Embeddings of manifolds with boundary: classification
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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$1, $\S$3]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. | For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$3]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. | ||
− | If the category is omitted, then we assume the smooth (DIFF) category. Denote by $\mathrm{lk}$ the linking coefficient \cite[$\S$ 77]{Seifert&Threlfall1980} of two disjoint cycles. | + | If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$ the linking coefficient \cite[$\S$ 77]{Seifert&Threlfall1980} of two disjoint cycles. |
We state the simplest results. These results 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. | We state the simplest results. These results 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. | ||
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</wikitex> | </wikitex> | ||
− | == Example | + | == Example of non-isotopic embeddings == |
<wikitex>; \label{sec::example} | <wikitex>; \label{sec::example} | ||
The following example is folklore. | The following example is folklore. | ||
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{{beginthm|Example}} \label{exm::linked_boundary} | {{beginthm|Example}} \label{exm::linked_boundary} | ||
Let $N=S^k\times [0, 1]$ be the cylinder over $S^k$. | Let $N=S^k\times [0, 1]$ be the cylinder over $S^k$. | ||
+ | |||
(a) Then there exist non-isotopic embeddings of $N$ into $\mathbb R^{2k+1}$. | (a) Then there exist non-isotopic embeddings of $N$ into $\mathbb R^{2k+1}$. | ||
(b) Then for each $a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$. | (b) Then for each $a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$. | ||
− | (c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is a bijection for $k | + | (c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$. |
{{endthm}} | {{endthm}} | ||
− | <div>''Proof of part (b).'' Informally speaking by twisting | + | <div>''Proof of part (b).'' Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$ be a map of degree $a$. (To prove part (a) it is sufficient to take as $h$ the identity map of $S^k$ as a map of degree one and the constant map as a map of degree zero.) |
Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$, where $1_k:=(1,0,\ldots,0)\in S^k$. | Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$, where $1_k:=(1,0,\ldots,0)\in S^k$. | ||
Let $f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation_and_conventions|the standard embedding]].Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$. | Let $f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation_and_conventions|the standard embedding]].Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$. | ||
− | ''Proof of part (c).'' By (b) $l$ is surjective. Now take any two embeddings $f_1, f_2$ such that $l([f_1]) = l([f_2])$. Since $k\geqslant 2$ [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that there exists an isotopy of $f_1(S^k\times 0)$ and $f_2(S^k\times 0)$. Since $l([f_1]) = l([f_2])$ it follows that $f_1$ and $f_2$ are isotopic. | + | ''Proof of part (c).'' Clearly $l$ is well-defined. By (b) $l$ is surjective. Now take any two embeddings $f_1, f_2$ such that $l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. |
+ | Since $k\geqslant 2$ [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that there exists an isotopy of $f_1(S^k\times 0)$ and $f_2(S^k\times 0)$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$ and $f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$ and $f_2$ are isotopic. | ||
{{endproof}} | {{endproof}} | ||
[[Image:Punctured_torus_embeddings.svg|thumb|450px|(a): Embeddings $f_1$ (top) and $f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$, $i=1,2$, so the ends of the vector field define the section $s_i$; (c): embedding $s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$.]] | [[Image:Punctured_torus_embeddings.svg|thumb|450px|(a): Embeddings $f_1$ (top) and $f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$, $i=1,2$, so the ends of the vector field define the section $s_i$; (c): embedding $s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$.]] | ||
{{beginthm|Example}} | {{beginthm|Example}} | ||
− | Let $N=S^k\times S^1$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. | + | Let $N=S^k\times S^1$. Assume $k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. |
{{endthm}} | {{endthm}} | ||
The surjectivity of $l$ is given analogously to Proposition \ref{exm::linked_boundary}(b). | The surjectivity of $l$ is given analogously to Proposition \ref{exm::linked_boundary}(b). | ||
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{{beginthm|Example}} | {{beginthm|Example}} | ||
− | (a) Let $N_0$ be the punctured 2-torus containing the meridian $x$ and the parallel $y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$ denote by $s$ the normal field of $\epsilon$- | + | (a) Let $N_0$ be the punctured 2-torus containing the meridian $x$ and the parallel $y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$ denote by $s$ the normal field of $\epsilon$-length vectors to $f(N_0)$ defined by orientation on $N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$. |
(b) Let $f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). | (b) Let $f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). | ||
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Let $N$ be a closed orientable connected $n$-manifold. | 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. | By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. | ||
− | |||
For a free Abelian group $A$, let $B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$ is even for each $x$ (the second condition automatically holds for n odd). | For a free Abelian group $A$, let $B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$ is even for each $x$ (the second condition automatically holds for n odd). | ||
