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
, and
, then every two embeddings
are isotopic. In this page we summarise the situation for
and some more general situations.
2 Classification
Theorem 1.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.](/images/math/8/0/9/809207b464d4e42615836c778b7a5d17.png)
is bijective if either or
and CAT=PL.
This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).
The classification of smooth embeddings of 3-manifolds in is more complicated.
For embeddings of -manifolds in
see the case of 4-manifolds, [Yasui1984], for
and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 1.1 is generalized to a description of for closed
-connected
-manifolds
.
3 Examples
Together with the Haefliger knotted sphere, examples of Hudson tori presented below 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 those below.)
For define the standard embedding
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection we construct, for and
, an embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
The reader might first consider the case (the `degenerate' case) and
(essentially the general case).
Definition 2.1.
Take the standard embeddings (where
means homothety with coefficient 2) and
. Let
.
Join embedded
-sphere and torus
![\displaystyle 2\partial D^{n+1}\times *\quad\text{and}\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/f/b/3/fb39e1a31583c68197f2d31a12accef4.png)
by an arc whose interior misses the two embedded manifolds.
The Hudson torus is the embedded connected sum of the two embeddings along this arc (compatible with the orientation).
Remark 2.2.
(a) This construction, as opposed to Definition 2.3, works for .
(b) Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.
Definition 2.3.
For instead of
we take
copies
(
) of
-sphere outside
`parallel' to
, with standard orientation for
or the opposite orientation for
. Then we make embedded connected sum by tubes joining each
-th copy to
-th copy.
We obtain an embedding
.
Let
be the linked embedded connected sum of this embedding
with the above standard embedding
.
Clearly, is isotopic to the standard embedding.
The original motivation for Hudson was that is not isotopic to
for each
(this is a particular case of Proposition 2.4 below).
One guesses that is not isotopic to
for
.
And that a
-valued invariant exists and is `realized' by the homotopy class of the abbreivation of embedding
from Definition 2.3
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{is}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/c/0/3/c03e7ae234eb7d76a88be0d32a010e11.png)
However, this is only true for odd.
Proposition 2.4.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 2.4 follows by the fifth part of Remark 3.1 and, for even, by Theorem 1.1. It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 2.4 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 embedding
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/b/4/4/b44063272e7093e36017894c775f40ff.png)
(See 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 left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the (left) 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 a closed connected orientable
-manifold
with
, we construct an embedding
from an embedding
.
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 even or N orientable)
Fix orientations on and, if
is odd, 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-1)})\cong H_1(N;\Zz_{(n-1)}).](/images/math/4/c/3/4c36e9ed4be725272b07c6bfcb1cf757.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 3.1.
- 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 an alternative one. 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
Let be a closed
-connected
-manifold.
We present description of
generalizing Theorem 1.1, and its generalization to
for
.
Examples are Hudson tori .
5.1 Classification
Theorem 4.1. 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 4.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically -connected manifolds. The proof works for homologically
-connected manifolds.
For this is covered by Theorem 1.1; for
it is not. The PL case of Theorem 4.1 gives nothing but the Unknotting Spheres Theorem for
.
E.g. by Theorem 4.1 the Whitney invariant is bijective for
. It is in fact a group isomorphism; the generator is the Hudson torus.
Because of the existence of knots the analogues of Theorem 4.1 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 only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].
Theorem 4.2. [Skopenkov2008] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
![\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)](/images/math/8/f/f/8ff4f04898ab29dfdb7e79d11cf52498.png)
is surjective and for each the Kreck invariant
![\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}](/images/math/d/7/5/d75d1c968abd8a5b117610e047314ad0.png)
is a 1-1 correspondence, where is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
.
E.g. by Theorem 4.2 the Whitney invariant is surjective and for each
there is a 1-1 correspondence
.
Theorem 4.3. [Becker&Glover1971] Let be a closed
-connected
-manifold embeddable into
,
and
. Then there is a 1-1 correspondence
![\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.](/images/math/d/9/4/d94be1cdb4405871db3c3856698008c9.png)
For this is covered by Theorem 4.1; for
it is not.
