Embeddings of manifolds with boundary: classification

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{{Authors|Mikhail Fedorov}}
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{{Authors|Mikhail Fedorov|Garaev Timur}}
{{Stub}}
{{Stub}}
== Introduction ==
== Introduction ==
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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.
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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.
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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.
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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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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. For some results we present direct proofs, which are easier than deduction from this criterion.
We do not claim the references we give are references to original proofs.
We do not claim the references we give are references to original proofs.
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== Embedding and unknotting theorems ==
== Embedding and unknotting theorems ==
<wikitex>; \label{sec::general_theorems}
<wikitex>; \label{sec::general_theorems}
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{{beginthm|Theorem}}
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{{beginthm|Theorem}}\label{thm::embed}
Assume that $N$ is a compact connected $n$-manifold.
Assume that $N$ is a compact connected $n$-manifold.
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(b) If $N$ has non-empty boundary, then $N$ embeds into $\mathbb R^{2n-1}$.
(b) If $N$ has non-empty boundary, then $N$ embeds into $\mathbb R^{2n-1}$.
{{endthm}}
{{endthm}}
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Part (a) is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]].
Part (a) is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]].
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The first proof of (b) presented below is essentially contained in \cite[Theorem 4.6]{Hirsch1961a} for the Diff case, and in references for Theorem \ref{thm::k_connect_embeds} below or in \cite[Theorem 5.2]{Horvatic1971} for the PL case.
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<div>'''Proof of part (b).''' By strong [[Wikipedia:Whitney_immersion_theorem|strong Whitney immersion theorem]] there exist an immersion $g\colon N\to\mathbb R^{2n-1}$. Since $N$ is connected and has non-empty boundary, it follows that $N$ collapses to an $(n-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $g|_{X}$ is an embedding, because $2(n-1) < 2n-1$. Since $g$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. The composition $g\circ h$ is an embedding of $N$.{{endproof}}This proof is essentially contained in \cite[Theorem 4.6]{Hirsch1961a} for the Diff case and in references for Theorem \ref{thm::k_connect_embeds} below or in \cite[Theorem 5.2]{Horvatic1971} for the PL case.
{{beginthm|Theorem}}\label{thm::unknotting}
{{beginthm|Theorem}}\label{thm::unknotting}
Assume that $N$ is a compact connected $n$-manifold and either
Assume that $N$ is a compact connected $n$-manifold and either
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Then any two embeddings of $N$ into $\mathbb R^m$ are isotopic.
Then any two embeddings of $N$ into $\mathbb R^m$ are isotopic.
{{endthm}}
{{endthm}}
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The part (a) is Whitney-Wu Unknotting Theorem, see \cite[$\S$ 2, Theorems 2.1, 2.2]{Skopenkov2016c}.
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Inequality in part (b) is sharp, see Proposition \ref{exm::linked_boundary}.
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Part (a) is Whitney-Wu Unknotting Theorem, see \cite[$\S$ 2, Theorems 2.1, 2.2]{Skopenkov2016c}.
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Part (b) in the case $n>2$ is proved in \cite[$\S$ 4, Corollary 5]{Edwards1968}. The case $n=1$ is clear. The case $n=2$ can be proved using the ideas presented below.
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Part (b) in case $n>2$ can be found in \cite[$\S$ 4, Corollary 5]{Edwards1968}. Case $n=1$ is clear.
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The inequality in part (b) is sharp by Proposition \ref{exm::linked_boundary}.
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Both parts of this theorem are special cases of the Theorem \ref{thm::k_connect_unknot}.
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Case $n=2$ can be proved using the following ideas.
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These basic results can be generalized to highly-connected manifolds (see $\S$\ref{sec::generalisations}).
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In particular, both parts of Theorem \ref{thm::embed} are special cases of Theorem \ref{thm::k_connect_unknot}.
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All theorems for manifolds with non-empty boundary stated in $\S$\ref{sec::general_theorems} and $\S$\ref{sec::generalisations} can be proved using
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* analogous results for immersions of manifolds stated in $\S$9, and general position ideas, see e.g. the first proof of Theorem \ref{thm::embed}.b in $\S$3.
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* handle decomposition, see e.g. the second proof of Theorem \ref{thm::embed}.b in $\S$3.
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Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
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</wikitex>
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== Proofs of Theorem \ref{thm::embed}.b and Theorem \ref{thm::unknotting}.b ==
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<wikitex>; \label{sec::pt}
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In this section we work only in smooth category. The first proof of Theorem \ref{thm::embed}.b uses immersions, while the second does not.
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<div>'''First proof of Theorem \ref{thm::embed}.b.''' By the [[Wikipedia:Whitney_immersion_theorem|strong Whitney immersion theorem]] there exist an immersion $g\colon N\to\mathbb R^{2n-1}$. Since $N$ is connected and has non-empty boundary, it follows that $N$ collapses to an $(n-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. Since $2(n-1) < 2n-1$, by general position we may assume that $g|_{X}$ is an embedding. Since $g$ is an immersion, it follows that $X$ has a sufficiently small tubular neighbourhood $M\supset X$ such that $g|_{M}$ is embedding. Since tubular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. The composition $g\circ h$ is an embedding of $N$.{{endproof}}
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For the second proof we need some lemmas.
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{{beginthm|Lemma}}\label{p:handle-decompose} \cite{Wall1966}
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Assume that $N$ is a closed connected $n$-manifold. Then $N_0$ have handle decomposition with indices of attaching maps at most $n-1$.
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{{endthm}}
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{{beginthm|Lemma}}\label{p:handle-add}
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Assume that $U$ is a compact $n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$, and $f:U \to \mathbb R^{m}$ is an embedding.
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$(a)$ If $m=2n-1$, then there is an extension of $f$ to an embedding of $U\cup_{\phi} D^i\times D^{n-i}$.
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$(b)$ Assume also that there is an embedding $g : D^i\times 0 \to \mathbb R^m$ such that $f\phi = g$ on $\partial D^i \times 0$. Suppose that on $g(D^i\times 0)$ there is a field of $n-i$ linear independed normal vectors whose restriction to $f(\partial D^i\times 0)$ is tangent to $f(U)$. Then $f\cup g$ extends to a embedding $U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m$.
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{{endthm}}
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<div>'''Second proof of Theorem \ref{thm::embed}.b assuming Lemma \ref{p:handle-decompose} and Lemma \ref{p:handle-add}.(a).''' By Lemma \ref{p:handle-decompose}.(a) there is a handle decomposition of $N_0$ with attaching maps $\phi_1,\ldots,\phi_s$ of indices at most $n-1$. Denote by $U^l$ the manifold obtained from $\emptyset$ by the attaching first $l$ handles. Take any embedding $F_1:U^1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$ of $U^l$ using an embedding $F_{l-1}$ of $U^{l-1}$. <!-- Define an embedding $F_l: U^l \rightarrow \R^{2n-1}$ recursively. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^l$ the $(l+1)$-th attaching map. Since $i\leq n$, --> Since the index $i$ of $\phi_l$ is smaller than $n$, by Lemma \ref{p:handle-add} there is extension of $F_{l-1}$ to an embedding $F_l:U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$, where $U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}$.
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{{endproof}}
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<div>'''Proof of Lemma \ref{p:handle-add}.(a) assuming Lemma \ref{p:handle-add}.(b).''' Since $i+n\leq 2n-1$ and $2i+1\leq 2n-1$, by general position there is an embedding $g: D^i\times 0\to \mathbb R^{2n-1}$ such that $f\phi = g$ on $\partial D^i \times 0$ and $f(\mbox{Int} U)$ has a finite number of intersections points with $g(\mbox{Int} D^i\times 0)$. Then by an isotopy $g_t$, where $g_0=g$, fixed on $\partial D^i\times 0$ we can "push out" the self-intersection points toward $\partial U$ so that $g_1(\mbox{Int} D^i\times 0)$ does not intersect $f(\mbox{Int} U)$. Then $f\cup g_1$ is an embedding.
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Denote by $A$ the $(n-i)\times n$ matrix whose rightmost $(n-i)\times (n-i)$ block is the identity matrix, and whose other elements are zeros. Denote by $v$ the field of $n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$ such that the $k$-th vector has coordinates equal to the $k$-th row in $A$. Then the vector field $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is tangent to $\partial U$. For $x\in g_1(\partial D^i\times 0)$ denote by $v'(x)$ the projection of $df(d\phi(v(x)))$ to the intersection of normal space to $g_1(\partial D^i\times 0)$ at $x$, and tangent space to $f(\partial U)$ at $x$. Since $i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$. Hence there is an extension of $v'$ to an linearly independent field of vectors normal to $g_1(D^i\times 0)$. Then by Lemma \ref{p:handle-add}.(b) there is an extension of $f\cup g_1$ to an embedding of $U \cup_{\phi} D^i\times D^{n-i}$.
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{{endproof}}
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In the proof of Theorem \ref{thm::unknotting} we will use Lemma \ref{p:handle-decompose} and Lemma \ref{p:handle-add2}.(a).
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{{beginthm|Lemma}}\label{p:handle-add2}
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Assume that $U$ is a compact $n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$ and $f_1|_U$.
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$(a)$ If $m\geq 2n$, then there is an extension of $F$ to a concordance between $f_0$ and $f_1$.
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$(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$ there is a field of $n-i$ linear independent normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$ are tangent to $$F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$ respectively. Then $F\cup G$ extends to a concordance between $f_0$ and $f_1$.
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{{endthm}}
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<div>'''Proof of the Theorem \ref{thm::unknotting} assuming Lemma \ref{p:handle-decompose} and Lemma \ref{p:handle-add2}.(a).''' Denote by $f_0, f_1$ any two embeddings of $N_0$ into $\mathbb{R}^m$.
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By the [http://www.map.mpim-bonn.mpg.de/Isotopy|a Concordance Implies Isotopy Theorem] it suffices to prove that there is a concordance between $f_0$ and $f_1$. In order to prove that observe that by Lemma \ref{p:handle-decompose} there is a handle decomposition of $N_0$ with attaching maps of indices at most $n-1$. Denote by $U^l$ the manifold obtained from $U^0=\emptyset$ by the attaching the first $l$ handles, starting with? $U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$ and $f_1|_{U^1}$. Let us define a concordance $F_l$ between $f_0|_{U^l}$ and $f_1|_{U^l}$ using a concordance $F_{l-1}$ of $U^{l-1}$ between $f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$. For this denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$-th attaching map. Since $i\leq n-1$, by Lemma \ref{p:handle-add2}.(a) there is an extension of $F_{l-1}$ to a concordance $$F_{l}:U^{l}\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$ between the restriction of $f_0$ and $f_1$ to $U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}$.
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{{endproof}}
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<div>'''Proof of Lemma \ref{p:handle-add2}.(a) assuming Lemma \ref{p:handle-add2}.(b).''' In the following text we identify $D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$. Define map $$\bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$ by the formula: $$\bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right. $$
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Since $$\dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\quad\text{and}\quad 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$ by general position there is an embedding $$G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$ such that $F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$, where $G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$. Then $F\cup G_1$ is an concordance between the restrictions of $f_0$ and $f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$.
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Denote by $A$ the $(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$ the field of $n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$-th vector has coordinates equal to the $k$-th row in $A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G_1(\partial (D^i\times 0\times [0, 1])$ denote by $v'(x)$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G_1(D^i\times 0\times [0, 1])$ at $x$, and tangent space to $F(\partial(U\times [0, 1]))$ at $x$. Since $i<m+1-(n-i)$, it follows that $\pi_{i}(V_{m+1, n-i})=0$. Hence there is an extension of $v'$ to a linear independent field of vectors normal to $G_1(D^i\times 0\times [0, 1])$. Then by Lemma \ref{p:handle-add2}.(b) there is an extension of $F\cup G_1$ to a concordance $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$.
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{{endproof}}
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These basic results can be generalized to the highly-connected manifolds (see $\S$\ref{sec::generalisations}).
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All stated theorems of $\S$\ref{sec::general_theorems} and $\S$\ref{sec::generalisations} for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.
</wikitex>
</wikitex>
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== Example on non-isotopic embeddings ==
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== 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|Exaple}} \label{exm::linked_boundary}
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{{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 $S^k$.
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(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 $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.
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(c) Then $l\colon N\to\mathbb R$
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(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}}
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{{beginproof}}
{{beginproof}}
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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.)
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''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.)
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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 $1_k:=(1,0,\ldots,0)\in S^k$.
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Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$.
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 $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$.
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{{endproof}}
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''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.
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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}}
[[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 $s_iy\sqcup f_ix$.]]
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Denote $1_k:=(1,0,\ldots,0)\in S^k$.
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{{beginthm|Example}}
{{beginthm|Example}}
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Let $N=S^k\times S^1$. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.
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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}}
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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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By twisting the $x$-ribbon one can obtain arbitrary value of linking coefficient.
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The injectivity of $l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$ and a cylinder.
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This example shows that Theorem \ref{thm::k_connect_classif} fails for $k=0$.
This example shows that Theorem \ref{thm::k_connect_classif} fails for $k=0$.
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{{beginthm|Example}}
{{beginthm|Example}}
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Let $N_0$ be the punctured 2-torus containing the meridian and the parallel of the torus. Denote by $x,y$ the meridian and the parallel of the torus. For each embedding $f\in\mathrm{Emb}^{3}N_0$ denote by $s$ a normal field to $f(N_0)$. Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$.
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(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 $l([f])=\mathrm{lk}(f(x), s(y))$.
−
{{endthm}}
+
−
{{beginthm|Example}}
+
(b) Let $f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a).
−
Let $f_1,f_2:N_0\to\R^3$ be two embeddings shown on figure (a).
+
Figure (c) shows that $l(f_1)=1$ and $l(f_2)=0$ which proves the intuitive fact that $f_1$ and $f_2$ are not isotopic.
Figure (c) shows that $l(f_1)=1$ and $l(f_2)=0$ which proves the intuitive fact that $f_1$ and $f_2$ are not isotopic.
(Notice that the restrictions of $f_1$ and $f_2$ on $x\cup y$ are isotopic!)
(Notice that the restrictions of $f_1$ and $f_2$ on $x\cup y$ are isotopic!)
−
If we use opposite normal vector field $s'=-s$, the values of $l(f_1)$ and $l(f_2)$ will change but will still be different (see figure (d)).
+
If we use the opposite normal vector field $s'=-s$, the values of $l(f_1)$ and $l(f_2)$ will change but will still be different (see figure (d)).
{{endthm}}
{{endthm}}
</wikitex>
</wikitex>
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== Seifert linking form ==
== Seifert linking form ==
<wikitex>; \label{sec::linking_form}
<wikitex>; \label{sec::linking_form}
−
Let $N$ be a closed orientable connected $n$-manifold.
+
For a simpler invariant see \cite{Skopenkov2022} and references therein.
−
By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere.
+
−
If the (co)homology coefficients are omitted, then we assume them to be $\mathbb Z$.
+
−
The following folklore result holds.
+
In this section assume that
−
{{beginthm|Lemma}}
+
−
Assume $N$ is a closed orientable connected $n$-manifold, $n$ is even and $H_1(N)$ is torsion free. Then for each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$.
+
−
{{endthm}}
+
−
{{beginproof}}
+
−
There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$.
+
−
A normal space to $f(N_0)$ at any point of $f(N_0)$ has dimension $n-1$. As $n$ is even thus $n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$. Since $H_1(N)$ is torsion free, it follows that $\bar e=0$.
+
* $N$ is any closed orientable connected $n$-manifold,
−
Since $N_0$ has non-empty boundary, we have that $N_0$ is homotopy equivalent to an $(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$ at any point of $f(N_0)$. Since $\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$.
+
* $f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,
−
{{endproof}}
+
−
Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients.
+
* if the (co)homology coefficients are omitted, then they are $\mathbb Z$,
−
{{beginthm|Definition}}
+
* $n$ is even and $H_1(N)$ is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).
−
For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote
+
+
By $N_0$ we denote the closure of the complement in $N$ to an closed $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere.
+
+
{{beginthm|Lemma}} There exists a nowhere vanishing normal vector field to $f(N_0)$.
+
{{endthm}}
+
+
This is essentially a folklore result, see an unpublished update of \cite{Tonkonog2010} and \cite[Lemma 5.1]{Fedorov2021}, cf. \cite[Lemma 4.1]{Saeki1999}.
+
+
Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients. Denote
$$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$
$$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$
where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$.
where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$.
−
{{endthm}}
{{beginthm|Lemma|($L$ is well-defined)}}\label{lmm:L_well_def}
{{beginthm|Lemma|($L$ is well-defined)}}\label{lmm:L_well_def}
−
For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$:
+
The integer $L(f)(x, y)$:
−
* is well-defined, i.e. does not change when $s$ is replaced by $s'$,
+
* is well-defined, i.e., does not change when $s$ is replaced by $s'$,
* does not change when $x$ or $y$ are changed to homologous cycles and,
* does not change when $x$ or $y$ are changed to homologous cycles and,
* does not change when $f$ is changed to an isotopic embedding.
* does not change when $f$ is changed to an isotopic embedding.
−
{{endthm}}
+
{{endthm}}
−
The first bullet was stated and proved in unpublished update of \cite{Tonkonog2010}, other two bullets are simple.
+
−
+
−
We will need the following supporting lemma.
+
−
{{beginthm|Lemma}}\label{lmm::saeki}
+
−
Let $f:N_0\to \mathbb R^{2n-1}$ be an embedding.
+
−
Let $s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$.
+
−
Then
+
−
$$\mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$
+
−
where $s(y)$ is the result of the shift of $f(y)$ by $s$, and $d(s,s')\in H_2(N_0)$
+
−
is (Poincare dual to) the first obstruction to $s,s'$ being homotopic in the class of the nowhere vanishing vector fields.
+
−
{{endthm}}
+
−
This Lemma is proved in \cite[Lemma 2.2]{Saeki1999} for $n=3$, but the proof is valid in all dimensions.
+
−
<div>'''Proof of Lemma \ref{lmm:L_well_def}.''' The first bullet point follows because:
+
−
$$
+
−
\begin{aligned}
+
−
\mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\
+
−
\mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\
+
−
&&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\
+
−
&&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0.
+
−
\end{aligned}
+
−
$$
+
−
Here the second equality follows from Lemma \ref{lmm::saeki}.
+
−
+
−
For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.
+
−
+
−
Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.
+
−
{{endproof}}
+
−
Lemma \ref{lmm:L_well_def} implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.
+
The first bullet was stated and proved in unpublished update of \cite{Tonkonog2010} and in \cite[Lemma 5.3]{Fedorov2021}, other two bullets are simple.
+
+
Lemma \ref{lmm:L_well_def} implies that $L(f)$ generates a bilinear form
+
$$L(f):H_{n-1}(N_0)\times H_{n-1}(N_0)\to\Z$$
+
denoted by the same letter and called '''Seifert linking form'''.
−
Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$.
+
Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$. Define the dual to [[Stiefel-Whitney_characteristic_classes|Stiefel-Whitney class]] $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent.
−
Define the dual to [[Stiefel-Whitney_characteristic_classes|Stiefel-Whitney class]] $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent.
{{beginthm|Lemma}} \label{lmm::L_equality}
{{beginthm|Lemma}} \label{lmm::L_equality}
−
Let $f:N_0\to \mathbb R^{2n-1}$ be an embedding.
+
For every $X, Y \in H_{n-1}(N_0)$ the following equality holds:
−
Then for every $X, Y \in H_{n-1}(N_0)$ the following equality holds:
+
$$\rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$
$$\rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$
{{endthm}}
{{endthm}}
−
This Lemma was stated in a unpublished update of \cite{Tonkonog2010}, the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in \cite[Lemma 2.2]{Crowley&Skopenkov2016}.
−
<div>'''Proof of Lemma \ref{lmm::L_equality}.''' Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get
+
This Lemma was stated in a unpublished update of \cite{Tonkonog2010}; a proof is presented in \cite[Lemma 6.1]{Fedorov2021} using the idea from that update. See also an analogous lemma for closed manifolds in \cite[Lemma 2.2]{Crowley&Skopenkov2016}.
−
$$
+
−
\begin{aligned}
+
−
L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\
+
−
&= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\
+
−
&= d(s, -s)\cap X\cap Y
+
−
.
+
−
\end{aligned}
+
−
$$
+
−
The first congruence is clear.
+
−
+
−
The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.
+
−
+
−
The third equality follows from Lemma \ref{lmm::saeki}.
+
−
+
−
Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. <!--( Note that this equality does not holds for homologies with coefficients in $\mathbb Z$. ) -->
+
−
Denote by $s'$ a general perturbation of $s$. We get:
+
−
$$
+
−
\rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).
+
−
$$
+
−
The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields.
+
−
Let us prove the second equality. The linear homotopy between $s'$ and $-s$ degenerates only at those points $x$ where $s'(x)=s(x)$. These points $x$ are exactly points where $s'(x)$ and $s(x)$ are linearly dependent. All those point $x$ form a $2$-cycle modulo two in $N_0$. The homotopy class of this $2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.
+
−
{{endproof}}
+
</wikitex>
</wikitex>
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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.
−
Denote $\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$ up to isotopy.
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}
−
In this paragraph we consider only punctured manifolds to simplify statements of the theorems, but the results seem to hold for manifolds with arbitrary non-empty boundary.
+
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 compact $k$-connected $n$-manifold.
+
Assume that $N$ is a closed $k$-connected $n$-manifold.
−
(a) If $n\geq 2n+3$, then $N$ embeds into $\mathbb R^{2n-k}$.
+
(a) If $n\geq 2k+3$, then $N$ embeds into $\mathbb R^{2n-k}$.
−
(b) Then $N_0$ embeds into $\mathbb R^{2n-k-1}$.
+
(b) If $n\geq 2k+2$ and $k\geq0$, then $N_0$ embeds into $\mathbb R^{2n-k-1}$.
−
{{endthm}}
+
{{endthm}}
−
The Diff case of part (a) is in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is in \cite[Theorem 1.1]{Penrose&Whitehead&Zeeman1961}, \cite[Corollary 1.3]{Irwin1965}.
+
−
The Diff case of part (b) is in \cite[Corollary 4.2]{Hirsch1961a}. For the PL case see \cite[Theorem 1.2]{Penrose&Whitehead&Zeeman1961}.
+
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 $n\ge2k + 2$, $m \ge 2n - k + 1$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic.
+
(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}}
−
For part (a) see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$ 2]{Skopenkov2016c}, or \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}.
+
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}.
−
For the PL case of part (b) see \cite[Theorem 10.3]{Hudson1969}, which is proved using [[Isotopy#Concordance|concordance implies isotopy theorem]].
+
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 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 $N_0$ 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}}
−
For $k>1$ part (b) is corollary of Theorem \ref{thm::k_connect_classif} below (but can be proved in a straightforward way). For $k=0$ part (b) coincides with Theorem \ref{thm::unknotting}b.
+
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]].
−
<div>'''Proof of Theorem \ref{thm::k_connect_unknot}(b) for $k=1$.''' By Theorem \ref{thm::isotop_unknot} below every two immersions of $N$ into $\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$ for each $x\in N$. Since $N$ is $1$-connected and has non-empty boundary, it follows that $N$ collapses to an $(n-2)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. 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 $h\colon N\to M$. 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}}
+
−
Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy.
{{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}. See also \cite[Theorem 2.1]{Vrabec1989}.
+
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>;
+
<wikitex>; \label{prt::arbitrary_boundary}
−
{{beginthm|Theorem}}[Smale-Hirsch]
+
{{beginthm|Theorem}}\label{thm::arbitarty_k_connect_embeds}
−
The space of immersions of a manifold in $\mathbb R^{m}$ is homotopically equivalent to the space of linear monomorphisms from $TM$ to $\mathrm R^{m}$.
+
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}}
{{endthm}}
−
See \cite{Hirsch1959} and \cite{Haefliger&Poenaru1964}.
+
Proof is similar to the proof of theorem \ref{thm::k_connect_unknot}.
−
{{beginthm|Theorem}}\label{thm::imm_dec_dim}
+
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.
−
If $N$ is immersible in $\mathbb R^{m+r}$ with a transversal $r$-field then it is immersible in $\mathbb R^{m}$.
+
{{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}}
{{endthm}}
−
This is \cite[Theorem 6.4]{Hirsch1959}.
+
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>;
+
+
{{beginthm|Theorem}}[Smale-Hirsch; \cite{Hirsch1959} and \cite{Haefliger&Poenaru1964}]
+
The space of immersions of a manifold in $\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$ to $\R^m$.
+
{{endthm}}
+
+
{{beginthm|Theorem}}[\cite[Theorem 6.4]{Hirsch1959}]\label{thm::imm_dec_dim}
+
If $N$ is immersible in $\R^{m+r}$ with a normal $r$-field, then $N$ is immersible in $\R^m$.
+
{{endthm}}
+
{{beginthm|Theorem}}
{{beginthm|Theorem}}
−
Every $n$-manifold $N$ with non-empty boundary is immersible in $\mathbb R^{2n-1}$.
+
Every $n$-manifold $N$ with non-empty boundary is immersible in $\R^{2n-1}$.
{{endthm}}
{{endthm}}
<!--{{beginproof}}
<!--{{beginproof}}
−
The obstruction to existence of immersion on $k$--skeleton of $N$ into $\mathbb R^{m}$ lies in $H_{n-k}(N; \pi_{k-1}(V_{m, n}))$. For $m=2n-1$ and $k<n$ we have $\pi_{k-1}(V_{m, k}) = 0$. Since $N$ has non-empty boundary it follows that the obstruction in $H_{n-1}(N;\pi_{n-1}(V_{2n-1, n})$ is zero. Thus $N$ is immersible into $\mathbb R^{2n-1}$.{{endproof}}-->
+
The obstruction to existence of immersion on $k$--skeleton of $N$ into $\mathbb R^{m}$ lies in $H_{n-k}(N; \pi_{k-1}(V_{m, n}))$. For $m=2n-1$ and $k<n$ we have $\pi_{k-1}(V_{m, k}) = 0$. Since $N$ has non-empty boundary it follows that the obstruction in $H_{n-1}(N;\pi_{n-1}(V_{2n-1, n})$ is zero. Thus $N$ is immersible into $\mathbb R^{2n-1}$.{{endproof}}
+
''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.-->
−
{{beginthm|Theorem}}[Whitney]
+
{{beginthm|Theorem}}[Whitney; \cite[Theorem 6.6]{Hirsch1961a}]
−
Every $n$-manifold $N$ is immersible in $\mathbb R^{2n-1}$.
+
Every $n$-manifold $N$ is immersible in $\R^{2n-1}$.
{{endthm}}
{{endthm}}
−
See \cite[Theorem 6.6]{Hirsch1961a}.
+
−
{{beginthm|Theorem}}
+
Denote by $V_{m,n}$ is Stiefel manifold of $n$-frames in $\R^m$.
−
Every $k$-connected $n$-manifold $N$ with non-empty boudary is immersible in $\mathbb R^{m}$ for each $m\geq2n-k-1$.
+
+
{{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 $\R^m$ for each $m\geq2n-k-1$.
{{endthm}}
{{endthm}}
+
{{beginproof}}
{{beginproof}}
It suffices to show that exists an immersion of $N$ in $\mathbb R^{2n-k-1}$.
It suffices to show that exists an immersion of $N$ in $\mathbb R^{2n-k-1}$.
−
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 $\mathbb R^{2n-k-1}$.
+
Let us construct such a linear monomorphism by skeleta of $N$.
+
It is clear that a 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 extend the linear monomorphism from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.
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 $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and has non-empty boundary.
−
For $r=n-k,\ldots, n$ we have $H_{n-r}(N; *) = 0$ since $N$ 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.
{{endproof}}
{{endproof}}
+
{{beginthm|Theorem}}\label{thm::isotop_unknot}
{{beginthm|Theorem}}\label{thm::isotop_unknot}
−
Suppose $N$ is a $k$-connected $n$-manifold with non-empty boudary and $m\geq2n-k$. Then every two immersions of $N$ in $\mathbb R^m$ are regulary homotopic.
+
Suppose $N$ is a connected $n$-manifold with non-empty boudary, $(N, \partial N)$ is $k$-connected and $m\geq2n-k$. Then every two immersions of $N$ in $\R^m$ 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 extend the homotopy from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.
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 $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and $N$ has non-empty boundary.
−
For $r=n-k,\ldots, n$ we have $H_{n-r}(N; *) = 0$ since $N$ is $k$-connected and 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.
{{endproof}}
{{endproof}}
</wikitex>
</wikitex>
−
<!--== Classification ==
<!--== Classification ==
<wikitex>;
<wikitex>;
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\cite{Becker&Glover1971}
\cite{Becker&Glover1971}
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== Acknowledgments ==
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== Further discussion ==
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−
<wikitex>;
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...
...
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</wikitex>-->
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<!-- == Acknowledgments ==
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== Footnotes ==
== Footnotes ==
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<references/> -->