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== A generalization to highly-connected manifolds == | == A generalization to highly-connected manifolds == | ||
<wikitex>; \label{sec::generalisations} | <wikitex>; \label{sec::generalisations} | ||
− | + | For simplicity in this paragraph we consider only punctured manifolds, see $\S$\ref{prt::arbitrary_boundary} for a generalization. | |
− | By $N_0$ denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. | + | Denote by $N$ a closed $n$-manifold. By $N_0$ denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. |
{{beginthm|Theorem}}\label{thm::k_connect_embeds} | {{beginthm|Theorem}}\label{thm::k_connect_embeds} | ||
− | Assume that $N$ is a closed | + | Assume that $N$ is a closed $k$-connected $n$-manifold. |
− | (a) If $n\geq | + | (a) If $n\geq 2k+3$, then $N$ embeds into $\mathbb R^{2n-k}$. |
− | (b) | + | (b) If $n\geq 2k+2$ and $k\geq0$, then $N_0$ embeds into $\mathbb R^{2n-k-1}$. |
{{endthm}} | {{endthm}} | ||
− | + | Part (a) is proved in \cite[Existence Theorem (a)]{Haefliger1961} for the Diff case and in \cite[Theorem 1.1]{Penrose&Whitehead&Zeeman1961}, \cite[Corollary 1.3]{Irwin1965} for PL case. | |
− | + | Part (b) is proved in \cite[Corollary 4.2]{Hirsch1961a} for the Diff case and in \cite[Theorem 1.2]{Penrose&Whitehead&Zeeman1961} for the PL case. | |
{{beginthm|Theorem}}\label{thm::k_connect_unknot} | {{beginthm|Theorem}}\label{thm::k_connect_unknot} | ||
<!--[The Haefliger-Zeeman Unknotting Theorem]--> | <!--[The Haefliger-Zeeman Unknotting Theorem]--> | ||
Assume that $N$ is a closed $k$-connected $n$-manifold. | Assume that $N$ is a closed $k$-connected $n$-manifold. | ||
− | (a) If | + | (a) If $m \ge 2n - k + 1$ and $n\ge2k + 2$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. |
− | (b) If $m \ge 2n - k$, then any two embeddings of $N_0$ into $\mathbb R^m$ are isotopic. | + | (b) If $m \ge 2n - k$ and $n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$ into $\mathbb R^m$ are isotopic. |
{{endthm}} | {{endthm}} | ||
− | + | Part (a) is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$ 2]{Skopenkov2016c}, and is proved in \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. | |
− | + | Part (b) is proved in \cite[Theorem 10.3]{Hudson1969} for the PL case, using [[Isotopy#Concordance|concordance implies isotopy theorem]]. | |
− | For $k>1$ part (b) is corollary of Theorem \ref{thm::k_connect_classif} below | + | For $k>1$ part (b) is a corollary of Theorem \ref{thm::k_connect_classif} below. For $k=0$ part (b) coincides with Theorem \ref{thm::unknotting}b. |
− | <div>'''Proof of Theorem \ref{thm::k_connect_unknot}(b) for $k=1$.''' By Theorem \ref{thm::isotop_unknot} below every two immersions of $ | + | |
+ | <div>'''Proof of Theorem \ref{thm::k_connect_unknot}(b) for $k=1$.''' By Theorem \ref{thm::isotop_unknot} below every two immersions of $N_0$ into $\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$ for each $x\in N_0$. It follows from Theorem \ref{thm::k_connect_is_spine} that $N_0$ collapses to an $(n-2)$-dimensional subcomplex $X\subset N_0$ of some triangulation of $N_0$. By general position we may assume that $F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$. Since $F$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$. It is clear that $f$ is isotopic to $f\circ h$ and $g$ is isotopic to $g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$ is a concordance of $f\circ h$ and $g\circ h$. By [[Isotopy#Concordance|concordance implies isotopy Theorem]] $f$ and $g$ are isotopic.{{endproof}} | ||
+ | |||
+ | {{beginthm|Conjecture}} | ||
+ | Assume that $N$ is a closed $1$-connected $4$-manifold. Then any two embeddings of $N_0$ in $\mathbb R^7$ are isotopic. | ||
+ | {{endthm}} | ||
+ | We may hope to get around the restrictions of Theorem \ref{thm::k_connect_is_spine} using the [[Some_calculations_involving_configuration_spaces_of_distinct_points|deleted product criterion]]. | ||
− | |||
{{beginthm|Theorem}}\label{thm::k_connect_classif} | {{beginthm|Theorem}}\label{thm::k_connect_classif} | ||
Assume $N$ is a closed $k$-connected $n$-manifold. Then for each $k\ge1$ there exists a bijection | Assume $N$ is a closed $k$-connected $n$-manifold. Then for each $k\ge1$ there exists a bijection | ||
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where $\mathbb Z_{(s)}$ denote $\mathbb Z$ for $s$ even and $\mathbb Z_2$ for $s$ odd. | where $\mathbb Z_{(s)}$ denote $\mathbb Z$ for $s$ even and $\mathbb Z_2$ for $s$ odd. | ||
{{endthm}} | {{endthm}} | ||
− | For definition of $W_0'$ and the proof of the latter Theorem see \cite[Lemma 2.2($W_0'$)]{Skopenkov2010}. | + | For definition of $W_0'$ and the proof of the latter Theorem see \cite[Lemma 2.2($W_0'$)]{Skopenkov2010}. The latter theorem was essentially proven in \cite[Theorem 2.1]{Vrabec1989}. |
Latter Theorem is essentially known result. Compare to the Theorem \ref{thm::punctured_class}, which describes $\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case. | Latter Theorem is essentially known result. Compare to the Theorem \ref{thm::punctured_class}, which describes $\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case. | ||
</wikitex> | </wikitex> | ||
− | == Comments == | + | == Comments on non-spherical boundary == |
+ | <wikitex>; \label{prt::arbitrary_boundary} | ||
+ | {{beginthm|Theorem}}\label{thm::arbitarty_k_connect_embeds} | ||
+ | Assume that $N$ is a compact $k$-connected $n$-manifold, $\partial N\neq\emptyset$, $(N, \partial N)$ is $k$-connected and $k< n - 3$. | ||
+ | Then $N$ embeds into $\mathbb R^{2n-k-1}$. | ||
+ | {{endthm}} | ||
+ | This is \cite[Theorem on p.567]{Wall1965}. | ||
