Embeddings of manifolds with boundary: classification
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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 theorems
Theorem 2.1.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold. Then
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n}](/images/math/b/9/2/b9289dcd4201c4a1b0ad79df027bda1d.png)
This is well-known Whitney embedding theorem.
Theorem 2.2.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold with nonempty boundary. Then
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-1}](/images/math/6/d/c/6dcc6fdbf961d308f4f506739919c281.png)
This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].
Theorem 2.3.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold and
![n>2k+2](/images/math/4/b/5/4b56b55e76b441b6762e6bb166287d75.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-k}](/images/math/7/a/f/7af7a4aaf4127af451bbcc7db28bc0bf.png)
Diff case of this result can be found in [Haefliger1961, Theoreme d'existance, a)], PL case of this result can be found in [Irwin1965, Corollary 1.3].
Theorem 2.4.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold with nonempty boundary,
![(N, \partial N)](/images/math/d/4/4/d448e6642d492b1a56bbe88023ad1bdb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![n\ge2k+2](/images/math/4/1/3/413096a3521440af17cabcf0c83c37c7.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-k-1}](/images/math/9/6/6/9666532d0ab6e26357abdcfd354f0340.png)
PL case of this result can be found in [Hudson1969, Theorem 8.3].
3 Unknotting Theorems
Theorem 3.1.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-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].
Theorem 3.2.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-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.
Theorem 3.3.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold. Then for each
![n\ge2k + 2](/images/math/9/6/b/96bd61dde76ff15c005b23b691dbc6a8.png)
![m \ge 2n - k + 1](/images/math/3/8/7/387a3daa72124095b8ce1df4f84f2575.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^m](/images/math/e/c/7/ec7854d70912266b9994b6e182b63f67.png)
See Theorem 2.4 of [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 3.4.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold with non-empty boundary.
Then for each and
any two embeddings of
into
are isotopic.
Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].
4 Examples
Observe that analog of Theorem 3.2 (a) fails for , i.e. for the first non trivial case. More precisely, the following folklore statement holds.
Proposition 4.1.
Let be the cylinder over
.
Then there exist non isotopic embeddings of
to
.
Proof.
Recall is the standard embedding.
Define
, where
is a fixed point.
Define .
Then embeddings
and
are not isotopic to each other.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
See also about the Hopf link in [Skopenkov2016h, 2].
5 Invariants
Denote by the linking coefficient ([Skopenkov2016h,
3, remark 3.2d]) of two cycles with disjoint support.
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold. Let
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
Tex syntax error-ball in
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N_0:=Cl(N-B^n)](/images/math/4/9/9/499730eb1c06341decd3c9eecd89c80b.png)
The following folklore result holds.
Lemma 5.1.
For each evenTex syntax errorand each embedding
![f\colon N_0 \to \mathbb R^{2n-1}](/images/math/b/0/d/b0dcdea51b0ba728a1e1d597163247e4.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
Tex syntax errorand every embedding
![f\colon N_0\to\mathbb R^{2n-1}](/images/math/f/3/f/f3fef2f7ebae3416f695c7d999e12db5.png)
![\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 are two homology classes, realized by closed connected orientation submanifolds of
,
is a nowhere vanishing normal field to
and
are the submanifolds
shifted by
.
Denote by reduction modulo
.
Denote by the set embeddings of
into
up to isotopy.
Define the dual to Steifel-Whitney class to be the class of the cycle on which two general position normal fields on
are linearly dependent.
Lemma 5.2.(O. Saeki)
Let be an embedding.
Let
be the boundary of a tubular neighborhood of
.
Given two homology classes
,
let
be two sections of
.
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 (Poincare dual to) the first obstruction
to
being homotopic as sections of
.
Lemma 5.3.
Let , then
![\displaystyle \rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.](/images/math/5/d/e/5ded55fdde5db827827cd521d540888a.png)
Proof.
Observe
Denote by the normal vector field opposite to
. If we shift the link
by
, we get the link
and the
will not change. Hence,
![\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).](/images/math/3/8/a/38a12a3cff4f3378c55345e6a5be290f.png)
By lemma 5.2
![\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.](/images/math/2/2/c/22c31c35229109dd771893d1cd4e5125.png)
Finally, let us show that .
If we generically perturb
it will become linearly dependent with
only on a 2--dimensional cycle
in
, such that
by definition.
On the other hand the linear homotopy of
to perturbed
degenerates on
.
Thus
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
6 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.
- [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
- [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.
- [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 theorems
Theorem 2.1.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold. Then
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n}](/images/math/b/9/2/b9289dcd4201c4a1b0ad79df027bda1d.png)
This is well-known Whitney embedding theorem.