E.g. by Theorem 4.3 there is a 1-1 correspondence ,
, for
,
and
. For a generalization see Knotted tori [Skopenkov2002].
Some estimations of for a closed
-connected
-manifold
are presented in [Skopenkov2010].
5.2 The Whitney invariant
Let be an
-manifold and
embeddings. Roughly speaking,
is defined as the homology class of the self-intersection set
of a general position homotopy
between
and
. We present an accurate definition in the smooth category for
when either
is even or
is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.
Fix orientations on and on
. Take embeddings
. Take a general position homotopy
between
and
. By general position the closure
of the self-intersection set has codimension 2 singularities and so carries a homology class with
coefficients. (Note that
can be assumed to be a submanifold for
.) For
odd it has a natural orientation and so carries a homology class with
coefficients. Define the Whitney invariant
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})](/images/math/c/2/0/c2080faf9c540d3f0171dc3cb9a43994.png)
by Analogously to [Skopenkov2006], \S2.4, this is well-defined.
6 An orientation on the self-intersection set
Let be a general position smooth map of an orientable
-manifold
. Assume that
so that the closure
of the self-intersection set of
has codimension 2 singularities. Then
- (1)
has a natural orientation.
- (2) the natural orientation on
need not extend to
.
- (3) the natural orientation on
extend to
if
is odd [Hudson1969], Lemma 11.4.
- (4)
has a natural orientation if
is even.
Fix an orientation on and on
.
Let us prove (1). Take points outside singularities of
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is orientable, we can take positive
-bases
and
at
and
normal to
and to
. If the base
of
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and
.
(Note that a change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.)
We can see that (2) holds by considering the cone over a general position map
having only one self-intersection point.
Let us prove (4). Take a -base
at a point
outside singularities of
. Since
is orientable, we can take a positive
-base
normal to
in one sheet of
. Analogously construct an
-base
for the other sheet of
. If
is even, then the orientation of the base
of
does not depend on choosing the first and the other sheet of
. If the base
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and so of
.
(Note that a change of the orientation of
forces changes of the signs of
and so does not change the orientation of
.)
7 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
- [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).
- [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
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [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
- [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.
- [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)
2 Classification
Theorem 1.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.](/images/math/8/0/9/809207b464d4e42615836c778b7a5d17.png)
is bijective if either or
and CAT=PL.
This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).
The classification of smooth embeddings of 3-manifolds in is more complicated.
For embeddings of -manifolds in
see the case of 4-manifolds, [Yasui1984], for
and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 1.1 is generalized to a description of for closed
-connected
-manifolds
.
3 Examples
Together with the Haefliger knotted sphere, examples of Hudson tori presented below 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 those below.)
For define the standard embedding
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection we construct, for and
, an embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
The reader might first consider the case (the `degenerate' case) and
(essentially the general case).
Definition 2.1.
Take the standard embeddings (where
means homothety with coefficient 2) and
. Let
.
Join embedded
-sphere and torus
![\displaystyle 2\partial D^{n+1}\times *\quad\text{and}\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/f/b/3/fb39e1a31583c68197f2d31a12accef4.png)
by an arc whose interior misses the two embedded manifolds.
The Hudson torus is the embedded connected sum of the two embeddings along this arc (compatible with the orientation).
Remark 2.2.
(a) This construction, as opposed to Definition 2.3, works for .
(b) Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.
Definition 2.3.
For instead of
we take
copies
(
) of
-sphere outside
`parallel' to
, with standard orientation for
or the opposite orientation for
. Then we make embedded connected sum by tubes joining each
-th copy to
-th copy.
We obtain an embedding
.
Let
be the linked embedded connected sum of this embedding
with the above standard embedding
.
Clearly, is isotopic to the standard embedding.
The original motivation for Hudson was that is not isotopic to
for each
(this is a particular case of Proposition 2.4 below).
One guesses that is not isotopic to
for
.