+ | {{beginproof}} By Theorem \ref{thm::k_immersible} below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$. Since $N$ is $k$-connected it follows from Theorem \ref{thm::k_connect_is_spine} that $N$ collapses to an $(n-k-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $f|_X$ is an embedding, because $2(n-k) < 2n-k-1$. Since $f$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. It is clear that $f\circ h$ is an embedding.{{endproof}} | ||
+ | {{beginthm|Theorem}}\label{thm::arbitrary_k_connect_unknotting} | ||
+ | Assume that $N$ is a $n$-manifold. If $N$ has $(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$, $m \ge 2n - k$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. | ||
+ | {{endthm}} | ||
+ | Proof is similar to the proof of theorem \ref{thm::k_connect_unknot}. | ||
+ | |||
+ | For a compact connected $n$-manifold with boundary, the property of having an $(n − k − 1)$-dimensional spine is close to $k$-connectedness. Indeed, the following theorem holds. | ||
+ | {{beginthm|Theorem}}\label{thm::k_connect_is_spine} | ||
+ | Every compact connected $n$-manifold $N$ with boundary for which $(N, \partial N)$ is $k$-connected, $\pi_1(\partial N)=0$, | ||
+ | $k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$-dimensional spine. | ||
+ | {{endthm}} | ||
+ | For this result see \cite[Theorem 5.5]{Wall1964a} and \cite[Lemma 5.1 and Remark 5.2]{Horvatic1969}. See also valuable remarks in \cite{Levine&Lidman2018} and \cite[Skopenkov2019]. | ||
+ | </wikitex> | ||
+ | |||
+ | == Comments on immersions == | ||
<wikitex>; | <wikitex>; | ||
{{beginthm|Theorem}}[Smale-Hirsch] | {{beginthm|Theorem}}[Smale-Hirsch] | ||
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{{endthm}} | {{endthm}} | ||
See \cite[Theorem 6.6]{Hirsch1961a}. | See \cite[Theorem 6.6]{Hirsch1961a}. | ||
− | {{beginthm|Theorem}} | + | {{beginthm|Theorem}}\label{thm::k_immersible} |
− | + | Suppose $N$ is a $n$-manifold with non-empty boudary, $(N,\partial N)$ is $k$-connected. Then $N$ is immersible in $\mathbb R^{m}$ for each $m\geq2n-k-1$. | |
{{endthm}} | {{endthm}} | ||
{{beginproof}} | {{beginproof}} | ||
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It suffices to show that exists a linear monomorphism from $TM$ to $\mathbb R^{2n-k-1}$. Lets cunstruct such linear monomorphist on each $r$-skeleton of $N$. It is clear that linear monomorphism exists on $0$-skeleton of $N$. | It suffices to show that exists a linear monomorphism from $TM$ to $\mathbb R^{2n-k-1}$. Lets cunstruct such linear monomorphist on each $r$-skeleton of $N$. It is clear that linear monomorphism exists on $0$-skeleton of $N$. | ||
− | The obstruction to continue the linear monomorphism from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N; \pi_{r-1}(V_{2n-k-1, n}))$, where $V_{2n-k, n}$ is Stiefel manifold of $n$-frames in $\mathbb R^{2n-k}$. | + | The obstruction to continue the linear monomorphism from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1, n}))$, where $V_{2n-k, n}$ is Stiefel manifold of $n$-frames in $\mathbb R^{2n-k}$. |
For $r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$. | For $r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$. | ||
− | For $r=n-k,\ldots, n$ we have $H_{n-r}(N; *) = 0$ since $N$ is $k$-connected and has non-empty boundary. | + | For $r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and has non-empty boundary. |
Thus the obstruction is always zero and such linear monomorphism exists. | Thus the obstruction is always zero and such linear monomorphism exists. | ||
− | ''Other variant''. By theorem \ref{thm::imm_dec_dim} it suffies to show that that there exists an immersion of $N$ into $\mathbb R^{2n}$ with $k$ tranversal linearly independent fields. It is true because $N$ is $k$-connected. | + | ''Other variant''. By theorem \ref{thm::imm_dec_dim} it suffies to show that that there exists an immersion of $N$ into $\mathbb R^{2n}$ with $k$ tranversal linearly independent fields. It is true because $(N,\partial N)$ is $k$-connected. |
{{endproof}} | {{endproof}} | ||
{{beginthm|Theorem}}\label{thm::isotop_unknot} | {{beginthm|Theorem}}\label{thm::isotop_unknot} | ||
− | Suppose $N$ is a | + | Suppose $N$ is a $n$-manifold with non-empty boudary, (N, \partial N) is $k$-connected and $m\geq2n-k$. Then every two immersions of $N$ in $\mathbb R^m$ are regulary homotopic. |
{{endthm}} | {{endthm}} | ||
{{beginproof}} | {{beginproof}} | ||
It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$ to $\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$-skeleton of $N$. It is clear that homotopy exists on $0$-skeleton of $N$. | It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$ to $\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$-skeleton of $N$. It is clear that homotopy exists on $0$-skeleton of $N$. | ||
− | The obstruction to continue the homotopy from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N; \pi_r(V_{2n-k, n}))$, where $V_{2n-k, n}$ is Stiefel manifold of $n$-frames in $\mathbb R^{2n-k}$. | + | The obstruction to continue the homotopy from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$, where $V_{2n-k, n}$ is Stiefel manifold of $n$-frames in $\mathbb R^{2n-k}$. |
For $r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$. | For $r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$. | ||
− | For $r=n-k,\ldots, n$ we have $H_{n-r}(N; *) = 0$ since $N$ is $k$-connected and has non-empty boundary. | + | For $r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and $N$ has non-empty boundary. |
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists. | Thus the obstruction is always zero and such homotopy of linear monomorphisms exists. |
Revision as of 11:16, 12 April 2021
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Contents |
1 Introduction
In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space.
