Embeddings of manifolds with boundary: classification

Revision as of 17:46, 7 June 2020 (view source) Mikhail Fedorov (Talk | contribs) (→Seifert linking form) ← Older edit		Revision as of 10:43, 11 June 2020 (view source) Mikhail Fedorov (Talk | contribs) (→Seifert linking form) Newer edit →
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	The following folklore result holds.		The following folklore result holds.
	{{beginthm|Lemma}}		{{beginthm|Lemma}}
−	Suppose $H_1(N; \mathbb Z)$ is torsion free. For each even $n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$.	+	Assume $N$ is a closed orientable connected $n$-manifold, $n$ is even and $H_1(N; \mathbb Z)$ is torsion free. Then for each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$.
	{{endthm}}		{{endthm}}
	{{beginproof}}		{{beginproof}}
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	{{beginthm|Definition}}		{{beginthm|Definition}}
−	For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote by	+	For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote
	$$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$		$$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$
	where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$.		where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$.
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	For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.		For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.
−	Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}$ it follows that . Linking coefficient preserves under isotopy.	+	Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}$ it follows that this isopoty restricts to isotopy of link $f(x)\sqcup s(y)$. Linking coefficient preserves under isotopy.
	{{endproof}}		{{endproof}}

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Revision as of 10:43, 11 June 2020

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Contents 1 Introduction 2 Embedding and unknotting theorems 3 Example on non-isotopic embeddings 4 Seifert linking form 5 Classification theorems 6 A generalization to highly-connected manifolds 7 References

1 Introduction

In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In $\S$\ref{sec::example} we give an example of non-isotopic embeddings of a cylinder over $(n-1)$-sphere. In $\S$\ref{sec::linking_form} we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. In $\S$\ref{sec::generalisations} which is independent from $\S$\ref{sec::example} and $\S$\ref{sec::linking_form} we state generalisations of theorems from $\S$\ref{sec::general_theorems} to highly-connected manifolds. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$3 we give an example of non-isotopic embeddings of a cylinder over $(n-1)$-sphere. In $\S$4 we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. In $\S$6 which is independent from $\S$3 and $\S$4 we state generalisations of theorems from $\S$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, $\S$1, $\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state only the results that can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$] for the PL case. Usually there exist easier direct proofs than deduction from this criterion. Sometimes we give references to such direct proofs but we do not claim these are original proofs.

2 Embedding and unknotting theorems

This is well-known strong Whitney embedding theorem.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, $\S$ 2].

Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.

Part (a) of this theorem in case $n>2$ can be found in [Edwards1968, $\S$ 4, Corollary 5]. Case $n=1$ is clear.

This theorem is a special case of the Theorem 6.4 .

Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$6).

3 Example on non-isotopic embeddings

The following example is folklore.

This construction is analogous to the Hopf link, see [Skopenkov2016h, $\S$ 2].

4 Seifert linking form

Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere.

The following folklore result holds.

Denote by $\mathrm{lk}$ the linking coefficient [Skopenkov2016h, $\S$ 3, remark 3.2d] of two disjoint cycles.

Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$, but the proof is valid in all dimensions.

The latter Lemma implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0;\mathbb Z)\times H_{n-1}(N_0;\mathbb Z)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent.

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.5 Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.2.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get:

The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. The second equality holds because the linear homotopy of $s'$ and $-s$ degenerates on a $2$-cycle in $N_0$ on which $s'$ and $-s$ are linearly dependent.

5 Classification theorems

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy.

This is the main Theorem of [Tonkonog2010]

6 A generalization to highly-connected manifolds

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

For the Diff case see [Haefliger1961, $\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).

See Theorem 2.4 of the survey [Skopenkov2016c, $\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 References

• [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
• [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
• [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
• [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
• [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
• [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
• [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
• [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).

, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. If the category is omitted, then we assume the smooth (DIFF) category. We state only the results that can be deduced from [[Some_calculations_involving_configuration_spaces_of_distinct_points|the Haefliger-Weber deleted product criterion]] \cite[$\S$ 5]{Skopenkov2006}, see \cite[6.4]{Haefliger1963}, \cite[Theorem 1.1$\alpha\partial$]{Skopenkov2002} for the DIFF case and \cite[Theorem 1.3$\alpha\partial$]{Skopenkov2002} for the PL case. Usually there exist easier direct proofs than deduction from this criterion. Sometimes we give references to such direct proofs but we do not claim these are original proofs. == Embedding and unknotting theorems == ; \label{sec::general_theorems} {{beginthm|Theorem}} Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\R^{2n}$. {{endthm}} This is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]]. {{beginthm|Theorem}} Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\R^{2n-1}$. {{endthm}} The Diff case of this result is proved in \cite[Theorem 4.6]{Hirsch1961a}. For the PL case see references for Theorem \ref{thm::k_connect_boundary} below and \cite[Theorem 5.2]{Horvatic1971}. {{beginthm|Theorem}}\label{thm::closed_unknotting} Assume that $N$ is a compact $n$-manifold and either (a) $m \ge 2n+2$ or (b) $N$ is connected and $m \ge 2n+1 \ge 5$. Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorems 2.1 and 2.2]] respectively of \cite[$\S$ 2]{Skopenkov2016c}. Note that inequality in part (a) is sharp, which is shown by the construction of [[High_codimension_links#Examples|the Hopf link]]. {{beginthm|Theorem}} \label{thm::special_Haef_Zem} Assume that $N$ is a compact $n$-manifold with non-empty boundary and either (a) $m \ge 2n$ or (b) $N$ is \S3 we give an example of non-isotopic embeddings of a cylinder over $(n-1)$-sphere. In $\S$4 we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. In $\S$6 which is independent from $\S$3 and $\S$4 we state generalisations of theorems from $\S$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, $\S$1, $\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state only the results that can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$] for the PL case. Usually there exist easier direct proofs than deduction from this criterion. Sometimes we give references to such direct proofs but we do not claim these are original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\R^{2n}$.

This is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\R^{2n-1}$.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. Assume that $N$ is a compact $n$-manifold and either

(a) $m \ge 2n+2$ or

(b) $N$ is connected and $m \ge 2n+1 \ge 5$.

Then any two embeddings of $N$ into $\R^m$ are isotopic.

The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, $\S$ 2].

Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.

Theorem 2.4. Assume that $N$ is a compact $n$-manifold with non-empty boundary and either

(a) $m \ge 2n$ or

(b) $N$ is $1$-connected, $m \ge 2n - 1\ge3$.

Then any two embeddings of $N$ into $\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$ can be found in [Edwards1968, $\S$ 4, Corollary 5]. Case $n=1$ is clear.

This theorem is a special case of the Theorem 6.4 .

Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$6).

3 Example on non-isotopic embeddings

The following example is folklore.

Proposition 3.1. Let $N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$. Then there exist non-isotopic embeddings of $N$ to $\mathbb R^{2n-1}$.

Proof. Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$, where $1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$.

Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$ and $\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are not linked while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are linked [Skopenkov2016h, $\S$ 3, remark 3.2d].

$\square$

This construction is analogous to the Hopf link, see [Skopenkov2016h, $\S$ 2].

4 Seifert linking form

Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere.

The following folklore result holds.

Lemma 4.1. Assume $N$ is a closed orientable connected $n$-manifold, $n$ is even and $H_1(N; \mathbb Z)$ is torsion free. Then for each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0; \mathbb Z)\cong H_1(N_0, \partial N_0; \mathbb Z)\cong H_1(N; \mathbb Z)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$.

A normal space to $f(N_0)$ at any point of $f(N_0)$ has dimension $n-1$. As $n$ is even thus $n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$. Since $H_1(N; \mathbb Z)$ is torsion free, it follows that $\bar e=0$.

Since $N_0$ has non-empty boundary, we have that $N_0$ is homotopy equivalent to an $(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$ at any point of $f(N_0)$. Since $\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$.

$\square$

Denote by $\mathrm{lk}$ the linking coefficient [Skopenkov2016h, $\S$ 3, remark 3.2d] of two disjoint cycles.

Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients.

Lemma 4.2. Let $f:N_0\to \R^{2n-1}$ be an embedding. Let $s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$. Then

$$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$ is the result of the shift of $f(y)$ by $s$, and $d(s,s')\in H_2(N_0; \mathbb Z)$ is (Poincare dual to) the first obstruction to $s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$, but the proof is valid in all dimensions.

Definition 4.3. For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote

$$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$.

