
-sphere
, and then we set
.
Take a vector field on normal to
. Extend
along this vector field to a smooth map
. Since
and
, by general position we may assume that
is an embedding and
misses
. Since
, we have
.
Hence the standard framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}](/images/math/4/0/f/40f08ff9a14fa1038d146656cb8fbf18.png)
![\displaystyle \mbox{Let}\qquad \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.](/images/math/9/8/5/985515b95fa1549044a22b99def3232e.png)
This construction generalizes the construction of (from
).
Clearly, is
or
. Thus unless
and CAT=DIFF
- all isotopy classes of embedings
can be obtained (from a certain given embedding
) by the above construction;
- the above construction defines an action
.
4 The Whitney invariant (for either n odd or N orientable)
Fix orientations on and, if
is even, on
. Fix an embedding
. For an embedding
the restrictions of
and
to
are regular homotopic [Hirsch1959]. Since
has an
-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of
and assume that
on
. Take a general position homotopy
relative to
between the restrictions of
and
to
. Then
(i.e. `the intersection of this homotopy with
') is a 1-manifold (possibly non-compact) without boundary. Define
to be the homology class of the closure of this 1-manifold:
![\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).](/images/math/2/b/b/2bbc331d0f1495918467d3ff52161652.png)
The orientation on is defined for
orientable as follows. (This orientation is defined for each
but used only for odd
.) For each point
take a vector at
tangent to
. Complete this vector to a positive base tangent to
. Since
, by general position there is a unique point
such that
. The tangent vector at
thus gives a tangent vector at
to
. Complete this vector to a positive base tangent to
, where the orientation on
comes from
. The union of the images of the constructed two bases is a base at
of
. If this base is positive, then call the initial vector of
positive. Since a change of the orientation on
forces a change of the orientation of the latter base of
, it follows that this condition indeed defines an orientation on
.
Remark 4.1.
- The Whitney invariant is well-defined, i.e. independent of the choice of
and of the isotopy making
outside
. This is so because the above definition is clearly equivalent to the following:
is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set
of a general position homotopy
between
and
. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need
-coefficients when
is even.
- Clearly,
. The definition of
depends on the choice of
, but we write
not
for brevity.
- Since a change of the orientation on
forces a change of the orientation on
, the class
is independent of the choice of the orientation on
. For the reflection
with respect to a hyperplane we have
(because we may assume that
on
and because a change of the orientation of
forces a change of the orientation of
).
- The above definition makes sense for each
, not only for
.
- Clearly,
is
or
for
for the Hudson tori.
for each embeddings
and
.
5 A generalization to highly-connected manifolds
Examples are the above Hudson tori . See also [Milgram&Rees1971].
5.1 Classification
Theorem 5.1. Let be a closed orientable homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})](/images/math/6/e/f/6ef58c45d6f2c416944ab02e70082041.png)
is a bijection, provided or
in the PL or DIFF categories, respectively.
Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977]
-connected manifolds. The proof works for
-connected manifolds.
E.g. by Theorem 5.1 we obtain that the Whitney invariant is bijective for
.
It is in fact a group isomorphism; the generator is the Hudson torus.
The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for .
Analogously to Theorem 5.1 it may be proved that
- if
is a closed connected non-orientable
-manifold, then
![\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},](/images/math/9/9/f/99f142acee2cec8c57f33cae640c57d3.png)
provided or
in the PL or DIFF categories, respectively.
This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).
Because of the existence of knots the analogues of Theorem 5.1 for in the PL case, and for
in the smooth case are false. So for the smooth category and
a classification is much harder: for 40 years the
known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.
Theorem 5.2. [Skopenkov2006] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
, where
is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
. E.g. by Theorem 5.2 we obtain that
- the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
.
5.2 The Whitney invariant
6 References
- [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
- [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).
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
- [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.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [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
- [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
This page has not been refereed. The information given here might be incomplete or provisional. |
![n>1](/images/math/f/3/2/f32ba7f7c2722280ef2c82ff05d61be1.png)
![N \to \Rr^m](/images/math/5/7/8/5783af70788b1a317af63b4be92832ae.png)
![m \geq 2n + 1](/images/math/3/3/8/338ad0c0ef7cedda6ae1e913ae3b5973.png)
![m = 2n\ge8](/images/math/7/b/5/7b5f4bcb390563f9db21e9fec11cdf91.png)
![m=2n-1\ge9](/images/math/3/e/c/3ecf9f1d79d81bb67c392fc47332094c.png)
For notation and conventions see high codimension embeddings.