In 4 we introduce an invariant of embedding of a
-manifold in
-space for even
.
In
6 which is independent from
3,
4 and
5 we state generalisations of theorems from
2 to highly-connected manifolds.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1,
3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category. 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. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a compact connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem.
![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)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.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)
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.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
Both parts of this theorem are special cases of the Theorem 6.2.
Case
can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6).
All stated theorems of
2 and
6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example of non-isotopic embeddings
The following example is folklore.
Example 3.1.
Let be the cylinder over
.
(a) Then there exist non-isotopic embeddings of into
.
(b) Then for each there exist an embedding
such that
.
(c) Then defined by the formula
is well-defined and is a bijection for
.
![h\colon S^k\to S^k](/images/math/e/8/4/e843b9ddce449a08c5623d1d6d8dd554.png)
![a](/images/math/5/2/d/52d5b5e885b21331cfd2304be571de0b.png)
![h](/images/math/6/0/1/60169cd1c47b7a7a85ab44f884635e41.png)
![S^k](/images/math/6/a/6/6a6a394959c43308ba7ce796df28123e.png)
Define by the formula
, where
.
Let , where
is the standard embedding.Thus
.
Proof of part (c). Clearly is well-defined. By (b)
is surjective. Now take any two embeddings
such that
. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since
Unknotting Spheres Theorem implies that there exists an isotopy of
and
. Thus we can assume
. Since
it follows that normal fields on
and
are homotopic in class of normal fields. This implies
and
are isotopic.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Example 3.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 3.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 6.4 fails for .
Example 3.3.
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 3.4.
(a) Let be the punctured 2-torus containing the meridian
and the parallel
of the torus. For each embedding
denote by
the normal field of
-length vectors to
defined by orientation on
(see figure (b)). Then there exists a surjection
defined by the formula
.
(b) Let be two embeddings shown on figure (a).
Figure (c) shows that
and
which proves the intuitive fact that
and
are not isotopic.
(Notice that the restrictions of
and
on
are isotopic!)
If we use the opposite normal vector field
, the values of
and
will change but will still be different (see figure (d)).
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
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 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
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.
![\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}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\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], 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].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\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}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
For a free Abelian group
, let
be the group of bilinear forms
such that
and
is even for each
(the second condition automatically holds for n odd).
Definition 5.1.
For each even define an invariant
. For each embedding
construct any PL embedding
by adding a cone over
. Now let
, where
is Whitney invariant, [Skopenkov2016e,
5].
Lemma 5.2.
The invariant is well-defined for
.
Proof.
Note that Unknotting Spheres Theorem implies that unknots in
. Thus
can be extended to embedding of an
-ball
into
. Unknotting Spheres Theorem implies that
-sphere unknots in
. Thus all extensions of
are isotopic in PL category.
Note also that if
and
are isotopic then their extensions are isotopic as well.
And Whitney invariant
is invariant for PL embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 5.3 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 5.4.
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 5.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 5.4:
Theorem 5.6.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 7 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 6.1.
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 6.2.
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 6.4 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 6.3.
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 7.3 using the deleted product criterion.
Theorem 6.4.
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 5.6, which describes
and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
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 7.2.
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 6.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
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 \cite[Skopenkov2019].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch]
The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from
to
.
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2.
If is immersible in
with a transversal
-field then it is immersible in
.
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 8.4.[Whitney]
Every -manifold
is immersible in
.
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
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
. Lets cunstruct such linear monomorphist on each
-skeleton of
. It is clear that linear monomorphism exists on
-skeleton of
.
The obstruction to continue the linear monomorphism from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into
with
tranversal linearly independent fields. It is true because
is
-connected.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.6.
Suppose is a
-manifold with non-empty boudary, (N, \partial N) 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 continue the homotopy from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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)
9 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [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.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![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. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a compact connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem.
![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)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.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)
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.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
Both parts of this theorem are special cases of the Theorem 6.2.
Case
can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6).
All stated theorems of
2 and
6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example of non-isotopic embeddings
The following example is folklore.
Example 3.1.
Let be the cylinder over
.
(a) Then there exist non-isotopic embeddings of into
.
(b) Then for each there exist an embedding
such that
.
(c) Then defined by the formula
is well-defined and is a bijection for
.
![h\colon S^k\to S^k](/images/math/e/8/4/e843b9ddce449a08c5623d1d6d8dd554.png)
![a](/images/math/5/2/d/52d5b5e885b21331cfd2304be571de0b.png)
![h](/images/math/6/0/1/60169cd1c47b7a7a85ab44f884635e41.png)
![S^k](/images/math/6/a/6/6a6a394959c43308ba7ce796df28123e.png)
Define by the formula
, where
.
Let , where
is the standard embedding.Thus
.
