Embeddings of manifolds with boundary: classification
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Contents |
1 Introduction
In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space.
In 4 we introduce an invariant of embedding of a
-manifold in
-space for even
.
In
6 which is independent from
3,
4 and
5 we state generalisations of theorems from
2 to highly-connected manifolds.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1
] for the DIFF case and [Skopenkov2002, Theorem 1.3
] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
Sometimes we give references to such direct proofs but we do not claim these are original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a closed compact
-manifold. Then
embeds into
.
This is well-known strong Whitney embedding theorem.
Theorem 2.2.
Assume that is a compact
-manifold with nonempty boundary. Then
embeds into
.
The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].
Theorem 2.3.
Assume that is a compact
-manifold and either
(a) or
(b) is connected and
.
Then any two embeddings of into
are isotopic.
The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, 2].
Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.
Theorem 2.4.
Assume that is a compact
-manifold with non-empty boundary and either
(a) or
(b) is
-connected,
.
Then any two embeddings of into
are isotopic.
Part (a) of this theorem in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
This theorem is a special case of the Theorem 6.4 .
Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
The following example is folklore.
Proposition 3.1.
Let be the cylinder over
.
Then there exist non-isotopic embeddings of
to
.
Proof.
Define by the formula
, where
. Define
by the formula
.
![\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}](/images/math/6/b/3/6b3975d440818eb9a945b0233464a028.png)
![\mathrm ig_1](/images/math/8/0/a/80aee5fbd9f1820e8e7c60594114bdab.png)
![\mathrm ig_2](/images/math/3/e/e/3ee2c6e0a8cd1ba74a69e581c539ce92.png)
![\mathrm ig_1(S^{n-1}\times \{0, 1\})](/images/math/7/d/5/7d5695103c9fc9eb431b1340d26c5297.png)
![\mathrm ig_2(S^{n-1}\times \{0, 1\})](/images/math/7/3/5/735af2164330b444f9ef79ce4f6d863d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by the linking coefficient [Skopenkov2016h,
3, remark 3.2d] of two disjoint cycles.
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and prooved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
Let
be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
![\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are niether unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Definition 5.1 of if
is even and
is torsion-free.
Take a collection
such that
, where the embedding
be obtained by adding a cone over
in
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Theorem 5.2.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times \bar L:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/d/6/7/d6702d6274ed35eda6c8af4dc344aa87.png)
is one-to-one.
Here the element is such that for every
holds
.
An equivalemt statement of this theorem:
Theorem 5.3.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1.
Assume that is a closed compact
-connected
-manifold and
. Then
embeds into
.
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].
Theorem 6.2.
Assume that is a compact
-manifold with nonempty boundary,
is
-connected and
. Then
embeds into
.
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).
Theorem 6.3.
Assume that is a closed
-connected
-manifold. Then for each
,
any two embeddings of
into
are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4.
Assume that is a
-connected
-manifold with non-empty boundary.
Then for each
and
any two embeddings of
into
are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Denote by
the set embeddings of
into
up to isotopy.
Theorem 6.5.
Assume is a closed orientable
-connected manifold embeddable in
. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)].
Latter Theorem is essetialy known result which can be considered as generalization of the Theorem 5.3.
7 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [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.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1
] for the DIFF case and [Skopenkov2002, Theorem 1.3
] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
Sometimes we give references to such direct proofs but we do not claim these are original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a closed compact
-manifold. Then
embeds into
.
This is well-known strong Whitney embedding theorem.
Theorem 2.2.
Assume that is a compact
-manifold with nonempty boundary. Then
embeds into
.
The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].
Theorem 2.3.
Assume that is a compact
-manifold and either
(a) or
(b) is connected and
.
Then any two embeddings of into
are isotopic.
The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, 2].
Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.
Theorem 2.4.
Assume that is a compact
-manifold with non-empty boundary and either
(a) or
(b) is
-connected,
.
Then any two embeddings of into
are isotopic.
Part (a) of this theorem in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
This theorem is a special case of the Theorem 6.4 .
Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
The following example is folklore.
Proposition 3.1.
Let be the cylinder over
.
Then there exist non-isotopic embeddings of
to
.
Proof.
Define by the formula
, where
. Define
by the formula
.
![\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}](/images/math/6/b/3/6b3975d440818eb9a945b0233464a028.png)
![\mathrm ig_1](/images/math/8/0/a/80aee5fbd9f1820e8e7c60594114bdab.png)
![\mathrm ig_2](/images/math/3/e/e/3ee2c6e0a8cd1ba74a69e581c539ce92.png)
![\mathrm ig_1(S^{n-1}\times \{0, 1\})](/images/math/7/d/5/7d5695103c9fc9eb431b1340d26c5297.png)
![\mathrm ig_2(S^{n-1}\times \{0, 1\})](/images/math/7/3/5/735af2164330b444f9ef79ce4f6d863d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by the linking coefficient [Skopenkov2016h,
3, remark 3.2d] of two disjoint cycles.
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and prooved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
Let
be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
![\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are niether unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Definition 5.1 of if
is even and
is torsion-free.
Take a collection
such that
, where the embedding
be obtained by adding a cone over
in
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Theorem 5.2.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times \bar L:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/d/6/7/d6702d6274ed35eda6c8af4dc344aa87.png)
is one-to-one.
Here the element is such that for every
holds
.
An equivalemt statement of this theorem:
Theorem 5.3.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1.
Assume that is a closed compact
-connected
-manifold and
. Then
embeds into
.
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].
Theorem 6.2.
Assume that is a compact
-manifold with nonempty boundary,
is
-connected and
. Then
embeds into
.
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).
Theorem 6.3.
Assume that is a closed
-connected
-manifold. Then for each
,
any two embeddings of
into
are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4.
Assume that is a
-connected
-manifold with non-empty boundary.
Then for each
and
any two embeddings of
into
are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Denote by
the set embeddings of
into
up to isotopy.
Theorem 6.5.
Assume is a closed orientable
-connected manifold embeddable in
. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)].
Latter Theorem is essetialy known result which can be considered as generalization of the Theorem 5.3.
7 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [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.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1
] for the DIFF case and [Skopenkov2002, Theorem 1.3
] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
Sometimes we give references to such direct proofs but we do not claim these are original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a closed compact
-manifold. Then
embeds into
.
This is well-known strong Whitney embedding theorem.
Theorem 2.2.
Assume that is a compact
-manifold with nonempty boundary. Then
embeds into
.
The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].
Theorem 2.3.
Assume that is a compact
-manifold and either
(a) or
(b) is connected and
.
Then any two embeddings of into
are isotopic.
The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, 2].
Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.
Theorem 2.4.
Assume that is a compact
-manifold with non-empty boundary and either
(a) or
(b) is
-connected,
.
Then any two embeddings of into
are isotopic.
Part (a) of this theorem in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
This theorem is a special case of the Theorem 6.4 .
Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
The following example is folklore.
Proposition 3.1.
Let be the cylinder over
.
Then there exist non-isotopic embeddings of
to
.
Proof.
Define by the formula
, where
. Define
by the formula
.
![\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}](/images/math/6/b/3/6b3975d440818eb9a945b0233464a028.png)
![\mathrm ig_1](/images/math/8/0/a/80aee5fbd9f1820e8e7c60594114bdab.png)
![\mathrm ig_2](/images/math/3/e/e/3ee2c6e0a8cd1ba74a69e581c539ce92.png)
![\mathrm ig_1(S^{n-1}\times \{0, 1\})](/images/math/7/d/5/7d5695103c9fc9eb431b1340d26c5297.png)
![\mathrm ig_2(S^{n-1}\times \{0, 1\})](/images/math/7/3/5/735af2164330b444f9ef79ce4f6d863d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by the linking coefficient [Skopenkov2016h,
3, remark 3.2d] of two disjoint cycles.
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and prooved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
Let
be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
![\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are niether unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Definition 5.1 of if
is even and
is torsion-free.
