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 5 we introduce an invariant of embedding of a
-manifold in
-space for even
.
In
7 which is independent from
4,
5 and
6 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 the set of all embeddings
up to isotopy. We 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. For some results we present direct proofs, which are easier 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 connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2.
Assume that is a compact connected
-manifold and either
(a) or
(b) has non-empty boundary and
.
Then any two embeddings of into
are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Part (b) in the case
is proved in [Edwards1968,
4, Corollary 5]. The case
is clear. The case
can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7).
In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and
7 can be proved using
- analogous results for immersions of manifolds stated in
9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in
3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in
3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
The first proof of Theorem 2.1.b uses immersions, while the second does not.
![g\colon N\to\mathbb R^{2n-1}](/images/math/8/e/6/8e6a5cb7a3e2550f77ea7422924c4c36.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.png)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![g|_{M}](/images/math/5/9/f/59f4f568d65d9448dd1940c1c1c54ccd.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966]
Assume that is a closed connected smooth
-manifold. Then
have handle decomposition with indices of attaching map at most
.
Lemma 3.2.
Assume that is a closed smooth
-manifold and
is an attaching map such that
. If there is embedding
, then
extends to an embedding of
.
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\phi_1,\ldots,\phi_s](/images/math/7/8/8/78859dd0fd89584cca15fd5cdfe87ad3.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![F_1:U_1 \cong D^n\to \R^{2n-1}](/images/math/9/f/8/9f8bbbb595ac2731aafcf78e1b4ec49d.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U_{l-1}](/images/math/d/7/b/d7bff18cf35e8b3f6294db53d54ae41a.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.png)
![\phi_l](/images/math/c/4/8/c48fb2f519dccf14afbd1e51cf3bc0ba.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}](/images/math/1/7/4/17424d98822d3296bb1df2e028c79fd2.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 3.3.
Let be a closed smooth
-manifold and
,
,
, are smooth embeddings such that
on
. Suppose that on
there is a field of
pairwise orthogonal normal vectors whose restriction to
is tangent to
. Then
extends to a smooth embedding
.
![2i+1\leq 2n-1](/images/math/5/0/7/5073858d0affb278fddded337c402bbf.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}](/images/math/0/9/c/09c06ccd69fd3dbfdac9dbcd4ef19e08.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![(n-i)\times n](/images/math/6/c/5/6c5fc3faa3ba16c183aee22582d452f6.png)
![(n-i)\times (n-i)](/images/math/2/d/f/2dfb60822310b2de38bc6249af8ad647.png)
![v](/images/math/a/3/d/a3d52e52a48936cde0f5356bb08652f2.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![\partial D^i\times 0\subset D^i\times D^{n-i}](/images/math/0/5/6/0565bdb60731bb6c99eaf13740e919fb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))](/images/math/a/4/9/a49035d328b3ba4dbc7d4a9e13d0f328.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![(df\circ d\phi)|_{\partial D^i\times 0}(v)](/images/math/a/7/e/a7e51ba475de09c9f31bf7eb9d8d29c4.png)
![g(\partial D^i\times 0)](/images/math/2/f/5/2f50bc11df5df7d9f9a2b88b073110c4.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![i-1<2n-1 - (n-i)](/images/math/3/6/7/3676c9c806cf335df19ec68165e853d1.png)
![\pi_{i-1}(V_{2n-1, n-i})=0](/images/math/f/8/9/f89f268f7e633aaa8196d938e6b4460f.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![g(D^i\times 0)](/images/math/4/b/7/4b7f27b73221b0c232472ba2fbc76873.png)
![f\cup g|_{D^i\times 0}](/images/math/7/5/1/751535d1daf30058bb6824f69056b13b.png)
![N \cup_{\phi_i} D^i\times D^{n-i}](/images/math/e/4/9/e4964e9d2fb9ffbc1400a28d511c7e84.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).
Lemma 3.4.
Assume that is a compact
-manifold,
is an embedding with
,
are embeddings and
is a concordance between
and
.
If also that
, then there is an extension of
to a concordance between
and
.
![(b)](/images/math/4/6/2/462dc277171bc47412ce4e6895fc5f71.png)
![G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]](/images/math/7/6/7/7673e4c9b3f109db6df179275aa770ea.png)
![f_0|_{D^i\times 0}](/images/math/f/a/6/fa6d0deae25bc2b3a2a08245df0fca47.png)
![f_1|_{D^i\times 0}](/images/math/1/3/7/1379becb65ff170614469725aead7078.png)
![G(D^i\times 0\times [0, 1])](/images/math/f/c/a/fca2beb45e712aea38b8468ffe73aa06.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![G(\partial D^i\times 0\times [0, 1])](/images/math/9/c/c/9cc2f3be82a33e109eee7e7e9cabc33e.png)
![G(D^i\times 0\times 0)](/images/math/e/c/e/ece7d2a9be32d5837d835b478c0ac691.png)
![G(D^i\times 0\times 1)](/images/math/6/4/f/64f40e39ee385a4ccc8c81dbadcb3248.png)
![\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,](/images/math/f/f/d/ffd7ad38f5cca1a0734fe05216e65aac.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0, f_1](/images/math/7/2/e/72ebfdcb110d47ba960b40061831d498.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb{R}^m](/images/math/d/5/8/d5893f855c0893746999626e43f403e8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![n-1](/images/math/9/9/9/99911c3ea3e1da2d10de72c8066d422d.png)
![U^l](/images/math/5/3/a/53acb6cefb187b0fe8f6487904d7204d.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![U^1\cong D^n](/images/math/6/5/b/65b7ae4e1956bf59dc0bb7cb2fd6662c.png)
![F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/c/8/f/c8fc688383c938817e2af703f09348bd.png)
![F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/5/d/a/5da41d0f9627f07b25eed4568694be9f.png)
![f_0|_{U^1}](/images/math/d/5/2/d528f2ec62cdb6e65600036a46c2f61e.png)
![f_1|_{U^1}](/images/math/3/5/2/352833bf922051b2c5d90eba01f0bb98.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![f_0|_{U^l}](/images/math/f/4/b/f4b1c5c13db70a3bedd1a3fea9bd3d88.png)
![f_1|_{U^l}](/images/math/6/1/6/616cdbab5567ebaedef7a8cec15d9c53.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U^{l-1}](/images/math/a/2/9/a2991e29987d56c8015bda65b5b3eef9.png)
![f_0|_{U^{l-1}}](/images/math/1/1/d/11df57b884288f12be41ee4d282a2076.png)
![f_1|_{U^{l-1}}](/images/math/8/7/6/876110e463d693d6c47005faa12a872e.png)
![\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}](/images/math/c/c/3/cc359a5b5c2e6e045aff646818537d90.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![i\leq n-1](/images/math/2/1/e/21efcdb306da63a9827111b061c527ca.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]](/images/math/5/e/c/5ec7d9235de5dc82bd67cc00872a24e0.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U^{l-1}\cup_\phi D^i\times D^{n-i}](/images/math/4/7/3/4730d8f990eef5bf92f13894bada0cb1.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![D^i\times D^{n-i}\times [0, 1]](/images/math/8/8/c/88cd3df672a5115775ffedfb544d5bb9.png)
![D^i\times [0, 1]\times D^{n-i}](/images/math/e/c/c/ecc8260a7d0e17d2ea80f735dd1f84b9.png)
![\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1](/images/math/0/7/2/072e9c7b08fb27a39c991a4f39e66c90.png)
Tex syntax error
![\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,](/images/math/a/1/5/a15823ae1406826bc52a87fbfc5b1f5b.png)
![\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]](/images/math/c/9/d/c9d97bc144c9fb07e80ab6036a323230.png)
![F\bar{\phi} = G](/images/math/d/1/2/d12fa134bcaa9731bbeaaed9f1a77edc.png)
![\partial (D^i \times 0\times [0, 1])](/images/math/4/7/5/47580101248722766a17873fe2537b54.png)
![F(\mbox{Int} (U\times [0, 1]))](/images/math/4/1/0/4107ba155251147427656118445b1e23.png)
![G(\mbox{Int}( D^i\times 0\times [0, 1]))](/images/math/f/a/a/faab5cd0d220270034ac52657c616897.png)
![G_t](/images/math/d/3/b/d3b2bdacad293d6efe6d78f5766b3af9.png)
![G_0=G](/images/math/3/c/e/3ce2590599de94d8a0af6f58bc30f92d.png)
![\partial (D^i\times 0\times [0, 1])](/images/math/f/e/2/fe2c72ed3c40eb59b3c08c33b008f134.png)
![F(\partial (U\times [0, 1]))](/images/math/2/b/2/2b2fc6ab2e202f22d54317673c6d2997.png)
![G_1(D^i\times 0\times [0, 1])](/images/math/8/9/a/89aa9e32478b85ce4b2076ccb36ec156.png)
![G(U\times [0, 1])](/images/math/8/8/e/88e8188653e195ec591958c50130d42b.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0](/images/math/2/e/5/2e5488ed59b0b412a6c955ca10d3b295.png)
Denote by the
matrix whose rightmost
submatrix is the identity matrix, and whose other elements are zeroes. Denote by
the field of
normal vectors on
whose
-th vector has coordinates equal to the
-th row in
. Then
is the vector field tangent to
. For
denote by
the projection of
to the intersection of normal space to
, and tangent space to
. Since
, it follows that
. Hence there is an extension of
to a linear independent field of vectors normal to
. Then by Lemma 3.4.(b) there is an extension of
to a concordance
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1.
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
.
Proof.
Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree
. (To prove part (a) it is sufficient to take as
the identity map of
as a map of degree one and the constant map as a map of degree zero.)
Define
by the formula
.
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.
![k\geqslant 2](/images/math/2/c/b/2cb2015a51de7c9716f0dc2bc92c5268.png)
![f_1|_{S^k\times 0}](/images/math/c/1/6/c16e77e36d34660fbb7758ba5b44d442.png)
![f_2|_{S^k\times 0}](/images/math/5/d/b/5dbeac55457a6521596b48f699af75a4.png)
![f_1|_{S^k\times 0} = f_2|_{S^k\times 0}](/images/math/e/d/9/ed9a3ee4cda2bc53c52adf8ed0077f9e.png)
![l([f_1]) = l([f_2])](/images/math/c/5/3/c5301a619ec25514327c3ce383e5828f.png)
![f_1(S^k\times 0)](/images/math/e/0/2/e02f33e5695c28b4d3aa1d0024357b52.png)
![f_2(S^k\times 0)](/images/math/b/d/c/bdc8efad4137b267f347affef67219bb.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_2](/images/math/e/7/3/e73368a1436351fb0d11fdfb8cf3b3bf.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote .
Example 4.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 4.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3.
Let be the connected sum of two tori. Then there exists a surjection
defined by the formula
.
To prove the surjectivity of it is sufficient to take linked
-spheres in
and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4.
(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)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
-
is any closed orientable connected
-manifold,
-
is any embedding,
- if the (co)homology coefficients are omitted, then they are
,
-
is even and
is torsion free (these two assumptions are not required in Lemma 5.3).
By we denote the closure of the complement in
to an closed
-ball. Thus
is the
-sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint
-cycles in
with integer coefficients. Denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 5.2 ( is well-defined).
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] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.3.
Let be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
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 5.4.
For every the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 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.
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 6.1.
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 6.2.
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.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 6.3 of if
is even and
is torsion-free.
Take a collection
such that
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Note also that depends on choice of collection
. The following Theorems hold for any choice of
.
Theorem 6.4.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/f/4/c/f4cfe3a2b4cc5a3cc782084295a3bb04.png)
is one-to-one.
Lemma 6.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 6.4:
Theorem 6.6.
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]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 7.1.
Assume that is a closed
-connected
-manifold.
(a) If , then
embeds into
.
(b) If and
, then
embeds into
.