Theorem 2.2.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold with nonempty boundary. Then
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-1}](/images/math/6/d/c/6dcc6fdbf961d308f4f506739919c281.png)
This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].
Theorem 2.3.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold and
![n>2k+2](/images/math/4/b/5/4b56b55e76b441b6762e6bb166287d75.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-k}](/images/math/7/a/f/7af7a4aaf4127af451bbcc7db28bc0bf.png)
Diff case of this result can be found in [Haefliger1961, Theoreme d'existance, a)], PL case of this result can be found in [Irwin1965, Corollary 1.3].
Theorem 2.4.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold with nonempty boundary,
![(N, \partial N)](/images/math/d/4/4/d448e6642d492b1a56bbe88023ad1bdb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![n\ge2k+2](/images/math/4/1/3/413096a3521440af17cabcf0c83c37c7.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-k-1}](/images/math/9/6/6/9666532d0ab6e26357abdcfd354f0340.png)
PL case of this result can be found in [Hudson1969, Theorem 8.3].
3 Unknotting Theorems
Theorem 3.1.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-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].
Theorem 3.2.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-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.
Theorem 3.3.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold. Then for each
![n\ge2k + 2](/images/math/9/6/b/96bd61dde76ff15c005b23b691dbc6a8.png)
![m \ge 2n - k + 1](/images/math/3/8/7/387a3daa72124095b8ce1df4f84f2575.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^m](/images/math/e/c/7/ec7854d70912266b9994b6e182b63f67.png)
See Theorem 2.4 of [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 3.4.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold with non-empty boundary.
Then for each and
any two embeddings of
into
are isotopic.
Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].
4 Examples
Observe that analog of Theorem 3.2 (a) fails for , i.e. for the first non trivial case. More precisely, the following folklore statement holds.
Proposition 4.1.
Let be the cylinder over
.
Then there exist non isotopic embeddings of
to
.
Proof.
Recall is the standard embedding.
Define
, where
is a fixed point.
Define .
Then embeddings
and
are not isotopic to each other.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
See also about the Hopf link in [Skopenkov2016h, 2].
5 Invariants
Denote by the linking coefficient ([Skopenkov2016h,
3, remark 3.2d]) of two cycles with disjoint support.
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold. Let
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
Tex syntax error-ball in
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N_0:=Cl(N-B^n)](/images/math/4/9/9/499730eb1c06341decd3c9eecd89c80b.png)
The following folklore result holds.
Lemma 5.1.
For each evenTex syntax errorand each embedding
![f\colon N_0 \to \mathbb R^{2n-1}](/images/math/b/0/d/b0dcdea51b0ba728a1e1d597163247e4.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
Tex syntax errorand every embedding
![f\colon N_0\to\mathbb R^{2n-1}](/images/math/f/3/f/f3fef2f7ebae3416f695c7d999e12db5.png)
![\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 are two homology classes, realized by closed connected orientation submanifolds of
,
is a nowhere vanishing normal field to
and
are the submanifolds
shifted by
.
Denote by reduction modulo
.
Denote by the set embeddings of
into
up to isotopy.
Define the dual to Steifel-Whitney class to be the class of the cycle on which two general position normal fields on
are linearly dependent.
Lemma 5.2.(O. Saeki)
Let be an embedding.
Let
be the boundary of a tubular neighborhood of
.
Given two homology classes
,
let
be two sections of
.
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 (Poincare dual to) the first obstruction
to
being homotopic as sections of
.
Lemma 5.3.
Let , then
![\displaystyle \rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.](/images/math/5/d/e/5ded55fdde5db827827cd521d540888a.png)
Proof.
Observe
Denote by the normal vector field opposite to
. If we shift the link
by
, we get the link
and the
will not change. Hence,
![\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).](/images/math/3/8/a/38a12a3cff4f3378c55345e6a5be290f.png)
By lemma 5.2
![\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.](/images/math/2/2/c/22c31c35229109dd771893d1cd4e5125.png)
Finally, let us show that .
If we generically perturb
it will become linearly dependent with
only on a 2--dimensional cycle
in
, such that
by definition.
On the other hand the linear homotopy of
to perturbed
degenerates on
.
Thus
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
6 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.
- [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
- [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.
- [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 theorems
Theorem 2.1.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold. Then
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n}](/images/math/b/9/2/b9289dcd4201c4a1b0ad79df027bda1d.png)
This is well-known Whitney embedding theorem.
Theorem 2.2.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold with nonempty boundary. Then
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-1}](/images/math/6/d/c/6dcc6fdbf961d308f4f506739919c281.png)
This result is a special case of [Hudson1969, Theorem 8.3]. See also [Horvatic1971, Theorem 5.2].