Latest revision as of 14:26, 21 May 2023

The users responsible for this page are: Mikhail Fedorov, Garaev Timur. No other users may edit this page at present.

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In \S5 we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S7 which is independent from \S4, \S5 and \S6 we state generalisations of theorems from \S2 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, \S1, \S3]. In those pages mostly results for closed manifolds are stated.

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 [Seifert&Threlfall1980, \S 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that N is a compact connected n-manifold.

(a) Then N embeds into \mathbb R^{2n}.

(b) If N has non-empty boundary, then N embeds into \mathbb R^{2n-1}.

Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

Theorem 2.2. Assume that N is a compact connected n-manifold and either

(a) m \ge 2n+1 \ge 5 or

(b) N has non-empty boundary and m\geq 2n.

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

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, \S 2, Theorems 2.1, 2.2]. Part (b) in the case n>2 is proved in [Edwards1968, \S 4, Corollary 5]. The case n=1 is clear. The case n=2 can be proved using the ideas presented below.

The inequality in part (b) is sharp by Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see \S7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.

All theorems for manifolds with non-empty boundary stated in \S2 and \S7 can be proved using

  • analogous results for immersions of manifolds stated in \S9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in \S3.
  • handle decomposition, see e.g. the second proof of Theorem 2.1.b in \S3.

Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion g\colon N\to\mathbb R^{2n-1}. Since N is connected and has non-empty boundary, it follows that N collapses to an (n-1)-dimensional subcomplex X\subset N of some triangulation of N. Since 2(n-1) < 2n-1, by general position we may assume that g|_{X} is an embedding. Since g is an immersion, it follows that X has a sufficiently small tubular neighbourhood M\supset X such that g|_{M} is embedding. Since tubular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. The composition g\circ h is an embedding of N.
\square

For the second proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that N is a closed connected n-manifold. Then N_0 have handle decomposition with indices of attaching maps at most n-1.