And that a
-valued invariant exists and is `realized' by the homotopy class of the abbreivation of embedding
from Definition 2.3
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{is}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/c/0/3/c03e7ae234eb7d76a88be0d32a010e11.png)
However, this is only true for odd.
Proposition 2.4.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 2.4 follows by the fifth part of Remark 3.1 and, for even, by Theorem 1.1. It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 2.4 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 embedding
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/b/4/4/b44063272e7093e36017894c775f40ff.png)
(See 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 left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the (left) 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 a closed connected orientable
-manifold
with
, we construct an embedding
from an embedding
.
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 even or N orientable)
Fix orientations on and, if
is odd, 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-1)})\cong H_1(N;\Zz_{(n-1)}).](/images/math/4/c/3/4c36e9ed4be725272b07c6bfcb1cf757.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 3.1.
- 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 an alternative one. 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
Let be a closed
-connected
-manifold.
We present description of
generalizing Theorem 1.1, and its generalization to
for
.
Examples are Hudson tori .
5.1 Classification
Theorem 4.1. 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 4.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically -connected manifolds. The proof works for homologically
-connected manifolds.
For this is covered by Theorem 1.1; for
it is not. The PL case of Theorem 4.1 gives nothing but the Unknotting Spheres Theorem for
.
E.g. by Theorem 4.1 the Whitney invariant is bijective for
. It is in fact a group isomorphism; the generator is the Hudson torus.
Because of the existence of knots the analogues of Theorem 4.1 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 only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].
Theorem 4.2. [Skopenkov2008] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
![\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)](/images/math/8/f/f/8ff4f04898ab29dfdb7e79d11cf52498.png)
is surjective and for each the Kreck invariant
![\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}](/images/math/d/7/5/d75d1c968abd8a5b117610e047314ad0.png)
is a 1-1 correspondence, where is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
.
E.g. by Theorem 4.2 the Whitney invariant is surjective and for each
there is a 1-1 correspondence
.
Theorem 4.3. [Becker&Glover1971] Let be a closed
-connected
-manifold embeddable into
,
and
. Then there is a 1-1 correspondence
![\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.](/images/math/d/9/4/d94be1cdb4405871db3c3856698008c9.png)
For this is covered by Theorem 4.1; for
it is not.
E.g. by Theorem 4.3 there is a 1-1 correspondence ,
, for
,
and
. For a generalization see Knotted tori [Skopenkov2002].
Some estimations of for a closed
-connected
-manifold
are presented in [Skopenkov2010].
5.2 The Whitney invariant
Let be an
-manifold and
embeddings. Roughly speaking,
is defined as the homology class of the self-intersection set
of a general position homotopy
between
and
. We present an accurate definition in the smooth category for
when either
is even or
is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.
Fix orientations on and on
. Take embeddings
. Take a general position homotopy
between
and
. By general position the closure
of the self-intersection set has codimension 2 singularities and so carries a homology class with
coefficients. (Note that
can be assumed to be a submanifold for
.) For
odd it has a natural orientation and so carries a homology class with
coefficients. Define the Whitney invariant
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})](/images/math/c/2/0/c2080faf9c540d3f0171dc3cb9a43994.png)
by Analogously to [Skopenkov2006], \S2.4, this is well-defined.
6 An orientation on the self-intersection set
Let be a general position smooth map of an orientable
-manifold
. Assume that
so that the closure
of the self-intersection set of
has codimension 2 singularities. Then
- (1)
has a natural orientation.
- (2) the natural orientation on
need not extend to
.
- (3) the natural orientation on
extend to
if
is odd [Hudson1969], Lemma 11.4.
- (4)
has a natural orientation if
is even.
Fix an orientation on and on
.
Let us prove (1). Take points outside singularities of
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is orientable, we can take positive
-bases
and
at
and
normal to
and to
. If the base
of
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and
.
(Note that a change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.)
We can see that (2) holds by considering the cone over a general position map
having only one self-intersection point.
