Embeddings of manifolds with boundary: classification
m |
|||
Line 1: | Line 1: | ||
− | + | {{Authors|Mikhail Fedorov}} | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | {{ | + | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
{{Stub}} | {{Stub}} | ||
== Introduction == | == Introduction == |
Revision as of 07:01, 19 November 2020
The user responsible for this page is Mikhail Fedorov. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In 4 we introduce an invariant of embedding of a -manifold in -space for even . In 6 which is independent from 3, 4 and 5 we state generalisations of theorems from 2 to highly-connected manifolds.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to 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 .
This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [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 connected -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. Both parts of this theorem are special cases of the Theorem 6.4.
Inequality in part (a) is sharp, see Proposition 3.1.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
Denote by the linking coefficient of two disjoint cycles with integer coefficient.
The following example is folklore.
Proposition 3.1. Let be the cylinder over . (a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
Proof. Let be a map of degree . To prove part (a) it is sufficient to take the identity map of as a map of degree one and the constant map as a map of degree zero. Define by the formula , where .
Let , where is the standard embedding.Thus
4 Seifert linking form
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. If the (co)homology coefficients are omitted, then we assume them to be .
Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
Example 4.1. For and each there exists a bijection given by .
The surjectivity of is given by Proposition 3.1(b).
The following folklore result holds.
Lemma 4.2. Assume is a closed orientable connected -manifold, is even and is torsion free. Then for each embedding there exists a nowhere vanishing normal vector field to .
Proof. There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to .
A normal space to at any point of has dimension . As is even thus is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore . Since is torsion free, it follows that .
Since has non-empty boundary, we have that is homotopy equivalent to an -complex. The dimension of this complex equals the dimension of normal space to at any point of . Since , it follows that there exists a nowhere vanishing normal vector field to .
Denote by two disjoint -cycles in with integer coefficients.
Definition 4.3. For even and every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.4 ( is well-defined). For even and every embedding the integer :
- is well-defined, i.e. does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.5. Let be an embedding. Let be two nowhere vanishing normal vector fields to . Then
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.
Here the second equality follows from Lemma 4.5.
For each two homologous -cycles in , the image of the homology between and is a -chain of such that . Since is a nowhere vanishing normal field to , this implies that the supports of and are disjoint. Hence .
Since isotopy of is a map from to , it follows that this isotopy gives an isotopy of the link . Now the third bullet point follows because the linking coefficient is preserved under isotopy.
Lemma 4.4 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.6. Let be an embedding. Then for every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
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 4.5.
Thus it is sufficient to show that . Denote by a general perturbation of . We get:
The first equality holds because and are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between and degenerates only at those points where . These points are exactly points where and are linearly dependent. All those point form a -cycle modulo two in . The homotopy class of this -cycle is by the definition of Stiefel-Whitney class.
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. Denote the set of all embeddings up to isotopy. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 5.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 5.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 5.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 5.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 5.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 5.4:
Theorem 5.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1. Assume that is a closed compact -connected -manifold and . Then embeds into .
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
Theorem 6.2. Assume that is a compact -manifold with nonempty boundary, is -connected and . Then embeds into .
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.3. Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4. Assume that is a -connected -manifold with non-empty boundary. Then for each and any two embeddings of into are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in to an open -ball. Thus is the -sphere. Denote by the set embeddings of into up to isotopy.
Theorem 6.5. Assume is a closed orientable -connected manifold embeddable into . Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248--342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, 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).
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to 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 .
This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [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 connected -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. Both parts of this theorem are special cases of the Theorem 6.4.
Inequality in part (a) is sharp, see Proposition 3.1.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
Denote by the linking coefficient of two disjoint cycles with integer coefficient.
The following example is folklore.
Proposition 3.1. Let be the cylinder over . (a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
Proof. Let be a map of degree . To prove part (a) it is sufficient to take the identity map of as a map of degree one and the constant map as a map of degree zero. Define by the formula , where .
Let , where is the standard embedding.Thus
4 Seifert linking form
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. If the (co)homology coefficients are omitted, then we assume them to be .
Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
Example 4.1. For and each there exists a bijection given by .
The surjectivity of is given by Proposition 3.1(b).
The following folklore result holds.