Lemma 4.4 ($L$ is well-defined, from unpublished update of [Tonkonog2010]). For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$ is replaced by $s'$,
• does not change when $x$ or $y$ are changed to homologous cycles and,
• does not change when $f$ is changed to an isotopic embedding.

Proof. The first bullet point follows because:

$$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 4.2.

For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}$ it follows that this isopoty restricts to isotopy of link $f(x)\sqcup s(y)$. Linking coefficient preserves under isotopy.

$\square$

The latter Lemma implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0;\mathbb Z)\times H_{n-1}(N_0;\mathbb Z)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent.

Lemma 4.5. Let $f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0; \mathbb Z)$ the following equality holds:

$$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.5 Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get

$$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.2.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get:

$$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. The second equality holds because the linear homotopy of $s'$ and $-s$ degenerates on a $2$-cycle in $N_0$ on which $s'$ and $-s$ are linearly dependent.

5 Classification theorems

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy.

Theorem 5.1. Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$ is the standard pairing.

This is the main Theorem of [Tonkonog2010]

6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n>2k+2$. Then $N$ embeds into $\R^{2n-k}$.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

Theorem 6.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary, $(N, \partial N)$ is $k$-connected and $n\ge2k+2$. Then $N$ embeds into $\R^{2n-k-1}$.

For the Diff case see [Haefliger1961, $\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).

Theorem 6.3. Assume that $N$ is a closed $k$-connected $n$-manifold. Then for each $n\ge2k + 2$, $m \ge 2n - k + 1$ any two embeddings of $N$ into $\R^m$ are isotopic.

See Theorem 2.4 of the survey [Skopenkov2016c, $\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 6.4. Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. Then for each $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\R^m$ are isotopic.

For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 References

• [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
• [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
• [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
• [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
• [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
• [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
• [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
• [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).

$-connected, $m \ge 2n - 1\ge3$. Then any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} Part (a) of this theorem in case $n>2$ can be found in \cite[$\S$ 4, Corollary 5]{Edwards1968}. Case $n=1$ is clear. This theorem is a special case of the Theorem \ref{thm::highly_connected_boundary_unknotting} . Inequality in part (a) is sharp, see Proposition \ref{exm::linked_boundary}. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem \ref{thm::closed_unknotting}. These basic results can be generalized to the highly-connected manifolds (see $\S$\ref{sec::generalisations}). == Example on non-isotopic embeddings == ; \label{sec::example} The following example is folklore. {{beginthm|Proposition}} \label{exm::linked_boundary} Let $N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$. Then there exist non-isotopic embeddings of $N$ to $\mathbb R^{2n-1}$. {{endthm}} {{beginproof}} Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$, where \S3 we give an example of non-isotopic embeddings of a cylinder over $(n-1)$-sphere. In $\S$4 we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. In $\S$6 which is independent from $\S$3 and $\S$4 we state generalisations of theorems from $\S$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, $\S$1, $\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state only the results that can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$] for the PL case. Usually there exist easier direct proofs than deduction from this criterion. Sometimes we give references to such direct proofs but we do not claim these are original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$ is a closed compact $n$-manifold. Then $N$ embeds into $\R^{2n}$.

This is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary. Then $N$ embeds into $\R^{2n-1}$.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

Theorem 2.3. Assume that $N$ is a compact $n$-manifold and either

(a) $m \ge 2n+2$ or

(b) $N$ is connected and $m \ge 2n+1 \ge 5$.

Then any two embeddings of $N$ into $\R^m$ are isotopic.

The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, $\S$ 2].

Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.

Theorem 2.4. Assume that $N$ is a compact $n$-manifold with non-empty boundary and either

(a) $m \ge 2n$ or

(b) $N$ is $1$-connected, $m \ge 2n - 1\ge3$.

Then any two embeddings of $N$ into $\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$ can be found in [Edwards1968, $\S$ 4, Corollary 5]. Case $n=1$ is clear.

This theorem is a special case of the Theorem 6.4 .

Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$6).

3 Example on non-isotopic embeddings

The following example is folklore.

Proposition 3.1. Let $N=S^{n-1}\times [0, 1]$ be the cylinder over $S^{n-1}$. Then there exist non-isotopic embeddings of $N$ to $\mathbb R^{2n-1}$.

Proof. Define $g_1\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_1(x, t) = (x, t1_{n-1})$, where $1_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$.

Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$ and $\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are not linked while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are linked [Skopenkov2016h, $\S$ 3, remark 3.2d].

