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 Classification 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 cases 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
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 Invariants
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 and by
their homology classes.
Lemma 4.2.
Let be an embedding.
Let
be two 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 shift of
by
and
is (Poincare dual to) the first obstruction
to
being homotopic in the class of the non-zero vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Denote by the reduction modulo
.
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 result of shift of
by
.
Note that does not change when
or
are changed to homologic cycles or when
is changed to an isotopic embedding. Thus
is a bilinear form on
, i.e.
generates a bilinear form denoted by the same letter.
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 not published version of[Tonkonog2010], the proof given here is covering a minor gap.
Proof of Lemma 4.4
Let be tho 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.
If we shift the link by
, we get the link
. The linking coefficient will not change after this shift.
The third line follows from Lemma 4.2.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \bar w_{n-2}(N_0) \underset{2}\equiv d(s', -s) = d(s, -s) .](/images/math/0/0/b/00b98dd606e714dff3a8b44debea2e3d.png)
is linearly dependent with
only on a 2--dimensional cycle
in
. And
by definition.
On the other hand the linear homotopy of
and
degenerates on
.
Thus
.
5 Classification
Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.
Denote by the set embeddings of
into
up to isotopy.
Theorem 5.1.
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, and
is the standard pairing.
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
.
The PL case of this result is proved in [Hudson1969, Theorem 8.3]. 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 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.
See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
7 References
- [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 Classification 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 cases 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
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 Invariants
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 and by
their homology classes.
Lemma 4.2.
Let be an embedding.
Let
be two 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 shift of
by
and
is (Poincare dual to) the first obstruction
to
being homotopic in the class of the non-zero vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Denote by the reduction modulo
.
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 result of shift of
by
.
Note that does not change when
or
are changed to homologic cycles or when
is changed to an isotopic embedding. Thus
is a bilinear form on
, i.e.
generates a bilinear form denoted by the same letter.
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 not published version of[Tonkonog2010], the proof given here is covering a minor gap.
Proof of Lemma 4.4
Let be tho 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.
If we shift the link by
, we get the link
. The linking coefficient will not change after this shift.
The third line follows from Lemma 4.2.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \bar w_{n-2}(N_0) \underset{2}\equiv d(s', -s) = d(s, -s) .](/images/math/0/0/b/00b98dd606e714dff3a8b44debea2e3d.png)
is linearly dependent with
only on a 2--dimensional cycle
in
. And
by definition.
On the other hand the linear homotopy of
and
degenerates on
.
Thus
.
5 Classification
Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.
Denote by the set embeddings of
into
up to isotopy.
Theorem 5.1.
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, and
is the standard pairing.
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
.
The PL case of this result is proved in [Hudson1969, Theorem 8.3]. 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 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.
See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
7 References
- [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 Classification 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 cases 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
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 Invariants
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 and by
their homology classes.
Lemma 4.2.
Let be an embedding.
Let
be two 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 shift of
by
and
is (Poincare dual to) the first obstruction
to
being homotopic in the class of the non-zero vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Denote by the reduction modulo
.
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 result of shift of
by
.
Note that does not change when
or
are changed to homologic cycles or when
is changed to an isotopic embedding. Thus
is a bilinear form on
, i.e.
generates a bilinear form denoted by the same letter.
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 not published version of[Tonkonog2010], the proof given here is covering a minor gap.
Proof of Lemma 4.4
Let be tho 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.
If we shift the link by
, we get the link
. The linking coefficient will not change after this shift.
The third line follows from Lemma 4.2.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \bar w_{n-2}(N_0) \underset{2}\equiv d(s', -s) = d(s, -s) .](/images/math/0/0/b/00b98dd606e714dff3a8b44debea2e3d.png)
is linearly dependent with
only on a 2--dimensional cycle
in
. And
by definition.
On the other hand the linear homotopy of
and
degenerates on
.
Thus
.
5 Classification
Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.
Denote by the set embeddings of
into
up to isotopy.
Theorem 5.1.
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, and
is the standard pairing.
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
.
The PL case of this result is proved in [Hudson1969, Theorem 8.3]. 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 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.
See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
7 References
- [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 Classification 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 cases 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
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 Invariants
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 and by
their homology classes.
Lemma 4.2.
Let be an embedding.
Let
be two 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 shift of
by
and
is (Poincare dual to) the first obstruction
to
being homotopic in the class of the non-zero vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Denote by the reduction modulo
.
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 result of shift of
by
.
Note that does not change when
or
are changed to homologic cycles or when
is changed to an isotopic embedding. Thus
is a bilinear form on
, i.e.
generates a bilinear form denoted by the same letter.
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 not published version of[Tonkonog2010], the proof given here is covering a minor gap.
Proof of Lemma 4.4
Let be tho 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.
If we shift the link by
, we get the link
. The linking coefficient will not change after this shift.
The third line follows from Lemma 4.2.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \bar w_{n-2}(N_0) \underset{2}\equiv d(s', -s) = d(s, -s) .](/images/math/0/0/b/00b98dd606e714dff3a8b44debea2e3d.png)
is linearly dependent with
only on a 2--dimensional cycle
in
. And
by definition.
On the other hand the linear homotopy of
and
degenerates on
.
Thus
.
5 Classification
Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.
Denote by the set embeddings of
into
up to isotopy.
Theorem 5.1.
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, and
is the standard pairing.
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
.
The PL case of this result is proved in [Hudson1969, Theorem 8.3]. 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 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.
See also [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
7 References
- [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).