Let us prove (4). Take a -base
at a point
outside singularities of
. Since
is orientable, we can take a positive
-base
normal to
in one sheet of
. Analogously construct an
-base
for the other sheet of
. If
is even, then the orientation of the base
of
does not depend on choosing the first and the other sheet of
. If the base
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and so of
.
(Note that a change of the orientation of
forces changes of the signs of
and so does not change the orientation of
.)
7 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
- [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).
- [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
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [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
- [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.
- [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)
2 Classification
Theorem 1.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.](/images/math/8/0/9/809207b464d4e42615836c778b7a5d17.png)
is bijective if either or
and CAT=PL.
This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).
The classification of smooth embeddings of 3-manifolds in is more complicated.
For embeddings of -manifolds in
see the case of 4-manifolds, [Yasui1984], for
and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 1.1 is generalized to a description of for closed
-connected
-manifolds
.
3 Examples
Together with the Haefliger knotted sphere, examples of Hudson tori presented below 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 those below.)
For define the standard embedding
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection we construct, for and
, an embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
The reader might first consider the case (the `degenerate' case) and
(essentially the general case).
Definition 2.1.
Take the standard embeddings (where
means homothety with coefficient 2) and
. Let
.
Join embedded
-sphere and torus
![\displaystyle 2\partial D^{n+1}\times *\quad\text{and}\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/f/b/3/fb39e1a31583c68197f2d31a12accef4.png)
by an arc whose interior misses the two embedded manifolds.
The Hudson torus is the embedded connected sum of the two embeddings along this arc (compatible with the orientation).
Remark 2.2.
(a) This construction, as opposed to Definition 2.3, works for .
(b) Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.
Definition 2.3.
For instead of
we take
copies
(
) of
-sphere outside
`parallel' to
, with standard orientation for
or the opposite orientation for
. Then we make embedded connected sum by tubes joining each
-th copy to
-th copy.
We obtain an embedding
.
Let
be the linked embedded connected sum of this embedding
with the above standard embedding
.
Clearly, is isotopic to the standard embedding.
The original motivation for Hudson was that is not isotopic to
for each
(this is a particular case of Proposition 2.4 below).
One guesses that is not isotopic to
for
.
And that a
-valued invariant exists and is `realized' by the homotopy class of the abbreivation of embedding
from Definition 2.3
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{is}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/c/0/3/c03e7ae234eb7d76a88be0d32a010e11.png)
However, this is only true for odd.
Proposition 2.4.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 2.4 follows by the fifth part of Remark 3.1 and, for even, by Theorem 1.1. It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 2.4 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 embedding
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/b/4/4/b44063272e7093e36017894c775f40ff.png)
(See 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 left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the (left) 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 a closed connected orientable
-manifold
with
, we construct an embedding
from an embedding
.
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 even or N orientable)
Fix orientations on and, if
is odd, 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-1)})\cong H_1(N;\Zz_{(n-1)}).](/images/math/4/c/3/4c36e9ed4be725272b07c6bfcb1cf757.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 3.1.
- 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 an alternative one. 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
Let be a closed
-connected
-manifold.
We present description of
generalizing Theorem 1.1, and its generalization to
for
.
Examples are Hudson tori .
5.1 Classification
Theorem 4.1. 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 4.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically -connected manifolds. The proof works for homologically
-connected manifolds.
For this is covered by Theorem 1.1; for
it is not. The PL case of Theorem 4.1 gives nothing but the Unknotting Spheres Theorem for
.
E.g. by Theorem 4.1 the Whitney invariant is bijective for
. It is in fact a group isomorphism; the generator is the Hudson torus.
Because of the existence of knots the analogues of Theorem 4.1 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 only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].
Theorem 4.2. [Skopenkov2008] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
![\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)](/images/math/8/f/f/8ff4f04898ab29dfdb7e79d11cf52498.png)
is surjective and for each the Kreck invariant
![\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}](/images/math/d/7/5/d75d1c968abd8a5b117610e047314ad0.png)
is a 1-1 correspondence, where is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
.