Lemma 3.2. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, and f:U \to \mathbb R^{m} is an embedding.

(a) If m=2n-1, then there is an extension of f to an embedding of U\cup_{\phi} D^i\times D^{n-i}.

(b) Assume also that there is an embedding g : D^i\times 0 \to \mathbb R^m such that f\phi = g on \partial D^i \times 0. Suppose that on g(D^i\times 0) there is a field of n-i linear independed normal vectors whose restriction to f(\partial D^i\times 0) is tangent to f(U). Then f\cup g extends to a embedding U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m.

Second proof of Theorem 2.1.b assuming Lemma 3.1 and Lemma 3.2.(a). By Lemma 3.1.(a) there is a handle decomposition of N_0 with attaching maps \phi_1,\ldots,\phi_s of indices at most n-1. Denote by U^l the manifold obtained from \emptyset by the attaching first l handles. Take any embedding F_1:U^1 \cong D^n\to \R^{2n-1}. Let us define an embedding F_l of U^l using an embedding F_{l-1} of U^{l-1}. Since the index i of \phi_l is smaller than n, by Lemma 3.2 there is extension of F_{l-1} to an embedding F_l:U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}, where U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}.
\square
Proof of Lemma 3.2.(a) assuming Lemma 3.2.(b). Since i+n\leq 2n-1 and 2i+1\leq 2n-1, by general position there is an embedding g: D^i\times 0\to \mathbb R^{2n-1} such that f\phi = g on \partial D^i \times 0 and f(\mbox{Int} U) has a finite number of intersections points with g(\mbox{Int} D^i\times 0). Then by an isotopy g_t, where g_0=g, fixed on \partial D^i\times 0 we can "push out" the self-intersection points toward \partial U so that g_1(\mbox{Int} D^i\times 0) does not intersect f(\mbox{Int} U). Then f\cup g_1 is an embedding.

Denote by A the (n-i)\times n matrix whose rightmost (n-i)\times (n-i) block is the identity matrix, and whose other elements are zeros. Denote by v the field of n-i normal vectors on \partial D^i\times 0\subset D^i\times D^{n-i} such that the k-th vector has coordinates equal to the k-th row in A. Then the vector field d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i})) is tangent to \partial U. For x\in g_1(\partial D^i\times 0) denote by v'(x) the projection of df(d\phi(v(x))) to the intersection of normal space to g_1(\partial D^i\times 0) at x, and tangent space to f(\partial U) at x. Since i-1<2n-1 - (n-i), it follows that \pi_{i-1}(V_{2n-1, n-i})=0. Hence there is an extension of v' to an linearly independent field of vectors normal to g_1(D^i\times 0). Then by Lemma 3.2.(b) there is an extension of f\cup g_1 to an embedding of U \cup_{\phi} D^i\times D^{n-i}.

\square

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).

Lemma 3.3. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m are embeddings and F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1] is a concordance between f_0|_U and f_1|_U.

(a) If m\geq 2n, then there is an extension of F to a concordance between f_0 and f_1.

(b) Assume also that there is a concordance G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1] between f_0|_{D^i\times 0} and f_1|_{D^i\times 0}, and on G(D^i\times 0\times [0, 1]) there is a field of n-i linear independent normal vectors whose restrictions to G(\partial D^i\times 0\times [0, 1]), to G(D^i\times 0\times 0), and to G(D^i\times 0\times 1) are tangent to
\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,
respectively. Then F\cup G extends to a concordance between f_0 and f_1.
Proof of the Theorem 2.2 assuming Lemma 3.1 and Lemma 3.3.(a). Denote by f_0, f_1 any two embeddings of N_0 into \mathbb{R}^m. By the Concordance Implies Isotopy Theorem it suffices to prove that there is a concordance between f_0 and f_1. In order to prove that observe that by Lemma 3.1 there is a handle decomposition of N_0 with attaching maps of indices at most n-1. Denote by U^l the manifold obtained from U^0=\emptyset by the attaching the first l handles, starting with? U^1\cong D^n. Define a concordance F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1] recursively. Take any concordance F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1] between f_0|_{U^1} and f_1|_{U^1}. Let us define a concordance F_l between f_0|_{U^l} and f_1|_{U^l} using a concordance F_{l-1} of U^{l-1} between f_0|_{U^{l-1}} and f_1|_{U^{l-1}}. For this denote by \phi:\partial D^i\times D^{n-i}\to \partial U^{l-1} the l-th attaching map. Since i\leq n-1, by Lemma 3.3.(a) there is an extension of F_{l-1} to a concordance
\displaystyle F_{l}:U^{l}\times [0, 1]\to\mathbb{R}^m\times [0, 1]
between the restriction of f_0 and f_1 to U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}.
\square
Proof of Lemma 3.3.(a) assuming Lemma 3.3.(b). In the following text we identify D^i\times D^{n-i}\times [0, 1] and D^i\times [0, 1]\times D^{n-i}. Define map
\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1
by the formula: 
Tex syntax error
Since
\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\quad\text{and}\quad 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,
by general position there is an embedding
\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]
such that F\bar{\phi} = G on \partial (D^i \times 0\times [0, 1]) and F(\mbox{Int} (U\times [0, 1])) has a finite number of intersection points with G(\mbox{Int}( D^i\times 0\times [0, 1])). Then by an isotopy G_t, where G_0=G, fixed on \partial (D^i\times 0\times [0, 1]) we can "push out" the self-intersection points toward F(\partial (U\times [0, 1])) so that G_1(D^i\times 0\times [0, 1]) does not intersect G(U\times [0, 1]). Then F\cup G_1 is an concordance between the restrictions of f_0 and f_1 on U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0.

Denote by A the (n-i)\times (n+1) matrix whose rightmost (n-i)\times (n-i) submatrix is the identity matrix, and whose other elements are zeroes. Denote by v the field of n-i normal vectors on \partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1] whose k-th vector has coordinates equal to the k-th row in A. Then d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i})) is the vector field tangent to \bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i}). For x\in G_1(\partial (D^i\times 0\times [0, 1]) denote by v'(x) the projection of d(F\cup f_0\cup f_1)d\phi (v(x)) to the intersection of normal space to G_1(D^i\times 0\times [0, 1]) at x, and tangent space to F(\partial(U\times [0, 1])) at x. Since i<m+1-(n-i), it follows that \pi_{i}(V_{m+1, n-i})=0. Hence there is an extension of v' to a linear independent field of vectors normal to G_1(D^i\times 0\times [0, 1]). Then by Lemma 3.3.(b) there is an extension of F\cup G_1 to a concordance (U \cup_{\phi} D^i\times D^{n-i})\times [0, 1].

\square


4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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}.

(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 well-defined and is a bijection for k\geqslant2.

Proof. 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).

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 the standard embedding.Thus \mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a.

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 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.
\square
(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.

Denote 1_k:=(1,0,\ldots,0)\in S^k.

Example 4.2. 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).

The surjectivity of l is given analogously to Proposition 4.1(b). The injectivity of l follows from forgetful bijection \mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1] between embeddings of N_0 and a cylinder.

This example shows that Theorem 7.4 fails for k=0.

Example 4.3. Let N=S^k_a\times S^1 \# S^k_b\times S^1 be the connected sum of two tori. Then there exists a surjection l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z defined by the formula l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1)).

To prove the surjectivity of l it is sufficient to take linked k-spheres in \mathbb R^{2k+1} and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (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). Figure (c) shows that l(f_1)=1 and l(f_2)=0 which proves the intuitive fact that f_1 and f_2 are not isotopic. (Notice that the restrictions of f_1 and f_2 on x\cup y are isotopic!) If we use the opposite normal vector field s'=-s, the values of l(f_1) and l(f_2) will change but will still be different (see figure (d)).

5 Seifert linking form

For a simpler invariant see [Skopenkov2022] and references therein.

In this section assume that

  • N is any closed orientable connected n-manifold,
  • f\colon N_0 \to \mathbb R^{2n-1} is any embedding,
  • if the (co)homology coefficients are omitted, then they are \mathbb Z,
  • n is even and H_1(N) is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).

By N_0 we denote the closure of the complement in N to an closed n-ball. Thus \partial N_0 is the (n-1)-sphere.

Lemma 5.1. There exists a nowhere vanishing normal vector field to f(N_0).

This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].

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

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

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

Lemma 5.2 (L is well-defined). The integer L(f)(x, y):

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

The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.

Lemma 5.2 implies that L(f) generates a bilinear form

\displaystyle L(f):H_{n-1}(N_0)\times H_{n-1}(N_0)\to\Z

denoted by the same letter and called Seifert linking form.

Denote by \rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2) the reduction modulo 2. Define the dual to Stiefel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2) to be the class of the cycle on which two general position normal fields to f(N_0) are linearly dependent.

Lemma 5.3. For every X, Y \in H_{n-1}(N_0) the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let 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. 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).

Definition 6.1. For each even n define an invariant W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2). For each embedding f\colon N_0\to\mathbb R^{2n-1} construct any PL embedding g\colon N\to\mathbb R^{2n} by adding a cone over f(\partial N_0). Now let W\Lambda([f]) = W(g), where W is Whitney invariant, [Skopenkov2016e, \S5].

Lemma 6.2. The invariant W\Lambda is well-defined for n\ge4.

Proof. Note that Unknotting Spheres Theorem implies that \partial N_0 unknots in \mathbb R^{2n}. Thus f|_{\partial N_0} can be extended to embedding of an n-ball B^n into \mathbb R^{2n}. Unknotting Spheres Theorem implies that n-sphere unknots in \mathbb R^{2n}. Thus all extensions of f are isotopic in PL category. Note also that if f and g are isotopic then their extensions are isotopic as well. And Whitney invariant W is invariant for PL embeddings.

\square

Definition 6.3 of G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) if n is even and H_1(N) is torsion-free. Take a collection \{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0 such that W\Lambda(f_z)=z. For each f such that W\Lambda(f)=z define

\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)

where x,y\in H_{n-1}(N_0).

Note also that G depends on choice of collection \{f_z\}. The following Theorems hold for any choice of \{f_z\}.

Theorem 6.4. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n 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}),

is one-to-one.

Lemma 6.5. For each even n\in H_{n-1}(N) and each x the following equality holds: W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right).

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. Then

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

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

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see \S8 for a generalization.

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.

Theorem 7.1. Assume that N is a closed k-connected n-manifold.

(a) If n\geq 2k+3, then N embeds into \mathbb R^{2n-k}.

(b) If n\geq 2k+2 and k\geq0, then N_0 embeds into \mathbb R^{2n-k-1}.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that N is a closed k-connected n-manifold.

(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 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.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, \S 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 k>1 part (b) is a corollary of Theorem 7.4 below. For k=0 part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for k=1. By Theorem 9.6 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 8.3 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 concordance implies isotopy Theorem f and g are isotopic.
\square

Conjecture 7.3. Assume that N is a closed 1-connected 4-manifold. Then any two embeddings of N_0 in \mathbb R^7 are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume N is a closed k-connected n-manifold. Then for each k\ge1 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)}),

where \mathbb Z_{(s)} denote \mathbb Z for s even and \mathbb Z_2 for s odd.

For definition of W_0' and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(W_0')]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes \mathrm{Emb}^{2n-1}(N_0) and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. 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}.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion f\colon N\to\mathbb R^{2n-k-1}. Since N is k-connected it follows from Theorem 8.3 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.
\square

Theorem 8.2. 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.

Proof is similar to the proof of theorem 7.2.

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.

Theorem 8.3. 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.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in \R^m is homotopy equivalent to the space of linear monomorphisms from TM to \R^m.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]]

If N is immersible in \R^{m+r} with a normal 
Tex syntax error
-field, then N is immersible in \R^m.

Theorem 9.3. Every n-manifold N with non-empty boundary is immersible in \R^{2n-1}.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every n-manifold N is immersible in \R^{2n-1}.

Denote by V_{m,n} is Stiefel manifold of n-frames in \R^m.

Theorem 9.5. Suppose N is a n-manifold with non-empty boudary, (N,\partial N) is k-connected. Then N is immersible in \R^m for each m\geq2n-k-1.

Proof. It suffices to show that exists an immersion of N in \mathbb R^{2n-k-1}. It suffices to show that exists a linear monomorphism from TM to \mathbb R^{2n-k-1}. Let us construct such a linear monomorphism by skeleta of N. It is clear that a linear monomorphism exists on 0-skeleton of N.

The obstruction to extend the linear monomorphism from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n})).

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, \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.

\square

Theorem 9.6. Suppose N is a connected n-manifold with non-empty boudary, (N, \partial N) is k-connected and m\geq2n-k. Then every two immersions of N in \R^m are regulary homotopic.

Proof.

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 
Tex syntax error
-skeleton of N. It is clear that homotopy exists on 0-skeleton of N. The obstruction to extend the homotopy from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n})).

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, \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.

\square

10 References

, $\S]{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. 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 do not claim the references we give are references to original proofs. == Embedding and unknotting theorems == ; \label{sec::general_theorems} {{beginthm|Theorem}} Assume that $N$ is a compact connected $n$-manifold. (a) Then $N$ embeds into $\mathbb R^{2n}$. (b) If $N$ has non-empty boundary, then $N$ embeds into $\mathbb R^{2n-1}$. {{endthm}} Part (a) is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]].
'''Proof of part (b).''' By strong [[Wikipedia:Whitney_immersion_theorem|strong Whitney immersion theorem]] there exist an immersion $g\colon N\to\mathbb R^{2n-1}$. Since $N$ is connected and has non-empty boundary, it follows that $N$ collapses to an $(n-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $g|_{X}$ is an embedding, because (n-1) < 2n-1$. Since $g$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. The composition $g\circ h$ is an embedding of $N$.{{endproof}}This proof is essentially contained in \cite[Theorem 4.6]{Hirsch1961a} for the Diff case and in references for Theorem \ref{thm::k_connect_embeds} below or in \cite[Theorem 5.2]{Horvatic1971} for the PL case. {{beginthm|Theorem}}\label{thm::unknotting} Assume that $N$ is a compact connected $n$-manifold and either (a) $m \ge 2n+1 \ge 5$ or (b) $N$ has non-empty boundary and $m\geq 2n$. Then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. {{endthm}} The part (a) is Whitney-Wu Unknotting Theorem, see \cite[$\S$ 2, Theorems 2.1, 2.2]{Skopenkov2016c}. Inequality in part (b) is sharp, see Proposition \ref{exm::linked_boundary}. Part (b) in case $n>2$ can be found in \cite[$\S$ 4, Corollary 5]{Edwards1968}. Case $n=1$ is clear. Both parts of this theorem are special cases of the Theorem \ref{thm::k_connect_unknot}. Case $n=2$ can be proved using the following ideas. These basic results can be generalized to the highly-connected manifolds (see $\S$\ref{sec::generalisations}). All stated theorems of $\S$\ref{sec::general_theorems} and $\S$\ref{sec::generalisations} for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas. == Example on non-isotopic embeddings == ; \label{sec::example} The following example is folklore. {{beginthm|Exaple}} \label{exm::linked_boundary} Let $N=S^k\times [0, 1]$ be the cylinder over $S^k$. (a) Then there exist non-isotopic embeddings of $N$ into $\mathbb R^{2k+1}$. (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 $l\colon N\to\mathbb R$ {{endthm}} {{beginproof}} 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 \S5 we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S7 which is independent from \S4, \S5 and \S6 we state generalisations of theorems from \S2 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, \S1, \S3]. In those pages mostly results for closed manifolds are stated.