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2.
Assume that is a closed
-connected
-manifold.
(a) If and
, then any two embeddings of
into
are isotopic.
(b) If and
and
then any two embeddings of
into
are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For
part (b) coincides with Theorem 2.2b.
![k=1](/images/math/a/6/f/a6f0672a50348fdc06bc34fdc560cae9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb R^{2n-1}](/images/math/3/9/c/39cdeb2c39f9f962228626f6a37d3148.png)
![f,g\colon N_0\to\mathbb R^{2n-1}](/images/math/5/5/f/55fb87ba3a74da42f8d1bfa4c3c0707d.png)
![F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]](/images/math/7/9/6/7963693062b9ca3ce3d3f649be11320a.png)
![F(x, 0) = (f(x), 0)](/images/math/6/6/1/6614f0070021c0c42008192554b7fb76.png)
![F(x, 1)=(g(x), 1)](/images/math/d/0/c/d0c83013a79bb31d635845e51035fc3d.png)
![x\in N_0](/images/math/c/2/0/c201ff39d50a65d45853d1e389c1d27b.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![(n-2)](/images/math/9/d/d/9dd8fe07fe7b65ff961c61d8c07f8424.png)
![X\subset N_0](/images/math/4/1/8/418f89a952ba2cd648a0dde2d76d6659.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![F|_{X\times[0,1]}](/images/math/8/c/5/8c58be4ffc545b6549527ed446a6de3a.png)
![2(n-1) < 2n](/images/math/d/1/b/d1bb9ca106010aa3b768454fb0ec6a3f.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![h\colon N_0\to M](/images/math/f/9/6/f9661c49d810e6cd5240a2c78ce76cf6.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Conjecture 7.3.
Assume that is a closed
-connected
-manifold. Then any two embeddings of
in
are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4.
Assume is a closed
-connected
-manifold. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1].
Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes
and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1.
Assume that is a compact
-connected
-manifold,
,
is
-connected and
.
Then
embeds into
.
This is [Wall1965, Theorem on p.567].
![f\colon N\to\mathbb R^{2n-k-1}](/images/math/e/f/7/ef7e5dbe129d03b62f57644b8dc55634.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-k-1)](/images/math/9/b/9/9b98561fa8ab10c4b891ab69108aee75.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f|_X](/images/math/c/0/b/c0b6464917fb6186467774799040cd45.png)
![2(n-k) < 2n-k-1](/images/math/2/9/7/2977a324af64a5bc37d5330638c439a4.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![f|_{M}](/images/math/3/7/7/377d6c2f9743bedc11e1db159583b466.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.2.
Assume that is a
-manifold. If
has
-dimensional spine,
,
, then any two embeddings of
into
are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 8.3.
Every compact connected -manifold
with boundary for which
is
-connected,
,
and
, has an
-dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]]
The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from
to
.
Theorem 9.2.[[Hirsch1959, Theorem 6.4]]
If is immersible in
with a normal
-field, then
is immersible in
.
Theorem 9.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]]
Every -manifold
is immersible in
.
Denote by is Stiefel manifold of
-frames in
.
Theorem 9.5.
Suppose is a
-manifold with non-empty boudary,
is
-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
.
Let us construct such a linear monomorphism by skeleta of
.
It is clear that a linear monomorphism exists on
-skeleton of
.
The obstruction to extend the linear monomorphism from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 9.6.
Suppose is a connected
-manifold with non-empty boudary,
is
-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 extend the homotopy from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and
has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
10 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
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [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
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [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.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [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
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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 the set of all embeddings
up to isotopy. We 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. For some results we present direct proofs, which are easier 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 connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2.
Assume that is a compact connected
-manifold and either
(a) or
(b) has non-empty boundary and
.
Then any two embeddings of into
are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Part (b) in the case
is proved in [Edwards1968,
4, Corollary 5]. The case
is clear. The case
can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7).
In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and
7 can be proved using
- analogous results for immersions of manifolds stated in
9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in
3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in
3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
The first proof of Theorem 2.1.b uses immersions, while the second does not.
![g\colon N\to\mathbb R^{2n-1}](/images/math/8/e/6/8e6a5cb7a3e2550f77ea7422924c4c36.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.png)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![g|_{M}](/images/math/5/9/f/59f4f568d65d9448dd1940c1c1c54ccd.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966]
Assume that is a closed connected smooth
-manifold. Then
have handle decomposition with indices of attaching map at most
.
Lemma 3.2.
Assume that is a closed smooth
-manifold and
is an attaching map such that
. If there is embedding
, then
extends to an embedding of
.
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\phi_1,\ldots,\phi_s](/images/math/7/8/8/78859dd0fd89584cca15fd5cdfe87ad3.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![F_1:U_1 \cong D^n\to \R^{2n-1}](/images/math/9/f/8/9f8bbbb595ac2731aafcf78e1b4ec49d.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U_{l-1}](/images/math/d/7/b/d7bff18cf35e8b3f6294db53d54ae41a.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.png)
![\phi_l](/images/math/c/4/8/c48fb2f519dccf14afbd1e51cf3bc0ba.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}](/images/math/1/7/4/17424d98822d3296bb1df2e028c79fd2.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 3.3.
Let be a closed smooth
-manifold and
,
,
, are smooth embeddings such that
on
. Suppose that on
there is a field of
pairwise orthogonal normal vectors whose restriction to
is tangent to
. Then
extends to a smooth embedding
.
![2i+1\leq 2n-1](/images/math/5/0/7/5073858d0affb278fddded337c402bbf.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}](/images/math/0/9/c/09c06ccd69fd3dbfdac9dbcd4ef19e08.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![(n-i)\times n](/images/math/6/c/5/6c5fc3faa3ba16c183aee22582d452f6.png)
![(n-i)\times (n-i)](/images/math/2/d/f/2dfb60822310b2de38bc6249af8ad647.png)
![v](/images/math/a/3/d/a3d52e52a48936cde0f5356bb08652f2.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![\partial D^i\times 0\subset D^i\times D^{n-i}](/images/math/0/5/6/0565bdb60731bb6c99eaf13740e919fb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))](/images/math/a/4/9/a49035d328b3ba4dbc7d4a9e13d0f328.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![(df\circ d\phi)|_{\partial D^i\times 0}(v)](/images/math/a/7/e/a7e51ba475de09c9f31bf7eb9d8d29c4.png)
![g(\partial D^i\times 0)](/images/math/2/f/5/2f50bc11df5df7d9f9a2b88b073110c4.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![i-1<2n-1 - (n-i)](/images/math/3/6/7/3676c9c806cf335df19ec68165e853d1.png)
![\pi_{i-1}(V_{2n-1, n-i})=0](/images/math/f/8/9/f89f268f7e633aaa8196d938e6b4460f.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![g(D^i\times 0)](/images/math/4/b/7/4b7f27b73221b0c232472ba2fbc76873.png)
![f\cup g|_{D^i\times 0}](/images/math/7/5/1/751535d1daf30058bb6824f69056b13b.png)
![N \cup_{\phi_i} D^i\times D^{n-i}](/images/math/e/4/9/e4964e9d2fb9ffbc1400a28d511c7e84.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).
Lemma 3.4.
Assume that is a compact
-manifold,
is an embedding with
,
are embeddings and
is a concordance between
and
.
If also that
, then there is an extension of
to a concordance between
and
.
![(b)](/images/math/4/6/2/462dc277171bc47412ce4e6895fc5f71.png)
![G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]](/images/math/7/6/7/7673e4c9b3f109db6df179275aa770ea.png)
![f_0|_{D^i\times 0}](/images/math/f/a/6/fa6d0deae25bc2b3a2a08245df0fca47.png)
![f_1|_{D^i\times 0}](/images/math/1/3/7/1379becb65ff170614469725aead7078.png)
![G(D^i\times 0\times [0, 1])](/images/math/f/c/a/fca2beb45e712aea38b8468ffe73aa06.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![G(\partial D^i\times 0\times [0, 1])](/images/math/9/c/c/9cc2f3be82a33e109eee7e7e9cabc33e.png)
![G(D^i\times 0\times 0)](/images/math/e/c/e/ece7d2a9be32d5837d835b478c0ac691.png)
![G(D^i\times 0\times 1)](/images/math/6/4/f/64f40e39ee385a4ccc8c81dbadcb3248.png)
![\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,](/images/math/f/f/d/ffd7ad38f5cca1a0734fe05216e65aac.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0, f_1](/images/math/7/2/e/72ebfdcb110d47ba960b40061831d498.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb{R}^m](/images/math/d/5/8/d5893f855c0893746999626e43f403e8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![n-1](/images/math/9/9/9/99911c3ea3e1da2d10de72c8066d422d.png)
![U^l](/images/math/5/3/a/53acb6cefb187b0fe8f6487904d7204d.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![U^1\cong D^n](/images/math/6/5/b/65b7ae4e1956bf59dc0bb7cb2fd6662c.png)
![F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/c/8/f/c8fc688383c938817e2af703f09348bd.png)
![F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/5/d/a/5da41d0f9627f07b25eed4568694be9f.png)
![f_0|_{U^1}](/images/math/d/5/2/d528f2ec62cdb6e65600036a46c2f61e.png)
![f_1|_{U^1}](/images/math/3/5/2/352833bf922051b2c5d90eba01f0bb98.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![f_0|_{U^l}](/images/math/f/4/b/f4b1c5c13db70a3bedd1a3fea9bd3d88.png)
![f_1|_{U^l}](/images/math/6/1/6/616cdbab5567ebaedef7a8cec15d9c53.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U^{l-1}](/images/math/a/2/9/a2991e29987d56c8015bda65b5b3eef9.png)
![f_0|_{U^{l-1}}](/images/math/1/1/d/11df57b884288f12be41ee4d282a2076.png)
![f_1|_{U^{l-1}}](/images/math/8/7/6/876110e463d693d6c47005faa12a872e.png)
![\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}](/images/math/c/c/3/cc359a5b5c2e6e045aff646818537d90.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![i\leq n-1](/images/math/2/1/e/21efcdb306da63a9827111b061c527ca.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]](/images/math/5/e/c/5ec7d9235de5dc82bd67cc00872a24e0.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U^{l-1}\cup_\phi D^i\times D^{n-i}](/images/math/4/7/3/4730d8f990eef5bf92f13894bada0cb1.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![D^i\times D^{n-i}\times [0, 1]](/images/math/8/8/c/88cd3df672a5115775ffedfb544d5bb9.png)
![D^i\times [0, 1]\times D^{n-i}](/images/math/e/c/c/ecc8260a7d0e17d2ea80f735dd1f84b9.png)
![\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1](/images/math/0/7/2/072e9c7b08fb27a39c991a4f39e66c90.png)
Tex syntax error
![\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,](/images/math/a/1/5/a15823ae1406826bc52a87fbfc5b1f5b.png)
![\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]](/images/math/c/9/d/c9d97bc144c9fb07e80ab6036a323230.png)
![F\bar{\phi} = G](/images/math/d/1/2/d12fa134bcaa9731bbeaaed9f1a77edc.png)
![\partial (D^i \times 0\times [0, 1])](/images/math/4/7/5/47580101248722766a17873fe2537b54.png)
![F(\mbox{Int} (U\times [0, 1]))](/images/math/4/1/0/4107ba155251147427656118445b1e23.png)
![G(\mbox{Int}( D^i\times 0\times [0, 1]))](/images/math/f/a/a/faab5cd0d220270034ac52657c616897.png)
![G_t](/images/math/d/3/b/d3b2bdacad293d6efe6d78f5766b3af9.png)
![G_0=G](/images/math/3/c/e/3ce2590599de94d8a0af6f58bc30f92d.png)
![\partial (D^i\times 0\times [0, 1])](/images/math/f/e/2/fe2c72ed3c40eb59b3c08c33b008f134.png)
![F(\partial (U\times [0, 1]))](/images/math/2/b/2/2b2fc6ab2e202f22d54317673c6d2997.png)
![G_1(D^i\times 0\times [0, 1])](/images/math/8/9/a/89aa9e32478b85ce4b2076ccb36ec156.png)
![G(U\times [0, 1])](/images/math/8/8/e/88e8188653e195ec591958c50130d42b.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0](/images/math/2/e/5/2e5488ed59b0b412a6c955ca10d3b295.png)
Denote by the
matrix whose rightmost
submatrix is the identity matrix, and whose other elements are zeroes. Denote by
the field of
normal vectors on
whose
-th vector has coordinates equal to the
-th row in
. Then
is the vector field tangent to
. For
denote by
the projection of
to the intersection of normal space to
, and tangent space to
. Since
, it follows that
. Hence there is an extension of
to a linear independent field of vectors normal to
. Then by Lemma 3.4.(b) there is an extension of
to a concordance
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1.