E.g. by Theorem 4.2 the Whitney invariant is surjective and for each
there is a 1-1 correspondence
.
Theorem 4.3. [Becker&Glover1971] Let be a closed
-connected
-manifold embeddable into
,
and
. Then there is a 1-1 correspondence
![\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.](/images/math/d/9/4/d94be1cdb4405871db3c3856698008c9.png)
For this is covered by Theorem 4.1; for
it is not.
E.g. by Theorem 4.3 there is a 1-1 correspondence ,
, for
,
and
. For a generalization see Knotted tori [Skopenkov2002].
Some estimations of for a closed
-connected
-manifold
are presented in [Skopenkov2010].
5.2 The Whitney invariant
Let be an
-manifold and
embeddings. Roughly speaking,
is defined as the homology class of the self-intersection set
of a general position homotopy
between
and
. We present an accurate definition in the smooth category for
when either
is even or
is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.
Fix orientations on and on
. Take embeddings
. Take a general position homotopy
between
and
. By general position the closure
of the self-intersection set has codimension 2 singularities and so carries a homology class with
coefficients. (Note that
can be assumed to be a submanifold for
.) For
odd it has a natural orientation and so carries a homology class with
coefficients. Define the Whitney invariant
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})](/images/math/c/2/0/c2080faf9c540d3f0171dc3cb9a43994.png)
by Analogously to [Skopenkov2006], \S2.4, this is well-defined.
6 An orientation on the self-intersection set
Let be a general position smooth map of an orientable
-manifold
. Assume that
so that the closure
of the self-intersection set of
has codimension 2 singularities. Then
- (1)
has a natural orientation.
- (2) the natural orientation on
need not extend to
.
- (3) the natural orientation on
extend to
if
is odd [Hudson1969], Lemma 11.4.
- (4)
has a natural orientation if
is even.
Fix an orientation on and on
.
Let us prove (1). Take points outside singularities of
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is orientable, we can take positive
-bases
and
at
and
normal to
and to
. If the base
of
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and
.
(Note that a change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.)
We can see that (2) holds by considering the cone over a general position map
having only one self-intersection point.
Let us prove (4). Take a -base
at a point
outside singularities of
. Since
is orientable, we can take a positive
-base
normal to
in one sheet of
. Analogously construct an
-base
for the other sheet of
. If
is even, then the orientation of the base
of
does not depend on choosing the first and the other sheet of
. If the base
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and so of
.
(Note that a change of the orientation of
forces changes of the signs of
and so does not change the orientation of
.)
7 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
- [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).
- [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
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [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
- [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.
- [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)
2 Classification
Theorem 1.1.
Let be a closed connected
-manifold. The Whitney invariant
![\displaystyle W:E^{2n}(N)\to\left\{\begin{array}{cc} H_1(N;\Zz_{(n-1)})& \mbox{either $n$ is odd or $N$ is orientable}\\ \Zz\oplus\Zz_2^{s-1}&\mbox{$n$ is even, $N$ is non-orientable and } H_1(N,\Zz_2)\cong\Zz_2^s\end{array}\right.](/images/math/8/0/9/809207b464d4e42615836c778b7a5d17.png)
is bijective if either or
and CAT=PL.
This is proved in [Haefliger1962b], [Haefliger&Hirsch1963], [Weber1967], [Bausum1975], [Vrabec1977] (a minor miscalculation for the non-orientable case being corrected only in [Vrabec1977]).
The classification of smooth embeddings of 3-manifolds in is more complicated.
For embeddings of -manifolds in
see the case of 4-manifolds, [Yasui1984], for
and [Saeki1999], [Skopenkov2010], [Tonkonog2010] for non-closed manifolds.
Theorem 1.1 is generalized to a description of for closed
-connected
-manifolds
.
3 Examples
Together with the Haefliger knotted sphere, examples of Hudson tori presented below 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 those below.)
For define the standard embedding
as the composition
of standard embeddings.