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 [Seifert&Threlfall1980, \S 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that N is a compact connected n-manifold.

(a) Then N embeds into \mathbb R^{2n}.

(b) If N has non-empty boundary, then N embeds into \mathbb R^{2n-1}.

Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

Theorem 2.2. Assume that N is a compact connected n-manifold and either

(a) m \ge 2n+1 \ge 5 or

(b) N has non-empty boundary and m\geq 2n.

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

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, \S 2, Theorems 2.1, 2.2]. Part (b) in the case n>2 is proved in [Edwards1968, \S 4, Corollary 5]. The case n=1 is clear. The case n=2 can be proved using the ideas presented below.

The inequality in part (b) is sharp by Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see \S7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.

All theorems for manifolds with non-empty boundary stated in \S2 and \S7 can be proved using

  • analogous results for immersions of manifolds stated in \S9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in \S3.
  • handle decomposition, see e.g. the second proof of Theorem 2.1.b in \S3.

Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion g\colon N\to\mathbb R^{2n-1}. Since N is connected and has non-empty boundary, it follows that N collapses to an (n-1)-dimensional subcomplex X\subset N of some triangulation of N. Since 2(n-1) < 2n-1, by general position we may assume that g|_{X} is an embedding. Since g is an immersion, it follows that X has a sufficiently small tubular neighbourhood M\supset X such that g|_{M} is embedding. Since tubular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. The composition g\circ h is an embedding of N.
\square

For the second proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that N is a closed connected n-manifold. Then N_0 have handle decomposition with indices of attaching maps at most n-1.

Lemma 3.2. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, and f:U \to \mathbb R^{m} is an embedding.

(a) If m=2n-1, then there is an extension of f to an embedding of U\cup_{\phi} D^i\times D^{n-i}.

(b) Assume also that there is an embedding g : D^i\times 0 \to \mathbb R^m such that f\phi = g on \partial D^i \times 0. Suppose that on g(D^i\times 0) there is a field of n-i linear independed normal vectors whose restriction to f(\partial D^i\times 0) is tangent to f(U). Then f\cup g extends to a embedding U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m.

Second proof of Theorem 2.1.b assuming Lemma 3.1 and Lemma 3.2.(a). By Lemma 3.1.(a) there is a handle decomposition of N_0 with attaching maps \phi_1,\ldots,\phi_s of indices at most n-1. Denote by U^l the manifold obtained from \emptyset by the attaching first l handles. Take any embedding F_1:U^1 \cong D^n\to \R^{2n-1}. Let us define an embedding F_l of U^l using an embedding F_{l-1} of U^{l-1}. Since the index i of \phi_l is smaller than n, by Lemma 3.2 there is extension of F_{l-1} to an embedding F_l:U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}, where U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}.
\square
Proof of Lemma 3.2.(a) assuming Lemma 3.2.(b). Since i+n\leq 2n-1 and 2i+1\leq 2n-1, by general position there is an embedding g: D^i\times 0\to \mathbb R^{2n-1} such that f\phi = g on \partial D^i \times 0 and f(\mbox{Int} U) has a finite number of intersections points with g(\mbox{Int} D^i\times 0). Then by an isotopy g_t, where g_0=g, fixed on \partial D^i\times 0 we can "push out" the self-intersection points toward \partial U so that g_1(\mbox{Int} D^i\times 0) does not intersect f(\mbox{Int} U). Then f\cup g_1 is an embedding.

Denote by A the (n-i)\times n matrix whose rightmost (n-i)\times (n-i) block is the identity matrix, and whose other elements are zeros. Denote by v the field of n-i normal vectors on \partial D^i\times 0\subset D^i\times D^{n-i} such that the k-th vector has coordinates equal to the k-th row in A. Then the vector field d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i})) is tangent to \partial U. For x\in g_1(\partial D^i\times 0) denote by v'(x) the projection of df(d\phi(v(x))) to the intersection of normal space to g_1(\partial D^i\times 0) at x, and tangent space to f(\partial U) at x. Since i-1<2n-1 - (n-i), it follows that \pi_{i-1}(V_{2n-1, n-i})=0. Hence there is an extension of v' to an linearly independent field of vectors normal to g_1(D^i\times 0). Then by Lemma 3.2.(b) there is an extension of f\cup g_1 to an embedding of U \cup_{\phi} D^i\times D^{n-i}.

\square

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).

Lemma 3.3. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m are embeddings and F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1] is a concordance between f_0|_U and f_1|_U.

(a) If m\geq 2n, then there is an extension of F to a concordance between f_0 and f_1.

(b) Assume also that there is a concordance G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1] between f_0|_{D^i\times 0} and f_1|_{D^i\times 0}, and on G(D^i\times 0\times [0, 1]) there is a field of n-i linear independent normal vectors whose restrictions to G(\partial D^i\times 0\times [0, 1]), to G(D^i\times 0\times 0), and to G(D^i\times 0\times 1) are tangent to
\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,
respectively. Then F\cup G extends to a concordance between f_0 and f_1.
Proof of the Theorem 2.2 assuming Lemma 3.1 and Lemma 3.3.(a). Denote by f_0, f_1 any two embeddings of N_0 into \mathbb{R}^m. By the Concordance Implies Isotopy Theorem it suffices to prove that there is a concordance between f_0 and f_1. In order to prove that observe that by Lemma 3.1 there is a handle decomposition of N_0 with attaching maps of indices at most n-1. Denote by U^l the manifold obtained from U^0=\emptyset by the attaching the first l handles, starting with? U^1\cong D^n. Define a concordance F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1] recursively. Take any concordance F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1] between f_0|_{U^1} and f_1|_{U^1}. Let us define a concordance F_l between f_0|_{U^l} and f_1|_{U^l} using a concordance F_{l-1} of U^{l-1} between f_0|_{U^{l-1}} and f_1|_{U^{l-1}}. For this denote by \phi:\partial D^i\times D^{n-i}\to \partial U^{l-1} the l-th attaching map. Since i\leq n-1, by Lemma 3.3.(a) there is an extension of F_{l-1} to a concordance
\displaystyle F_{l}:U^{l}\times [0, 1]\to\mathbb{R}^m\times [0, 1]
between the restriction of f_0 and f_1 to U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}.
\square
Proof of Lemma 3.3.(a) assuming Lemma 3.3.(b). In the following text we identify D^i\times D^{n-i}\times [0, 1] and D^i\times [0, 1]\times D^{n-i}. Define map
\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1
by the formula: 
Tex syntax error
Since
\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\quad\text{and}\quad 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,
by general position there is an embedding
\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]
such that F\bar{\phi} = G on \partial (D^i \times 0\times [0, 1]) and F(\mbox{Int} (U\times [0, 1])) has a finite number of intersection points with G(\mbox{Int}( D^i\times 0\times [0, 1])). Then by an isotopy G_t, where G_0=G, fixed on \partial (D^i\times 0\times [0, 1]) we can "push out" the self-intersection points toward F(\partial (U\times [0, 1])) so that G_1(D^i\times 0\times [0, 1]) does not intersect G(U\times [0, 1]). Then F\cup G_1 is an concordance between the restrictions of f_0 and f_1 on U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0.

Denote by A the (n-i)\times (n+1) matrix whose rightmost (n-i)\times (n-i) submatrix is the identity matrix, and whose other elements are zeroes. Denote by v the field of n-i normal vectors on \partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1] whose k-th vector has coordinates equal to the k-th row in A. Then d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i})) is the vector field tangent to \bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i}). For x\in G_1(\partial (D^i\times 0\times [0, 1]) denote by v'(x) the projection of d(F\cup f_0\cup f_1)d\phi (v(x)) to the intersection of normal space to G_1(D^i\times 0\times [0, 1]) at x, and tangent space to F(\partial(U\times [0, 1])) at x. Since i<m+1-(n-i), it follows that \pi_{i}(V_{m+1, n-i})=0. Hence there is an extension of v' to a linear independent field of vectors normal to G_1(D^i\times 0\times [0, 1]). Then by Lemma 3.3.(b) there is an extension of F\cup G_1 to a concordance (U \cup_{\phi} D^i\times D^{n-i})\times [0, 1].

\square


4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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}.

(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 well-defined and is a bijection for k\geqslant2.

Proof. 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).

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 the standard embedding.Thus \mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a.

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 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.
\square
(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.

Denote 1_k:=(1,0,\ldots,0)\in S^k.

Example 4.2. 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).

The surjectivity of l is given analogously to Proposition 4.1(b). The injectivity of l follows from forgetful bijection \mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1] between embeddings of N_0 and a cylinder.

This example shows that Theorem 7.4 fails for k=0.

Example 4.3. Let N=S^k_a\times S^1 \# S^k_b\times S^1 be the connected sum of two tori. Then there exists a surjection l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z defined by the formula l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1)).

To prove the surjectivity of l it is sufficient to take linked k-spheres in \mathbb R^{2k+1} and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (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). Figure (c) shows that l(f_1)=1 and l(f_2)=0 which proves the intuitive fact that f_1 and f_2 are not isotopic. (Notice that the restrictions of f_1 and f_2 on x\cup y are isotopic!) If we use the opposite normal vector field s'=-s, the values of l(f_1) and l(f_2) will change but will still be different (see figure (d)).

5 Seifert linking form

For a simpler invariant see [Skopenkov2022] and references therein.

In this section assume that

  • N is any closed orientable connected n-manifold,
  • f\colon N_0 \to \mathbb R^{2n-1} is any embedding,
  • if the (co)homology coefficients are omitted, then they are \mathbb Z,
  • n is even and H_1(N) is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).

By N_0 we denote the closure of the complement in N to an closed n-ball. Thus \partial N_0 is the (n-1)-sphere.

Lemma 5.1. There exists a nowhere vanishing normal vector field to f(N_0).

This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].

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

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

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

Lemma 5.2 (L is well-defined). The integer L(f)(x, y):

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

The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.

Lemma 5.2 implies that L(f) generates a bilinear form

\displaystyle L(f):H_{n-1}(N_0)\times H_{n-1}(N_0)\to\Z

denoted by the same letter and called Seifert linking form.

Denote by \rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2) the reduction modulo 2. Define the dual to Stiefel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2) to be the class of the cycle on which two general position normal fields to f(N_0) are linearly dependent.

Lemma 5.3. For every X, Y \in H_{n-1}(N_0) the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let 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. 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).

Definition 6.1. For each even n define an invariant W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2). For each embedding f\colon N_0\to\mathbb R^{2n-1} construct any PL embedding g\colon N\to\mathbb R^{2n} by adding a cone over f(\partial N_0). Now let W\Lambda([f]) = W(g), where W is Whitney invariant, [Skopenkov2016e, \S5].

Lemma 6.2. The invariant W\Lambda is well-defined for n\ge4.

Proof. Note that Unknotting Spheres Theorem implies that \partial N_0 unknots in \mathbb R^{2n}. Thus f|_{\partial N_0} can be extended to embedding of an n-ball B^n into \mathbb R^{2n}. Unknotting Spheres Theorem implies that n-sphere unknots in \mathbb R^{2n}. Thus all extensions of f are isotopic in PL category. Note also that if f and g are isotopic then their extensions are isotopic as well. And Whitney invariant W is invariant for PL embeddings.

\square

Definition 6.3 of G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) if n is even and H_1(N) is torsion-free. Take a collection \{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0 such that W\Lambda(f_z)=z. For each f such that W\Lambda(f)=z define

\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)

where x,y\in H_{n-1}(N_0).

Note also that G depends on choice of collection \{f_z\}. The following Theorems hold for any choice of \{f_z\}.

Theorem 6.4. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n 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}),

is one-to-one.

Lemma 6.5. For each even n\in H_{n-1}(N) and each x the following equality holds: W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right).

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. Then

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

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

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see \S8 for a generalization.

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.

Theorem 7.1. Assume that N is a closed k-connected n-manifold.

(a) If n\geq 2k+3, then N embeds into \mathbb R^{2n-k}.

(b) If n\geq 2k+2 and k\geq0, then N_0 embeds into \mathbb R^{2n-k-1}.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that N is a closed k-connected n-manifold.

(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 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.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, \S 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 k>1 part (b) is a corollary of Theorem 7.4 below. For k=0 part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for k=1. By Theorem 9.6 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 8.3 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 concordance implies isotopy Theorem f and g are isotopic.
\square

Conjecture 7.3. Assume that N is a closed 1-connected 4-manifold. Then any two embeddings of N_0 in \mathbb R^7 are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume N is a closed k-connected n-manifold. Then for each k\ge1 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)}),

where \mathbb Z_{(s)} denote \mathbb Z for s even and \mathbb Z_2 for s odd.

For definition of W_0' and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(W_0')]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes \mathrm{Emb}^{2n-1}(N_0) and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. 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}.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion f\colon N\to\mathbb R^{2n-k-1}. Since N is k-connected it follows from Theorem 8.3 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.
\square

Theorem 8.2. 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.

Proof is similar to the proof of theorem 7.2.

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.

Theorem 8.3. 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.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in \R^m is homotopy equivalent to the space of linear monomorphisms from TM to \R^m.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]]

If N is immersible in \R^{m+r} with a normal 
Tex syntax error
-field, then N is immersible in \R^m.

Theorem 9.3. Every n-manifold N with non-empty boundary is immersible in \R^{2n-1}.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every n-manifold N is immersible in \R^{2n-1}.

Denote by V_{m,n} is Stiefel manifold of n-frames in \R^m.

Theorem 9.5. Suppose N is a n-manifold with non-empty boudary, (N,\partial N) is k-connected. Then N is immersible in \R^m for each m\geq2n-k-1.

Proof. It suffices to show that exists an immersion of N in \mathbb R^{2n-k-1}. It suffices to show that exists a linear monomorphism from TM to \mathbb R^{2n-k-1}. Let us construct such a linear monomorphism by skeleta of N. It is clear that a linear monomorphism exists on 0-skeleton of N.

The obstruction to extend the linear monomorphism from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n})).

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, \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.

\square

Theorem 9.6. Suppose N is a connected n-manifold with non-empty boudary, (N, \partial N) is k-connected and m\geq2n-k. Then every two immersions of N in \R^m are regulary homotopic.

Proof.

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 
Tex syntax error
-skeleton of N. It is clear that homotopy exists on 0-skeleton of N. The obstruction to extend the homotopy from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n})).

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, \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.