Theorem 2.3.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold and
![n>2k+2](/images/math/4/b/5/4b56b55e76b441b6762e6bb166287d75.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-k}](/images/math/7/a/f/7af7a4aaf4127af451bbcc7db28bc0bf.png)
Diff case of this result can be found in [Haefliger1961, Theoreme d'existance, a)], PL case of this result can be found in [Irwin1965, Corollary 1.3].
Theorem 2.4.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold with nonempty boundary,
![(N, \partial N)](/images/math/d/4/4/d448e6642d492b1a56bbe88023ad1bdb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![n\ge2k+2](/images/math/4/1/3/413096a3521440af17cabcf0c83c37c7.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^{2n-k-1}](/images/math/9/6/6/9666532d0ab6e26357abdcfd354f0340.png)
PL case of this result can be found in [Hudson1969, Theorem 8.3].
3 Unknotting Theorems
Theorem 3.1.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-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].
Theorem 3.2.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-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.
Theorem 3.3.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold. Then for each
![n\ge2k + 2](/images/math/9/6/b/96bd61dde76ff15c005b23b691dbc6a8.png)
![m \ge 2n - k + 1](/images/math/3/8/7/387a3daa72124095b8ce1df4f84f2575.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\R^m](/images/math/e/c/7/ec7854d70912266b9994b6e182b63f67.png)
See Theorem 2.4 of [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 3.4.
Assume that![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
Tex syntax error-manifold with non-empty boundary.
Then for each and
any two embeddings of
into
are isotopic.
Theorem 3.2 is a special cases of the latter result. See also [Hudson1969, Theorem 10.3].
4 Examples
Observe that analog of Theorem 3.2 (a) fails for , i.e. for the first non trivial case. More precisely, the following folklore statement holds.
Proposition 4.1.
Let be the cylinder over
.
Then there exist non isotopic embeddings of
to
.
Proof.
Recall is the standard embedding.
Define
, where
is a fixed point.
Define .
Then embeddings
and
are not isotopic to each other.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
See also about the Hopf link in [Skopenkov2016h, 2].
5 Invariants
Denote by the linking coefficient ([Skopenkov2016h,
3, remark 3.2d]) of two cycles with disjoint support.
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
Tex syntax error-manifold. Let
![B^n](/images/math/0/d/5/0d5ee235988a4f4261c4b6a69521a856.png)
Tex syntax error-ball in
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N_0:=Cl(N-B^n)](/images/math/4/9/9/499730eb1c06341decd3c9eecd89c80b.png)
The following folklore result holds.
Lemma 5.1.
For each evenTex syntax errorand each embedding
![f\colon N_0 \to \mathbb R^{2n-1}](/images/math/b/0/d/b0dcdea51b0ba728a1e1d597163247e4.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
Tex syntax errorand every embedding
![f\colon N_0\to\mathbb R^{2n-1}](/images/math/f/3/f/f3fef2f7ebae3416f695c7d999e12db5.png)
![\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 are two homology classes, realized by closed connected orientation submanifolds of
,
is a nowhere vanishing normal field to
and
are the submanifolds
shifted by
.
Denote by reduction modulo
.
Denote by the set embeddings of
into
up to isotopy.
Define the dual to Steifel-Whitney class to be the class of the cycle on which two general position normal fields on
are linearly dependent.
Lemma 5.2.(O. Saeki)
Let be an embedding.
Let
be the boundary of a tubular neighborhood of
.
Given two homology classes
,
let
be two sections of
.
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 (Poincare dual to) the first obstruction
to
being homotopic as sections of
.
Lemma 5.3.
Let , then
![\displaystyle \rho_2(L(f)(x, y)) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2x\cap\rho_2y.](/images/math/5/d/e/5ded55fdde5db827827cd521d540888a.png)
Proof.
Observe
Denote by the normal vector field opposite to
. If we shift the link
by
, we get the link
and the
will not change. Hence,
![\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(s(x), f(y)) = \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)).](/images/math/3/8/a/38a12a3cff4f3378c55345e6a5be290f.png)
By lemma 5.2
![\displaystyle \mathrm{lk}(f(x), s(y)) - \mathrm{lk}(f(x), -s(y)) = d(s, -s)\cap x\cap y.](/images/math/2/2/c/22c31c35229109dd771893d1cd4e5125.png)
Finally, let us show that .
If we generically perturb
it will become linearly dependent with
only on a 2--dimensional cycle
in
, such that
by definition.
On the other hand the linear homotopy of
to perturbed
degenerates on
.
Thus
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
6 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.
- [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
- [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.
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).