2 Classification
Classification Theorem 2.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s,\end{array}\right.](/images/math/9/7/7/977eb87ba150391f256831cd135a87cc.png)
is bijective if either or
and CAT=PL [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]
The Whitney invariant is defined below.
The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of see Embeddings of highly-connected manifolds. Some estimations of
for a closed
-connected
-manifold
and
(including
) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.
3 Examples
Together with the Haefliger knotted sphere , Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).
For define the
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection, we recall for and
. Hudson's construction of embedings
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
Take the standard embeddings (where
means homothety with coefficient 2) and
.
Fix a point
.
The Hudson torus
is the embedded connected sum of
![\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.](/images/math/5/8/2/582d83ee56b6551b010d068fd3f28ee3.png)
(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)
For instead of an embedded
-sphere
we can take
copies
(
) of
-sphere outside
`parallel' to
.
Then we join these spheres by tubes so that the homotopy class of the resulting embedding
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/5/e/4/5e4f894b5aa579c4e813954dd0f3454f.png)
Let be the connected sum of this embedding with the above standard
embedding
.
Clearly, is isotopic to the standard embedding.
Proposition 3.1.
For odd
is isotopic to
if and only
if
.
For even
is isotopic to
if and only if
.
In particular, is not isotopic to
for each
(this was the original motivation for Hudson).
Proposition 3.1 follows by Remark 5.e and, for even, by Theorem 4, both results from classification just below the stable range.
It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of
Proposition 3.1 for
.
3.2 Hudson tori 2
In this subsection we give, for and
another construction of embeddings
![\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/f/0/3/f03bdba139558d9b127d28e84c53d4b5.png)
Define a map to be the constant
on one
component
and the standard embedding
on the other component.
This map gives an
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset \Rr^{2n}.](/images/math/7/a/7/7a7bb377fc8f6be96b696d21d08e3952.png)
(See Figure 2.2 of [Skopenkov2006].)
Each disk intersects the image of this embedding at two
points lying in
.
Extend this embedding
for each
to an
embedding
.
(See Figure 2.3 of [Skopenkov2006].)
Thus we obtain the Hudson torus
![\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/4/4/d/44d0d089254c0e5a3b52719760ea1909.png)
The embedding is obtained in the same way starting from a map
of degree
.
The same proposition as above holds with replaced to
.
3.3 Remarks
We have is PL isotopic to
[Skopenkov2006a].
It would be interesting to prove the smooth analogue of this result.
For these construction give what we call the
Hudson torus.
The
Hudson torus is constructed analogously and is the
composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the Hudson torus
for
or, more generally,
for
and
for
.
3.4 An action of the first homology group on embeddings
In this subsection, for and for orientable
with
, we construct an embedding
from an embedding
.
For , represent
by an embedding
. Since any orientable bundle over
is trivial,
. Identify
with
. It remains to make an embedded surgery of
to obtain an
-sphere
, and then we set
.
Take a vector field on normal to
. Extend
along this vector field to a smooth map
. Since
and
, by general position we may assume that
is an embedding and
misses
. Since
, we have
.
Hence the standard framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}](/images/math/4/0/f/40f08ff9a14fa1038d146656cb8fbf18.png)
![\displaystyle \mbox{Let}\qquad \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.](/images/math/9/8/5/985515b95fa1549044a22b99def3232e.png)
This construction generalizes the construction of (from
).
Clearly, is
or
. Thus unless
and CAT=DIFF
- all isotopy classes of embedings
can be obtained (from a certain given embedding
) by the above construction;
- the above construction defines an action
.
4 The Whitney invariant (for either n odd or N orientable)
Fix orientations on and, if
is even, on
. Fix an embedding
. For an embedding
the restrictions of
and
to
are regular homotopic [Hirsch1959]. Since
has an
-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of
and assume that
on
. Take a general position homotopy
relative to
between the restrictions of
and
to
. Then
(i.e. `the intersection of this homotopy with
') is a 1-manifold (possibly non-compact) without boundary. Define
to be the homology class of the closure of this 1-manifold:
![\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).](/images/math/2/b/b/2bbc331d0f1495918467d3ff52161652.png)
The orientation on is defined for
orientable as follows. (This orientation is defined for each
but used only for odd
.) For each point
take a vector at
tangent to
. Complete this vector to a positive base tangent to
. Since
, by general position there is a unique point
such that
. The tangent vector at
thus gives a tangent vector at
to
. Complete this vector to a positive base tangent to
, where the orientation on
comes from
. The union of the images of the constructed two bases is a base at
of
. If this base is positive, then call the initial vector of
positive. Since a change of the orientation on
forces a change of the orientation of the latter base of
, it follows that this condition indeed defines an orientation on
.