\square

10 References

_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 $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$. {{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 $s_iy\sqcup f_ix$.]] {{beginthm|Example}} Let $N=S^k\times S^1$. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. {{endthm}} The surjectivity of $l$ is given analogously to Proposition \ref{exm::linked_boundary}(b). By twisting the $x$-ribbon one can obtain arbitrary value of linking coefficient. This example shows that Theorem \ref{thm::k_connect_classif} fails for $k=0$. {{beginthm|Example}} Let $N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$. {{endthm}} To prove the surjectivity of $l$ it is sufficient to take linked $k$-spheres in $\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres. {{beginthm|Example}} Let $N_0$ be the punctured 2-torus containing the meridian and the parallel of the torus. Denote by $x,y$ the meridian and the parallel of the torus. For each embedding $f\in\mathrm{Emb}^{3}N_0$ denote by $s$ a normal field to $f(N_0)$. Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$. {{endthm}} {{beginthm|Example}} Let $f_1,f_2:N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$ and $l(f_2)=0$ which proves the intuitive fact that $f_1$ and $f_2$ are not isotopic. (Notice that the restrictions of $f_1$ and $f_2$ on $x\cup y$ are isotopic!) If we use opposite normal vector field $s'=-s$, the values of $l(f_1)$ and $l(f_2)$ will change but will still be different (see figure (d)). {{endthm}} == Seifert linking form == ; \label{sec::linking_form} Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. If the (co)homology coefficients are omitted, then we assume them to be $\mathbb Z$. The following folklore result holds. {{beginthm|Lemma}} Assume $N$ is a closed orientable connected $n$-manifold, $n$ is even and $H_1(N)$ is torsion free. Then for each embedding $f\colon N_0 \to \mathbb R^{2n-1}$ there exists a nowhere vanishing normal vector field to $f(N_0)$. {{endthm}} {{beginproof}} There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$. A normal space to $f(N_0)$ at any point of $f(N_0)$ has dimension $n-1$. As $n$ is even thus $n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$. Since $H_1(N)$ is torsion free, it follows that $\bar e=0$. Since $N_0$ has non-empty boundary, we have that $N_0$ is homotopy equivalent to an $(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$ at any point of $f(N_0)$. Since $\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$. {{endproof}} Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients. {{beginthm|Definition}} For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ denote $$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$ where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$. {{endthm}} {{beginthm|Lemma|($L$ is well-defined)}}\label{lmm:L_well_def} For even $n$ and every embedding $f\colon N_0\to\mathbb R^{2n-1}$ the integer $L(f)(x, y)$: * is well-defined, i.e. does not change when $s$ is replaced by $s'$, * does not change when $x$ or $y$ are changed to homologous cycles and, * does not change when $f$ is changed to an isotopic embedding. {{endthm}} The first bullet was stated and proved in unpublished update of \cite{Tonkonog2010}, other two bullets are simple. We will need the following supporting lemma. {{beginthm|Lemma}}\label{lmm::saeki} Let $f:N_0\to \mathbb R^{2n-1}$ be an embedding. Let $s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$. Then $$\mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$ where $s(y)$ is the result of the shift of $f(y)$ by $s$, and $d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$ being homotopic in the class of the nowhere vanishing vector fields. {{endthm}} This Lemma is proved in \cite[Lemma 2.2]{Saeki1999} for $n=3$, but the proof is valid in all dimensions.
'''Proof of Lemma \ref{lmm:L_well_def}.''' The first bullet point follows because: $$ \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned} $$ Here the second equality follows from Lemma \ref{lmm::saeki}. For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$. Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy. {{endproof}} Lemma \ref{lmm:L_well_def} implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter. Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $. Define the dual to [[Stiefel-Whitney_characteristic_classes|Stiefel-Whitney class]] $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent. {{beginthm|Lemma}} \label{lmm::L_equality} Let $f:N_0\to \mathbb R^{2n-1}$ be an embedding. Then for every $X, Y \in H_{n-1}(N_0)$ the following equality holds: $$\rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$ {{endthm}} This Lemma was stated in a unpublished update of \cite{Tonkonog2010}, the following proof is obtained by M. Fedorov using the idea from that update. See also an analogous lemma for closed manifolds in \cite[Lemma 2.2]{Crowley&Skopenkov2016}.
'''Proof of Lemma \ref{lmm::L_equality}.''' Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get $$ \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \ &= d(s, -s)\cap X\cap Y . \end{aligned} $$ The first congruence is clear. The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift. The third equality follows from Lemma \ref{lmm::saeki}. Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get: $$ \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0). $$ The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$ and $-s$ degenerates only at those points $x$ where $s'(x)=s(x)$. These points $x$ are exactly points where $s'(x)$ and $s(x)$ are linearly dependent. All those point $x$ form a $-cycle modulo two in $N_0$. The homotopy class of this $-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class. {{endproof}} == Classification theorems == ; \label{sec::classification} Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary. 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. Denote $\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$ up to isotopy. 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). {{beginthm|Definition}} For each even $n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$, where $W$ is [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|Whitney invariant]], \cite[$\S]{Skopenkov2016e}. {{endthm}} {{beginthm|Lemma}} The invariant $W\Lambda$ is well-defined for $n\ge4$. {{endthm}} {{beginproof}} Note that [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that $\partial N_0$ unknots in $\mathbb R^{2n}$. Thus $f|_{\partial N_0}$ can be extended to embedding of an $n$-ball $B^n$ into $\mathbb R^{2n}$. [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that $n$-sphere unknots in $\mathbb R^{2n}$. Thus all extensions of $f$ are isotopic in PL category. Note also that if $f$ and $g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$ is invariant for PL embeddings. {{endproof}} {{beginthm|Definition|of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$ is even and $H_1(N)$ is torsion-free}}\label{DefG} Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$. For each $f$ such that $W\Lambda(f)=z$ define $$ G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right) $$ where $x,y\in H_{n-1}(N_0)$. {{endthm}} Note also that $G$ depends on choice of collection $\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$. {{beginthm|Theorem}}\label{Tlink} Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. The map $$ G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}), $$ is one-to-one. {{endthm}} {{beginthm|Lemma}} For each even $n\in H_{n-1}(N)$ and each $x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$. {{endthm}} An equivalemt statement of Theorem \ref{Tlink}: {{beginthm|Theorem}}\label{thm::punctured_class} Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. Then (a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection. (b) The image of $L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$ is the normal Stiefel-Whitney class. {{endthm}} This is the main Theorem of \cite{Tonkonog2010} == A generalization to highly-connected manifolds == ; \label{sec::generalisations} In this paragraph we consider only punctured manifolds to simplify statements of the theorems, but the results seem to hold for manifolds with arbitrary non-empty boundary. 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} Assume that $N$ is a closed compact $k$-connected $n$-manifold. (a) If $n\geq 2n+3$, then $N$ embeds into $\mathbb R^{2n-k}$. (b) Then $N_0$ embeds into $\mathbb R^{2n-k-1}$. {{endthm}} The Diff case of part (a) is in \cite[Existence Theorem (a)]{Haefliger1961}, the PL case of this result is in \cite[Theorem 1.1]{Penrose&Whitehead&Zeeman1961}, \cite[Corollary 1.3]{Irwin1965}. The Diff case of part (b) is in \cite[Corollary 4.2]{Hirsch1961a}. For the PL case see \cite[Theorem 1.2]{Penrose&Whitehead&Zeeman1961}. {{beginthm|Theorem}}\label{thm::k_connect_unknot} Assume that $N$ is a closed $k$-connected $n$-manifold. (a) If $n\ge2k + 2$, $m \ge 2n - k + 1$, 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. {{endthm}} For part (a) see [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$ 2]{Skopenkov2016c}, or \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. For the PL case of part (b) see \cite[Theorem 10.3]{Hudson1969}, which is proved using [[Isotopy#Concordance|concordance implies isotopy theorem]]. For $k>1$ part (b) is corollary of Theorem \ref{thm::k_connect_classif} below (but can be proved in a straightforward way). For $k=0$ part (b) coincides with Theorem \ref{thm::unknotting}b.
'''Proof of Theorem \ref{thm::k_connect_unknot}(b) for $k=1$.''' By Theorem \ref{thm::isotop_unknot} below every two immersions of $N$ into $\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$ for each $x\in N$. Since $N$ is \S5 we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S7 which is independent from \S4, \S5 and \S6 we state generalisations of theorems from \S2 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, \S1, \S3]. In those pages mostly results for closed manifolds are stated.

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 [Seifert&Threlfall1980, \S 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that N is a compact connected n-manifold.

(a) Then N embeds into \mathbb R^{2n}.

(b) If N has non-empty boundary, then N embeds into \mathbb R^{2n-1}.

Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

Theorem 2.2. Assume that N is a compact connected n-manifold and either

(a) m \ge 2n+1 \ge 5 or

(b) N has non-empty boundary and m\geq 2n.

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

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, \S 2, Theorems 2.1, 2.2]. Part (b) in the case n>2 is proved in [Edwards1968, \S 4, Corollary 5]. The case n=1 is clear. The case n=2 can be proved using the ideas presented below.

The inequality in part (b) is sharp by Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see \S7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.

All theorems for manifolds with non-empty boundary stated in \S2 and \S7 can be proved using

  • analogous results for immersions of manifolds stated in \S9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in \S3.
  • handle decomposition, see e.g. the second proof of Theorem 2.1.b in \S3.

Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion g\colon N\to\mathbb R^{2n-1}. Since N is connected and has non-empty boundary, it follows that N collapses to an (n-1)-dimensional subcomplex X\subset N of some triangulation of N. Since 2(n-1) < 2n-1, by general position we may assume that g|_{X} is an embedding. Since g is an immersion, it follows that X has a sufficiently small tubular neighbourhood M\supset X such that g|_{M} is embedding. Since tubular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. The composition g\circ h is an embedding of N.
\square

For the second proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that N is a closed connected n-manifold. Then N_0 have handle decomposition with indices of attaching maps at most n-1.

Lemma 3.2. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, and f:U \to \mathbb R^{m} is an embedding.

(a) If m=2n-1, then there is an extension of f to an embedding of U\cup_{\phi} D^i\times D^{n-i}.

(b) Assume also that there is an embedding g : D^i\times 0 \to \mathbb R^m such that f\phi = g on \partial D^i \times 0. Suppose that on g(D^i\times 0) there is a field of n-i linear independed normal vectors whose restriction to f(\partial D^i\times 0) is tangent to f(U). Then f\cup g extends to a embedding U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m.

Second proof of Theorem 2.1.b assuming Lemma 3.1 and Lemma 3.2.(a). By Lemma 3.1.(a) there is a handle decomposition of N_0 with attaching maps \phi_1,\ldots,\phi_s of indices at most n-1. Denote by U^l the manifold obtained from \emptyset by the attaching first l handles. Take any embedding F_1:U^1 \cong D^n\to \R^{2n-1}. Let us define an embedding F_l of U^l using an embedding F_{l-1} of U^{l-1}. Since the index i of \phi_l is smaller than n, by Lemma 3.2 there is extension of F_{l-1} to an embedding F_l:U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}, where U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}.
\square
Proof of Lemma 3.2.(a) assuming Lemma 3.2.(b). Since i+n\leq 2n-1 and 2i+1\leq 2n-1, by general position there is an embedding g: D^i\times 0\to \mathbb R^{2n-1} such that f\phi = g on \partial D^i \times 0 and f(\mbox{Int} U) has a finite number of intersections points with g(\mbox{Int} D^i\times 0). Then by an isotopy g_t, where g_0=g, fixed on \partial D^i\times 0 we can "push out" the self-intersection points toward \partial U so that g_1(\mbox{Int} D^i\times 0) does not intersect f(\mbox{Int} U). Then f\cup g_1 is an embedding.

Denote by A the (n-i)\times n matrix whose rightmost (n-i)\times (n-i) block is the identity matrix, and whose other elements are zeros. Denote by v the field of n-i normal vectors on \partial D^i\times 0\subset D^i\times D^{n-i} such that the k-th vector has coordinates equal to the k-th row in A. Then the vector field d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i})) is tangent to \partial U. For x\in g_1(\partial D^i\times 0) denote by v'(x) the projection of df(d\phi(v(x))) to the intersection of normal space to g_1(\partial D^i\times 0) at x, and tangent space to f(\partial U) at x. Since i-1<2n-1 - (n-i), it follows that \pi_{i-1}(V_{2n-1, n-i})=0. Hence there is an extension of v' to an linearly independent field of vectors normal to g_1(D^i\times 0). Then by Lemma 3.2.(b) there is an extension of f\cup g_1 to an embedding of U \cup_{\phi} D^i\times D^{n-i}.

\square

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).

Lemma 3.3. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m are embeddings and F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1] is a concordance between f_0|_U and f_1|_U.

(a) If m\geq 2n, then there is an extension of F to a concordance between f_0 and f_1.

(b) Assume also that there is a concordance G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1] between f_0|_{D^i\times 0} and f_1|_{D^i\times 0}, and on G(D^i\times 0\times [0, 1]) there is a field of n-i linear independent normal vectors whose restrictions to G(\partial D^i\times 0\times [0, 1]), to G(D^i\times 0\times 0), and to G(D^i\times 0\times 1) are tangent to
\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,
respectively. Then F\cup G extends to a concordance between f_0 and f_1.
Proof of the Theorem 2.2 assuming Lemma 3.1 and Lemma 3.3.(a). Denote by f_0, f_1 any two embeddings of N_0 into \mathbb{R}^m. By the Concordance Implies Isotopy Theorem it suffices to prove that there is a concordance between f_0 and f_1. In order to prove that observe that by Lemma 3.1 there is a handle decomposition of N_0 with attaching maps of indices at most n-1. Denote by U^l the manifold obtained from U^0=\emptyset by the attaching the first l handles, starting with? U^1\cong D^n. Define a concordance F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1] recursively. Take any concordance F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1] between f_0|_{U^1} and f_1|_{U^1}. Let us define a concordance F_l between f_0|_{U^l} and f_1|_{U^l} using a concordance F_{l-1} of U^{l-1} between f_0|_{U^{l-1}} and f_1|_{U^{l-1}}. For this denote by \phi:\partial D^i\times D^{n-i}\to \partial U^{l-1} the l-th attaching map. Since i\leq n-1, by Lemma 3.3.(a) there is an extension of F_{l-1} to a concordance
\displaystyle F_{l}:U^{l}\times [0, 1]\to\mathbb{R}^m\times [0, 1]
between the restriction of f_0 and f_1 to U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}.
\square
Proof of Lemma 3.3.(a) assuming Lemma 3.3.(b). In the following text we identify D^i\times D^{n-i}\times [0, 1] and D^i\times [0, 1]\times D^{n-i}. Define map
\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1
by the formula: 
Tex syntax error
Since
\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\quad\text{and}\quad 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,
by general position there is an embedding
\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]
such that F\bar{\phi} = G on \partial (D^i \times 0\times [0, 1]) and F(\mbox{Int} (U\times [0, 1])) has a finite number of intersection points with G(\mbox{Int}( D^i\times 0\times [0, 1])). Then by an isotopy G_t, where G_0=G, fixed on \partial (D^i\times 0\times [0, 1]) we can "push out" the self-intersection points toward F(\partial (U\times [0, 1])) so that G_1(D^i\times 0\times [0, 1]) does not intersect G(U\times [0, 1]). Then F\cup G_1 is an concordance between the restrictions of f_0 and f_1 on U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0.

Denote by A the (n-i)\times (n+1) matrix whose rightmost (n-i)\times (n-i) submatrix is the identity matrix, and whose other elements are zeroes. Denote by v the field of n-i normal vectors on \partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1] whose k-th vector has coordinates equal to the k-th row in A. Then d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i})) is the vector field tangent to \bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i}). For x\in G_1(\partial (D^i\times 0\times [0, 1]) denote by v'(x) the projection of d(F\cup f_0\cup f_1)d\phi (v(x)) to the intersection of normal space to G_1(D^i\times 0\times [0, 1]) at x, and tangent space to F(\partial(U\times [0, 1])) at x. Since i<m+1-(n-i), it follows that \pi_{i}(V_{m+1, n-i})=0. Hence there is an extension of v' to a linear independent field of vectors normal to G_1(D^i\times 0\times [0, 1]). Then by Lemma 3.3.(b) there is an extension of F\cup G_1 to a concordance (U \cup_{\phi} D^i\times D^{n-i})\times [0, 1].

