Embeddings of manifolds with boundary: classification
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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 aTex syntax error-manifold in -space for even
Tex syntax error.
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. Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
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 compact connectedTex syntax error-manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem.
Tex syntax errorhas a sufficiently small regular neighbourhood such that is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . The composition is an embedding of .
Theorem 2.2.
Assume that is a compact connectedTex syntax error-manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968, 4, Corollary 5]. Case is clear. Both parts of this theorem are special cases of the Theorem 6.2. Case can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6). All stated theorems of 2 and 6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example on non-isotopic embeddings
The following example is folklore.
Example 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 .
(c) Then defined by the formula is well-defined and is a bijection for .
Define by the formula , where .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.
Example 3.2. Let . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 3.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 6.3 fails for .
Example 3.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
Tex syntax error-spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 3.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
4 Seifert linking form
Tex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-ball. Thus is the -sphere.
If the (co)homology coefficients are omitted, then we assume them to be .
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connectedTex syntax error-manifold,
Tex syntax erroris 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 . AsTex syntax erroris 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 4.2.
For evenTex syntax errorand every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.3 ( is well-defined).
For evenTex syntax errorand 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
Tex syntax error
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 4.4. 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 4.4.
For each two homologous -cycles in , the image of the homology between and is aTex syntax error-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
Tex syntax erroris 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 4.3 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.5. Let be an embedding. Then for every the following equality holds:
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 4.4.
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 connectedTex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-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 evenTex syntax errordefine an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where
Tex syntax erroris 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 anTex syntax error-ball into . Unknotting Spheres Theorem implies that
Tex syntax error-sphere unknots in . Thus all extensions of
Tex syntax errorare isotopic in PL category. Note also that if
Tex syntax errorand are isotopic then their extensions are isotopic as well. And Whitney invariant
Tex syntax erroris invariant for PL embeddings.
Tex syntax erroris even and is torsion-free.
Take a collection such that .
For eachTex syntax errorsuch 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
Tex syntax error-manifold with torsion-free, ,
Tex syntax erroreven.
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 orientableTex syntax error-manifold with torsion-free, ,
Tex syntax erroreven. 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
For simplicity in this paragraph we consider only punctured manifolds, see 7 for generalization for.
By denote the complement in to an openTex syntax error-ball. Thus is the -sphere.
Denote by the set embeddings of into up to isotopy.
Theorem 6.1.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If , then embeds into .
(b) Then embeds into .
The Diff case of part (a) is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
The Diff case of part (b) is in [Hirsch1961a, Corollary 4.2]. For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.2.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
For part (a) 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].
For the PL case of part (b) see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
For part (b) is corollary of Theorem 6.3 below. For part (b) coincides with Theorem 2.2b.
Tex syntax error-connected it follows that collapses to an -dimensional subcomplex of some triangulation of . By general position we may assume that is an embedding, because . Since is an immersion, it follows that has a sufficiently small regular neighbourhood such that is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . It is clear that
Tex syntax erroris isotopic to and is isotopic to . Thus the restriction is a concordance of and . By concordance implies isotopy Theorem
Tex syntax errorand are isotopic.
Theorem 6.3.
Assume is a closedTex syntax error-connected
Tex syntax error-manifold. 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()]. See also [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
Assume that is a compactTex syntax error-connected
Tex syntax error-manifold, , is
Tex syntax error-connected and .
Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 7.2.
Assume that is aTex syntax error-manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 6.2.
For a compact connectedTex syntax error-manifold with boundary, the property of having an -dimensional spine is close to
Tex syntax error-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
Every compact connectedTex syntax error-manifold with boundary for which is
Tex syntax error-connected, ,
and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch] The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from to .
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2. If is immersible in with a transversal -field then it is immersible in .
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
EveryTex syntax error-manifold with non-empty boundary is immersible in .
Theorem 8.4.[Whitney]
EveryTex syntax error-manifold is immersible in .
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
Suppose is aTex syntax error-manifold with non-empty boudary, is
Tex syntax error-connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Lets cunstruct such linear monomorphist on each -skeleton of . It is clear that linear monomorphism exists on -skeleton of .
The obstruction to continue the linear monomorphism from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into withTex syntax errortranversal linearly independent fields. It is true because is
Tex syntax error-connected.
Theorem 8.6.
Suppose is aTex syntax error-manifold with non-empty boudary, (N, \partial N) is
Tex syntax error-connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to continue the homotopy from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
9 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
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [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.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [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
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
Tex syntax error-manifold in -space for even
Tex syntax error.