Remark 4.1.
- The Whitney invariant is well-defined, i.e. independent of the choice of
and of the isotopy making
outside
. This is so because the above definition is clearly equivalent to the following:
is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set
of a general position homotopy
between
and
. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need
-coefficients when
is even.
- Clearly,
. The definition of
depends on the choice of
, but we write
not
for brevity.
- Since a change of the orientation on
forces a change of the orientation on
, the class
is independent of the choice of the orientation on
. For the reflection
with respect to a hyperplane we have
(because we may assume that
on
and because a change of the orientation of
forces a change of the orientation of
).
- The above definition makes sense for each
, not only for
.
- Clearly,
is
or
for
for the Hudson tori.
for each embeddings
and
.
5 A generalization to highly-connected manifolds
Examples are the above Hudson tori . See also [Milgram&Rees1971].
5.1 Classification
Theorem 5.1. Let be a closed orientable homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})](/images/math/6/e/f/6ef58c45d6f2c416944ab02e70082041.png)
is a bijection, provided or
in the PL or DIFF categories, respectively.
Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977]
-connected manifolds. The proof works for
-connected manifolds.
E.g. by Theorem 5.1 we obtain that the Whitney invariant is bijective for
.
It is in fact a group isomorphism; the generator is the Hudson torus.
The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for .
Analogously to Theorem 5.1 it may be proved that
- if
is a closed connected non-orientable
-manifold, then
![\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},](/images/math/9/9/f/99f142acee2cec8c57f33cae640c57d3.png)
provided or
in the PL or DIFF categories, respectively.
This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).
Because of the existence of knots the analogues of Theorem 5.1 for in the PL case, and for
in the smooth case are false. So for the smooth category and
a classification is much harder: for 40 years the
known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.
Theorem 5.2. [Skopenkov2006] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
, where
is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
. E.g. by Theorem 5.2 we obtain that
- the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
.
5.2 The Whitney invariant
6 References
- [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
- [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).
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hudson1963] J. F. P. Hudson, Knotted tori, Topology 2 (1963), 11–22. MR0146829 (26 #4349) Zbl 0151.32503
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Milgram&Rees1971] R. Milgram and E. Rees, On the normal bundle to an embedding., Topology 10 (1971), 299-308. MR0290391 (44 #7572) Zbl 0207.22302
- [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.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [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
- [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
This page has not been refereed. The information given here might be incomplete or provisional. |
![n>1](/images/math/f/3/2/f32ba7f7c2722280ef2c82ff05d61be1.png)
![N \to \Rr^m](/images/math/5/7/8/5783af70788b1a317af63b4be92832ae.png)
![m \geq 2n + 1](/images/math/3/3/8/338ad0c0ef7cedda6ae1e913ae3b5973.png)
![m = 2n\ge8](/images/math/7/b/5/7b5f4bcb390563f9db21e9fec11cdf91.png)
![m=2n-1\ge9](/images/math/3/e/c/3ecf9f1d79d81bb67c392fc47332094c.png)
For notation and conventions see high codimension embeddings.
2 Classification
Classification Theorem 2.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s,\end{array}\right.](/images/math/9/7/7/977eb87ba150391f256831cd135a87cc.png)
is bijective if either or
and CAT=PL [Haefliger&Hirsch1963], [Bausum1975], [Vrabec1977], cf. [Hudson1969]
The Whitney invariant is defined below.
The classification of smooth embeddings of 3-manifolds in 6-space is more complicated. For analogous classification of see Embeddings of highly-connected manifolds. Some estimations of
for a closed
-connected
-manifold
and
(including
) are presented in [Skopenkov2010]. See also Embeddings of 4-manifolds in 7-space.
3 Examples
Together with the Haefliger knotted sphere , Hudson's examples were the first examples of embeddings in codimension greater than 2 not isotopic to the standard embedding. (Hudson's construction [Hudson1963] was not as explicit as the one below).