\square


4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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}.

(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 well-defined and is a bijection for k\geqslant2.

Proof. 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).

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 the standard embedding.Thus \mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a.

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 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.
\square
(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.

Denote 1_k:=(1,0,\ldots,0)\in S^k.

Example 4.2. 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).

The surjectivity of l is given analogously to Proposition 4.1(b). The injectivity of l follows from forgetful bijection \mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1] between embeddings of N_0 and a cylinder.

This example shows that Theorem 7.4 fails for k=0.

Example 4.3. Let N=S^k_a\times S^1 \# S^k_b\times S^1 be the connected sum of two tori. Then there exists a surjection l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z defined by the formula l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1)).

To prove the surjectivity of l it is sufficient to take linked k-spheres in \mathbb R^{2k+1} and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (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). Figure (c) shows that l(f_1)=1 and l(f_2)=0 which proves the intuitive fact that f_1 and f_2 are not isotopic. (Notice that the restrictions of f_1 and f_2 on x\cup y are isotopic!) If we use the opposite normal vector field s'=-s, the values of l(f_1) and l(f_2) will change but will still be different (see figure (d)).

5 Seifert linking form

For a simpler invariant see [Skopenkov2022] and references therein.

In this section assume that

  • N is any closed orientable connected n-manifold,
  • f\colon N_0 \to \mathbb R^{2n-1} is any embedding,
  • if the (co)homology coefficients are omitted, then they are \mathbb Z,
  • n is even and H_1(N) is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).

By N_0 we denote the closure of the complement in N to an closed n-ball. Thus \partial N_0 is the (n-1)-sphere.

Lemma 5.1. There exists a nowhere vanishing normal vector field to f(N_0).

This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].

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

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

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

Lemma 5.2 (L is well-defined). The integer L(f)(x, y):

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

The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.

Lemma 5.2 implies that L(f) generates a bilinear form

\displaystyle L(f):H_{n-1}(N_0)\times H_{n-1}(N_0)\to\Z

denoted by the same letter and called Seifert linking form.

Denote by \rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2) the reduction modulo 2. Define the dual to Stiefel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2) to be the class of the cycle on which two general position normal fields to f(N_0) are linearly dependent.

Lemma 5.3. For every X, Y \in H_{n-1}(N_0) the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let 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. 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).

Definition 6.1. For each even n define an invariant W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2). For each embedding f\colon N_0\to\mathbb R^{2n-1} construct any PL embedding g\colon N\to\mathbb R^{2n} by adding a cone over f(\partial N_0). Now let W\Lambda([f]) = W(g), where W is Whitney invariant, [Skopenkov2016e, \S5].

Lemma 6.2. The invariant W\Lambda is well-defined for n\ge4.

Proof. Note that Unknotting Spheres Theorem implies that \partial N_0 unknots in \mathbb R^{2n}. Thus f|_{\partial N_0} can be extended to embedding of an n-ball B^n into \mathbb R^{2n}. Unknotting Spheres Theorem implies that n-sphere unknots in \mathbb R^{2n}. Thus all extensions of f are isotopic in PL category. Note also that if f and g are isotopic then their extensions are isotopic as well. And Whitney invariant W is invariant for PL embeddings.

\square

Definition 6.3 of G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) if n is even and H_1(N) is torsion-free. Take a collection \{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0 such that W\Lambda(f_z)=z. For each f such that W\Lambda(f)=z define

\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)

where x,y\in H_{n-1}(N_0).

Note also that G depends on choice of collection \{f_z\}. The following Theorems hold for any choice of \{f_z\}.

Theorem 6.4. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n 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}),

is one-to-one.

Lemma 6.5. For each even n\in H_{n-1}(N) and each x the following equality holds: W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right).

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. Then

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

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

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see \S8 for a generalization.

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.

Theorem 7.1. Assume that N is a closed k-connected n-manifold.

(a) If n\geq 2k+3, then N embeds into \mathbb R^{2n-k}.

(b) If n\geq 2k+2 and k\geq0, then N_0 embeds into \mathbb R^{2n-k-1}.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that N is a closed k-connected n-manifold.

(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 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.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, \S 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 k>1 part (b) is a corollary of Theorem 7.4 below. For k=0 part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for k=1. By Theorem 9.6 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 8.3 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 concordance implies isotopy Theorem f and g are isotopic.
\square

Conjecture 7.3. Assume that N is a closed 1-connected 4-manifold. Then any two embeddings of N_0 in \mathbb R^7 are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume N is a closed k-connected n-manifold. Then for each k\ge1 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)}),

where \mathbb Z_{(s)} denote \mathbb Z for s even and \mathbb Z_2 for s odd.

For definition of W_0' and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(W_0')]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes \mathrm{Emb}^{2n-1}(N_0) and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. 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}.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion f\colon N\to\mathbb R^{2n-k-1}. Since N is k-connected it follows from Theorem 8.3 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.
\square

Theorem 8.2. 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.

Proof is similar to the proof of theorem 7.2.

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.

Theorem 8.3. 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.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in \R^m is homotopy equivalent to the space of linear monomorphisms from TM to \R^m.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]]

If N is immersible in \R^{m+r} with a normal 
Tex syntax error
-field, then N is immersible in \R^m.

Theorem 9.3. Every n-manifold N with non-empty boundary is immersible in \R^{2n-1}.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every n-manifold N is immersible in \R^{2n-1}.

Denote by V_{m,n} is Stiefel manifold of n-frames in \R^m.

Theorem 9.5. Suppose N is a n-manifold with non-empty boudary, (N,\partial N) is k-connected. Then N is immersible in \R^m for each m\geq2n-k-1.

Proof. It suffices to show that exists an immersion of N in \mathbb R^{2n-k-1}. It suffices to show that exists a linear monomorphism from TM to \mathbb R^{2n-k-1}. Let us construct such a linear monomorphism by skeleta of N. It is clear that a linear monomorphism exists on 0-skeleton of N.

The obstruction to extend the linear monomorphism from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n})).

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, \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.

\square

Theorem 9.6. Suppose N is a connected n-manifold with non-empty boudary, (N, \partial N) is k-connected and m\geq2n-k. Then every two immersions of N in \R^m are regulary homotopic.

Proof.

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 
Tex syntax error
-skeleton of N. It is clear that homotopy exists on 0-skeleton of N. The obstruction to extend the homotopy from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n})).

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, \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.

\square

10 References

$-connected and has non-empty boundary, it follows that $N$ collapses to an $(n-2)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $F|_{X\times[0,1]}$ is an embedding, because (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 $h\colon N\to M$. 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}} Denote by $\mathrm{Emb}^{m}N_0$ the set embeddings of $N_0$ into $\mathbb R^{m}$ up to isotopy. {{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 $$ W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}), $$ where $\mathbb Z_{(s)}$ denote $\mathbb Z$ for $s$ even and $\mathbb Z_2$ for $s$ odd. {{endthm}} For definition of $W_0'$ and the proof of the latter Theorem see \cite[Lemma 2.2($W_0'$)]{Skopenkov2010}. See also \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. == Comments == ; {{beginthm|Theorem}}[Smale-Hirsch] The space of immersions of a manifold in $\mathbb R^{m}$ is homotopically equivalent to the space of linear monomorphisms from $TM$ to $\mathrm R^{m}$. {{endthm}} See \cite{Hirsch1959} and \cite{Haefliger&Poenaru1964}. {{beginthm|Theorem}}\label{thm::imm_dec_dim} If $N$ is immersible in $\mathbb R^{m+r}$ with a transversal $r$-field then it is immersible in $\mathbb R^{m}$. {{endthm}} This is \cite[Theorem 6.4]{Hirsch1959}. {{beginthm|Theorem}} Every $n$-manifold $N$ with non-empty boundary is immersible in $\mathbb R^{2n-1}$. {{endthm}} {{beginthm|Theorem}}[Whitney] Every $n$-manifold $N$ is immersible in $\mathbb R^{2n-1}$. {{endthm}} See \cite[Theorem 6.6]{Hirsch1961a}. {{beginthm|Theorem}} Every $k$-connected $n$-manifold $N$ with non-empty boudary is immersible in $\mathbb R^{m}$ for each $m\geq2n-k-1$. {{endthm}} {{beginproof}} It suffices to show that exists an immersion of $N$ in $\mathbb R^{2n-k-1}$. 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 5 we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S7 which is independent from \S4, \S5 and \S6 we state generalisations of theorems from \S2 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, \S1, \S3]. In those pages mostly results for closed manifolds are stated.

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 [Seifert&Threlfall1980, \S 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that N is a compact connected n-manifold.

(a) Then N embeds into \mathbb R^{2n}.

(b) If N has non-empty boundary, then N embeds into \mathbb R^{2n-1}.

Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

Theorem 2.2. Assume that N is a compact connected n-manifold and either

(a) m \ge 2n+1 \ge 5 or

(b) N has non-empty boundary and m\geq 2n.

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

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, \S 2, Theorems 2.1, 2.2]. Part (b) in the case n>2 is proved in [Edwards1968, \S 4, Corollary 5]. The case n=1 is clear. The case n=2 can be proved using the ideas presented below.

The inequality in part (b) is sharp by Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see \S7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.

All theorems for manifolds with non-empty boundary stated in \S2 and \S7 can be proved using

  • analogous results for immersions of manifolds stated in \S9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in \S3.
  • handle decomposition, see e.g. the second proof of Theorem 2.1.b in \S3.

Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion g\colon N\to\mathbb R^{2n-1}. Since N is connected and has non-empty boundary, it follows that N collapses to an (n-1)-dimensional subcomplex X\subset N of some triangulation of N. Since 2(n-1) < 2n-1, by general position we may assume that g|_{X} is an embedding. Since g is an immersion, it follows that X has a sufficiently small tubular neighbourhood M\supset X such that g|_{M} is embedding. Since tubular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. The composition g\circ h is an embedding of N.
\square

For the second proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that N is a closed connected n-manifold. Then N_0 have handle decomposition with indices of attaching maps at most n-1.

Lemma 3.2. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, and f:U \to \mathbb R^{m} is an embedding.

(a) If m=2n-1, then there is an extension of f to an embedding of U\cup_{\phi} D^i\times D^{n-i}.

(b) Assume also that there is an embedding g : D^i\times 0 \to \mathbb R^m such that f\phi = g on \partial D^i \times 0. Suppose that on g(D^i\times 0) there is a field of n-i linear independed normal vectors whose restriction to f(\partial D^i\times 0) is tangent to f(U). Then f\cup g extends to a embedding U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m.

Second proof of Theorem 2.1.b assuming Lemma 3.1 and Lemma 3.2.(a). By Lemma 3.1.(a) there is a handle decomposition of N_0 with attaching maps \phi_1,\ldots,\phi_s of indices at most n-1. Denote by U^l the manifold obtained from \emptyset by the attaching first l handles. Take any embedding F_1:U^1 \cong D^n\to \R^{2n-1}. Let us define an embedding F_l of U^l using an embedding F_{l-1} of U^{l-1}. Since the index i of \phi_l is smaller than n, by Lemma 3.2 there is extension of F_{l-1} to an embedding F_l:U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}, where U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}.
\square
Proof of Lemma 3.2.(a) assuming Lemma 3.2.(b). Since i+n\leq 2n-1 and 2i+1\leq 2n-1, by general position there is an embedding g: D^i\times 0\to \mathbb R^{2n-1} such that f\phi = g on \partial D^i \times 0 and f(\mbox{Int} U) has a finite number of intersections points with g(\mbox{Int} D^i\times 0). Then by an isotopy g_t, where g_0=g, fixed on \partial D^i\times 0 we can "push out" the self-intersection points toward \partial U so that g_1(\mbox{Int} D^i\times 0) does not intersect f(\mbox{Int} U). Then f\cup g_1 is an embedding.

Denote by A the (n-i)\times n matrix whose rightmost (n-i)\times (n-i) block is the identity matrix, and whose other elements are zeros. Denote by v the field of n-i normal vectors on \partial D^i\times 0\subset D^i\times D^{n-i} such that the k-th vector has coordinates equal to the k-th row in A. Then the vector field d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i})) is tangent to \partial U. For x\in g_1(\partial D^i\times 0) denote by v'(x) the projection of df(d\phi(v(x))) to the intersection of normal space to g_1(\partial D^i\times 0) at x, and tangent space to f(\partial U) at x. Since i-1<2n-1 - (n-i), it follows that \pi_{i-1}(V_{2n-1, n-i})=0. Hence there is an extension of v' to an linearly independent field of vectors normal to g_1(D^i\times 0). Then by Lemma 3.2.(b) there is an extension of f\cup g_1 to an embedding of U \cup_{\phi} D^i\times D^{n-i}.

\square

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).

Lemma 3.3. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m are embeddings and F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1] is a concordance between f_0|_U and f_1|_U.

(a) If m\geq 2n, then there is an extension of F to a concordance between f_0 and f_1.

(b) Assume also that there is a concordance G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1] between f_0|_{D^i\times 0} and f_1|_{D^i\times 0}, and on G(D^i\times 0\times [0, 1]) there is a field of n-i linear independent normal vectors whose restrictions to G(\partial D^i\times 0\times [0, 1]), to G(D^i\times 0\times 0), and to G(D^i\times 0\times 1) are tangent to
\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,
respectively. Then F\cup G extends to a concordance between f_0 and f_1.
Proof of the Theorem 2.2 assuming Lemma 3.1 and Lemma 3.3.(a). Denote by f_0, f_1 any two embeddings of N_0 into \mathbb{R}^m. By the Concordance Implies Isotopy Theorem it suffices to prove that there is a concordance between f_0 and f_1. In order to prove that observe that by Lemma 3.1 there is a handle decomposition of N_0 with attaching maps of indices at most n-1. Denote by U^l the manifold obtained from U^0=\emptyset by the attaching the first l handles, starting with? U^1\cong D^n. Define a concordance F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1] recursively. Take any concordance F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1] between f_0|_{U^1} and f_1|_{U^1}. Let us define a concordance F_l between f_0|_{U^l} and f_1|_{U^l} using a concordance F_{l-1} of U^{l-1} between f_0|_{U^{l-1}} and f_1|_{U^{l-1}}. For this denote by \phi:\partial D^i\times D^{n-i}\to \partial U^{l-1} the l-th attaching map. Since i\leq n-1, by Lemma 3.3.(a) there is an extension of F_{l-1} to a concordance
\displaystyle F_{l}:U^{l}\times [0, 1]\to\mathbb{R}^m\times [0, 1]
between the restriction of f_0 and f_1 to U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}.
\square
Proof of Lemma 3.3.(a) assuming Lemma 3.3.(b). In the following text we identify D^i\times D^{n-i}\times [0, 1] and D^i\times [0, 1]\times D^{n-i}. Define map
\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1
by the formula: 
Tex syntax error
Since
\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\quad\text{and}\quad 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,
by general position there is an embedding
\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]
such that F\bar{\phi} = G on \partial (D^i \times 0\times [0, 1]) and F(\mbox{Int} (U\times [0, 1])) has a finite number of intersection points with G(\mbox{Int}( D^i\times 0\times [0, 1])). Then by an isotopy G_t, where G_0=G, fixed on \partial (D^i\times 0\times [0, 1]) we can "push out" the self-intersection points toward F(\partial (U\times [0, 1])) so that G_1(D^i\times 0\times [0, 1]) does not intersect G(U\times [0, 1]). Then F\cup G_1 is an concordance between the restrictions of f_0 and f_1 on U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0.