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. Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
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 compact connectedTex syntax error-manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem.
Tex syntax errorhas a sufficiently small regular neighbourhood such that is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . The composition is an embedding of .
Theorem 2.2.
Assume that is a compact connectedTex syntax error-manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968, 4, Corollary 5]. Case is clear. Both parts of this theorem are special cases of the Theorem 6.2. Case can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6). All stated theorems of 2 and 6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example on non-isotopic embeddings
The following example is folklore.
Example 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 .
(c) Then defined by the formula is well-defined and is a bijection for .
Define by the formula , where .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.
Example 3.2. Let . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 3.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 6.3 fails for .
Example 3.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
Tex syntax error-spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 3.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
4 Seifert linking form
Tex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-ball. Thus is the -sphere.
If the (co)homology coefficients are omitted, then we assume them to be .
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connectedTex syntax error-manifold,
Tex syntax erroris 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 . AsTex syntax erroris 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 4.2.
For evenTex syntax errorand every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.3 ( is well-defined).
For evenTex syntax errorand 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
Tex syntax error
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 4.4. 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 4.4.
For each two homologous -cycles in , the image of the homology between and is aTex syntax error-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
Tex syntax erroris 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 4.3 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.5. Let be an embedding. Then for every the following equality holds:
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 4.4.
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 connectedTex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-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 evenTex syntax errordefine an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where
Tex syntax erroris 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 anTex syntax error-ball into . Unknotting Spheres Theorem implies that
Tex syntax error-sphere unknots in . Thus all extensions of
Tex syntax errorare isotopic in PL category. Note also that if
Tex syntax errorand are isotopic then their extensions are isotopic as well. And Whitney invariant
Tex syntax erroris invariant for PL embeddings.
Tex syntax erroris even and is torsion-free.
Take a collection such that .
For eachTex syntax errorsuch 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
Tex syntax error-manifold with torsion-free, ,
Tex syntax erroreven.
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 orientableTex syntax error-manifold with torsion-free, ,
Tex syntax erroreven. 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
For simplicity in this paragraph we consider only punctured manifolds, see 7 for generalization for.
By denote the complement in to an openTex syntax error-ball. Thus is the -sphere.
Denote by the set embeddings of into up to isotopy.
Theorem 6.1.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If , then embeds into .
(b) Then embeds into .
The Diff case of part (a) is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
The Diff case of part (b) is in [Hirsch1961a, Corollary 4.2]. For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.2.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
For part (a) 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].
For the PL case of part (b) see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
For part (b) is corollary of Theorem 6.3 below. For part (b) coincides with Theorem 2.2b.
Tex syntax error-connected it follows that collapses to an -dimensional subcomplex of some triangulation of . By general position we may assume that is an embedding, because . Since is an immersion, it follows that has a sufficiently small regular neighbourhood such that is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . It is clear that
Tex syntax erroris isotopic to and is isotopic to . Thus the restriction is a concordance of and . By concordance implies isotopy Theorem
Tex syntax errorand are isotopic.
Theorem 6.3.
Assume is a closedTex syntax error-connected
Tex syntax error-manifold. 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()]. See also [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
Assume that is a compactTex syntax error-connected
Tex syntax error-manifold, , is
Tex syntax error-connected and .
Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 7.2.
Assume that is aTex syntax error-manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 6.2.
For a compact connectedTex syntax error-manifold with boundary, the property of having an -dimensional spine is close to
Tex syntax error-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
Every compact connectedTex syntax error-manifold with boundary for which is
Tex syntax error-connected, ,
and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch] The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from to .
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2. If is immersible in with a transversal -field then it is immersible in .
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
EveryTex syntax error-manifold with non-empty boundary is immersible in .
Theorem 8.4.[Whitney]
EveryTex syntax error-manifold is immersible in .
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
Suppose is aTex syntax error-manifold with non-empty boudary, is
Tex syntax error-connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Lets cunstruct such linear monomorphist on each -skeleton of . It is clear that linear monomorphism exists on -skeleton of .
The obstruction to continue the linear monomorphism from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into withTex syntax errortranversal linearly independent fields. It is true because is
Tex syntax error-connected.
Theorem 8.6.
Suppose is aTex syntax error-manifold with non-empty boudary, (N, \partial N) is
Tex syntax error-connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to continue the homotopy from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
9 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
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [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.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [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
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
Tex syntax error-manifold in -space for even
Tex syntax error.
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. Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
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 compact connectedTex syntax error-manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem.
