Embeddings of manifolds with boundary: classification
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Contents |
1 Introduction
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.
In this page we present results peculiar for manifold with non-empty boundary.
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
We state only the results that 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 5.2 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 5.4 .
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 5).
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 connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
The following folklore result holds.
Lemma 4.1.
Suppose is torsion free. For each even
and 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.
Lemma 4.2.
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.
Definition 4.3.
For even and every embedding
denote by
![\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
.
Note that does not change when
or
are changed to homologous cycles or when
is changed to an isotopic embedding. Thus
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 4.4.
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 by M. Fedorov is obtained using the idea from that update.
See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
Proof of Lemma 4.4
Let be the normal field to
opposite to
. We get
![\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 4.2.
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.
The second equality holds because the linear homotopy of
and
degenerates on a
-cycle in
on which
and
are linearly dependent.
5 A generalization to highly-connected manifolds
Theorem 5.1.
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 5.2.
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 5.3.
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 5.4.
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.
6 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.
- [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).
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
We state only the results that 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 5.2 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 5.4 .
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 5).
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 connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
The following folklore result holds.
Lemma 4.1.
Suppose is torsion free. For each even
and 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.
Lemma 4.2.
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.
Definition 4.3.
For even and every embedding
denote by
![\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
.
Note that does not change when
or
are changed to homologous cycles or when
is changed to an isotopic embedding. Thus
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 4.4.
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 by M. Fedorov is obtained using the idea from that update.
See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
Proof of Lemma 4.4
Let be the normal field to
opposite to
. We get
![\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 4.2.
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.
The second equality holds because the linear homotopy of
and
degenerates on a
-cycle in
on which
and
are linearly dependent.
5 A generalization to highly-connected manifolds
Theorem 5.1.
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 5.2.
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 5.3.
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 5.4.
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.
6 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.
- [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).
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
We state only the results that 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 5.2 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 5.4 .
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 5).
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 connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
The following folklore result holds.
Lemma 4.1.
Suppose is torsion free. For each even
and 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.
Lemma 4.2.
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.
Definition 4.3.
For even and every embedding
denote by
![\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
.
Note that does not change when
or
are changed to homologous cycles or when
is changed to an isotopic embedding. Thus
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 4.4.
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 by M. Fedorov is obtained using the idea from that update.
See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
Proof of Lemma 4.4
Let be the normal field to
opposite to
. We get
![\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 4.2.
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.
The second equality holds because the linear homotopy of
and
degenerates on a
-cycle in
on which
and
are linearly dependent.
5 A generalization to highly-connected manifolds
Theorem 5.1.
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 5.2.
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 5.3.
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 5.4.
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.
6 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.
- [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).
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
We state only the results that 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 5.2 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 5.4 .
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 5).
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 connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
The following folklore result holds.
Lemma 4.1.
Suppose is torsion free. For each even
and 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.
Lemma 4.2.
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.
Definition 4.3.
For even and every embedding
denote by
![\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
.
Note that does not change when
or
are changed to homologous cycles or when
is changed to an isotopic embedding. Thus
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 4.4.
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 by M. Fedorov is obtained using the idea from that update.
See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
Proof of Lemma 4.4
Let be the normal field to
opposite to
. We get
![\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 4.2.
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.
The second equality holds because the linear homotopy of
and
degenerates on a
-cycle in
on which
and
are linearly dependent.
5 A generalization to highly-connected manifolds
Theorem 5.1.
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 5.2.
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 5.3.
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 5.4.
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.
6 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.
- [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).