Proof of part (c). Clearly is well-defined. By (b)
is surjective. Now take any two embeddings
such that
. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since
Unknotting Spheres Theorem implies that there exists an isotopy of
and
. Thus we can assume
. Since
it follows that normal fields on
and
are homotopic in class of normal fields. This implies
and
are isotopic.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Example 3.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 3.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 6.4 fails for .
Example 3.3.
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 3.4.
(a) Let be the punctured 2-torus containing the meridian
and the parallel
of the torus. For each embedding
denote by
the normal field of
-length vectors to
defined by orientation on
(see figure (b)). Then there exists a surjection
defined by the formula
.
(b) Let be two embeddings shown on figure (a).
Figure (c) shows that
and
which proves the intuitive fact that
and
are not isotopic.
(Notice that the restrictions of
and
on
are isotopic!)
If we use the opposite normal vector field
, the values of
and
will change but will still be different (see figure (d)).
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
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 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
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.
![\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}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\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], 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].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\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}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
For a free Abelian group
, let
be the group of bilinear forms
such that
and
is even for each
(the second condition automatically holds for n odd).
Definition 5.1.
For each even define an invariant
. For each embedding
construct any PL embedding
by adding a cone over
. Now let
, where
is Whitney invariant, [Skopenkov2016e,
5].
Lemma 5.2.
The invariant is well-defined for
.
Proof.
Note that Unknotting Spheres Theorem implies that unknots in
. Thus
can be extended to embedding of an
-ball
into
. Unknotting Spheres Theorem implies that
-sphere unknots in
. Thus all extensions of
are isotopic in PL category.
Note also that if
and
are isotopic then their extensions are isotopic as well.
And Whitney invariant
is invariant for PL embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 5.3 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 5.4.
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 5.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 5.4:
Theorem 5.6.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 7 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 6.1.
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 6.2.
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 6.4 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 6.3.
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 7.3 using the deleted product criterion.
Theorem 6.4.
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 5.6, which describes
and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
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 7.2.
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 6.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
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 \cite[Skopenkov2019].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch]
The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from
to
.
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2.
If is immersible in
with a transversal
-field then it is immersible in
.
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 8.4.[Whitney]
Every -manifold
is immersible in
.
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
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
. Lets cunstruct such linear monomorphist on each
-skeleton of
. It is clear that linear monomorphism exists on
-skeleton of
.
The obstruction to continue the linear monomorphism from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into
with
tranversal linearly independent fields. It is true because
is
-connected.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.6.
Suppose is a
-manifold with non-empty boudary, (N, \partial N) 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 continue the homotopy from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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)
9 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [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.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![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. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a compact connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem.
![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)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.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)
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.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
Both parts of this theorem are special cases of the Theorem 6.2.
Case
can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6).
All stated theorems of
2 and
6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example of non-isotopic embeddings
The following example is folklore.
Example 3.1.
Let be the cylinder over
.
(a) Then there exist non-isotopic embeddings of into
.
(b) Then for each there exist an embedding
such that
.
(c) Then defined by the formula
is well-defined and is a bijection for
.
![h\colon S^k\to S^k](/images/math/e/8/4/e843b9ddce449a08c5623d1d6d8dd554.png)
![a](/images/math/5/2/d/52d5b5e885b21331cfd2304be571de0b.png)
![h](/images/math/6/0/1/60169cd1c47b7a7a85ab44f884635e41.png)
![S^k](/images/math/6/a/6/6a6a394959c43308ba7ce796df28123e.png)
Define by the formula
, where
.
Let , where
is the standard embedding.Thus
.
Proof of part (c). Clearly is well-defined. By (b)
is surjective. Now take any two embeddings
such that
. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since
Unknotting Spheres Theorem implies that there exists an isotopy of
and
. Thus we can assume
. Since
it follows that normal fields on
and
are homotopic in class of normal fields. This implies
and
are isotopic.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Example 3.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 3.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 6.4 fails for .
Example 3.3.
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 3.4.
(a) Let be the punctured 2-torus containing the meridian
and the parallel
of the torus. For each embedding
denote by
the normal field of
-length vectors to
defined by orientation on
(see figure (b)). Then there exists a surjection
defined by the formula
.
(b) Let be two embeddings shown on figure (a).
Figure (c) shows that
and
which proves the intuitive fact that
and
are not isotopic.
(Notice that the restrictions of
and
on
are isotopic!)
If we use the opposite normal vector field
, the values of
and
will change but will still be different (see figure (d)).
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
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 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
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.
![\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}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\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], 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].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\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}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
For a free Abelian group
, let
be the group of bilinear forms
such that
and
is even for each
(the second condition automatically holds for n odd).
Definition 5.1.
For each even define an invariant
. For each embedding
construct any PL embedding
by adding a cone over
. Now let
, where
is Whitney invariant, [Skopenkov2016e,
5].
Lemma 5.2.
The invariant is well-defined for
.
Proof.
Note that Unknotting Spheres Theorem implies that unknots in
. Thus
can be extended to embedding of an
-ball
into
. Unknotting Spheres Theorem implies that
-sphere unknots in
. Thus all extensions of
are isotopic in PL category.
Note also that if
and
are isotopic then their extensions are isotopic as well.
And Whitney invariant
is invariant for PL embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 5.3 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 5.4.
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 5.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 5.4:
Theorem 5.6.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 7 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 6.1.
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 6.2.
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 6.4 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 6.3.
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 7.3 using the deleted product criterion.
Theorem 6.4.
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 5.6, which describes
and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
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 7.2.