Denote by A the (n-i)\times (n+1) matrix whose rightmost (n-i)\times (n-i) submatrix is the identity matrix, and whose other elements are zeroes. Denote by v the field of n-i normal vectors on \partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1] whose k-th vector has coordinates equal to the k-th row in A. Then d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i})) is the vector field tangent to \bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i}). For x\in G_1(\partial (D^i\times 0\times [0, 1]) denote by v'(x) the projection of d(F\cup f_0\cup f_1)d\phi (v(x)) to the intersection of normal space to G_1(D^i\times 0\times [0, 1]) at x, and tangent space to F(\partial(U\times [0, 1])) at x. Since i<m+1-(n-i), it follows that \pi_{i}(V_{m+1, n-i})=0. Hence there is an extension of v' to a linear independent field of vectors normal to G_1(D^i\times 0\times [0, 1]). Then by Lemma 3.3.(b) there is an extension of F\cup G_1 to a concordance (U \cup_{\phi} D^i\times D^{n-i})\times [0, 1].

\square


4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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}.

(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 well-defined and is a bijection for k\geqslant2.

Proof. 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).

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 the standard embedding.Thus \mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a.

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 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.
\square
(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.

Denote 1_k:=(1,0,\ldots,0)\in S^k.

Example 4.2. 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).

The surjectivity of l is given analogously to Proposition 4.1(b). The injectivity of l follows from forgetful bijection \mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1] between embeddings of N_0 and a cylinder.

This example shows that Theorem 7.4 fails for k=0.

Example 4.3. Let N=S^k_a\times S^1 \# S^k_b\times S^1 be the connected sum of two tori. Then there exists a surjection l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z defined by the formula l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1)).

To prove the surjectivity of l it is sufficient to take linked k-spheres in \mathbb R^{2k+1} and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (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). Figure (c) shows that l(f_1)=1 and l(f_2)=0 which proves the intuitive fact that f_1 and f_2 are not isotopic. (Notice that the restrictions of f_1 and f_2 on x\cup y are isotopic!) If we use the opposite normal vector field s'=-s, the values of l(f_1) and l(f_2) will change but will still be different (see figure (d)).

5 Seifert linking form

For a simpler invariant see [Skopenkov2022] and references therein.

In this section assume that

  • N is any closed orientable connected n-manifold,
  • f\colon N_0 \to \mathbb R^{2n-1} is any embedding,
  • if the (co)homology coefficients are omitted, then they are \mathbb Z,
  • n is even and H_1(N) is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).

By N_0 we denote the closure of the complement in N to an closed n-ball. Thus \partial N_0 is the (n-1)-sphere.

Lemma 5.1. There exists a nowhere vanishing normal vector field to f(N_0).

This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].

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

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

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

Lemma 5.2 (L is well-defined). The integer L(f)(x, y):

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

The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.

Lemma 5.2 implies that L(f) generates a bilinear form

\displaystyle L(f):H_{n-1}(N_0)\times H_{n-1}(N_0)\to\Z

denoted by the same letter and called Seifert linking form.

Denote by \rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2) the reduction modulo 2. Define the dual to Stiefel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2) to be the class of the cycle on which two general position normal fields to f(N_0) are linearly dependent.

Lemma 5.3. For every X, Y \in H_{n-1}(N_0) the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let 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. 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).

Definition 6.1. For each even n define an invariant W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2). For each embedding f\colon N_0\to\mathbb R^{2n-1} construct any PL embedding g\colon N\to\mathbb R^{2n} by adding a cone over f(\partial N_0). Now let W\Lambda([f]) = W(g), where W is Whitney invariant, [Skopenkov2016e, \S5].

Lemma 6.2. The invariant W\Lambda is well-defined for n\ge4.

Proof. Note that Unknotting Spheres Theorem implies that \partial N_0 unknots in \mathbb R^{2n}. Thus f|_{\partial N_0} can be extended to embedding of an n-ball B^n into \mathbb R^{2n}. Unknotting Spheres Theorem implies that n-sphere unknots in \mathbb R^{2n}. Thus all extensions of f are isotopic in PL category. Note also that if f and g are isotopic then their extensions are isotopic as well. And Whitney invariant W is invariant for PL embeddings.

\square

Definition 6.3 of G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) if n is even and H_1(N) is torsion-free. Take a collection \{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0 such that W\Lambda(f_z)=z. For each f such that W\Lambda(f)=z define

\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)

where x,y\in H_{n-1}(N_0).

Note also that G depends on choice of collection \{f_z\}. The following Theorems hold for any choice of \{f_z\}.

Theorem 6.4. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n 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}),

is one-to-one.

Lemma 6.5. For each even n\in H_{n-1}(N) and each x the following equality holds: W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right).

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. Then

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

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

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see \S8 for a generalization.

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.

Theorem 7.1. Assume that N is a closed k-connected n-manifold.

(a) If n\geq 2k+3, then N embeds into \mathbb R^{2n-k}.

(b) If n\geq 2k+2 and k\geq0, then N_0 embeds into \mathbb R^{2n-k-1}.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that N is a closed k-connected n-manifold.

(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 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.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, \S 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 k>1 part (b) is a corollary of Theorem 7.4 below. For k=0 part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for k=1. By Theorem 9.6 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 8.3 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 concordance implies isotopy Theorem f and g are isotopic.
\square

Conjecture 7.3. Assume that N is a closed 1-connected 4-manifold. Then any two embeddings of N_0 in \mathbb R^7 are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume N is a closed k-connected n-manifold. Then for each k\ge1 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)}),

where \mathbb Z_{(s)} denote \mathbb Z for s even and \mathbb Z_2 for s odd.

For definition of W_0' and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(W_0')]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes \mathrm{Emb}^{2n-1}(N_0) and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. 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}.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion f\colon N\to\mathbb R^{2n-k-1}. Since N is k-connected it follows from Theorem 8.3 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.
\square

Theorem 8.2. 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.

Proof is similar to the proof of theorem 7.2.

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.

Theorem 8.3. 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.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in \R^m is homotopy equivalent to the space of linear monomorphisms from TM to \R^m.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]]

If N is immersible in \R^{m+r} with a normal 
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-field, then N is immersible in \R^m.

Theorem 9.3. Every n-manifold N with non-empty boundary is immersible in \R^{2n-1}.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every n-manifold N is immersible in \R^{2n-1}.

Denote by V_{m,n} is Stiefel manifold of n-frames in \R^m.

Theorem 9.5. Suppose N is a n-manifold with non-empty boudary, (N,\partial N) is k-connected. Then N is immersible in \R^m for each m\geq2n-k-1.

Proof. It suffices to show that exists an immersion of N in \mathbb R^{2n-k-1}. It suffices to show that exists a linear monomorphism from TM to \mathbb R^{2n-k-1}. Let us construct such a linear monomorphism by skeleta of N. It is clear that a linear monomorphism exists on 0-skeleton of N.

The obstruction to extend the linear monomorphism from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n})).

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, \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.

\square

Theorem 9.6. Suppose N is a connected n-manifold with non-empty boudary, (N, \partial N) is k-connected and m\geq2n-k. Then every two immersions of N in \R^m are regulary homotopic.

Proof.

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 
Tex syntax error
-skeleton of N. It is clear that homotopy exists on 0-skeleton of N. The obstruction to extend the homotopy from (r-1)-skeleton to 
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-skeleton lies in H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n})).

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, \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.

\square

10 References

$-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 $\mathbb R^{2n-k}$. 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. 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. {{endproof}} {{beginthm|Theorem}}\label{thm::isotop_unknot} Suppose $N$ is a $k$-connected $n$-manifold with non-empty boudary and $m\geq2n-k$. Then every two immersions of $N$ in $\mathbb R^m$ are regulary homotopic. {{endthm}} {{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 5 we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S7 which is independent from \S4, \S5 and \S6 we state generalisations of theorems from \S2 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, \S1, \S3]. In those pages mostly results for closed manifolds are stated.

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 [Seifert&Threlfall1980, \S 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that N is a compact connected n-manifold.

(a) Then N embeds into \mathbb R^{2n}.

(b) If N has non-empty boundary, then N embeds into \mathbb R^{2n-1}.

Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

Theorem 2.2. Assume that N is a compact connected n-manifold and either

(a) m \ge 2n+1 \ge 5 or

(b) N has non-empty boundary and m\geq 2n.

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

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, \S 2, Theorems 2.1, 2.2]. Part (b) in the case n>2 is proved in [Edwards1968, \S 4, Corollary 5]. The case n=1 is clear. The case n=2 can be proved using the ideas presented below.

The inequality in part (b) is sharp by Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see \S7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.

All theorems for manifolds with non-empty boundary stated in \S2 and \S7 can be proved using

  • analogous results for immersions of manifolds stated in \S9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in \S3.
  • handle decomposition, see e.g. the second proof of Theorem 2.1.b in \S3.

Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion g\colon N\to\mathbb R^{2n-1}. Since N is connected and has non-empty boundary, it follows that N collapses to an (n-1)-dimensional subcomplex X\subset N of some triangulation of N. Since 2(n-1) < 2n-1, by general position we may assume that g|_{X} is an embedding. Since g is an immersion, it follows that X has a sufficiently small tubular neighbourhood M\supset X such that g|_{M} is embedding. Since tubular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. The composition g\circ h is an embedding of N.
\square

For the second proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that N is a closed connected n-manifold. Then N_0 have handle decomposition with indices of attaching maps at most n-1.

Lemma 3.2. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, and f:U \to \mathbb R^{m} is an embedding.

(a) If m=2n-1, then there is an extension of f to an embedding of U\cup_{\phi} D^i\times D^{n-i}.

(b) Assume also that there is an embedding g : D^i\times 0 \to \mathbb R^m such that f\phi = g on \partial D^i \times 0. Suppose that on g(D^i\times 0) there is a field of n-i linear independed normal vectors whose restriction to f(\partial D^i\times 0) is tangent to f(U). Then f\cup g extends to a embedding U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m.

Second proof of Theorem 2.1.b assuming Lemma 3.1 and Lemma 3.2.(a). By Lemma 3.1.(a) there is a handle decomposition of N_0 with attaching maps \phi_1,\ldots,\phi_s of indices at most n-1. Denote by U^l the manifold obtained from \emptyset by the attaching first l handles. Take any embedding F_1:U^1 \cong D^n\to \R^{2n-1}. Let us define an embedding F_l of U^l using an embedding F_{l-1} of U^{l-1}. Since the index i of \phi_l is smaller than n, by Lemma 3.2 there is extension of F_{l-1} to an embedding F_l:U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}, where U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}.
\square
Proof of Lemma 3.2.(a) assuming Lemma 3.2.(b). Since i+n\leq 2n-1 and 2i+1\leq 2n-1, by general position there is an embedding g: D^i\times 0\to \mathbb R^{2n-1} such that f\phi = g on \partial D^i \times 0 and f(\mbox{Int} U) has a finite number of intersections points with g(\mbox{Int} D^i\times 0). Then by an isotopy g_t, where g_0=g, fixed on \partial D^i\times 0 we can "push out" the self-intersection points toward \partial U so that g_1(\mbox{Int} D^i\times 0) does not intersect f(\mbox{Int} U). Then f\cup g_1 is an embedding.

Denote by A the (n-i)\times n matrix whose rightmost (n-i)\times (n-i) block is the identity matrix, and whose other elements are zeros. Denote by v the field of n-i normal vectors on \partial D^i\times 0\subset D^i\times D^{n-i} such that the k-th vector has coordinates equal to the k-th row in A. Then the vector field d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i})) is tangent to \partial U. For x\in g_1(\partial D^i\times 0) denote by v'(x) the projection of df(d\phi(v(x))) to the intersection of normal space to g_1(\partial D^i\times 0) at x, and tangent space to f(\partial U) at x. Since i-1<2n-1 - (n-i), it follows that \pi_{i-1}(V_{2n-1, n-i})=0. Hence there is an extension of v' to an linearly independent field of vectors normal to g_1(D^i\times 0). Then by Lemma 3.2.(b) there is an extension of f\cup g_1 to an embedding of U \cup_{\phi} D^i\times D^{n-i}.

\square

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).

Lemma 3.3. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m are embeddings and F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1] is a concordance between f_0|_U and f_1|_U.

(a) If m\geq 2n, then there is an extension of F to a concordance between f_0 and f_1.

(b) Assume also that there is a concordance G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1] between f_0|_{D^i\times 0} and f_1|_{D^i\times 0}, and on G(D^i\times 0\times [0, 1]) there is a field of n-i linear independent normal vectors whose restrictions to G(\partial D^i\times 0\times [0, 1]), to G(D^i\times 0\times 0), and to G(D^i\times 0\times 1) are tangent to
\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,
respectively. Then F\cup G extends to a concordance between f_0 and f_1.
Proof of the Theorem 2.2 assuming Lemma 3.1 and Lemma 3.3.(a). Denote by f_0, f_1 any two embeddings of N_0 into \mathbb{R}^m. By the Concordance Implies Isotopy Theorem it suffices to prove that there is a concordance between f_0 and f_1. In order to prove that observe that by Lemma 3.1 there is a handle decomposition of N_0 with attaching maps of indices at most n-1. Denote by U^l the manifold obtained from U^0=\emptyset by the attaching the first l handles, starting with? U^1\cong D^n. Define a concordance F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1] recursively. Take any concordance F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1] between f_0|_{U^1} and f_1|_{U^1}. Let us define a concordance F_l between f_0|_{U^l} and f_1|_{U^l} using a concordance F_{l-1} of U^{l-1} between f_0|_{U^{l-1}} and f_1|_{U^{l-1}}. For this denote by \phi:\partial D^i\times D^{n-i}\to \partial U^{l-1} the l-th attaching map. Since i\leq n-1, by Lemma 3.3.(a) there is an extension of F_{l-1} to a concordance
\displaystyle F_{l}:U^{l}\times [0, 1]\to\mathbb{R}^m\times [0, 1]
between the restriction of f_0 and f_1 to U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}.
\square
Proof of Lemma 3.3.(a) assuming Lemma 3.3.(b). In the following text we identify D^i\times D^{n-i}\times [0, 1] and D^i\times [0, 1]\times D^{n-i}. Define map
\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1
by the formula: 
Tex syntax error
Since
\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\quad\text{and}\quad 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,
by general position there is an embedding
\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]
such that F\bar{\phi} = G on \partial (D^i \times 0\times [0, 1]) and F(\mbox{Int} (U\times [0, 1])) has a finite number of intersection points with G(\mbox{Int}( D^i\times 0\times [0, 1])). Then by an isotopy G_t, where G_0=G, fixed on \partial (D^i\times 0\times [0, 1]) we can "push out" the self-intersection points toward F(\partial (U\times [0, 1])) so that G_1(D^i\times 0\times [0, 1]) does not intersect G(U\times [0, 1]). Then F\cup G_1 is an concordance between the restrictions of f_0 and f_1 on U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0.