Lemma 4.2. Assume is a closed orientable connected -manifold, is even and is torsion free. Then for each embedding there exists a nowhere vanishing normal vector field to .
Proof. There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to .
A normal space to at any point of has dimension . As is even thus is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore . Since is torsion free, it follows that .
Since has non-empty boundary, we have that is homotopy equivalent to an -complex. The dimension of this complex equals the dimension of normal space to at any point of . Since , it follows that there exists a nowhere vanishing normal vector field to .
Denote by two disjoint -cycles in with integer coefficients.
Definition 4.3. For even and every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.4 ( is well-defined). For even and every embedding the integer :
- is well-defined, i.e. does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.5. Let be an embedding. Let be two nowhere vanishing normal vector fields to . Then
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.
Here the second equality follows from Lemma 4.5.
For each two homologous -cycles in , the image of the homology between and is a -chain of such that . Since is a nowhere vanishing normal field to , this implies that the supports of and are disjoint. Hence .
Since isotopy of is a map from to , it follows that this isotopy gives an isotopy of the link . Now the third bullet point follows because the linking coefficient is preserved under isotopy.
Lemma 4.4 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.6. Let be an embedding. Then for every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
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 4.5.
Thus it is sufficient to show that . Denote by a general perturbation of . We get:
The first equality holds because and are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between and degenerates only at those points where . These points are exactly points where and are linearly dependent. All those point form a -cycle modulo two in . The homotopy class of this -cycle is by the definition of Stiefel-Whitney class.
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. Denote the set of all embeddings up to isotopy. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 5.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 5.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 5.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 5.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 5.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 5.4:
Theorem 5.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1. Assume that is a closed compact -connected -manifold and . Then embeds into .
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
Theorem 6.2. Assume that is a compact -manifold with nonempty boundary, is -connected and . Then embeds into .
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.3. Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4. Assume that is a -connected -manifold with non-empty boundary. Then for each and any two embeddings of into are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in to an open -ball. Thus is the -sphere. Denote by the set embeddings of into up to isotopy.
Theorem 6.5. Assume is a closed orientable -connected manifold embeddable into . Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248--342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, 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).
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to 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 .
This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [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 connected -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. Both parts of this theorem are special cases of the Theorem 6.4.
Inequality in part (a) is sharp, see Proposition 3.1.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
Denote by the linking coefficient of two disjoint cycles with integer coefficient.
The following example is folklore.
Proposition 3.1. Let be the cylinder over . (a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
Proof. Let be a map of degree . To prove part (a) it is sufficient to take the identity map of as a map of degree one and the constant map as a map of degree zero. Define by the formula , where .
Let , where is the standard embedding.Thus
4 Seifert linking form
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. If the (co)homology coefficients are omitted, then we assume them to be .
Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
Example 4.1. For and each there exists a bijection given by .
The surjectivity of is given by Proposition 3.1(b).
The following folklore result holds.
Lemma 4.2. Assume is a closed orientable connected -manifold, is even and is torsion free. Then for each embedding there exists a nowhere vanishing normal vector field to .
Proof. There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to .
A normal space to at any point of has dimension . As is even thus is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore . Since is torsion free, it follows that .
Since has non-empty boundary, we have that is homotopy equivalent to an -complex. The dimension of this complex equals the dimension of normal space to at any point of . Since , it follows that there exists a nowhere vanishing normal vector field to .
Denote by two disjoint -cycles in with integer coefficients.
Definition 4.3. For even and every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.4 ( is well-defined). For even and every embedding the integer :
- is well-defined, i.e. does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.5. Let be an embedding. Let be two nowhere vanishing normal vector fields to . Then
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.
Here the second equality follows from Lemma 4.5.
For each two homologous -cycles in , the image of the homology between and is a -chain of such that . Since is a nowhere vanishing normal field to , this implies that the supports of and are disjoint. Hence .
Since isotopy of is a map from to , it follows that this isotopy gives an isotopy of the link . Now the third bullet point follows because the linking coefficient is preserved under isotopy.
Lemma 4.4 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.6. Let be an embedding. Then for every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
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 4.5.