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 6.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
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 \cite[Skopenkov2019].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch]
The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from
to
.
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2.
If is immersible in
with a transversal
-field then it is immersible in
.
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 8.4.[Whitney]
Every -manifold
is immersible in
.
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
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
. Lets cunstruct such linear monomorphist on each
-skeleton of
. It is clear that linear monomorphism exists on
-skeleton of
.
The obstruction to continue the linear monomorphism from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into
with
tranversal linearly independent fields. It is true because
is
-connected.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.6.
Suppose is a
-manifold with non-empty boudary, (N, \partial N) 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 continue the homotopy from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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)
9 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [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.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![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. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a compact connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem.
![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)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.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)
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.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
Both parts of this theorem are special cases of the Theorem 6.2.
Case
can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6).
All stated theorems of
2 and
6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example of non-isotopic embeddings
The following example is folklore.
Example 3.1.
Let be the cylinder over
.
(a) Then there exist non-isotopic embeddings of into
.
(b) Then for each there exist an embedding
such that
.
(c) Then defined by the formula
is well-defined and is a bijection for
.
![h\colon S^k\to S^k](/images/math/e/8/4/e843b9ddce449a08c5623d1d6d8dd554.png)
![a](/images/math/5/2/d/52d5b5e885b21331cfd2304be571de0b.png)
![h](/images/math/6/0/1/60169cd1c47b7a7a85ab44f884635e41.png)
![S^k](/images/math/6/a/6/6a6a394959c43308ba7ce796df28123e.png)
Define by the formula
, where
.
Let , where
is the standard embedding.Thus
.
Proof of part (c). Clearly is well-defined. By (b)
is surjective. Now take any two embeddings
such that
. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since
Unknotting Spheres Theorem implies that there exists an isotopy of
and
. Thus we can assume
. Since
it follows that normal fields on
and
are homotopic in class of normal fields. This implies
and
are isotopic.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Example 3.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 3.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 6.4 fails for .
Example 3.3.
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 3.4.
(a) Let be the punctured 2-torus containing the meridian
and the parallel
of the torus. For each embedding
denote by
the normal field of
-length vectors to
defined by orientation on
(see figure (b)). Then there exists a surjection
defined by the formula
.
(b) Let be two embeddings shown on figure (a).
Figure (c) shows that
and
which proves the intuitive fact that
and
are not isotopic.
(Notice that the restrictions of
and
on
are isotopic!)
If we use the opposite normal vector field
, the values of
and
will change but will still be different (see figure (d)).
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
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 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
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.
![\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}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\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], 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].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\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}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
For a free Abelian group
, let
be the group of bilinear forms
such that
and
is even for each
(the second condition automatically holds for n odd).
Definition 5.1.
For each even define an invariant
. For each embedding
construct any PL embedding
by adding a cone over
. Now let
, where
is Whitney invariant, [Skopenkov2016e,
5].
Lemma 5.2.
The invariant is well-defined for
.
Proof.
Note that Unknotting Spheres Theorem implies that unknots in
. Thus
can be extended to embedding of an
-ball
into
. Unknotting Spheres Theorem implies that
-sphere unknots in
. Thus all extensions of
are isotopic in PL category.
Note also that if
and
are isotopic then their extensions are isotopic as well.
And Whitney invariant
is invariant for PL embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 5.3 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 5.4.
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 5.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 5.4:
Theorem 5.6.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 7 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 6.1.
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 6.2.
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 6.4 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 6.3.
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 7.3 using the deleted product criterion.
Theorem 6.4.
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 5.6, which describes
and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
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 7.2.
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 6.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
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 \cite[Skopenkov2019].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch]
The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from
to
.
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2.
If is immersible in
with a transversal
-field then it is immersible in
.
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 8.4.[Whitney]
Every -manifold
is immersible in
.
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
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
. Lets cunstruct such linear monomorphist on each
-skeleton of
. It is clear that linear monomorphism exists on
-skeleton of
.
The obstruction to continue the linear monomorphism from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into
with
tranversal linearly independent fields. It is true because
is
-connected.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.6.
Suppose is a
-manifold with non-empty boudary, (N, \partial N) 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 continue the homotopy from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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)
9 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [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.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![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. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a compact connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem.
![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)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.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)
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.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
Both parts of this theorem are special cases of the Theorem 6.2.
Case
can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6).
All stated theorems of
2 and
6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example of non-isotopic embeddings
The following example is folklore.
Example 3.1.
Let be the cylinder over
.
(a) Then there exist non-isotopic embeddings of into
.
(b) Then for each there exist an embedding
such that
.
(c) Then defined by the formula
is well-defined and is a bijection for
.
![h\colon S^k\to S^k](/images/math/e/8/4/e843b9ddce449a08c5623d1d6d8dd554.png)
![a](/images/math/5/2/d/52d5b5e885b21331cfd2304be571de0b.png)
![h](/images/math/6/0/1/60169cd1c47b7a7a85ab44f884635e41.png)
![S^k](/images/math/6/a/6/6a6a394959c43308ba7ce796df28123e.png)
Define by the formula
, where
.
Let , where
is the standard embedding.Thus
.
Proof of part (c). Clearly is well-defined. By (b)
is surjective. Now take any two embeddings
such that
. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since
Unknotting Spheres Theorem implies that there exists an isotopy of
and
. Thus we can assume
. Since
it follows that normal fields on
and
are homotopic in class of normal fields. This implies
and
are isotopic.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Example 3.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 3.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 6.4 fails for .