3.1 Hudson tori 1
In this subsection we construct, for and
, an embedding
![\displaystyle \Hud_n(a):S^1\times S^{n-1}\to\Rr^{2n}.](/images/math/0/e/9/0e91c3bf0d283c4df6842ac2b40fcaf6.png)
The reader might first consider the case (the `degenerate' case) and
(essentially the general case).
Definition 2.1.
Take the standard embeddings (where
means homothety with coefficient 2) and
. Let
.
Join embedded
-sphere and torus
![\displaystyle 2\partial D^{n+1}\times *\quad\text{and}\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/f/b/3/fb39e1a31583c68197f2d31a12accef4.png)
by an arc whose interior misses the two embedded manifolds.
The Hudson torus is the embedded connected sum of the two embeddings along this arc (compatible with the orientation).
Remark 2.2.
(a) This construction, as opposed to Definition 2.3, works for .
(b) Unlike the unlinked embedded connected sum this is a linked embedded connected sum, i.e. connected sum of two embeddings whose images are not contained in disjoint cubes.
Definition 2.3.
For instead of
we take
copies
(
) of
-sphere outside
`parallel' to
, with standard orientation for
or the opposite orientation for
. Then we make embedded connected sum by tubes joining each
-th copy to
-th copy.
We obtain an embedding
.
Let
be the linked embedded connected sum of this embedding
with the above standard embedding
.
Clearly, is isotopic to the standard embedding.
The original motivation for Hudson was that is not isotopic to
for each
(this is a particular case of Proposition 2.4 below).
One guesses that is not isotopic to
for
.
And that a
-valued invariant exists and is `realized' by the homotopy class of the abbreivation of embedding
from Definition 2.3
![\displaystyle S^n\to S^{2n}-D^{n+1}\times S^{n-1}\simeq S^{2n}-S^{n-1}\simeq S^n \quad\text{is}\quad a\in\pi_n(S^n)\cong\Zz.](/images/math/c/0/3/c03e7ae234eb7d76a88be0d32a010e11.png)
However, this is only true for odd.
Proposition 2.4.
For odd
is isotopic to
if and only if
.
For even
is isotopic to
if and only if
.
Proposition 2.4 follows by the fifth part of Remark 3.1 and, for even, by Theorem 1.1. It would be interesting to find an explicit construction of an isotopy between
and
(cf. [Vrabec1977], \S6) and to prove the analogue of Proposition 2.4 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 embedding
![\displaystyle S^0\times S^{n-1}\to D^n\times S^{n-1}\subset D^{n+1}\times S^{n-1}\subset\Rr^{2n}.](/images/math/b/4/4/b44063272e7093e36017894c775f40ff.png)
(See 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 left Hudson torus. The right Hudson torus is constructed analogously and is the composition of the left Hudson torus and the exchanging factors
autodiffeomorphism of
.
Analogously one constructs the (left) 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 a closed connected orientable
-manifold
with
, we construct an embedding
from an embedding
.
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 even or N orientable)
Fix orientations on and, if
is odd, 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-1)})\cong H_1(N;\Zz_{(n-1)}).](/images/math/4/c/3/4c36e9ed4be725272b07c6bfcb1cf757.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 3.1.
- 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 an alternative one. 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
Let be a closed
-connected
-manifold.
We present description of
generalizing Theorem 1.1, and its generalization to
for
.
Examples are Hudson tori .
5.1 Classification
Theorem 4.1. 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 4.1 was proved in [Haefliger&Hirsch1963], [Hudson1969], \S11, [Boechat&Haefliger1970], [Boechat1971], [Vrabec1977] homotopically -connected manifolds. The proof works for homologically
-connected manifolds.
For this is covered by Theorem 1.1; for
it is not. The PL case of Theorem 4.1 gives nothing but the Unknotting Spheres Theorem for
.
E.g. by Theorem 4.1 the Whitney invariant is bijective for
. It is in fact a group isomorphism; the generator is the Hudson torus.
Because of the existence of knots the analogues of Theorem 4.1 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 only known concrete complete classification results were for spheres. The following result was obtained using the Bo\'echat-Haefliger formula for the smoothing obstruction [Boechat1971].