Take a collection
such that
, where the embedding
be obtained by adding a cone over
in
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Theorem 5.2.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times \bar L:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/d/6/7/d6702d6274ed35eda6c8af4dc344aa87.png)
is one-to-one.
Here the element is such that for every
holds
.
An equivalemt statement of this theorem:
Theorem 5.3.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1.
Assume that is a closed compact
-connected
-manifold and
. Then
embeds into
.
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].
Theorem 6.2.
Assume that is a compact
-manifold with nonempty boundary,
is
-connected and
. Then
embeds into
.
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).
Theorem 6.3.
Assume that is a closed
-connected
-manifold. Then for each
,
any two embeddings of
into
are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4.
Assume that is a
-connected
-manifold with non-empty boundary.
Then for each
and
any two embeddings of
into
are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Denote by
the set embeddings of
into
up to isotopy.
Theorem 6.5.
Assume is a closed orientable
-connected manifold embeddable in
. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)].
Latter Theorem is essetialy known result which can be considered as generalization of the Theorem 5.3.
7 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [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.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1
] for the DIFF case and [Skopenkov2002, Theorem 1.3
] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
Sometimes we give references to such direct proofs but we do not claim these are original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a closed compact
-manifold. Then
embeds into
.
This is well-known strong Whitney embedding theorem.
Theorem 2.2.
Assume that is a compact
-manifold with nonempty boundary. Then
embeds into
.
The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].
Theorem 2.3.
Assume that is a compact
-manifold and either
(a) or
(b) is connected and
.
Then any two embeddings of into
are isotopic.
The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, 2].
Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.
Theorem 2.4.
Assume that is a compact
-manifold with non-empty boundary and either
(a) or
(b) is
-connected,
.
Then any two embeddings of into
are isotopic.
Part (a) of this theorem in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
This theorem is a special case of the Theorem 6.4 .
Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
The following example is folklore.
Proposition 3.1.
Let be the cylinder over
.
Then there exist non-isotopic embeddings of
to
.
Proof.
Define by the formula
, where
. Define
by the formula
.
![\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}](/images/math/6/b/3/6b3975d440818eb9a945b0233464a028.png)
![\mathrm ig_1](/images/math/8/0/a/80aee5fbd9f1820e8e7c60594114bdab.png)
![\mathrm ig_2](/images/math/3/e/e/3ee2c6e0a8cd1ba74a69e581c539ce92.png)
![\mathrm ig_1(S^{n-1}\times \{0, 1\})](/images/math/7/d/5/7d5695103c9fc9eb431b1340d26c5297.png)
![\mathrm ig_2(S^{n-1}\times \{0, 1\})](/images/math/7/3/5/735af2164330b444f9ef79ce4f6d863d.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by the linking coefficient [Skopenkov2016h,
3, remark 3.2d] of two disjoint cycles.
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and prooved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
Let
be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
![\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are niether unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Definition 5.1 of if
is even and
is torsion-free.
Take a collection
such that
, where the embedding
be obtained by adding a cone over
in
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Theorem 5.2.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times \bar L:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/d/6/7/d6702d6274ed35eda6c8af4dc344aa87.png)
is one-to-one.
Here the element is such that for every
holds
.
An equivalemt statement of this theorem:
Theorem 5.3.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1.
Assume that is a closed compact
-connected
-manifold and
. Then
embeds into
.
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].
Theorem 6.2.
Assume that is a compact
-manifold with nonempty boundary,
is
-connected and
. Then
embeds into
.
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).
Theorem 6.3.
Assume that is a closed
-connected
-manifold. Then for each
,
any two embeddings of
into
are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4.
Assume that is a
-connected
-manifold with non-empty boundary.
Then for each
and
any two embeddings of
into
are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in
to an open
-ball. Thus
is the
-sphere.
Denote by
the set embeddings of
into
up to isotopy.
Theorem 6.5.
Assume is a closed orientable
-connected manifold embeddable in
. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)].
Latter Theorem is essetialy known result which can be considered as generalization of the Theorem 5.3.
7 References
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, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
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