$\square$

This construction is analogous to the Hopf link, see [Skopenkov2016h, $\S$ 2].

4 Seifert linking form

Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere.

The following folklore result holds.

Lemma 4.1. Assume $N$ is a closed orientable connected $n$-manifold, $n$ is even and $H_1(N; \mathbb Z)$ is torsion free. Then for each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0; \mathbb Z)\cong H_1(N_0, \partial N_0; \mathbb Z)\cong H_1(N; \mathbb Z)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$.

A normal space to $f(N_0)$ at any point of $f(N_0)$ has dimension $n-1$. As $n$ is even thus $n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$. Since $H_1(N; \mathbb Z)$ is torsion free, it follows that $\bar e=0$.

Since $N_0$ has non-empty boundary, we have that $N_0$ is homotopy equivalent to an $(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$ at any point of $f(N_0)$. Since $\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$.

$\square$

Denote by $\mathrm{lk}$ the linking coefficient [Skopenkov2016h, $\S$ 3, remark 3.2d] of two disjoint cycles.

Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients.

Lemma 4.2. Let $f:N_0\to \R^{2n-1}$ be an embedding. Let $s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$. Then

$$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$ is the result of the shift of $f(y)$ by $s$, and $d(s,s')\in H_2(N_0; \mathbb Z)$ is (Poincare dual to) the first obstruction to $s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$, but the proof is valid in all dimensions.

Definition 4.3. For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote

$$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$.

Lemma 4.4 ($L$ is well-defined, from unpublished update of [Tonkonog2010]). For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$ is replaced by $s'$,
• does not change when $x$ or $y$ are changed to homologous cycles and,
• does not change when $f$ is changed to an isotopic embedding.

Proof. The first bullet point follows because:

$$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 4.2.

For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}$ it follows that this isopoty restricts to isotopy of link $f(x)\sqcup s(y)$. Linking coefficient preserves under isotopy.

$\square$

The latter Lemma implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0;\mathbb Z)\times H_{n-1}(N_0;\mathbb Z)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent.

Lemma 4.5. Let $f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0; \mathbb Z)$ the following equality holds:

$$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.5 Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get

$$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.2.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get:

$$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. The second equality holds because the linear homotopy of $s'$ and $-s$ degenerates on a $2$-cycle in $N_0$ on which $s'$ and $-s$ are linearly dependent.

5 Classification theorems

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy.

Theorem 5.1. Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$ is the standard pairing.

This is the main Theorem of [Tonkonog2010]

6 A generalization to highly-connected manifolds

Theorem 6.1. Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n>2k+2$. Then $N$ embeds into $\R^{2n-k}$.

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

Theorem 6.2. Assume that $N$ is a compact $n$-manifold with nonempty boundary, $(N, \partial N)$ is $k$-connected and $n\ge2k+2$. Then $N$ embeds into $\R^{2n-k-1}$.

For the Diff case see [Haefliger1961, $\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).

Theorem 6.3. Assume that $N$ is a closed $k$-connected $n$-manifold. Then for each $n\ge2k + 2$, $m \ge 2n - k + 1$ any two embeddings of $N$ into $\R^m$ are isotopic.

See Theorem 2.4 of the survey [Skopenkov2016c, $\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Theorem 6.4. Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. Then for each $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\R^m$ are isotopic.

For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 References

• [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
• [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
• [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
• [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
• [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
• [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
• [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)

• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
• [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
• [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).