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
.
Proof.
Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree
. (To prove part (a) it is sufficient to take as
the identity map of
as a map of degree one and the constant map as a map of degree zero.)
Define
by the formula
.
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.
![k\geqslant 2](/images/math/2/c/b/2cb2015a51de7c9716f0dc2bc92c5268.png)
![f_1|_{S^k\times 0}](/images/math/c/1/6/c16e77e36d34660fbb7758ba5b44d442.png)
![f_2|_{S^k\times 0}](/images/math/5/d/b/5dbeac55457a6521596b48f699af75a4.png)
![f_1|_{S^k\times 0} = f_2|_{S^k\times 0}](/images/math/e/d/9/ed9a3ee4cda2bc53c52adf8ed0077f9e.png)
![l([f_1]) = l([f_2])](/images/math/c/5/3/c5301a619ec25514327c3ce383e5828f.png)
![f_1(S^k\times 0)](/images/math/e/0/2/e02f33e5695c28b4d3aa1d0024357b52.png)
![f_2(S^k\times 0)](/images/math/b/d/c/bdc8efad4137b267f347affef67219bb.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_2](/images/math/e/7/3/e73368a1436351fb0d11fdfb8cf3b3bf.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote .
Example 4.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 4.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3.
Let be the connected sum of two tori. Then there exists a surjection
defined by the formula
.
To prove the surjectivity of it is sufficient to take linked
-spheres in
and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4.
(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)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
-
is any closed orientable connected
-manifold,
-
is any embedding,
- if the (co)homology coefficients are omitted, then they are
,
-
is even and
is torsion free (these two assumptions are not required in Lemma 5.3).
By we denote the closure of the complement in
to an closed
-ball. Thus
is the
-sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint
-cycles in
with integer coefficients. Denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 5.2 ( is well-defined).
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] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.3.
Let be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
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 5.4.
For every the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 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.
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 6.1.
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 6.2.
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.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 6.3 of if
is even and
is torsion-free.
Take a collection
such that
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Note also that depends on choice of collection
. The following Theorems hold for any choice of
.
Theorem 6.4.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/f/4/c/f4cfe3a2b4cc5a3cc782084295a3bb04.png)
is one-to-one.
Lemma 6.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 6.4:
Theorem 6.6.
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]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 7.1.
Assume that is a closed
-connected
-manifold.
(a) If , then
embeds into
.
(b) If and
, then
embeds into
.
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2.
Assume that is a closed
-connected
-manifold.
(a) If and
, then any two embeddings of
into
are isotopic.
(b) If and
and
then any two embeddings of
into
are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For
part (b) coincides with Theorem 2.2b.
![k=1](/images/math/a/6/f/a6f0672a50348fdc06bc34fdc560cae9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb R^{2n-1}](/images/math/3/9/c/39cdeb2c39f9f962228626f6a37d3148.png)
![f,g\colon N_0\to\mathbb R^{2n-1}](/images/math/5/5/f/55fb87ba3a74da42f8d1bfa4c3c0707d.png)
![F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]](/images/math/7/9/6/7963693062b9ca3ce3d3f649be11320a.png)
![F(x, 0) = (f(x), 0)](/images/math/6/6/1/6614f0070021c0c42008192554b7fb76.png)
![F(x, 1)=(g(x), 1)](/images/math/d/0/c/d0c83013a79bb31d635845e51035fc3d.png)
![x\in N_0](/images/math/c/2/0/c201ff39d50a65d45853d1e389c1d27b.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![(n-2)](/images/math/9/d/d/9dd8fe07fe7b65ff961c61d8c07f8424.png)
![X\subset N_0](/images/math/4/1/8/418f89a952ba2cd648a0dde2d76d6659.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![F|_{X\times[0,1]}](/images/math/8/c/5/8c58be4ffc545b6549527ed446a6de3a.png)
![2(n-1) < 2n](/images/math/d/1/b/d1bb9ca106010aa3b768454fb0ec6a3f.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![h\colon N_0\to M](/images/math/f/9/6/f9661c49d810e6cd5240a2c78ce76cf6.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Conjecture 7.3.
Assume that is a closed
-connected
-manifold. Then any two embeddings of
in
are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4.
Assume is a closed
-connected
-manifold. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1].
Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes
and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1.
Assume that is a compact
-connected
-manifold,
,
is
-connected and
.
Then
embeds into
.
This is [Wall1965, Theorem on p.567].
![f\colon N\to\mathbb R^{2n-k-1}](/images/math/e/f/7/ef7e5dbe129d03b62f57644b8dc55634.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-k-1)](/images/math/9/b/9/9b98561fa8ab10c4b891ab69108aee75.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f|_X](/images/math/c/0/b/c0b6464917fb6186467774799040cd45.png)
![2(n-k) < 2n-k-1](/images/math/2/9/7/2977a324af64a5bc37d5330638c439a4.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![f|_{M}](/images/math/3/7/7/377d6c2f9743bedc11e1db159583b466.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.2.
Assume that is a
-manifold. If
has
-dimensional spine,
,
, then any two embeddings of
into
are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 8.3.
Every compact connected -manifold
with boundary for which
is
-connected,
,
and
, has an
-dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]]
The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from
to
.
Theorem 9.2.[[Hirsch1959, Theorem 6.4]]
If is immersible in
with a normal
-field, then
is immersible in
.
Theorem 9.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]]
Every -manifold
is immersible in
.
Denote by is Stiefel manifold of
-frames in
.
Theorem 9.5.
Suppose is a
-manifold with non-empty boudary,
is
-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
.
Let us construct such a linear monomorphism by skeleta of
.
It is clear that a linear monomorphism exists on
-skeleton of
.
The obstruction to extend the linear monomorphism from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 9.6.
Suppose is a connected
-manifold with non-empty boudary,
is
-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 extend the homotopy from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and
has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
10 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
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [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
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [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.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [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
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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 the set of all embeddings
up to isotopy. We 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. For some results we present direct proofs, which are easier 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 connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2.
Assume that is a compact connected
-manifold and either
(a) or
(b) has non-empty boundary and
.
Then any two embeddings of into
are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Part (b) in the case
is proved in [Edwards1968,
4, Corollary 5]. The case
is clear. The case
can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7).
In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and
7 can be proved using
- analogous results for immersions of manifolds stated in
9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in
3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in
3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
The first proof of Theorem 2.1.b uses immersions, while the second does not.
![g\colon N\to\mathbb R^{2n-1}](/images/math/8/e/6/8e6a5cb7a3e2550f77ea7422924c4c36.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.png)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![g|_{M}](/images/math/5/9/f/59f4f568d65d9448dd1940c1c1c54ccd.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966]
Assume that is a closed connected smooth
-manifold. Then
have handle decomposition with indices of attaching map at most
.
Lemma 3.2.
Assume that is a closed smooth
-manifold and
is an attaching map such that
. If there is embedding
, then
extends to an embedding of
.
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\phi_1,\ldots,\phi_s](/images/math/7/8/8/78859dd0fd89584cca15fd5cdfe87ad3.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![F_1:U_1 \cong D^n\to \R^{2n-1}](/images/math/9/f/8/9f8bbbb595ac2731aafcf78e1b4ec49d.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U_{l-1}](/images/math/d/7/b/d7bff18cf35e8b3f6294db53d54ae41a.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.png)
![\phi_l](/images/math/c/4/8/c48fb2f519dccf14afbd1e51cf3bc0ba.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}](/images/math/1/7/4/17424d98822d3296bb1df2e028c79fd2.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 3.3.
Let be a closed smooth
-manifold and
,
,
, are smooth embeddings such that
on
. Suppose that on
there is a field of
pairwise orthogonal normal vectors whose restriction to
is tangent to
. Then
extends to a smooth embedding
.
![2i+1\leq 2n-1](/images/math/5/0/7/5073858d0affb278fddded337c402bbf.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}](/images/math/0/9/c/09c06ccd69fd3dbfdac9dbcd4ef19e08.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![(n-i)\times n](/images/math/6/c/5/6c5fc3faa3ba16c183aee22582d452f6.png)
![(n-i)\times (n-i)](/images/math/2/d/f/2dfb60822310b2de38bc6249af8ad647.png)
![v](/images/math/a/3/d/a3d52e52a48936cde0f5356bb08652f2.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![\partial D^i\times 0\subset D^i\times D^{n-i}](/images/math/0/5/6/0565bdb60731bb6c99eaf13740e919fb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))](/images/math/a/4/9/a49035d328b3ba4dbc7d4a9e13d0f328.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![(df\circ d\phi)|_{\partial D^i\times 0}(v)](/images/math/a/7/e/a7e51ba475de09c9f31bf7eb9d8d29c4.png)
![g(\partial D^i\times 0)](/images/math/2/f/5/2f50bc11df5df7d9f9a2b88b073110c4.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![i-1<2n-1 - (n-i)](/images/math/3/6/7/3676c9c806cf335df19ec68165e853d1.png)
![\pi_{i-1}(V_{2n-1, n-i})=0](/images/math/f/8/9/f89f268f7e633aaa8196d938e6b4460f.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![g(D^i\times 0)](/images/math/4/b/7/4b7f27b73221b0c232472ba2fbc76873.png)
![f\cup g|_{D^i\times 0}](/images/math/7/5/1/751535d1daf30058bb6824f69056b13b.png)
![N \cup_{\phi_i} D^i\times D^{n-i}](/images/math/e/4/9/e4964e9d2fb9ffbc1400a28d511c7e84.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).
Lemma 3.4.
Assume that is a compact
-manifold,
is an embedding with
,
are embeddings and
is a concordance between
and
.
If also that
, then there is an extension of
to a concordance between
and
.