For define the
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection, we recall for and
. Hudson's construction of embedings
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
Take the standard embeddings (where
means homothety with coefficient 2) and
.
Fix a point
.
The Hudson torus
is the embedded connected sum of
![\displaystyle 2\partial D^{n+1}\times x\quad\text{with}\quad \partial D^2\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset 2D^{n+1} \times S^{n-1}\subset\Rr^{2n}.](/images/math/5/8/2/582d83ee56b6551b010d068fd3f28ee3.png)
(Unlike the connected sum mentioned in embedded conntected sum this is a `linked' connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.)
For instead of an embedded
-sphere
we can take
copies
(
) of
-sphere outside
`parallel' to
.
Then we join these spheres by tubes so that the homotopy class of the resulting embedding
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{will be}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/5/e/4/5e4f894b5aa579c4e813954dd0f3454f.png)
Let be the connected sum of this embedding with the above standard
embedding
.
Clearly, is isotopic to the standard embedding.
Proposition 3.1.
For odd
is isotopic to
if and only
if
.
For even
is isotopic to
if and only if
.
In particular, is not isotopic to
for each
(this was the original motivation for Hudson).
Proposition 3.1 follows by Remark 5.e and, for even, by Theorem 4, both results from classification just below the stable range.
It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of
Proposition 3.1 for
.
3.2 Hudson tori 2
In this subsection we give, for and
another construction of embeddings
![\displaystyle \Hud_n'(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/f/0/3/f03bdba139558d9b127d28e84c53d4b5.png)
Define a map to be the constant
on one
component
and the standard embedding
on the other component.
This map gives an
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset \Rr^{2n}.](/images/math/7/a/7/7a7bb377fc8f6be96b696d21d08e3952.png)
(See Figure 2.2 of [Skopenkov2006].)
Each disk intersects the image of this embedding at two
points lying in
.
Extend this embedding
for each
to an
embedding
.
(See Figure 2.3 of [Skopenkov2006].)
Thus we obtain the Hudson torus
![\displaystyle \Hud_n'(1):S^1\times S^{n-1}\to D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/4/4/d/44d0d089254c0e5a3b52719760ea1909.png)
The embedding is obtained in the same way starting from a map
of degree
.
The same proposition as above holds with replaced to
.
3.3 Remarks
We have is PL isotopic to
[Skopenkov2006a].
It would be interesting to prove the smooth analogue of this result.
For these construction give what we call the
Hudson torus.
The
Hudson torus is constructed analogously and is the
composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the Hudson torus
for
or, more generally,
for
and
for
.
3.4 An action of the first homology group on embeddings
In this subsection, for and for orientable
with
, we construct an embedding
from an embedding
.
For , represent
by an embedding
. Since any orientable bundle over
is trivial,
. Identify
with
. It remains to make an embedded surgery of
to obtain an
-sphere
, and then we set
.
Take a vector field on normal to
. Extend
along this vector field to a smooth map
. Since
and
, by general position we may assume that
is an embedding and
misses
. Since
, we have
.
Hence the standard framing of
in
extends to an
-framing on
in
.
Thus
extends to an embedding
![\displaystyle \widehat a:D^2\times D^{n-1}\to C_{f_0}\quad\text{such that}\quad \widehat a(\partial D^2\times D^{n-1})\subset S^1\times S^{n-1}](/images/math/4/0/f/40f08ff9a14fa1038d146656cb8fbf18.png)
![\displaystyle \mbox{Let}\qquad \Sigma:\ =\ S^1\times S^{n-1}-\widehat a(\partial D^2\times Int D^{n-1}) \bigcup\limits_{\widehat a(\partial D^2\times\partial D^{n-1})} \widehat a(D^2\times\partial D^{n-1})\ \cong\ S^n.](/images/math/9/8/5/985515b95fa1549044a22b99def3232e.png)
This construction generalizes the construction of (from
).
Clearly, is
or
. Thus unless
and CAT=DIFF
- all isotopy classes of embedings
can be obtained (from a certain given embedding
) by the above construction;
- the above construction defines an action
.