_{n-1}:=(1,0,\ldots,0)\in S^{n-1}$. Define $g_2\colon S^{n-1}\times [0, 1] \to D^n\times S^{n-1}$ by the formula $g_2(x, t) = (x, tx)$. Recall that $\mathrm i=\mathrm i_{2n-1,n-1}\colon D^n\times S^{n-1} \to \R^{2n-1}$ is the standard embedding. Then embeddings $\mathrm ig_1$ and $\mathrm ig_2$ are not isotopic. Indeed, the components of $\mathrm ig_1(S^{n-1}\times \{0, 1\})$ are [[High_codimension_links#The_linking_coefficient|not linked]] while the components of $\mathrm ig_2(S^{n-1}\times \{0, 1\})$ are [[High_codimension_links#The_linking_coefficient|linked]] \cite[$\S$ 3, remark 3.2d]{Skopenkov2016h}.{{endproof}}This construction is analogous to [[High_codimension_links#Examples|the Hopf link]], see \cite[$\S$ 2]{Skopenkov2016h}. == Seifert linking form == ; \label{sec::linking_form} Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. The following folklore result holds. {{beginthm|Lemma}} Suppose $H_1(N; \mathbb Z)$ is torsion free. For each even $n$ and each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$. {{endthm}} {{beginproof}} There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0; \mathbb Z)\cong H_1(N_0, \partial N_0; \mathbb Z)\cong H_1(N; \mathbb Z)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$. A normal space to $f(N_0)$ at any point of $f(N_0)$ has dimension $n-1$. As $n$ is even thus $n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$. Since $H_1(N; \mathbb Z)$ is torsion free, it follows that $\bar e=0$. Since $N_0$ has non-empty boundary, we have that $N_0$ is homotopy equivalent to an $(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$ at any point of $f(N_0)$. Since $\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$. {{endproof}} Denote by $\mathrm{lk}$ [[High_codimension_links#The_linking_coefficient|the linking coefficient]] \cite[$\S$ 3, remark 3.2d]{Skopenkov2016h} of two disjoint cycles. Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients. {{beginthm|Lemma}}\label{lmm::saeki} Let $f:N_0\to \R^{2n-1}$ be an embedding. Let $s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$. Then $$\mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$ where $s(y)$ is the result of the shift of $f(y)$ by $s$, and $d(s,s')\in H_2(N_0; \mathbb Z)$ is (Poincare dual to) the first obstruction to $s,s'$ being homotopic in the class of the nowhere vanishing vector fields. {{endthm}} This Lemma is proved in \cite[Lemma 2.2]{Saeki1999} for $n=3$, but the proof is valid in all dimensions. {{beginthm|Definition}} For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote by $$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$ where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$. {{endthm}} {{beginthm|Lemma|($L$ is well-defined, from unpublished update of \cite{Tonkonog2010})}} For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$: * is well-defined, i.e. does not change when $s$ is replaced by $s'$, * does not change when $x$ or $y$ are changed to homologous cycles and, * does not change when $f$ is changed to an isotopic embedding. {{endthm}} {{beginproof}} The first bullet point follows because: $$ \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned} $$ Here the second equality follows from Lemma \ref{lmm::saeki}. For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$. Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}$ it follows that . Linking coefficient preserves under isotopy. {{endproof}} The latter Lemma implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0;\mathbb Z)\times H_{n-1}(N_0;\mathbb Z)\to\mathbb Z$ denoted by the same letter. Denote by $\rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2)$ the reduction modulo $. Define the dual to [[Stiefel-Whitney_characteristic_classes|Stiefel-Whitney class]] $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent. {{beginthm|Lemma}} \label{lmm::L_equality} Let $f:N_0\to \R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0; \mathbb Z)$ the following equality holds: $$\rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$ {{endthm}} This Lemma was stated in a unpublished update of \cite{Tonkonog2010}, the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in \cite[Lemma 2.2]{Crowley&Skopenkov2016}. '''Proof of Lemma \ref{lmm::L_equality}''' Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get $$ \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \ &= d(s, -s)\cap X\cap Y . \end{aligned} $$ The first congruence is clear. The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift. The third equality follows from Lemma \ref{lmm::saeki}. Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get: $$ \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0). $$ The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. The second equality holds because the linear homotopy of $s'$ and $-s$ degenerates on a $-cycle in $N_0$ on which $s'$ and $-s$ are linearly dependent. == Classification theorems == ; \label{sec::classification} Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions. Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy. {{beginthm|Theorem}} Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. Then (a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection. (b) The image of $L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)=\langle \bar w_2(N_0),\rho_2(x\cap y)\rangle$. Here $\bar w_2(N_0)$ is the normal Stiefel-Whitney class, and $\langle\cdot,\cdot\rangle$ is the standard pairing. {{endthm}} This is the main Theorem of \cite{Tonkonog2010} == A generalization to highly-connected manifolds == ; \label{sec::generalisations} {{beginthm|Theorem}} Assume that $N$ is a closed compact $k$-connected $n$-manifold and $n>2k+2$. Then $N$ embeds into $\R^{2n-k}$. {{endthm}} The Diff case of this result is in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is in \cite[Corollary 1.3]{Irwin1965}. {{beginthm|Theorem}}\label{thm::k_connect_boundary} Assume that $N$ is a compact $n$-manifold with nonempty boundary, $(N, \partial N)$ is $k$-connected and $n\ge2k+2$. Then $N$ embeds into $\R^{2n-k-1}$. {{endthm}} For the Diff case see \cite[$\S$ 1.7, remark 2]{Haefliger1961} (there Haefliger proposes to use the deleted product criterion to obtain this result). {{beginthm|Theorem}}\label{thm::k_connect_closed_unknot} Assume that $N$ is a closed $k$-connected $n$-manifold. Then for each $n\ge2k + 2$, $m \ge 2n - k + 1$ any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} See [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$ 2]{Skopenkov2016c}, or \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. {{beginthm|Theorem}}\label{thm::highly_connected_boundary_unknotting} Assume that $N$ is a $k$-connected $n$-manifold with non-empty boundary. Then for each $n\ge k+3$ and $m\ge2n-k$ any two embeddings of $N$ into $\R^m$ are isotopic. {{endthm}} For the PL case of this result see \cite[Theorem 10.3]{Hudson1969}, which is proved using [[Isotopy#Concordance|concordance implies isotopy theorem]]. == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S3 we give an example of non-isotopic embeddings of a cylinder over $(n-1)$-sphere. In $\S$4 we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. In $\S$6 which is independent from $\S$3 and $\S$4 we state generalisations of theorems from $\S$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, $\S$1, $\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category.