![(b)](/images/math/4/6/2/462dc277171bc47412ce4e6895fc5f71.png)
![G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]](/images/math/7/6/7/7673e4c9b3f109db6df179275aa770ea.png)
![f_0|_{D^i\times 0}](/images/math/f/a/6/fa6d0deae25bc2b3a2a08245df0fca47.png)
![f_1|_{D^i\times 0}](/images/math/1/3/7/1379becb65ff170614469725aead7078.png)
![G(D^i\times 0\times [0, 1])](/images/math/f/c/a/fca2beb45e712aea38b8468ffe73aa06.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![G(\partial D^i\times 0\times [0, 1])](/images/math/9/c/c/9cc2f3be82a33e109eee7e7e9cabc33e.png)
![G(D^i\times 0\times 0)](/images/math/e/c/e/ece7d2a9be32d5837d835b478c0ac691.png)
![G(D^i\times 0\times 1)](/images/math/6/4/f/64f40e39ee385a4ccc8c81dbadcb3248.png)
![\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,](/images/math/f/f/d/ffd7ad38f5cca1a0734fe05216e65aac.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0, f_1](/images/math/7/2/e/72ebfdcb110d47ba960b40061831d498.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb{R}^m](/images/math/d/5/8/d5893f855c0893746999626e43f403e8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![n-1](/images/math/9/9/9/99911c3ea3e1da2d10de72c8066d422d.png)
![U^l](/images/math/5/3/a/53acb6cefb187b0fe8f6487904d7204d.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![U^1\cong D^n](/images/math/6/5/b/65b7ae4e1956bf59dc0bb7cb2fd6662c.png)
![F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/c/8/f/c8fc688383c938817e2af703f09348bd.png)
![F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/5/d/a/5da41d0f9627f07b25eed4568694be9f.png)
![f_0|_{U^1}](/images/math/d/5/2/d528f2ec62cdb6e65600036a46c2f61e.png)
![f_1|_{U^1}](/images/math/3/5/2/352833bf922051b2c5d90eba01f0bb98.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![f_0|_{U^l}](/images/math/f/4/b/f4b1c5c13db70a3bedd1a3fea9bd3d88.png)
![f_1|_{U^l}](/images/math/6/1/6/616cdbab5567ebaedef7a8cec15d9c53.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U^{l-1}](/images/math/a/2/9/a2991e29987d56c8015bda65b5b3eef9.png)
![f_0|_{U^{l-1}}](/images/math/1/1/d/11df57b884288f12be41ee4d282a2076.png)
![f_1|_{U^{l-1}}](/images/math/8/7/6/876110e463d693d6c47005faa12a872e.png)
![\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}](/images/math/c/c/3/cc359a5b5c2e6e045aff646818537d90.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![i\leq n-1](/images/math/2/1/e/21efcdb306da63a9827111b061c527ca.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]](/images/math/5/e/c/5ec7d9235de5dc82bd67cc00872a24e0.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U^{l-1}\cup_\phi D^i\times D^{n-i}](/images/math/4/7/3/4730d8f990eef5bf92f13894bada0cb1.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![D^i\times D^{n-i}\times [0, 1]](/images/math/8/8/c/88cd3df672a5115775ffedfb544d5bb9.png)
![D^i\times [0, 1]\times D^{n-i}](/images/math/e/c/c/ecc8260a7d0e17d2ea80f735dd1f84b9.png)
![\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1](/images/math/0/7/2/072e9c7b08fb27a39c991a4f39e66c90.png)
Tex syntax error
![\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,](/images/math/a/1/5/a15823ae1406826bc52a87fbfc5b1f5b.png)
![\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]](/images/math/c/9/d/c9d97bc144c9fb07e80ab6036a323230.png)
![F\bar{\phi} = G](/images/math/d/1/2/d12fa134bcaa9731bbeaaed9f1a77edc.png)
![\partial (D^i \times 0\times [0, 1])](/images/math/4/7/5/47580101248722766a17873fe2537b54.png)
![F(\mbox{Int} (U\times [0, 1]))](/images/math/4/1/0/4107ba155251147427656118445b1e23.png)
![G(\mbox{Int}( D^i\times 0\times [0, 1]))](/images/math/f/a/a/faab5cd0d220270034ac52657c616897.png)
![G_t](/images/math/d/3/b/d3b2bdacad293d6efe6d78f5766b3af9.png)
![G_0=G](/images/math/3/c/e/3ce2590599de94d8a0af6f58bc30f92d.png)
![\partial (D^i\times 0\times [0, 1])](/images/math/f/e/2/fe2c72ed3c40eb59b3c08c33b008f134.png)
![F(\partial (U\times [0, 1]))](/images/math/2/b/2/2b2fc6ab2e202f22d54317673c6d2997.png)
![G_1(D^i\times 0\times [0, 1])](/images/math/8/9/a/89aa9e32478b85ce4b2076ccb36ec156.png)
![G(U\times [0, 1])](/images/math/8/8/e/88e8188653e195ec591958c50130d42b.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0](/images/math/2/e/5/2e5488ed59b0b412a6c955ca10d3b295.png)
Denote by the
matrix whose rightmost
submatrix is the identity matrix, and whose other elements are zeroes. Denote by
the field of
normal vectors on
whose
-th vector has coordinates equal to the
-th row in
. Then
is the vector field tangent to
. For
denote by
the projection of
to the intersection of normal space to
, and tangent space to
. Since
, it follows that
. Hence there is an extension of
to a linear independent field of vectors normal to
. Then by Lemma 3.4.(b) there is an extension of
to a concordance
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1.
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
.
Proof.
Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree
. (To prove part (a) it is sufficient to take as
the identity map of
as a map of degree one and the constant map as a map of degree zero.)
Define
by the formula
.
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.
![k\geqslant 2](/images/math/2/c/b/2cb2015a51de7c9716f0dc2bc92c5268.png)
![f_1|_{S^k\times 0}](/images/math/c/1/6/c16e77e36d34660fbb7758ba5b44d442.png)
![f_2|_{S^k\times 0}](/images/math/5/d/b/5dbeac55457a6521596b48f699af75a4.png)
![f_1|_{S^k\times 0} = f_2|_{S^k\times 0}](/images/math/e/d/9/ed9a3ee4cda2bc53c52adf8ed0077f9e.png)
![l([f_1]) = l([f_2])](/images/math/c/5/3/c5301a619ec25514327c3ce383e5828f.png)
![f_1(S^k\times 0)](/images/math/e/0/2/e02f33e5695c28b4d3aa1d0024357b52.png)
![f_2(S^k\times 0)](/images/math/b/d/c/bdc8efad4137b267f347affef67219bb.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_2](/images/math/e/7/3/e73368a1436351fb0d11fdfb8cf3b3bf.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote .
Example 4.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 4.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3.
Let be the connected sum of two tori. Then there exists a surjection
defined by the formula
.
To prove the surjectivity of it is sufficient to take linked
-spheres in
and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4.
(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)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
-
is any closed orientable connected
-manifold,
-
is any embedding,
- if the (co)homology coefficients are omitted, then they are
,
-
is even and
is torsion free (these two assumptions are not required in Lemma 5.3).
By we denote the closure of the complement in
to an closed
-ball. Thus
is the
-sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint
-cycles in
with integer coefficients. Denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 5.2 ( is well-defined).
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] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.3.
Let be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
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 5.4.
For every the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 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.
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 6.1.
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 6.2.
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.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 6.3 of if
is even and
is torsion-free.
Take a collection
such that
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Note also that depends on choice of collection
. The following Theorems hold for any choice of
.
Theorem 6.4.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/f/4/c/f4cfe3a2b4cc5a3cc782084295a3bb04.png)
is one-to-one.
Lemma 6.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 6.4:
Theorem 6.6.
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]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 7.1.
Assume that is a closed
-connected
-manifold.
(a) If , then
embeds into
.
(b) If and
, then
embeds into
.
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2.
Assume that is a closed
-connected
-manifold.
(a) If and
, then any two embeddings of
into
are isotopic.
(b) If and
and
then any two embeddings of
into
are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For
part (b) coincides with Theorem 2.2b.
![k=1](/images/math/a/6/f/a6f0672a50348fdc06bc34fdc560cae9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb R^{2n-1}](/images/math/3/9/c/39cdeb2c39f9f962228626f6a37d3148.png)
![f,g\colon N_0\to\mathbb R^{2n-1}](/images/math/5/5/f/55fb87ba3a74da42f8d1bfa4c3c0707d.png)
![F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]](/images/math/7/9/6/7963693062b9ca3ce3d3f649be11320a.png)
![F(x, 0) = (f(x), 0)](/images/math/6/6/1/6614f0070021c0c42008192554b7fb76.png)
![F(x, 1)=(g(x), 1)](/images/math/d/0/c/d0c83013a79bb31d635845e51035fc3d.png)
![x\in N_0](/images/math/c/2/0/c201ff39d50a65d45853d1e389c1d27b.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![(n-2)](/images/math/9/d/d/9dd8fe07fe7b65ff961c61d8c07f8424.png)
![X\subset N_0](/images/math/4/1/8/418f89a952ba2cd648a0dde2d76d6659.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![F|_{X\times[0,1]}](/images/math/8/c/5/8c58be4ffc545b6549527ed446a6de3a.png)
![2(n-1) < 2n](/images/math/d/1/b/d1bb9ca106010aa3b768454fb0ec6a3f.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![h\colon N_0\to M](/images/math/f/9/6/f9661c49d810e6cd5240a2c78ce76cf6.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Conjecture 7.3.
Assume that is a closed
-connected
-manifold. Then any two embeddings of
in
are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4.
Assume is a closed
-connected
-manifold. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1].
Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes
and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1.
Assume that is a compact
-connected
-manifold,
,
is
-connected and
.
Then
embeds into
.
This is [Wall1965, Theorem on p.567].
![f\colon N\to\mathbb R^{2n-k-1}](/images/math/e/f/7/ef7e5dbe129d03b62f57644b8dc55634.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-k-1)](/images/math/9/b/9/9b98561fa8ab10c4b891ab69108aee75.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f|_X](/images/math/c/0/b/c0b6464917fb6186467774799040cd45.png)
![2(n-k) < 2n-k-1](/images/math/2/9/7/2977a324af64a5bc37d5330638c439a4.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![f|_{M}](/images/math/3/7/7/377d6c2f9743bedc11e1db159583b466.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.2.
Assume that is a
-manifold. If
has
-dimensional spine,
,
, then any two embeddings of
into
are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 8.3.
Every compact connected -manifold
with boundary for which
is
-connected,
,
and
, has an
-dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]]
The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from
to
.
Theorem 9.2.[[Hirsch1959, Theorem 6.4]]
If is immersible in
with a normal
-field, then
is immersible in
.
Theorem 9.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]]
Every -manifold
is immersible in
.
Denote by is Stiefel manifold of
-frames in
.
Theorem 9.5.
Suppose is a
-manifold with non-empty boudary,
is
-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
.
Let us construct such a linear monomorphism by skeleta of
.
It is clear that a linear monomorphism exists on
-skeleton of
.
The obstruction to extend the linear monomorphism from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 9.6.
Suppose is a connected
-manifold with non-empty boudary,
is
-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 extend the homotopy from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and
has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
10 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
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [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
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [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.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [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
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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 the set of all embeddings
up to isotopy. We 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. For some results we present direct proofs, which are easier 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 connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2.
Assume that is a compact connected
-manifold and either
(a) or
(b) has non-empty boundary and
.
Then any two embeddings of into
are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Part (b) in the case
is proved in [Edwards1968,
4, Corollary 5]. The case
is clear. The case
can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7).
In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and
7 can be proved using
- analogous results for immersions of manifolds stated in
9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in
3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in
3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
The first proof of Theorem 2.1.b uses immersions, while the second does not.
![g\colon N\to\mathbb R^{2n-1}](/images/math/8/e/6/8e6a5cb7a3e2550f77ea7422924c4c36.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.png)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![g|_{M}](/images/math/5/9/f/59f4f568d65d9448dd1940c1c1c54ccd.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966]
Assume that is a closed connected smooth
-manifold. Then
have handle decomposition with indices of attaching map at most
.
Lemma 3.2.
Assume that is a closed smooth
-manifold and
is an attaching map such that
. If there is embedding
, then
extends to an embedding of
.
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\phi_1,\ldots,\phi_s](/images/math/7/8/8/78859dd0fd89584cca15fd5cdfe87ad3.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![F_1:U_1 \cong D^n\to \R^{2n-1}](/images/math/9/f/8/9f8bbbb595ac2731aafcf78e1b4ec49d.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U_{l-1}](/images/math/d/7/b/d7bff18cf35e8b3f6294db53d54ae41a.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.png)
![\phi_l](/images/math/c/4/8/c48fb2f519dccf14afbd1e51cf3bc0ba.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}](/images/math/1/7/4/17424d98822d3296bb1df2e028c79fd2.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 3.3.