Tex syntax errorhas a sufficiently small regular neighbourhood such that is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . The composition is an embedding of .
Theorem 2.2.
Assume that is a compact connectedTex syntax error-manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968, 4, Corollary 5]. Case is clear. Both parts of this theorem are special cases of the Theorem 6.2. Case can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6). All stated theorems of 2 and 6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example on non-isotopic embeddings
The following example is folklore.
Example 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 .
(c) Then defined by the formula is well-defined and is a bijection for .
Define by the formula , where .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.
Example 3.2. Let . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 3.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 6.3 fails for .
Example 3.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
Tex syntax error-spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 3.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
4 Seifert linking form
Tex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-ball. Thus is the -sphere.
If the (co)homology coefficients are omitted, then we assume them to be .
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connectedTex syntax error-manifold,
Tex syntax erroris 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 . AsTex syntax erroris 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 4.2.
For evenTex syntax errorand every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.3 ( is well-defined).
For evenTex syntax errorand 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
Tex syntax error
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 4.4. 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 4.4.
For each two homologous -cycles in , the image of the homology between and is aTex syntax error-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
Tex syntax erroris 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 4.3 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.5. Let be an embedding. Then for every the following equality holds:
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 4.4.
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 connectedTex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-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 evenTex syntax errordefine an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where
Tex syntax erroris 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 anTex syntax error-ball into . Unknotting Spheres Theorem implies that
Tex syntax error-sphere unknots in . Thus all extensions of
Tex syntax errorare isotopic in PL category. Note also that if
Tex syntax errorand are isotopic then their extensions are isotopic as well. And Whitney invariant
Tex syntax erroris invariant for PL embeddings.
Tex syntax erroris even and is torsion-free.
Take a collection such that .
For eachTex syntax errorsuch 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
Tex syntax error-manifold with torsion-free, ,
Tex syntax erroreven.
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 orientableTex syntax error-manifold with torsion-free, ,
Tex syntax erroreven. 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
For simplicity in this paragraph we consider only punctured manifolds, see 7 for generalization for.
By denote the complement in to an openTex syntax error-ball. Thus is the -sphere.
Denote by the set embeddings of into up to isotopy.
Theorem 6.1.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If , then embeds into .
(b) Then embeds into .
The Diff case of part (a) is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
The Diff case of part (b) is in [Hirsch1961a, Corollary 4.2]. For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.2.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
For part (a) 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].
For the PL case of part (b) see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
For part (b) is corollary of Theorem 6.3 below. For part (b) coincides with Theorem 2.2b.
Tex syntax error-connected it follows that collapses to an -dimensional subcomplex of some triangulation of . By general position we may assume that is an embedding, because . Since is an immersion, it follows that has a sufficiently small regular neighbourhood such that is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . It is clear that
Tex syntax erroris isotopic to and is isotopic to . Thus the restriction is a concordance of and . By concordance implies isotopy Theorem
Tex syntax errorand are isotopic.
Theorem 6.3.
Assume is a closedTex syntax error-connected
Tex syntax error-manifold. 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()]. See also [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
Assume that is a compactTex syntax error-connected
Tex syntax error-manifold, , is
Tex syntax error-connected and .
Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 7.2.
Assume that is aTex syntax error-manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 6.2.
For a compact connectedTex syntax error-manifold with boundary, the property of having an -dimensional spine is close to
Tex syntax error-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
Every compact connectedTex syntax error-manifold with boundary for which is
Tex syntax error-connected, ,
and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch] The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from to .
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2. If is immersible in with a transversal -field then it is immersible in .
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
EveryTex syntax error-manifold with non-empty boundary is immersible in .
Theorem 8.4.[Whitney]
EveryTex syntax error-manifold is immersible in .
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
Suppose is aTex syntax error-manifold with non-empty boudary, is
Tex syntax error-connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Lets cunstruct such linear monomorphist on each -skeleton of . It is clear that linear monomorphism exists on -skeleton of .
The obstruction to continue the linear monomorphism from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into withTex syntax errortranversal linearly independent fields. It is true because is
Tex syntax error-connected.
Theorem 8.6.
Suppose is aTex syntax error-manifold with non-empty boudary, (N, \partial N) is
Tex syntax error-connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to continue the homotopy from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
9 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
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [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.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [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
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
Tex syntax error-manifold in -space for even
Tex syntax error.
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. Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
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 compact connectedTex syntax error-manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem.
Tex syntax errorhas a sufficiently small regular neighbourhood such that is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . The composition is an embedding of .
Theorem 2.2.