We state only the results that can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$] for the PL case. Usually there exist easier direct proofs than deduction from this criterion. Sometimes we give references to such direct proofs but we do not claim these are original proofs.

2 Embedding and unknotting theorems

This is well-known strong Whitney embedding theorem.

The Diff case of this result is proved in [Hirsch1961a, Theorem 4.6]. For the PL case see references for Theorem 6.2 below and [Horvatic1971, Theorem 5.2].

The condition (a) stands for General Position Theorem and the condition (b) stands for Whitney-Wu Unknotting Theorem, see Theorems 2.1 and 2.2 respectively of [Skopenkov2016c, $\S$ 2].

Note that inequality in part (a) is sharp, which is shown by the construction of the Hopf link.

Part (a) of this theorem in case $n>2$ can be found in [Edwards1968, $\S$ 4, Corollary 5]. Case $n=1$ is clear.

This theorem is a special case of the Theorem 6.4 .

Inequality in part (a) is sharp, see Proposition 3.1. Observe that inequality in part (a) is sharp not only for non-connected manifolds but even for connected manifolds. This differs from the case of closed manifolds, see Theorem 2.3.

These basic results can be generalized to the highly-connected manifolds (see $\S$6).

3 Example on non-isotopic embeddings

The following example is folklore.

This construction is analogous to the Hopf link, see [Skopenkov2016h, $\S$ 2].

4 Seifert linking form

Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere.

The following folklore result holds.

Denote by $\mathrm{lk}$ the linking coefficient [Skopenkov2016h, $\S$ 3, remark 3.2d] of two disjoint cycles.

Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$, but the proof is valid in all dimensions.

The latter Lemma implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0;\mathbb Z)\times H_{n-1}(N_0;\mathbb Z)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N; \mathbb Z)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent.

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 4.5 Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 4.2.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get:

The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. The second equality holds because the linear homotopy of $s'$ and $-s$ degenerates on a $2$-cycle in $N_0$ on which $s'$ and $-s$ are linearly dependent.

5 Classification theorems

Here we state all other results concerning embeddings of manifolds with boundary. One exception are some results when the classification of embeddiongs coinsides with the classification of immersions.

Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy.

This is the main Theorem of [Tonkonog2010]

6 A generalization to highly-connected manifolds

The Diff case of this result is in [Haefliger1961, Existence Theorem (a)], the PL case of this result is in [Irwin1965, Corollary 1.3].

For the Diff case see [Haefliger1961, $\S$ 1.7, remark 2] (there Haefliger proposes to use the deleted product criterion to obtain this result).

See Theorem 2.4 of the survey [Skopenkov2016c, $\S$ 2], or [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

For the PL case of this result see [Hudson1969, Theorem 10.3], which is proved using concordance implies isotopy theorem.

7 References

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• [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
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• [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
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• [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).