Let be a closed smooth
-manifold and
,
,
, are smooth embeddings such that
on
. Suppose that on
there is a field of
pairwise orthogonal normal vectors whose restriction to
is tangent to
. Then
extends to a smooth embedding
.
![2i+1\leq 2n-1](/images/math/5/0/7/5073858d0affb278fddded337c402bbf.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}](/images/math/0/9/c/09c06ccd69fd3dbfdac9dbcd4ef19e08.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![(n-i)\times n](/images/math/6/c/5/6c5fc3faa3ba16c183aee22582d452f6.png)
![(n-i)\times (n-i)](/images/math/2/d/f/2dfb60822310b2de38bc6249af8ad647.png)
![v](/images/math/a/3/d/a3d52e52a48936cde0f5356bb08652f2.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![\partial D^i\times 0\subset D^i\times D^{n-i}](/images/math/0/5/6/0565bdb60731bb6c99eaf13740e919fb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))](/images/math/a/4/9/a49035d328b3ba4dbc7d4a9e13d0f328.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![(df\circ d\phi)|_{\partial D^i\times 0}(v)](/images/math/a/7/e/a7e51ba475de09c9f31bf7eb9d8d29c4.png)
![g(\partial D^i\times 0)](/images/math/2/f/5/2f50bc11df5df7d9f9a2b88b073110c4.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![i-1<2n-1 - (n-i)](/images/math/3/6/7/3676c9c806cf335df19ec68165e853d1.png)
![\pi_{i-1}(V_{2n-1, n-i})=0](/images/math/f/8/9/f89f268f7e633aaa8196d938e6b4460f.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![g(D^i\times 0)](/images/math/4/b/7/4b7f27b73221b0c232472ba2fbc76873.png)
![f\cup g|_{D^i\times 0}](/images/math/7/5/1/751535d1daf30058bb6824f69056b13b.png)
![N \cup_{\phi_i} D^i\times D^{n-i}](/images/math/e/4/9/e4964e9d2fb9ffbc1400a28d511c7e84.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).
Lemma 3.4.
Assume that is a compact
-manifold,
is an embedding with
,
are embeddings and
is a concordance between
and
.
If also that
, then there is an extension of
to a concordance between
and
.
![(b)](/images/math/4/6/2/462dc277171bc47412ce4e6895fc5f71.png)
![G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]](/images/math/7/6/7/7673e4c9b3f109db6df179275aa770ea.png)
![f_0|_{D^i\times 0}](/images/math/f/a/6/fa6d0deae25bc2b3a2a08245df0fca47.png)
![f_1|_{D^i\times 0}](/images/math/1/3/7/1379becb65ff170614469725aead7078.png)
![G(D^i\times 0\times [0, 1])](/images/math/f/c/a/fca2beb45e712aea38b8468ffe73aa06.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![G(\partial D^i\times 0\times [0, 1])](/images/math/9/c/c/9cc2f3be82a33e109eee7e7e9cabc33e.png)
![G(D^i\times 0\times 0)](/images/math/e/c/e/ece7d2a9be32d5837d835b478c0ac691.png)
![G(D^i\times 0\times 1)](/images/math/6/4/f/64f40e39ee385a4ccc8c81dbadcb3248.png)
![\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,](/images/math/f/f/d/ffd7ad38f5cca1a0734fe05216e65aac.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0, f_1](/images/math/7/2/e/72ebfdcb110d47ba960b40061831d498.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb{R}^m](/images/math/d/5/8/d5893f855c0893746999626e43f403e8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![n-1](/images/math/9/9/9/99911c3ea3e1da2d10de72c8066d422d.png)
![U^l](/images/math/5/3/a/53acb6cefb187b0fe8f6487904d7204d.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![U^1\cong D^n](/images/math/6/5/b/65b7ae4e1956bf59dc0bb7cb2fd6662c.png)
![F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/c/8/f/c8fc688383c938817e2af703f09348bd.png)
![F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/5/d/a/5da41d0f9627f07b25eed4568694be9f.png)
![f_0|_{U^1}](/images/math/d/5/2/d528f2ec62cdb6e65600036a46c2f61e.png)
![f_1|_{U^1}](/images/math/3/5/2/352833bf922051b2c5d90eba01f0bb98.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![f_0|_{U^l}](/images/math/f/4/b/f4b1c5c13db70a3bedd1a3fea9bd3d88.png)
![f_1|_{U^l}](/images/math/6/1/6/616cdbab5567ebaedef7a8cec15d9c53.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U^{l-1}](/images/math/a/2/9/a2991e29987d56c8015bda65b5b3eef9.png)
![f_0|_{U^{l-1}}](/images/math/1/1/d/11df57b884288f12be41ee4d282a2076.png)
![f_1|_{U^{l-1}}](/images/math/8/7/6/876110e463d693d6c47005faa12a872e.png)
![\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}](/images/math/c/c/3/cc359a5b5c2e6e045aff646818537d90.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![i\leq n-1](/images/math/2/1/e/21efcdb306da63a9827111b061c527ca.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]](/images/math/5/e/c/5ec7d9235de5dc82bd67cc00872a24e0.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U^{l-1}\cup_\phi D^i\times D^{n-i}](/images/math/4/7/3/4730d8f990eef5bf92f13894bada0cb1.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![D^i\times D^{n-i}\times [0, 1]](/images/math/8/8/c/88cd3df672a5115775ffedfb544d5bb9.png)
![D^i\times [0, 1]\times D^{n-i}](/images/math/e/c/c/ecc8260a7d0e17d2ea80f735dd1f84b9.png)
![\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1](/images/math/0/7/2/072e9c7b08fb27a39c991a4f39e66c90.png)
Tex syntax error
![\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,](/images/math/a/1/5/a15823ae1406826bc52a87fbfc5b1f5b.png)
![\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]](/images/math/c/9/d/c9d97bc144c9fb07e80ab6036a323230.png)
![F\bar{\phi} = G](/images/math/d/1/2/d12fa134bcaa9731bbeaaed9f1a77edc.png)
![\partial (D^i \times 0\times [0, 1])](/images/math/4/7/5/47580101248722766a17873fe2537b54.png)
![F(\mbox{Int} (U\times [0, 1]))](/images/math/4/1/0/4107ba155251147427656118445b1e23.png)
![G(\mbox{Int}( D^i\times 0\times [0, 1]))](/images/math/f/a/a/faab5cd0d220270034ac52657c616897.png)
![G_t](/images/math/d/3/b/d3b2bdacad293d6efe6d78f5766b3af9.png)
![G_0=G](/images/math/3/c/e/3ce2590599de94d8a0af6f58bc30f92d.png)
![\partial (D^i\times 0\times [0, 1])](/images/math/f/e/2/fe2c72ed3c40eb59b3c08c33b008f134.png)
![F(\partial (U\times [0, 1]))](/images/math/2/b/2/2b2fc6ab2e202f22d54317673c6d2997.png)
![G_1(D^i\times 0\times [0, 1])](/images/math/8/9/a/89aa9e32478b85ce4b2076ccb36ec156.png)
![G(U\times [0, 1])](/images/math/8/8/e/88e8188653e195ec591958c50130d42b.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0](/images/math/2/e/5/2e5488ed59b0b412a6c955ca10d3b295.png)
Denote by the
matrix whose rightmost
submatrix is the identity matrix, and whose other elements are zeroes. Denote by
the field of
normal vectors on
whose
-th vector has coordinates equal to the
-th row in
. Then
is the vector field tangent to
. For
denote by
the projection of
to the intersection of normal space to
, and tangent space to
. Since
, it follows that
. Hence there is an extension of
to a linear independent field of vectors normal to
. Then by Lemma 3.4.(b) there is an extension of
to a concordance
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1.
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
.
Proof.
Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree
. (To prove part (a) it is sufficient to take as
the identity map of
as a map of degree one and the constant map as a map of degree zero.)
Define
by the formula
.
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.
![k\geqslant 2](/images/math/2/c/b/2cb2015a51de7c9716f0dc2bc92c5268.png)
![f_1|_{S^k\times 0}](/images/math/c/1/6/c16e77e36d34660fbb7758ba5b44d442.png)
![f_2|_{S^k\times 0}](/images/math/5/d/b/5dbeac55457a6521596b48f699af75a4.png)
![f_1|_{S^k\times 0} = f_2|_{S^k\times 0}](/images/math/e/d/9/ed9a3ee4cda2bc53c52adf8ed0077f9e.png)
![l([f_1]) = l([f_2])](/images/math/c/5/3/c5301a619ec25514327c3ce383e5828f.png)
![f_1(S^k\times 0)](/images/math/e/0/2/e02f33e5695c28b4d3aa1d0024357b52.png)
![f_2(S^k\times 0)](/images/math/b/d/c/bdc8efad4137b267f347affef67219bb.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_2](/images/math/e/7/3/e73368a1436351fb0d11fdfb8cf3b3bf.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote .
Example 4.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 4.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3.
Let be the connected sum of two tori. Then there exists a surjection
defined by the formula
.
To prove the surjectivity of it is sufficient to take linked
-spheres in
and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4.
(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)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
-
is any closed orientable connected
-manifold,
-
is any embedding,
- if the (co)homology coefficients are omitted, then they are
,
-
is even and
is torsion free (these two assumptions are not required in Lemma 5.3).
By we denote the closure of the complement in
to an closed
-ball. Thus
is the
-sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint
-cycles in
with integer coefficients. Denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 5.2 ( is well-defined).
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] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.3.
Let be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
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 5.4.
For every the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 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.
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 6.1.
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 6.2.
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.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 6.3 of if
is even and
is torsion-free.
Take a collection
such that
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Note also that depends on choice of collection
. The following Theorems hold for any choice of
.
Theorem 6.4.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/f/4/c/f4cfe3a2b4cc5a3cc782084295a3bb04.png)
is one-to-one.
Lemma 6.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 6.4:
Theorem 6.6.
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]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 7.1.
Assume that is a closed
-connected
-manifold.
(a) If , then
embeds into
.
(b) If and
, then
embeds into
.
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2.
Assume that is a closed
-connected
-manifold.
(a) If and
, then any two embeddings of
into
are isotopic.
(b) If and
and
then any two embeddings of
into
are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For
part (b) coincides with Theorem 2.2b.
![k=1](/images/math/a/6/f/a6f0672a50348fdc06bc34fdc560cae9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb R^{2n-1}](/images/math/3/9/c/39cdeb2c39f9f962228626f6a37d3148.png)
![f,g\colon N_0\to\mathbb R^{2n-1}](/images/math/5/5/f/55fb87ba3a74da42f8d1bfa4c3c0707d.png)
![F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]](/images/math/7/9/6/7963693062b9ca3ce3d3f649be11320a.png)
![F(x, 0) = (f(x), 0)](/images/math/6/6/1/6614f0070021c0c42008192554b7fb76.png)
![F(x, 1)=(g(x), 1)](/images/math/d/0/c/d0c83013a79bb31d635845e51035fc3d.png)
![x\in N_0](/images/math/c/2/0/c201ff39d50a65d45853d1e389c1d27b.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![(n-2)](/images/math/9/d/d/9dd8fe07fe7b65ff961c61d8c07f8424.png)
![X\subset N_0](/images/math/4/1/8/418f89a952ba2cd648a0dde2d76d6659.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![F|_{X\times[0,1]}](/images/math/8/c/5/8c58be4ffc545b6549527ed446a6de3a.png)
![2(n-1) < 2n](/images/math/d/1/b/d1bb9ca106010aa3b768454fb0ec6a3f.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![h\colon N_0\to M](/images/math/f/9/6/f9661c49d810e6cd5240a2c78ce76cf6.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Conjecture 7.3.
Assume that is a closed
-connected
-manifold. Then any two embeddings of
in
are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4.
Assume is a closed
-connected
-manifold. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1].
Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes
and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1.
Assume that is a compact
-connected
-manifold,
,
is
-connected and
.
Then
embeds into
.
This is [Wall1965, Theorem on p.567].
![f\colon N\to\mathbb R^{2n-k-1}](/images/math/e/f/7/ef7e5dbe129d03b62f57644b8dc55634.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-k-1)](/images/math/9/b/9/9b98561fa8ab10c4b891ab69108aee75.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f|_X](/images/math/c/0/b/c0b6464917fb6186467774799040cd45.png)
![2(n-k) < 2n-k-1](/images/math/2/9/7/2977a324af64a5bc37d5330638c439a4.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![f|_{M}](/images/math/3/7/7/377d6c2f9743bedc11e1db159583b466.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.2.
Assume that is a
-manifold. If
has
-dimensional spine,
,
, then any two embeddings of
into
are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 8.3.
Every compact connected -manifold
with boundary for which
is
-connected,
,
and
, has an
-dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]]
The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from
to
.
Theorem 9.2.[[Hirsch1959, Theorem 6.4]]
If is immersible in
with a normal
-field, then
is immersible in
.
Theorem 9.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]]
Every -manifold
is immersible in
.
Denote by is Stiefel manifold of
-frames in
.
Theorem 9.5.
Suppose is a
-manifold with non-empty boudary,
is
-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
.
Let us construct such a linear monomorphism by skeleta of
.
It is clear that a linear monomorphism exists on
-skeleton of
.
The obstruction to extend the linear monomorphism from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 9.6.
Suppose is a connected
-manifold with non-empty boudary,
is
-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 extend the homotopy from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and
has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
10 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
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [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
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [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.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [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
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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 the set of all embeddings
up to isotopy. We 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. For some results we present direct proofs, which are easier 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 connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2.
Assume that is a compact connected
-manifold and either
(a) or
(b) has non-empty boundary and
.
Then any two embeddings of into
are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Part (b) in the case
is proved in [Edwards1968,
4, Corollary 5]. The case
is clear. The case
can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7).
In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and
7 can be proved using
- analogous results for immersions of manifolds stated in
9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in
3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in
3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
The first proof of Theorem 2.1.b uses immersions, while the second does not.
![g\colon N\to\mathbb R^{2n-1}](/images/math/8/e/6/8e6a5cb7a3e2550f77ea7422924c4c36.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.png)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![g|_{M}](/images/math/5/9/f/59f4f568d65d9448dd1940c1c1c54ccd.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966]
Assume that is a closed connected smooth
-manifold. Then
have handle decomposition with indices of attaching map at most
.
Lemma 3.2.
Assume that is a closed smooth
-manifold and
is an attaching map such that
. If there is embedding
, then
extends to an embedding of
.
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\phi_1,\ldots,\phi_s](/images/math/7/8/8/78859dd0fd89584cca15fd5cdfe87ad3.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![F_1:U_1 \cong D^n\to \R^{2n-1}](/images/math/9/f/8/9f8bbbb595ac2731aafcf78e1b4ec49d.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U_{l-1}](/images/math/d/7/b/d7bff18cf35e8b3f6294db53d54ae41a.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.png)
![\phi_l](/images/math/c/4/8/c48fb2f519dccf14afbd1e51cf3bc0ba.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}](/images/math/1/7/4/17424d98822d3296bb1df2e028c79fd2.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 3.3.
Let be a closed smooth
-manifold and
,
,
, are smooth embeddings such that
on
. Suppose that on
there is a field of
pairwise orthogonal normal vectors whose restriction to
is tangent to
. Then
extends to a smooth embedding
.
![2i+1\leq 2n-1](/images/math/5/0/7/5073858d0affb278fddded337c402bbf.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}](/images/math/0/9/c/09c06ccd69fd3dbfdac9dbcd4ef19e08.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![(n-i)\times n](/images/math/6/c/5/6c5fc3faa3ba16c183aee22582d452f6.png)
![(n-i)\times (n-i)](/images/math/2/d/f/2dfb60822310b2de38bc6249af8ad647.png)
![v](/images/math/a/3/d/a3d52e52a48936cde0f5356bb08652f2.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![\partial D^i\times 0\subset D^i\times D^{n-i}](/images/math/0/5/6/0565bdb60731bb6c99eaf13740e919fb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))](/images/math/a/4/9/a49035d328b3ba4dbc7d4a9e13d0f328.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![(df\circ d\phi)|_{\partial D^i\times 0}(v)](/images/math/a/7/e/a7e51ba475de09c9f31bf7eb9d8d29c4.png)
![g(\partial D^i\times 0)](/images/math/2/f/5/2f50bc11df5df7d9f9a2b88b073110c4.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![i-1<2n-1 - (n-i)](/images/math/3/6/7/3676c9c806cf335df19ec68165e853d1.png)
![\pi_{i-1}(V_{2n-1, n-i})=0](/images/math/f/8/9/f89f268f7e633aaa8196d938e6b4460f.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![g(D^i\times 0)](/images/math/4/b/7/4b7f27b73221b0c232472ba2fbc76873.png)
![f\cup g|_{D^i\times 0}](/images/math/7/5/1/751535d1daf30058bb6824f69056b13b.png)
![N \cup_{\phi_i} D^i\times D^{n-i}](/images/math/e/4/9/e4964e9d2fb9ffbc1400a28d511c7e84.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).
Lemma 3.4.
Assume that is a compact
-manifold,
is an embedding with
,
are embeddings and
is a concordance between
and
.
If also that
, then there is an extension of
to a concordance between
and
.
![(b)](/images/math/4/6/2/462dc277171bc47412ce4e6895fc5f71.png)
![G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]](/images/math/7/6/7/7673e4c9b3f109db6df179275aa770ea.png)
![f_0|_{D^i\times 0}](/images/math/f/a/6/fa6d0deae25bc2b3a2a08245df0fca47.png)
![f_1|_{D^i\times 0}](/images/math/1/3/7/1379becb65ff170614469725aead7078.png)
![G(D^i\times 0\times [0, 1])](/images/math/f/c/a/fca2beb45e712aea38b8468ffe73aa06.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![G(\partial D^i\times 0\times [0, 1])](/images/math/9/c/c/9cc2f3be82a33e109eee7e7e9cabc33e.png)
![G(D^i\times 0\times 0)](/images/math/e/c/e/ece7d2a9be32d5837d835b478c0ac691.png)
![G(D^i\times 0\times 1)](/images/math/6/4/f/64f40e39ee385a4ccc8c81dbadcb3248.png)
![\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,](/images/math/f/f/d/ffd7ad38f5cca1a0734fe05216e65aac.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0, f_1](/images/math/7/2/e/72ebfdcb110d47ba960b40061831d498.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb{R}^m](/images/math/d/5/8/d5893f855c0893746999626e43f403e8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![n-1](/images/math/9/9/9/99911c3ea3e1da2d10de72c8066d422d.png)
![U^l](/images/math/5/3/a/53acb6cefb187b0fe8f6487904d7204d.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![U^1\cong D^n](/images/math/6/5/b/65b7ae4e1956bf59dc0bb7cb2fd6662c.png)
![F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/c/8/f/c8fc688383c938817e2af703f09348bd.png)
![F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/5/d/a/5da41d0f9627f07b25eed4568694be9f.png)
![f_0|_{U^1}](/images/math/d/5/2/d528f2ec62cdb6e65600036a46c2f61e.png)
![f_1|_{U^1}](/images/math/3/5/2/352833bf922051b2c5d90eba01f0bb98.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![f_0|_{U^l}](/images/math/f/4/b/f4b1c5c13db70a3bedd1a3fea9bd3d88.png)
![f_1|_{U^l}](/images/math/6/1/6/616cdbab5567ebaedef7a8cec15d9c53.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U^{l-1}](/images/math/a/2/9/a2991e29987d56c8015bda65b5b3eef9.png)
![f_0|_{U^{l-1}}](/images/math/1/1/d/11df57b884288f12be41ee4d282a2076.png)
![f_1|_{U^{l-1}}](/images/math/8/7/6/876110e463d693d6c47005faa12a872e.png)
![\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}](/images/math/c/c/3/cc359a5b5c2e6e045aff646818537d90.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![i\leq n-1](/images/math/2/1/e/21efcdb306da63a9827111b061c527ca.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]](/images/math/5/e/c/5ec7d9235de5dc82bd67cc00872a24e0.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U^{l-1}\cup_\phi D^i\times D^{n-i}](/images/math/4/7/3/4730d8f990eef5bf92f13894bada0cb1.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![D^i\times D^{n-i}\times [0, 1]](/images/math/8/8/c/88cd3df672a5115775ffedfb544d5bb9.png)
![D^i\times [0, 1]\times D^{n-i}](/images/math/e/c/c/ecc8260a7d0e17d2ea80f735dd1f84b9.png)
![\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1](/images/math/0/7/2/072e9c7b08fb27a39c991a4f39e66c90.png)
Tex syntax error
![\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,](/images/math/a/1/5/a15823ae1406826bc52a87fbfc5b1f5b.png)
![\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]](/images/math/c/9/d/c9d97bc144c9fb07e80ab6036a323230.png)
![F\bar{\phi} = G](/images/math/d/1/2/d12fa134bcaa9731bbeaaed9f1a77edc.png)
![\partial (D^i \times 0\times [0, 1])](/images/math/4/7/5/47580101248722766a17873fe2537b54.png)
![F(\mbox{Int} (U\times [0, 1]))](/images/math/4/1/0/4107ba155251147427656118445b1e23.png)
![G(\mbox{Int}( D^i\times 0\times [0, 1]))](/images/math/f/a/a/faab5cd0d220270034ac52657c616897.png)
![G_t](/images/math/d/3/b/d3b2bdacad293d6efe6d78f5766b3af9.png)
![G_0=G](/images/math/3/c/e/3ce2590599de94d8a0af6f58bc30f92d.png)
![\partial (D^i\times 0\times [0, 1])](/images/math/f/e/2/fe2c72ed3c40eb59b3c08c33b008f134.png)
![F(\partial (U\times [0, 1]))](/images/math/2/b/2/2b2fc6ab2e202f22d54317673c6d2997.png)
![G_1(D^i\times 0\times [0, 1])](/images/math/8/9/a/89aa9e32478b85ce4b2076ccb36ec156.png)
![G(U\times [0, 1])](/images/math/8/8/e/88e8188653e195ec591958c50130d42b.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0](/images/math/2/e/5/2e5488ed59b0b412a6c955ca10d3b295.png)
Denote by the
matrix whose rightmost
submatrix is the identity matrix, and whose other elements are zeroes. Denote by
the field of
normal vectors on
whose
-th vector has coordinates equal to the
-th row in
. Then
is the vector field tangent to
. For
denote by
the projection of
to the intersection of normal space to
, and tangent space to
. Since
, it follows that
. Hence there is an extension of
to a linear independent field of vectors normal to
. Then by Lemma 3.4.(b) there is an extension of
to a concordance
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1.
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
.
Proof.
Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree
. (To prove part (a) it is sufficient to take as
the identity map of
as a map of degree one and the constant map as a map of degree zero.)
Define
by the formula
.
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.
![k\geqslant 2](/images/math/2/c/b/2cb2015a51de7c9716f0dc2bc92c5268.png)
![f_1|_{S^k\times 0}](/images/math/c/1/6/c16e77e36d34660fbb7758ba5b44d442.png)
![f_2|_{S^k\times 0}](/images/math/5/d/b/5dbeac55457a6521596b48f699af75a4.png)
![f_1|_{S^k\times 0} = f_2|_{S^k\times 0}](/images/math/e/d/9/ed9a3ee4cda2bc53c52adf8ed0077f9e.png)
![l([f_1]) = l([f_2])](/images/math/c/5/3/c5301a619ec25514327c3ce383e5828f.png)
![f_1(S^k\times 0)](/images/math/e/0/2/e02f33e5695c28b4d3aa1d0024357b52.png)
![f_2(S^k\times 0)](/images/math/b/d/c/bdc8efad4137b267f347affef67219bb.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_2](/images/math/e/7/3/e73368a1436351fb0d11fdfb8cf3b3bf.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote .
Example 4.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 4.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3.
Let be the connected sum of two tori. Then there exists a surjection
defined by the formula
.
To prove the surjectivity of it is sufficient to take linked
-spheres in
and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4.
(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)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
-
is any closed orientable connected
-manifold,
-
is any embedding,
- if the (co)homology coefficients are omitted, then they are
,
-
is even and
is torsion free (these two assumptions are not required in Lemma 5.3).
By we denote the closure of the complement in
to an closed
-ball. Thus
is the
-sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint
-cycles in
with integer coefficients. Denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 5.2 ( is well-defined).
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] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.3.
Let be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
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 5.4.
For every the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 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.
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 6.1.
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 6.2.
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.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 6.3 of if
is even and
is torsion-free.
Take a collection
such that
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Note also that depends on choice of collection
. The following Theorems hold for any choice of
.
Theorem 6.4.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/f/4/c/f4cfe3a2b4cc5a3cc782084295a3bb04.png)
is one-to-one.
Lemma 6.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 6.4:
Theorem 6.6.
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]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 7.1.
Assume that is a closed
-connected
-manifold.
(a) If , then
embeds into
.
(b) If and
, then
embeds into
.
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2.
Assume that is a closed
-connected
-manifold.
(a) If and
, then any two embeddings of
into
are isotopic.
(b) If and
and
then any two embeddings of
into
are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For
part (b) coincides with Theorem 2.2b.
![k=1](/images/math/a/6/f/a6f0672a50348fdc06bc34fdc560cae9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb R^{2n-1}](/images/math/3/9/c/39cdeb2c39f9f962228626f6a37d3148.png)
![f,g\colon N_0\to\mathbb R^{2n-1}](/images/math/5/5/f/55fb87ba3a74da42f8d1bfa4c3c0707d.png)
![F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]](/images/math/7/9/6/7963693062b9ca3ce3d3f649be11320a.png)
![F(x, 0) = (f(x), 0)](/images/math/6/6/1/6614f0070021c0c42008192554b7fb76.png)
![F(x, 1)=(g(x), 1)](/images/math/d/0/c/d0c83013a79bb31d635845e51035fc3d.png)
![x\in N_0](/images/math/c/2/0/c201ff39d50a65d45853d1e389c1d27b.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![(n-2)](/images/math/9/d/d/9dd8fe07fe7b65ff961c61d8c07f8424.png)
![X\subset N_0](/images/math/4/1/8/418f89a952ba2cd648a0dde2d76d6659.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![F|_{X\times[0,1]}](/images/math/8/c/5/8c58be4ffc545b6549527ed446a6de3a.png)
![2(n-1) < 2n](/images/math/d/1/b/d1bb9ca106010aa3b768454fb0ec6a3f.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![h\colon N_0\to M](/images/math/f/9/6/f9661c49d810e6cd5240a2c78ce76cf6.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Conjecture 7.3.
Assume that is a closed
-connected
-manifold. Then any two embeddings of
in
are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4.
Assume is a closed
-connected
-manifold. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1].
Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes
and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1.
Assume that is a compact
-connected
-manifold,
,
is
-connected and
.
Then
embeds into
.
This is [Wall1965, Theorem on p.567].
![f\colon N\to\mathbb R^{2n-k-1}](/images/math/e/f/7/ef7e5dbe129d03b62f57644b8dc55634.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-k-1)](/images/math/9/b/9/9b98561fa8ab10c4b891ab69108aee75.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f|_X](/images/math/c/0/b/c0b6464917fb6186467774799040cd45.png)
![2(n-k) < 2n-k-1](/images/math/2/9/7/2977a324af64a5bc37d5330638c439a4.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![f|_{M}](/images/math/3/7/7/377d6c2f9743bedc11e1db159583b466.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.2.
Assume that is a
-manifold. If
has
-dimensional spine,
,
, then any two embeddings of
into
are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 8.3.
Every compact connected -manifold
with boundary for which
is
-connected,
,
and
, has an
-dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]]
The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from
to
.
Theorem 9.2.[[Hirsch1959, Theorem 6.4]]
If is immersible in
with a normal
-field, then
is immersible in
.
Theorem 9.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]]
Every -manifold
is immersible in
.
Denote by is Stiefel manifold of
-frames in
.
Theorem 9.5.
Suppose is a
-manifold with non-empty boudary,
is
-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
.
Let us construct such a linear monomorphism by skeleta of
.
It is clear that a linear monomorphism exists on
-skeleton of
.
The obstruction to extend the linear monomorphism from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 9.6.
Suppose is a connected
-manifold with non-empty boudary,
is
-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 extend the homotopy from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and
has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
10 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
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [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
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [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.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [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
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain
-manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
![\S](/images/math/9/0/3/9037f6609cc196a78441f1697f0f4c00.png)
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 the set of all embeddings
up to isotopy. We 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. For some results we present direct proofs, which are easier 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 connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2.
Assume that is a compact connected
-manifold and either
(a) or
(b) has non-empty boundary and
.
Then any two embeddings of into
are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Part (b) in the case
is proved in [Edwards1968,
4, Corollary 5]. The case
is clear. The case
can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7).
In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and
7 can be proved using
- analogous results for immersions of manifolds stated in
9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in
3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in
3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
The first proof of Theorem 2.1.b uses immersions, while the second does not.
![g\colon N\to\mathbb R^{2n-1}](/images/math/8/e/6/8e6a5cb7a3e2550f77ea7422924c4c36.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-1)](/images/math/1/1/a/11a59ed91d8f4cb860db22df0c49f391.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.png)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![g|_{M}](/images/math/5/9/f/59f4f568d65d9448dd1940c1c1c54ccd.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966]
Assume that is a closed connected smooth
-manifold. Then
have handle decomposition with indices of attaching map at most
.
Lemma 3.2.
Assume that is a closed smooth
-manifold and
is an attaching map such that
. If there is embedding
, then
extends to an embedding of
.
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\phi_1,\ldots,\phi_s](/images/math/7/8/8/78859dd0fd89584cca15fd5cdfe87ad3.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![F_1:U_1 \cong D^n\to \R^{2n-1}](/images/math/9/f/8/9f8bbbb595ac2731aafcf78e1b4ec49d.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![U_l](/images/math/3/3/f/33f44415bdcda9de858c378ea0cc58dd.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U_{l-1}](/images/math/d/7/b/d7bff18cf35e8b3f6294db53d54ae41a.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.png)
![\phi_l](/images/math/c/4/8/c48fb2f519dccf14afbd1e51cf3bc0ba.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}](/images/math/1/7/4/17424d98822d3296bb1df2e028c79fd2.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 3.3.
Let be a closed smooth
-manifold and
,
,
, are smooth embeddings such that
on
. Suppose that on
there is a field of
pairwise orthogonal normal vectors whose restriction to
is tangent to
. Then
extends to a smooth embedding
.
![2i+1\leq 2n-1](/images/math/5/0/7/5073858d0affb278fddded337c402bbf.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}](/images/math/0/9/c/09c06ccd69fd3dbfdac9dbcd4ef19e08.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![(n-i)\times n](/images/math/6/c/5/6c5fc3faa3ba16c183aee22582d452f6.png)
![(n-i)\times (n-i)](/images/math/2/d/f/2dfb60822310b2de38bc6249af8ad647.png)
![v](/images/math/a/3/d/a3d52e52a48936cde0f5356bb08652f2.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![\partial D^i\times 0\subset D^i\times D^{n-i}](/images/math/0/5/6/0565bdb60731bb6c99eaf13740e919fb.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))](/images/math/a/4/9/a49035d328b3ba4dbc7d4a9e13d0f328.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![(df\circ d\phi)|_{\partial D^i\times 0}(v)](/images/math/a/7/e/a7e51ba475de09c9f31bf7eb9d8d29c4.png)
![g(\partial D^i\times 0)](/images/math/2/f/5/2f50bc11df5df7d9f9a2b88b073110c4.png)
![\partial N](/images/math/0/3/c/03c3ed3c7f3335d6ad0b4549484d2ef8.png)
![i-1<2n-1 - (n-i)](/images/math/3/6/7/3676c9c806cf335df19ec68165e853d1.png)
![\pi_{i-1}(V_{2n-1, n-i})=0](/images/math/f/8/9/f89f268f7e633aaa8196d938e6b4460f.png)
![v'](/images/math/a/c/6/ac606f92d3c657b707d3b62be6f7935c.png)
![g(D^i\times 0)](/images/math/4/b/7/4b7f27b73221b0c232472ba2fbc76873.png)
![f\cup g|_{D^i\times 0}](/images/math/7/5/1/751535d1daf30058bb6824f69056b13b.png)
![N \cup_{\phi_i} D^i\times D^{n-i}](/images/math/e/4/9/e4964e9d2fb9ffbc1400a28d511c7e84.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).
Lemma 3.4.
Assume that is a compact
-manifold,
is an embedding with
,
are embeddings and
is a concordance between
and
.
If also that
, then there is an extension of
to a concordance between
and
.