4 The Whitney invariant (for either n odd or N orientable)
Fix orientations on and, if
is even, on
. Fix an embedding
. For an embedding
the restrictions of
and
to
are regular homotopic [Hirsch1959]. Since
has an
-dimensional spine, it follows that these restrictions are isotopic, cf. [Haefliger&Hirsch1963], 3.1.b, [Takase2006], Lemma 2.2. So we can make an isotopy of
and assume that
on
. Take a general position homotopy
relative to
between the restrictions of
and
to
. Then
(i.e. `the intersection of this homotopy with
') is a 1-manifold (possibly non-compact) without boundary. Define
to be the homology class of the closure of this 1-manifold:
![\displaystyle W(f):=[Cl(f\cap F)]\in H_1(N_0,\partial N_0;\Zz_{(n)})\cong H_1(N;\Zz_{(n)}).](/images/math/2/b/b/2bbc331d0f1495918467d3ff52161652.png)
The orientation on is defined for
orientable as follows. (This orientation is defined for each
but used only for odd
.) For each point
take a vector at
tangent to
. Complete this vector to a positive base tangent to
. Since
, by general position there is a unique point
such that
. The tangent vector at
thus gives a tangent vector at
to
. Complete this vector to a positive base tangent to
, where the orientation on
comes from
. The union of the images of the constructed two bases is a base at
of
. If this base is positive, then call the initial vector of
positive. Since a change of the orientation on
forces a change of the orientation of the latter base of
, it follows that this condition indeed defines an orientation on
.
Remark 4.1.
- The Whitney invariant is well-defined, i.e. independent of the choice of
and of the isotopy making
outside
. This is so because the above definition is clearly equivalent to the following:
is the homology class of the algebraic sum of the top-dimensional simplices of the self-intersection set
of a general position homotopy
between
and
. (For details and definition of the signs of the simplices see [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2, [Skopenkov2010].) It is for being well-defined that we need
-coefficients when
is even.
- Clearly,
. The definition of
depends on the choice of
, but we write
not
for brevity.
- Since a change of the orientation on
forces a change of the orientation on
, the class
is independent of the choice of the orientation on
. For the reflection
with respect to a hyperplane we have
(because we may assume that
on
and because a change of the orientation of
forces a change of the orientation of
).
- The above definition makes sense for each
, not only for
.
- Clearly,
is
or
for
for the Hudson tori.
for each embeddings
and
.
5 A generalization to highly-connected manifolds
Examples are the above Hudson tori . See also [Milgram&Rees1971].
5.1 Classification
Theorem 5.1. Let be a closed orientable homologically
-connected
-manifold,
. Then the Whitney invariant
![\displaystyle W:E^{2n-k}(N)\to H_{k+1}(N,\Zz_{(n-k-1)})](/images/math/6/e/f/6ef58c45d6f2c416944ab02e70082041.png)
is a bijection, provided or
in the PL or DIFF categories, respectively.
Theorem 5.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977]
-connected manifolds. The proof works for
-connected manifolds.
E.g. by Theorem 5.1 we obtain that the Whitney invariant is bijective for
.
It is in fact a group isomorphism; the generator is the Hudson torus.
The PL case of Theorem 5.1 gives nothing but the Unknotting Spheres Theorem for .
Analogously to Theorem 5.1 it may be proved that
- if
is a closed connected non-orientable
-manifold, then
![\displaystyle E^{2n}(N)=\begin{cases} H_1(N,\Zz_2)& n\text{ odd, }\\ \Zz\oplus\Zz_2^{s-1}&n\text{ even and }H_1(N,\Zz_2)\cong\Zz_2^s\end{cases},](/images/math/9/9/f/99f142acee2cec8c57f33cae640c57d3.png)
provided or
in the PL or DIFF categories, respectively.
This is proved in [Haefliger1962b], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation being corrected only in [Vrabec1977]).
Because of the existence of knots the analogues of Theorem 5.1 for in the PL case, and for
in the smooth case are false. So for the smooth category and
a classification is much harder: for 40 years the
known concrete complete classification results were for spheres. The following result was obtained using the Kreck modification of surgery theory.
Theorem 5.2. [Skopenkov2006] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
, where
is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
. E.g. by Theorem 5.2 we obtain that
- the Whitney invariant
is surjective and for each
there is a 1--1 correspondence
.
5.2 The Whitney invariant
6 References
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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
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, Comment. Math. Helv. 46 (1971), 141–161. MR0295373 (45 #4439) Zbl 0218.57016
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, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
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