Assume that is a compact connectedTex syntax error-manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968, 4, Corollary 5]. Case is clear. Both parts of this theorem are special cases of the Theorem 6.2. Case can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6). All stated theorems of 2 and 6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example on non-isotopic embeddings
The following example is folklore.
Example 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 .
(c) Then defined by the formula is well-defined and is a bijection for .
Define by the formula , where .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.
Example 3.2. Let . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 3.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 6.3 fails for .
Example 3.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
Tex syntax error-spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 3.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
4 Seifert linking form
Tex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-ball. Thus is the -sphere.
If the (co)homology coefficients are omitted, then we assume them to be .
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connectedTex syntax error-manifold,
Tex syntax erroris 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 . AsTex syntax erroris 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 4.2.
For evenTex syntax errorand every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.3 ( is well-defined).
For evenTex syntax errorand 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
Tex syntax error
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 4.4. 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 4.4.
For each two homologous -cycles in , the image of the homology between and is aTex syntax error-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
Tex syntax erroris 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 4.3 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.5. Let be an embedding. Then for every the following equality holds:
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 4.4.
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 connectedTex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-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 evenTex syntax errordefine an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where
Tex syntax erroris 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 anTex syntax error-ball into . Unknotting Spheres Theorem implies that
Tex syntax error-sphere unknots in . Thus all extensions of
Tex syntax errorare isotopic in PL category. Note also that if
Tex syntax errorand are isotopic then their extensions are isotopic as well. And Whitney invariant
Tex syntax erroris invariant for PL embeddings.
Tex syntax erroris even and is torsion-free.
Take a collection such that .
For eachTex syntax errorsuch 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
Tex syntax error-manifold with torsion-free, ,
Tex syntax erroreven.
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 orientableTex syntax error-manifold with torsion-free, ,
Tex syntax erroreven. 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
For simplicity in this paragraph we consider only punctured manifolds, see 7 for generalization for.
By denote the complement in to an openTex syntax error-ball. Thus is the -sphere.
Denote by the set embeddings of into up to isotopy.
Theorem 6.1.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If , then embeds into .
(b) Then embeds into .
The Diff case of part (a) is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
The Diff case of part (b) is in [Hirsch1961a, Corollary 4.2]. For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.2.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
For part (a) 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].
For the PL case of part (b) see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
For part (b) is corollary of Theorem 6.3 below. For part (b) coincides with Theorem 2.2b.
Tex syntax error-connected it follows that collapses to an -dimensional subcomplex of some triangulation of . By general position we may assume that is an embedding, because . Since is an immersion, it follows that has a sufficiently small regular neighbourhood such that is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . It is clear that
Tex syntax erroris isotopic to and is isotopic to . Thus the restriction is a concordance of and . By concordance implies isotopy Theorem
Tex syntax errorand are isotopic.
Theorem 6.3.
Assume is a closedTex syntax error-connected
Tex syntax error-manifold. 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()]. See also [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
Assume that is a compactTex syntax error-connected
Tex syntax error-manifold, , is
Tex syntax error-connected and .
Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 7.2.
Assume that is aTex syntax error-manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 6.2.
For a compact connectedTex syntax error-manifold with boundary, the property of having an -dimensional spine is close to
Tex syntax error-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
Every compact connectedTex syntax error-manifold with boundary for which is
Tex syntax error-connected, ,
and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch] The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from to .
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2. If is immersible in with a transversal -field then it is immersible in .
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
EveryTex syntax error-manifold with non-empty boundary is immersible in .
Theorem 8.4.[Whitney]
EveryTex syntax error-manifold is immersible in .
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
Suppose is aTex syntax error-manifold with non-empty boudary, is
Tex syntax error-connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Lets cunstruct such linear monomorphist on each -skeleton of . It is clear that linear monomorphism exists on -skeleton of .
The obstruction to continue the linear monomorphism from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into withTex syntax errortranversal linearly independent fields. It is true because is
Tex syntax error-connected.
Theorem 8.6.
Suppose is aTex syntax error-manifold with non-empty boudary, (N, \partial N) is
Tex syntax error-connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to continue the homotopy from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
9 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
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [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.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [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
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
Tex syntax error-manifold in -space for even
Tex syntax error.
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. Denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
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 compact connectedTex syntax error-manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem.
Tex syntax errorhas a sufficiently small regular neighbourhood such that is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . The composition is an embedding of .
Theorem 2.2.
Assume that is a compact connectedTex syntax error-manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968, 4, Corollary 5]. Case is clear. Both parts of this theorem are special cases of the Theorem 6.2. Case can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6). All stated theorems of 2 and 6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example on non-isotopic embeddings
The following example is folklore.