Example 3.3.
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 3.4.
(a) Let be the punctured 2-torus containing the meridian
and the parallel
of the torus. For each embedding
denote by
the normal field of
-length vectors to
defined by orientation on
(see figure (b)). Then there exists a surjection
defined by the formula
.
(b) Let be two embeddings shown on figure (a).
Figure (c) shows that
and
which proves the intuitive fact that
and
are not isotopic.
(Notice that the restrictions of
and
on
are isotopic!)
If we use the opposite normal vector field
, the values of
and
will change but will still be different (see figure (d)).
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
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 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
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.
![\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}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\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], 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].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\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}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
For a free Abelian group
, let
be the group of bilinear forms
such that
and
is even for each
(the second condition automatically holds for n odd).
Definition 5.1.
For each even define an invariant
. For each embedding
construct any PL embedding
by adding a cone over
. Now let
, where
is Whitney invariant, [Skopenkov2016e,
5].
Lemma 5.2.
The invariant is well-defined for
.
Proof.
Note that Unknotting Spheres Theorem implies that unknots in
. Thus
can be extended to embedding of an
-ball
into
. Unknotting Spheres Theorem implies that
-sphere unknots in
. Thus all extensions of
are isotopic in PL category.
Note also that if
and
are isotopic then their extensions are isotopic as well.
And Whitney invariant
is invariant for PL embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 5.3 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 5.4.
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 5.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 5.4:
Theorem 5.6.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 7 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 6.1.
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 6.2.
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 6.4 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 6.3.
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 7.3 using the deleted product criterion.
Theorem 6.4.
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 5.6, which describes
and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
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 7.2.
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 6.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
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 \cite[Skopenkov2019].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch]
The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from
to
.
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2.
If is immersible in
with a transversal
-field then it is immersible in
.
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 8.4.[Whitney]
Every -manifold
is immersible in
.
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
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
. Lets cunstruct such linear monomorphist on each
-skeleton of
. It is clear that linear monomorphism exists on
-skeleton of
.
The obstruction to continue the linear monomorphism from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into
with
tranversal linearly independent fields. It is true because
is
-connected.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.6.
Suppose is a
-manifold with non-empty boudary, (N, \partial N) 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 continue the homotopy from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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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- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into
, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
![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. Usually there exist easier direct proofs than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1.
Assume that is a compact connected
-manifold.
(a) Then embeds into
.
(b) If has non-empty boundary, then
embeds into
.
Part (a) is well-known strong Whitney embedding theorem.
![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)
![g|_{X}](/images/math/a/4/c/a4c7f84562c6ba9441e5cd9fa1ceb082.png)
![2(n-1) < 2n-1](/images/math/c/b/7/cb7e8721dfb70ee49e227a27da0e03b4.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)
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.
The part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2].
Inequality in part (b) is sharp, see Proposition 3.1.
Part (b) in case can be found in [Edwards1968,
4, Corollary 5]. Case
is clear.
Both parts of this theorem are special cases of the Theorem 6.2.
Case
can be proved using the following ideas.
These basic results can be generalized to the highly-connected manifolds (see 6).
All stated theorems of
2 and
6 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
3 Example of non-isotopic embeddings
The following example is folklore.
Example 3.1.
Let be the cylinder over
.
(a) Then there exist non-isotopic embeddings of into
.
(b) Then for each there exist an embedding
such that
.
(c) Then defined by the formula
is well-defined and is a bijection for
.
![h\colon S^k\to S^k](/images/math/e/8/4/e843b9ddce449a08c5623d1d6d8dd554.png)
![a](/images/math/5/2/d/52d5b5e885b21331cfd2304be571de0b.png)
![h](/images/math/6/0/1/60169cd1c47b7a7a85ab44f884635e41.png)
![S^k](/images/math/6/a/6/6a6a394959c43308ba7ce796df28123e.png)
Define by the formula
, where
.
Let , where
is the standard embedding.Thus
.
Proof of part (c). Clearly is well-defined. By (b)
is surjective. Now take any two embeddings
such that
. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since
Unknotting Spheres Theorem implies that there exists an isotopy of
and
. Thus we can assume
. Since
it follows that normal fields on
and
are homotopic in class of normal fields. This implies
and
are isotopic.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Example 3.2.
Let . Assume
. Then there exists a bijection
defined by the formula
.
The surjectivity of is given analogously to Proposition 3.1(b).
The injectivity of
follows from forgetful bijection
between embeddings of
and a cylinder.
This example shows that Theorem 6.4 fails for .
Example 3.3.
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 3.4.
(a) Let be the punctured 2-torus containing the meridian
and the parallel
of the torus. For each embedding
denote by
the normal field of
-length vectors to
defined by orientation on
(see figure (b)). Then there exists a surjection
defined by the formula
.
(b) Let be two embeddings shown on figure (a).
Figure (c) shows that
and
which proves the intuitive fact that
and
are not isotopic.
(Notice that the restrictions of
and
on
are isotopic!)
If we use the opposite normal vector field
, the values of
and
will change but will still be different (see figure (d)).
4 Seifert linking form
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
If the (co)homology coefficients are omitted, then we assume them to be
.
The following folklore result holds.
Lemma 4.1.
Assume is a closed orientable connected
-manifold,
is even and
is torsion free. Then for each embedding
there exists a nowhere vanishing normal vector field to
.
Proof.
There is an obstruction (Euler class) to existence of a nowhere vanishing normal vector field to
.