![(b)](/images/math/4/6/2/462dc277171bc47412ce4e6895fc5f71.png)
![G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]](/images/math/7/6/7/7673e4c9b3f109db6df179275aa770ea.png)
![f_0|_{D^i\times 0}](/images/math/f/a/6/fa6d0deae25bc2b3a2a08245df0fca47.png)
![f_1|_{D^i\times 0}](/images/math/1/3/7/1379becb65ff170614469725aead7078.png)
![G(D^i\times 0\times [0, 1])](/images/math/f/c/a/fca2beb45e712aea38b8468ffe73aa06.png)
![n-i](/images/math/9/d/7/9d7f39a6c4a80d5e5da71f98cb5b0907.png)
![G(\partial D^i\times 0\times [0, 1])](/images/math/9/c/c/9cc2f3be82a33e109eee7e7e9cabc33e.png)
![G(D^i\times 0\times 0)](/images/math/e/c/e/ece7d2a9be32d5837d835b478c0ac691.png)
![G(D^i\times 0\times 1)](/images/math/6/4/f/64f40e39ee385a4ccc8c81dbadcb3248.png)
![\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,](/images/math/f/f/d/ffd7ad38f5cca1a0734fe05216e65aac.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0, f_1](/images/math/7/2/e/72ebfdcb110d47ba960b40061831d498.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb{R}^m](/images/math/d/5/8/d5893f855c0893746999626e43f403e8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![n-1](/images/math/9/9/9/99911c3ea3e1da2d10de72c8066d422d.png)
![U^l](/images/math/5/3/a/53acb6cefb187b0fe8f6487904d7204d.png)
![\emptyset](/images/math/8/3/f/83f26ceba0b3c1ae6c539266698fd3cd.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![U^1\cong D^n](/images/math/6/5/b/65b7ae4e1956bf59dc0bb7cb2fd6662c.png)
![F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/c/8/f/c8fc688383c938817e2af703f09348bd.png)
![F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]](/images/math/5/d/a/5da41d0f9627f07b25eed4568694be9f.png)
![f_0|_{U^1}](/images/math/d/5/2/d528f2ec62cdb6e65600036a46c2f61e.png)
![f_1|_{U^1}](/images/math/3/5/2/352833bf922051b2c5d90eba01f0bb98.png)
![F_l](/images/math/d/4/3/d4358d74d78e92249820bb51825c732d.png)
![f_0|_{U^l}](/images/math/f/4/b/f4b1c5c13db70a3bedd1a3fea9bd3d88.png)
![f_1|_{U^l}](/images/math/6/1/6/616cdbab5567ebaedef7a8cec15d9c53.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![U^{l-1}](/images/math/a/2/9/a2991e29987d56c8015bda65b5b3eef9.png)
![f_0|_{U^{l-1}}](/images/math/1/1/d/11df57b884288f12be41ee4d282a2076.png)
![f_1|_{U^{l-1}}](/images/math/8/7/6/876110e463d693d6c47005faa12a872e.png)
![\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}](/images/math/c/c/3/cc359a5b5c2e6e045aff646818537d90.png)
![l](/images/math/2/4/d/24d59cd0b76a27b85f35d40a3cf6ec37.png)
![i\leq n-1](/images/math/2/1/e/21efcdb306da63a9827111b061c527ca.png)
![F_{l-1}](/images/math/b/5/b/b5b7f13d95588bd8853c354c87d920cb.png)
![\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]](/images/math/5/e/c/5ec7d9235de5dc82bd67cc00872a24e0.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U^{l-1}\cup_\phi D^i\times D^{n-i}](/images/math/4/7/3/4730d8f990eef5bf92f13894bada0cb1.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![D^i\times D^{n-i}\times [0, 1]](/images/math/8/8/c/88cd3df672a5115775ffedfb544d5bb9.png)
![D^i\times [0, 1]\times D^{n-i}](/images/math/e/c/c/ecc8260a7d0e17d2ea80f735dd1f84b9.png)
![\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1](/images/math/0/7/2/072e9c7b08fb27a39c991a4f39e66c90.png)
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![\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,](/images/math/a/1/5/a15823ae1406826bc52a87fbfc5b1f5b.png)
![\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]](/images/math/c/9/d/c9d97bc144c9fb07e80ab6036a323230.png)
![F\bar{\phi} = G](/images/math/d/1/2/d12fa134bcaa9731bbeaaed9f1a77edc.png)
![\partial (D^i \times 0\times [0, 1])](/images/math/4/7/5/47580101248722766a17873fe2537b54.png)
![F(\mbox{Int} (U\times [0, 1]))](/images/math/4/1/0/4107ba155251147427656118445b1e23.png)
![G(\mbox{Int}( D^i\times 0\times [0, 1]))](/images/math/f/a/a/faab5cd0d220270034ac52657c616897.png)
![G_t](/images/math/d/3/b/d3b2bdacad293d6efe6d78f5766b3af9.png)
![G_0=G](/images/math/3/c/e/3ce2590599de94d8a0af6f58bc30f92d.png)
![\partial (D^i\times 0\times [0, 1])](/images/math/f/e/2/fe2c72ed3c40eb59b3c08c33b008f134.png)
![F(\partial (U\times [0, 1]))](/images/math/2/b/2/2b2fc6ab2e202f22d54317673c6d2997.png)
![G_1(D^i\times 0\times [0, 1])](/images/math/8/9/a/89aa9e32478b85ce4b2076ccb36ec156.png)
![G(U\times [0, 1])](/images/math/8/8/e/88e8188653e195ec591958c50130d42b.png)
![F\cup G](/images/math/c/c/5/cc5fe8d704e6fd0b7e7802e0f18931a8.png)
![f_0](/images/math/0/7/e/07ed23691bfe5279d8b1eb9c83119baa.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0](/images/math/2/e/5/2e5488ed59b0b412a6c955ca10d3b295.png)
Denote by the
matrix whose rightmost
submatrix is the identity matrix, and whose other elements are zeroes. Denote by
the field of
normal vectors on
whose
-th vector has coordinates equal to the
-th row in
. Then
is the vector field tangent to
. For
denote by
the projection of
to the intersection of normal space to
, and tangent space to
. Since
, it follows that
. Hence there is an extension of
to a linear independent field of vectors normal to
. Then by Lemma 3.4.(b) there is an extension of
to a concordance
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1.
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
.
Proof.
Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree
. (To prove part (a) it is sufficient to take as
the identity map of
as a map of degree one and the constant map as a map of degree zero.)
Define
by the formula
.
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.
![k\geqslant 2](/images/math/2/c/b/2cb2015a51de7c9716f0dc2bc92c5268.png)
![f_1|_{S^k\times 0}](/images/math/c/1/6/c16e77e36d34660fbb7758ba5b44d442.png)
![f_2|_{S^k\times 0}](/images/math/5/d/b/5dbeac55457a6521596b48f699af75a4.png)
![f_1|_{S^k\times 0} = f_2|_{S^k\times 0}](/images/math/e/d/9/ed9a3ee4cda2bc53c52adf8ed0077f9e.png)
![l([f_1]) = l([f_2])](/images/math/c/5/3/c5301a619ec25514327c3ce383e5828f.png)
![f_1(S^k\times 0)](/images/math/e/0/2/e02f33e5695c28b4d3aa1d0024357b52.png)
![f_2(S^k\times 0)](/images/math/b/d/c/bdc8efad4137b267f347affef67219bb.png)
![f_1](/images/math/8/c/b/8cb6d88543008dcf8d30151a1f169f19.png)
![f_2](/images/math/e/7/3/e73368a1436351fb0d11fdfb8cf3b3bf.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote .
Example 4.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 4.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3.
Let be the connected sum of two tori. Then there exists a surjection
defined by the formula
.
To prove the surjectivity of it is sufficient to take linked
-spheres in
and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4.
(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)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
-
is any closed orientable connected
-manifold,
-
is any embedding,
- if the (co)homology coefficients are omitted, then they are
,
-
is even and
is torsion free (these two assumptions are not required in Lemma 5.3).
By we denote the closure of the complement in
to an closed
-ball. Thus
is the
-sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint
-cycles in
with integer coefficients. Denote
![\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),](/images/math/3/b/d/3bd3a1f6dd0417b213f6384693e531b0.png)
where is a nowhere vanishing normal field to
and
are the results of the shift of
by
.
Lemma 5.2 ( is well-defined).
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] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.3.
Let be two nowhere vanishing normal vector fields to
.
Then
![\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y](/images/math/a/2/5/a255beedf98b5b70a53573d6916bec48.png)
where is the result of the shift of
by
, and
is (Poincare dual to) the first obstruction to
being homotopic in the class of the nowhere vanishing vector fields.
This Lemma is proved in [Saeki1999, Lemma 2.2] for , but the proof is valid in all dimensions.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
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 5.4.
For every the following equality holds:
![\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.](/images/math/a/d/7/ad7732d8f7f47d81a740eeb9868d2bf3.png)
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 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.
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 6.1.
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 6.2.
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.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 6.3 of if
is even and
is torsion-free.
Take a collection
such that
.
For each
such that
define
![\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)](/images/math/2/3/7/237bdb005bb6a8987cc45be0cae8f657.png)
where
.
Note also that depends on choice of collection
. The following Theorems hold for any choice of
.
Theorem 6.4.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even.
The map
![\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),](/images/math/f/4/c/f4cfe3a2b4cc5a3cc782084295a3bb04.png)
is one-to-one.
Lemma 6.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 6.4:
Theorem 6.6.
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]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 7.1.
Assume that is a closed
-connected
-manifold.
(a) If , then
embeds into
.
(b) If and
, then
embeds into
.
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2.
Assume that is a closed
-connected
-manifold.
(a) If and
, then any two embeddings of
into
are isotopic.
(b) If and
and
then any two embeddings of
into
are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For
part (b) coincides with Theorem 2.2b.
![k=1](/images/math/a/6/f/a6f0672a50348fdc06bc34fdc560cae9.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![\mathbb R^{2n-1}](/images/math/3/9/c/39cdeb2c39f9f962228626f6a37d3148.png)
![f,g\colon N_0\to\mathbb R^{2n-1}](/images/math/5/5/f/55fb87ba3a74da42f8d1bfa4c3c0707d.png)
![F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]](/images/math/7/9/6/7963693062b9ca3ce3d3f649be11320a.png)
![F(x, 0) = (f(x), 0)](/images/math/6/6/1/6614f0070021c0c42008192554b7fb76.png)
![F(x, 1)=(g(x), 1)](/images/math/d/0/c/d0c83013a79bb31d635845e51035fc3d.png)
![x\in N_0](/images/math/c/2/0/c201ff39d50a65d45853d1e389c1d27b.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![(n-2)](/images/math/9/d/d/9dd8fe07fe7b65ff961c61d8c07f8424.png)
![X\subset N_0](/images/math/4/1/8/418f89a952ba2cd648a0dde2d76d6659.png)
![N_0](/images/math/a/e/7/ae77718b7e730be439983a5bf2348800.png)
![F|_{X\times[0,1]}](/images/math/8/c/5/8c58be4ffc545b6549527ed446a6de3a.png)
![2(n-1) < 2n](/images/math/d/1/b/d1bb9ca106010aa3b768454fb0ec6a3f.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![h\colon N_0\to M](/images/math/f/9/6/f9661c49d810e6cd5240a2c78ce76cf6.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![F|_{M\times[0,1]}](/images/math/2/b/a/2ba2ff85f9227591c2967fbb841c3b37.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![g\circ h](/images/math/8/2/3/82367e5c55ae541f2d0e7496f061ce07.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![g](/images/math/f/4/6/f46271e5c04cf1146670e9315ac9713d.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Conjecture 7.3.
Assume that is a closed
-connected
-manifold. Then any two embeddings of
in
are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4.
Assume is a closed
-connected
-manifold. Then for each
there exists a bijection
![\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),](/images/math/c/5/8/c58a212bdfba0cc5a91cf05f6d037192.png)
where denote
for
even and
for
odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(
)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1].
Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes
and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1.
Assume that is a compact
-connected
-manifold,
,
is
-connected and
.
Then
embeds into
.
This is [Wall1965, Theorem on p.567].
![f\colon N\to\mathbb R^{2n-k-1}](/images/math/e/f/7/ef7e5dbe129d03b62f57644b8dc55634.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![(n-k-1)](/images/math/9/b/9/9b98561fa8ab10c4b891ab69108aee75.png)
![X\subset N](/images/math/9/b/0/9b0c60d78ff8d1ef84bd81c0221a3154.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![f|_X](/images/math/c/0/b/c0b6464917fb6186467774799040cd45.png)
![2(n-k) < 2n-k-1](/images/math/2/9/7/2977a324af64a5bc37d5330638c439a4.png)
![f](/images/math/6/b/6/6b6e98cde8b33087a33e4d3a497bd86b.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![M\supset X](/images/math/5/8/3/583c7c1b2da2154deb5b9b2568e97c9c.png)
![f|_{M}](/images/math/3/7/7/377d6c2f9743bedc11e1db159583b466.png)
![h\colon N\to M](/images/math/b/c/1/bc18c27225aa9edc040f1d2d4fd85381.png)
![f\circ h](/images/math/2/0/7/2072263303b4d8ce8df29f37938a24d3.png)
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.2.
Assume that is a
-manifold. If
has
-dimensional spine,
,
, then any two embeddings of
into
are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 8.3.
Every compact connected -manifold
with boundary for which
is
-connected,
,
and
, has an
-dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]]
The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from
to
.
Theorem 9.2.[[Hirsch1959, Theorem 6.4]]
If is immersible in
with a normal
-field, then
is immersible in
.
Theorem 9.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]]
Every -manifold
is immersible in
.
Denote by is Stiefel manifold of
-frames in
.
Theorem 9.5.
Suppose is a
-manifold with non-empty boudary,
is
-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
.
Let us construct such a linear monomorphism by skeleta of
.
It is clear that a linear monomorphism exists on
-skeleton of
.
The obstruction to extend the linear monomorphism from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 9.6.
Suppose is a connected
-manifold with non-empty boudary,
is
-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 extend the homotopy from -skeleton to
-skeleton lies in
.
For we know
.
For
we have
since
is
-connected and
has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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