Theorem 4.2. [Skopenkov2008] Let be a closed homologically
-connected
-manifold. Then the Whitney invariant
![\displaystyle W:E^{6k}_D(N)\to H_{2k-1}(N)](/images/math/8/f/f/8ff4f04898ab29dfdb7e79d11cf52498.png)
is surjective and for each the Kreck invariant
![\displaystyle \eta_u:W^{-1}u\to\Zz_{d(u)}](/images/math/d/7/5/d75d1c968abd8a5b117610e047314ad0.png)
is a 1-1 correspondence, where is the divisibility of the projection of
to the free part of
.
Recall that the divisibility of zero is zero and the divisibility of is
.
E.g. by Theorem 4.2 the Whitney invariant is surjective and for each
there is a 1-1 correspondence
.
Theorem 4.3. [Becker&Glover1971] Let be a closed
-connected
-manifold embeddable into
,
and
. Then there is a 1-1 correspondence
![\displaystyle E^m(N)\to [N_0;V_{M,M+n-m+1}],\quad\mbox{where}\quad M>n.](/images/math/d/9/4/d94be1cdb4405871db3c3856698008c9.png)
For this is covered by Theorem 4.1; for
it is not.
E.g. by Theorem 4.3 there is a 1-1 correspondence ,
, for
,
and
. For a generalization see Knotted tori [Skopenkov2002].
Some estimations of for a closed
-connected
-manifold
are presented in [Skopenkov2010].
5.2 The Whitney invariant
Let be an
-manifold and
embeddings. Roughly speaking,
is defined as the homology class of the self-intersection set
of a general position homotopy
between
and
. We present an accurate definition in the smooth category for
when either
is even or
is orientable [Skopenkov2010]. The definition in the PL category is analogous [Hudson1969], \S12, [Vrabec1977], p. 145, [Skopenkov2006], \S2.4.
Fix orientations on and on
. Take embeddings
. Take a general position homotopy
between
and
. By general position the closure
of the self-intersection set has codimension 2 singularities and so carries a homology class with
coefficients. (Note that
can be assumed to be a submanifold for
.) For
odd it has a natural orientation and so carries a homology class with
coefficients. Define the Whitney invariant
![\displaystyle W:E^m(N)\to H_{2n-m+1}(N\times I;\Zz_{(m-n-1)})\cong H_{2n-m+1}(N;\Zz_{(m-n-1)})](/images/math/c/2/0/c2080faf9c540d3f0171dc3cb9a43994.png)
by Analogously to [Skopenkov2006], \S2.4, this is well-defined.
6 An orientation on the self-intersection set
Let be a general position smooth map of an orientable
-manifold
. Assume that
so that the closure
of the self-intersection set of
has codimension 2 singularities. Then
- (1)
has a natural orientation.
- (2) the natural orientation on
need not extend to
.
- (3) the natural orientation on
extend to
if
is odd [Hudson1969], Lemma 11.4.
- (4)
has a natural orientation if
is even.
Fix an orientation on and on
.
Let us prove (1). Take points outside singularities of
and such that
. Then a
-base
tangent to
at
gives a
-base
tangent to
at
. Since
is orientable, we can take positive
-bases
and
at
and
normal to
and to
. If the base
of
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and
.
(Note that a change of the orientation of
forces changes of the signs of
and
and so does not change the orientation of
.)
We can see that (2) holds by considering the cone over a general position map
having only one self-intersection point.
Let us prove (4). Take a -base
at a point
outside singularities of
. Since
is orientable, we can take a positive
-base
normal to
in one sheet of
. Analogously construct an
-base
for the other sheet of
. If
is even, then the orientation of the base
of
does not depend on choosing the first and the other sheet of
. If the base
is positive, then call the base
positive. This is well-defined because a change of the sign of
forces changes of the signs of
and so of
.
(Note that a change of the orientation of
forces changes of the signs of
and so does not change the orientation of
.)
7 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
- [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).
- [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
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [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
- [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.
- [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