A normal space to at any point of
has dimension
. As
is even thus
is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore
. Since
is torsion free, it follows that
.
Since has non-empty boundary, we have that
is homotopy equivalent to an
-complex. The dimension of this complex equals the dimension of normal space to
at any point of
. Since
, it follows that there exists a nowhere vanishing normal vector field to
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Denote by two disjoint
-cycles in
with integer coefficients.
Definition 4.2.
For even and every embedding
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 4.3 ( is well-defined).
For even
and every embedding
the integer
:
- is well-defined, i.e. does not change when
is replaced by
,
- does not change when
or
are changed to homologous cycles and,
- does not change when
is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.
We will need the following supporting lemma.
Lemma 4.4.
Let be an embedding.
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.
![\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}](/images/math/9/3/b/93bc3818bfeb0ad8aed9663263194f05.png)
Here the second equality follows from Lemma 4.4.
For each two homologous -cycles
in
, the image of the homology between
and
is a
-chain
of
such that
. Since
is a nowhere vanishing normal field to
, this implies that the supports of
and
are disjoint. Hence
.
Since isotopy of is a map from
to
, it follows that this isotopy gives an isotopy of the link
. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Lemma 4.3 implies that generates a bilinear form
denoted by the same letter.
Denote by the reduction modulo
.
Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to
are linearly dependent.
Lemma 4.5.
Let be an embedding.
Then for every
the following equality holds:
![\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], 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].
![-s](/images/math/7/0/b/70beda81154258e434c9d15d5590629c.png)
![f(N_0)](/images/math/0/b/a/0ba24a0c59c1e81e9588e5aa355efb41.png)
![s](/images/math/3/c/b/3cb9cdaed257453cfa56b9ef81b44c57.png)
![\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}](/images/math/4/d/c/4dce1ae4a5e1f1d2f1ac70a43e9d2013.png)
The first congruence is clear.
The second equality holds because if we shift the link by
, we get the link
and the linking coefficient will not change after this shift.
The third equality follows from Lemma 4.4.
Thus it is sufficient to show that .
Denote by
a general perturbation of
. We get:
![\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).](/images/math/9/b/b/9bb92431c6f8e1a43fee03db379bc85d.png)
The first equality holds because and
are homotopic in the class of nowhere vanishing normal vector fields.
Let us prove the second equality. The linear homotopy between
and
degenerates only at those points
where
. These points
are exactly points where
and
are linearly dependent. All those point
form a
-cycle modulo two in
. The homotopy class of this
-cycle is
by the definition of Stiefel-Whitney class.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
5 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected
-manifold.
By
we denote the complement in
to an open
-ball. Thus
is the
-sphere.
For a free Abelian group
, let
be the group of bilinear forms
such that
and
is even for each
(the second condition automatically holds for n odd).
Definition 5.1.
For each even define an invariant
. For each embedding
construct any PL embedding
by adding a cone over
. Now let
, where
is Whitney invariant, [Skopenkov2016e,
5].
Lemma 5.2.
The invariant is well-defined for
.
Proof.
Note that Unknotting Spheres Theorem implies that unknots in
. Thus
can be extended to embedding of an
-ball
into
. Unknotting Spheres Theorem implies that
-sphere unknots in
. Thus all extensions of
are isotopic in PL category.
Note also that if
and
are isotopic then their extensions are isotopic as well.
And Whitney invariant
is invariant for PL embeddings.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Definition 5.3 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 5.4.
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 5.5.
For each even and each
the following equality holds:
.
An equivalemt statement of Theorem 5.4:
Theorem 5.6.
Let be a closed connected orientable
-manifold with
torsion-free,
,
even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms
such that
. Here
is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
6 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 7 for a generalization.
Denote by a closed
-manifold. By
denote the complement in
to an open
-ball. Thus
is the
-sphere.
Theorem 6.1.
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 6.2.
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 6.4 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 6.3.
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 7.3 using the deleted product criterion.
Theorem 6.4.
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 5.6, which describes
and differs from the general case.
7 Comments on non-spherical boundary
Theorem 7.1.
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 7.2.
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 6.2.
For a compact connected -manifold with boundary, the property of having an
-dimensional spine is close to
-connectedness. Indeed, the following theorem holds.
Theorem 7.3.
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 \cite[Skopenkov2019].
8 Comments on immersions
Theorem 8.1.[Smale-Hirsch]
The space of immersions of a manifold in is homotopically equivalent to the space of linear monomorphisms from
to
.
See [Hirsch1959] and [Haefliger&Poenaru1964].
Theorem 8.2.
If is immersible in
with a transversal
-field then it is immersible in
.
This is [Hirsch1959, Theorem 6.4].
Theorem 8.3.
Every -manifold
with non-empty boundary is immersible in
.
Theorem 8.4.[Whitney]
Every -manifold
is immersible in
.
See [Hirsch1961a, Theorem 6.6].
Theorem 8.5.
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
. Lets cunstruct such linear monomorphist on each
-skeleton of
. It is clear that linear monomorphism exists on
-skeleton of
.
The obstruction to continue the linear monomorphism from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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.
Other variant. By theorem 8.2 it suffies to show that that there exists an immersion of into
with
tranversal linearly independent fields. It is true because
is
-connected.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Theorem 8.6.
Suppose is a
-manifold with non-empty boudary, (N, \partial N) 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 continue the homotopy from -skeleton to
-skeleton lies in
, where
is Stiefel manifold of
-frames 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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