Example 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 .
(c) Then defined by the formula is well-defined and is a bijection for .
Define by the formula , where .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.
Example 3.2. Let . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 3.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 6.3 fails for .
Example 3.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
Tex syntax error-spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 3.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
4 Seifert linking form
Tex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-ball. Thus is the -sphere.
If the (co)homology coefficients are omitted, then we assume them to be .
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connectedTex syntax error-manifold,
Tex syntax erroris 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 . AsTex syntax erroris 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 4.2.
For evenTex syntax errorand every embedding denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 4.3 ( is well-defined).
For evenTex syntax errorand 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
Tex syntax error
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 4.4. 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 4.4.
For each two homologous -cycles in , the image of the homology between and is aTex syntax error-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
Tex syntax erroris 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 4.3 implies that generates a bilinear form denoted by the same letter.
Denote by the reduction modulo .
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 4.5. Let be an embedding. Then for every the following equality holds:
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 4.4.
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 connectedTex syntax error-manifold. By we denote the complement in to an open
Tex syntax error-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 evenTex syntax errordefine an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where
Tex syntax erroris 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 anTex syntax error-ball into . Unknotting Spheres Theorem implies that
Tex syntax error-sphere unknots in . Thus all extensions of
Tex syntax errorare isotopic in PL category. Note also that if
Tex syntax errorand are isotopic then their extensions are isotopic as well. And Whitney invariant
Tex syntax erroris invariant for PL embeddings.
Tex syntax erroris even and is torsion-free.
Take a collection such that .
For eachTex syntax errorsuch 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
Tex syntax error-manifold with torsion-free, ,
Tex syntax erroreven.
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 orientableTex syntax error-manifold with torsion-free, ,
Tex syntax erroreven. 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
For simplicity in this paragraph we consider only punctured manifolds, see 7 for generalization for.
By denote the complement in to an openTex syntax error-ball. Thus is the -sphere.
Denote by the set embeddings of into up to isotopy.
Theorem 6.1.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If , then embeds into .
(b) Then embeds into .
The Diff case of part (a) is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3].
The Diff case of part (b) is in [Hirsch1961a, Corollary 4.2]. For the PL case see [Penrose&Whitehead&Zeeman1961, Theorem 1.2].
Theorem 6.2.
Assume that is a closedTex syntax error-connected
Tex syntax error-manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
For part (a) 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].
For the PL case of part (b) see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.
For part (b) is corollary of Theorem 6.3 below. For part (b) coincides with Theorem 2.2b.
Tex syntax error-connected it follows that collapses to an -dimensional subcomplex of some triangulation of . By general position we may assume that is an embedding, because . Since is an immersion, it follows that has a sufficiently small regular neighbourhood such that is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism . It is clear that
Tex syntax erroris isotopic to and is isotopic to . Thus the restriction is a concordance of and . By concordance implies isotopy Theorem
Tex syntax errorand are isotopic.
Theorem 6.3.
Assume is a closedTex syntax error-connected
Tex syntax error-manifold. 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()]. See also [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 5.6, which describes and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
Assume that is a compactTex syntax error-connected
Tex syntax error-manifold, , is
Tex syntax error-connected and .
Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 7.2.
Assume that is aTex syntax error-manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 6.2.
For a compact connectedTex syntax error-manifold with boundary, the property of having an -dimensional spine is close to
Tex syntax error-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
Every compact connectedTex syntax error-manifold with boundary for which is
Tex syntax error-connected, ,
and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch] The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from to .
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2. If is immersible in with a transversal -field then it is immersible in .
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
EveryTex syntax error-manifold with non-empty boundary is immersible in .
Theorem 8.4.[Whitney]
EveryTex syntax error-manifold is immersible in .
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
Suppose is aTex syntax error-manifold with non-empty boudary, is
Tex syntax error-connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Lets cunstruct such linear monomorphist on each -skeleton of . It is clear that linear monomorphism exists on -skeleton of .
The obstruction to continue the linear monomorphism from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into withTex syntax errortranversal linearly independent fields. It is true because is
Tex syntax error-connected.
Theorem 8.6.
Suppose is aTex syntax error-manifold with non-empty boudary, (N, \partial N) is
Tex syntax error-connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to continue the homotopy from -skeleton to -skeleton lies in , where is Stiefel manifold ofTex syntax error-frames in .
For we know .
For we have since isTex syntax error-connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
9 References
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