Thus it is sufficient to show that . Denote by a general perturbation of . We get:
The first equality holds because and are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between and degenerates only at those points where . These points are exactly points where and are linearly dependent. All those point form a -cycle modulo two in . The homotopy class of this -cycle is by the definition of Stiefel-Whitney class.
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. Denote the set of all embeddings up to isotopy. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 5.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 5.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 5.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 5.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 5.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 5.4:
Theorem 5.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1. Assume that is a closed compact -connected -manifold and . Then embeds into .
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
Theorem 6.2. Assume that is a compact -manifold with nonempty boundary, is -connected and . Then embeds into .
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.3. Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4. Assume that is a -connected -manifold with non-empty boundary. Then for each and any two embeddings of into are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in to an open -ball. Thus is the -sphere. Denote by the set embeddings of into up to isotopy.
Theorem 6.5. Assume is a closed orientable -connected manifold embeddable into . Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248--342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, 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).
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to 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 .
This theorem is a corollary of strong Whitney immersion theorem. For the Diff case of this result see [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 connected -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. Both parts of this theorem are special cases of the Theorem 6.4.
Inequality in part (a) is sharp, see Proposition 3.1.
These basic results can be generalized to the highly-connected manifolds (see 6).
3 Example on non-isotopic embeddings
Denote by the linking coefficient of two disjoint cycles with integer coefficient.
The following example is folklore.
Proposition 3.1. Let be the cylinder over . (a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
Proof. Let be a map of degree . To prove part (a) it is sufficient to take the identity map of as a map of degree one and the constant map as a map of degree zero. Define by the formula , where .
Let , where is the standard embedding.Thus
4 Seifert linking form
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. If the (co)homology coefficients are omitted, then we assume them to be .
Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
Example 4.1. For and each there exists a bijection given by .
The surjectivity of is given by Proposition 3.1(b).
The following folklore result holds.
Lemma 4.2. Assume is a closed orientable connected -manifold, is even and is torsion free. Then for each embedding there exists a nowhere vanishing normal vector field to .
Proof. There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to .
A normal space to at any point of has dimension . As is even thus is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore . Since is torsion free, it follows that .
Since has non-empty boundary, we have that is homotopy equivalent to an -complex. The dimension of this complex equals the dimension of normal space to at any point of . Since , it follows that there exists a nowhere vanishing normal vector field to .
Denote by two disjoint -cycles in with integer coefficients.
Definition 4.3. For even and every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.4 ( is well-defined). For even and every embedding the integer :
- is well-defined, i.e. does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.5. Let be an embedding. Let be two nowhere vanishing normal vector fields to . Then
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.
Here the second equality follows from Lemma 4.5.
For each two homologous -cycles in , the image of the homology between and is a -chain of such that . Since is a nowhere vanishing normal field to , this implies that the supports of and are disjoint. Hence .
Since isotopy of is a map from to , it follows that this isotopy gives an isotopy of the link . Now the third bullet point follows because the linking coefficient is preserved under isotopy.
Lemma 4.4 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.6. Let be an embedding. Then for every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
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 4.5.
Thus it is sufficient to show that . Denote by a general perturbation of . We get:
The first equality holds because and are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between and degenerates only at those points where . These points are exactly points where and are linearly dependent. All those point form a -cycle modulo two in . The homotopy class of this -cycle is by the definition of Stiefel-Whitney class.
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. Denote the set of all embeddings up to isotopy. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 5.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 5.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 5.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 5.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 5.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 5.4:
Theorem 5.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
Theorem 6.1. Assume that is a closed compact -connected -manifold and . Then embeds into .
The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
Theorem 6.2. Assume that is a compact -manifold with nonempty boundary, is -connected and . Then embeds into .
For the Diff case see [Haefliger1961, 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result). For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.3. Assume that is a closed -connected -manifold. Then for each , any two embeddings of into are isotopic.
See Theorem 2.4 of the survey [Skopenkov2016c, 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Theorem 6.4. Assume that is a -connected -manifold with non-empty boundary. Then for each and any two embeddings of into are isotopic.
For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
By we denote the complement in to an open -ball. Thus is the -sphere. Denote by the set embeddings of into up to isotopy.
Theorem 6.5. Assume is a closed orientable -connected manifold embeddable into . Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. Latter Theorem is essetialy known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248--342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, 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).