Denote by A the (n-i)\times (n+1) matrix whose rightmost (n-i)\times (n-i) submatrix is the identity matrix, and whose other elements are zeroes. Denote by v the field of n-i normal vectors on \partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1] whose k-th vector has coordinates equal to the k-th row in A. Then d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i})) is the vector field tangent to \bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i}). For x\in G_1(\partial (D^i\times 0\times [0, 1]) denote by v'(x) the projection of d(F\cup f_0\cup f_1)d\phi (v(x)) to the intersection of normal space to G_1(D^i\times 0\times [0, 1]) at x, and tangent space to F(\partial(U\times [0, 1])) at x. Since i<m+1-(n-i), it follows that \pi_{i}(V_{m+1, n-i})=0. Hence there is an extension of v' to a linear independent field of vectors normal to G_1(D^i\times 0\times [0, 1]). Then by Lemma 3.3.(b) there is an extension of F\cup G_1 to a concordance (U \cup_{\phi} D^i\times D^{n-i})\times [0, 1].

\square


4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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}.

(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 well-defined and is a bijection for k\geqslant2.

Proof. 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).

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 the standard embedding.Thus \mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a.

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 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.
\square
(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.

Denote 1_k:=(1,0,\ldots,0)\in S^k.

Example 4.2. 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).

The surjectivity of l is given analogously to Proposition 4.1(b). The injectivity of l follows from forgetful bijection \mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1] between embeddings of N_0 and a cylinder.

This example shows that Theorem 7.4 fails for k=0.

Example 4.3. Let N=S^k_a\times S^1 \# S^k_b\times S^1 be the connected sum of two tori. Then there exists a surjection l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z defined by the formula l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1)).

To prove the surjectivity of l it is sufficient to take linked k-spheres in \mathbb R^{2k+1} and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (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). Figure (c) shows that l(f_1)=1 and l(f_2)=0 which proves the intuitive fact that f_1 and f_2 are not isotopic. (Notice that the restrictions of f_1 and f_2 on x\cup y are isotopic!) If we use the opposite normal vector field s'=-s, the values of l(f_1) and l(f_2) will change but will still be different (see figure (d)).

5 Seifert linking form

For a simpler invariant see [Skopenkov2022] and references therein.

In this section assume that

  • N is any closed orientable connected n-manifold,
  • f\colon N_0 \to \mathbb R^{2n-1} is any embedding,
  • if the (co)homology coefficients are omitted, then they are \mathbb Z,
  • n is even and H_1(N) is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).

By N_0 we denote the closure of the complement in N to an closed n-ball. Thus \partial N_0 is the (n-1)-sphere.

Lemma 5.1. There exists a nowhere vanishing normal vector field to f(N_0).

This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].

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

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

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

Lemma 5.2 (L is well-defined). The integer L(f)(x, y):

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

The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.

Lemma 5.2 implies that L(f) generates a bilinear form

\displaystyle L(f):H_{n-1}(N_0)\times H_{n-1}(N_0)\to\Z

denoted by the same letter and called Seifert linking form.

Denote by \rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2) the reduction modulo 2. Define the dual to Stiefel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2) to be the class of the cycle on which two general position normal fields to f(N_0) are linearly dependent.

Lemma 5.3. For every X, Y \in H_{n-1}(N_0) the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let 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. 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).

Definition 6.1. For each even n define an invariant W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2). For each embedding f\colon N_0\to\mathbb R^{2n-1} construct any PL embedding g\colon N\to\mathbb R^{2n} by adding a cone over f(\partial N_0). Now let W\Lambda([f]) = W(g), where W is Whitney invariant, [Skopenkov2016e, \S5].

Lemma 6.2. The invariant W\Lambda is well-defined for n\ge4.

Proof. Note that Unknotting Spheres Theorem implies that \partial N_0 unknots in \mathbb R^{2n}. Thus f|_{\partial N_0} can be extended to embedding of an n-ball B^n into \mathbb R^{2n}. Unknotting Spheres Theorem implies that n-sphere unknots in \mathbb R^{2n}. Thus all extensions of f are isotopic in PL category. Note also that if f and g are isotopic then their extensions are isotopic as well. And Whitney invariant W is invariant for PL embeddings.

\square

Definition 6.3 of G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) if n is even and H_1(N) is torsion-free. Take a collection \{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0 such that W\Lambda(f_z)=z. For each f such that W\Lambda(f)=z define

\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)

where x,y\in H_{n-1}(N_0).

Note also that G depends on choice of collection \{f_z\}. The following Theorems hold for any choice of \{f_z\}.

Theorem 6.4. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n 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}),

is one-to-one.

Lemma 6.5. For each even n\in H_{n-1}(N) and each x the following equality holds: W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right).

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. Then

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

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

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see \S8 for a generalization.

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.

Theorem 7.1. Assume that N is a closed k-connected n-manifold.

(a) If n\geq 2k+3, then N embeds into \mathbb R^{2n-k}.

(b) If n\geq 2k+2 and k\geq0, then N_0 embeds into \mathbb R^{2n-k-1}.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that N is a closed k-connected n-manifold.

(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 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.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, \S 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 k>1 part (b) is a corollary of Theorem 7.4 below. For k=0 part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for k=1. By Theorem 9.6 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 8.3 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 concordance implies isotopy Theorem f and g are isotopic.
\square

Conjecture 7.3. Assume that N is a closed 1-connected 4-manifold. Then any two embeddings of N_0 in \mathbb R^7 are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume N is a closed k-connected n-manifold. Then for each k\ge1 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)}),

where \mathbb Z_{(s)} denote \mathbb Z for s even and \mathbb Z_2 for s odd.

For definition of W_0' and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(W_0')]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes \mathrm{Emb}^{2n-1}(N_0) and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. 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}.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion f\colon N\to\mathbb R^{2n-k-1}. Since N is k-connected it follows from Theorem 8.3 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.
\square

Theorem 8.2. 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.

Proof is similar to the proof of theorem 7.2.

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.

Theorem 8.3. 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.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in \R^m is homotopy equivalent to the space of linear monomorphisms from TM to \R^m.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]]

If N is immersible in \R^{m+r} with a normal 
Tex syntax error
-field, then N is immersible in \R^m.

Theorem 9.3. Every n-manifold N with non-empty boundary is immersible in \R^{2n-1}.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every n-manifold N is immersible in \R^{2n-1}.

Denote by V_{m,n} is Stiefel manifold of n-frames in \R^m.

Theorem 9.5. Suppose N is a n-manifold with non-empty boudary, (N,\partial N) is k-connected. Then N is immersible in \R^m for each m\geq2n-k-1.

Proof. It suffices to show that exists an immersion of N in \mathbb R^{2n-k-1}. It suffices to show that exists a linear monomorphism from TM to \mathbb R^{2n-k-1}. Let us construct such a linear monomorphism by skeleta of N. It is clear that a linear monomorphism exists on 0-skeleton of N.

The obstruction to extend the linear monomorphism from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n})).

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, \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.

\square

Theorem 9.6. Suppose N is a connected n-manifold with non-empty boudary, (N, \partial N) is k-connected and m\geq2n-k. Then every two immersions of N in \R^m are regulary homotopic.

Proof.

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 
Tex syntax error
-skeleton of N. It is clear that homotopy exists on 0-skeleton of N. The obstruction to extend the homotopy from (r-1)-skeleton to 
Tex syntax error
-skeleton lies in H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n})).

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, \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.

\square

10 References

$-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 $\mathbb R^{2n-k}$. 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. Thus the obstruction is always zero and such homotopy of linear monomorphisms exists. {{endproof}} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S5 we introduce an invariant of embedding of a n-manifold in (n-1)-space for even n. In \S7 which is independent from \S4, \S5 and \S6 we state generalisations of theorems from \S2 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, \S1, \S3]. In those pages mostly results for closed manifolds are stated.

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 [Seifert&Threlfall1980, \S 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, \S 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that N is a compact connected n-manifold.

(a) Then N embeds into \mathbb R^{2n}.

(b) If N has non-empty boundary, then N embeds into \mathbb R^{2n-1}.

Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

Theorem 2.2. Assume that N is a compact connected n-manifold and either

(a) m \ge 2n+1 \ge 5 or

(b) N has non-empty boundary and m\geq 2n.

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

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, \S 2, Theorems 2.1, 2.2]. Part (b) in the case n>2 is proved in [Edwards1968, \S 4, Corollary 5]. The case n=1 is clear. The case n=2 can be proved using the ideas presented below.

The inequality in part (b) is sharp by Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see \S7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.

All theorems for manifolds with non-empty boundary stated in \S2 and \S7 can be proved using

  • analogous results for immersions of manifolds stated in \S9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in \S3.
  • handle decomposition, see e.g. the second proof of Theorem 2.1.b in \S3.

Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion g\colon N\to\mathbb R^{2n-1}. Since N is connected and has non-empty boundary, it follows that N collapses to an (n-1)-dimensional subcomplex X\subset N of some triangulation of N. Since 2(n-1) < 2n-1, by general position we may assume that g|_{X} is an embedding. Since g is an immersion, it follows that X has a sufficiently small tubular neighbourhood M\supset X such that g|_{M} is embedding. Since tubular neighbourhood is unique up to homeomorphism, there exists a homeomorphism h\colon N\to M. The composition g\circ h is an embedding of N.
\square

For the second proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that N is a closed connected n-manifold. Then N_0 have handle decomposition with indices of attaching maps at most n-1.

Lemma 3.2. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, and f:U \to \mathbb R^{m} is an embedding.

(a) If m=2n-1, then there is an extension of f to an embedding of U\cup_{\phi} D^i\times D^{n-i}.

(b) Assume also that there is an embedding g : D^i\times 0 \to \mathbb R^m such that f\phi = g on \partial D^i \times 0. Suppose that on g(D^i\times 0) there is a field of n-i linear independed normal vectors whose restriction to f(\partial D^i\times 0) is tangent to f(U). Then f\cup g extends to a embedding U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m.

Second proof of Theorem 2.1.b assuming Lemma 3.1 and Lemma 3.2.(a). By Lemma 3.1.(a) there is a handle decomposition of N_0 with attaching maps \phi_1,\ldots,\phi_s of indices at most n-1. Denote by U^l the manifold obtained from \emptyset by the attaching first l handles. Take any embedding F_1:U^1 \cong D^n\to \R^{2n-1}. Let us define an embedding F_l of U^l using an embedding F_{l-1} of U^{l-1}. Since the index i of \phi_l is smaller than n, by Lemma 3.2 there is extension of F_{l-1} to an embedding F_l:U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}, where U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}.
\square
Proof of Lemma 3.2.(a) assuming Lemma 3.2.(b). Since i+n\leq 2n-1 and 2i+1\leq 2n-1, by general position there is an embedding g: D^i\times 0\to \mathbb R^{2n-1} such that f\phi = g on \partial D^i \times 0 and f(\mbox{Int} U) has a finite number of intersections points with g(\mbox{Int} D^i\times 0). Then by an isotopy g_t, where g_0=g, fixed on \partial D^i\times 0 we can "push out" the self-intersection points toward \partial U so that g_1(\mbox{Int} D^i\times 0) does not intersect f(\mbox{Int} U). Then f\cup g_1 is an embedding.

Denote by A the (n-i)\times n matrix whose rightmost (n-i)\times (n-i) block is the identity matrix, and whose other elements are zeros. Denote by v the field of n-i normal vectors on \partial D^i\times 0\subset D^i\times D^{n-i} such that the k-th vector has coordinates equal to the k-th row in A. Then the vector field d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i})) is tangent to \partial U. For x\in g_1(\partial D^i\times 0) denote by v'(x) the projection of df(d\phi(v(x))) to the intersection of normal space to g_1(\partial D^i\times 0) at x, and tangent space to f(\partial U) at x. Since i-1<2n-1 - (n-i), it follows that \pi_{i-1}(V_{2n-1, n-i})=0. Hence there is an extension of v' to an linearly independent field of vectors normal to g_1(D^i\times 0). Then by Lemma 3.2.(b) there is an extension of f\cup g_1 to an embedding of U \cup_{\phi} D^i\times D^{n-i}.

\square

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).

Lemma 3.3. Assume that U is a compact n-manifold, \phi:\partial D^i\times D^{n-i}\to \partial U is an embedding with i\leq n-1, f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m are embeddings and F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1] is a concordance between f_0|_U and f_1|_U.

(a) If m\geq 2n, then there is an extension of F to a concordance between f_0 and f_1.

(b) Assume also that there is a concordance G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1] between f_0|_{D^i\times 0} and f_1|_{D^i\times 0}, and on G(D^i\times 0\times [0, 1]) there is a field of n-i linear independent normal vectors whose restrictions to G(\partial D^i\times 0\times [0, 1]), to G(D^i\times 0\times 0), and to G(D^i\times 0\times 1) are tangent to
\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,
respectively. Then F\cup G extends to a concordance between f_0 and f_1.
Proof of the Theorem 2.2 assuming Lemma 3.1 and Lemma 3.3.(a). Denote by f_0, f_1 any two embeddings of N_0 into \mathbb{R}^m. By the Concordance Implies Isotopy Theorem it suffices to prove that there is a concordance between f_0 and f_1. In order to prove that observe that by Lemma 3.1 there is a handle decomposition of N_0 with attaching maps of indices at most n-1. Denote by U^l the manifold obtained from U^0=\emptyset by the attaching the first l handles, starting with? U^1\cong D^n. Define a concordance F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1] recursively. Take any concordance F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1] between f_0|_{U^1} and f_1|_{U^1}. Let us define a concordance F_l between f_0|_{U^l} and f_1|_{U^l} using a concordance F_{l-1} of U^{l-1} between f_0|_{U^{l-1}} and f_1|_{U^{l-1}}. For this denote by \phi:\partial D^i\times D^{n-i}\to \partial U^{l-1} the l-th attaching map. Since i\leq n-1, by Lemma 3.3.(a) there is an extension of F_{l-1} to a concordance
\displaystyle F_{l}:U^{l}\times [0, 1]\to\mathbb{R}^m\times [0, 1]
between the restriction of f_0 and f_1 to U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}.
\square
Proof of Lemma 3.3.(a) assuming Lemma 3.3.(b). In the following text we identify D^i\times D^{n-i}\times [0, 1] and D^i\times [0, 1]\times D^{n-i}. Define map
\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1
by the formula: 
Tex syntax error
Since
\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\quad\text{and}\quad 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,
by general position there is an embedding
\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]
such that F\bar{\phi} = G on \partial (D^i \times 0\times [0, 1]) and F(\mbox{Int} (U\times [0, 1])) has a finite number of intersection points with G(\mbox{Int}( D^i\times 0\times [0, 1])). Then by an isotopy G_t, where G_0=G, fixed on \partial (D^i\times 0\times [0, 1]) we can "push out" the self-intersection points toward F(\partial (U\times [0, 1])) so that G_1(D^i\times 0\times [0, 1]) does not intersect G(U\times [0, 1]). Then F\cup G_1 is an concordance between the restrictions of f_0 and f_1 on U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0.

Denote by A the (n-i)\times (n+1) matrix whose rightmost (n-i)\times (n-i) submatrix is the identity matrix, and whose other elements are zeroes. Denote by v the field of n-i normal vectors on \partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1] whose k-th vector has coordinates equal to the k-th row in A. Then d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i})) is the vector field tangent to \bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i}). For x\in G_1(\partial (D^i\times 0\times [0, 1]) denote by v'(x) the projection of d(F\cup f_0\cup f_1)d\phi (v(x)) to the intersection of normal space to G_1(D^i\times 0\times [0, 1]) at x, and tangent space to F(\partial(U\times [0, 1])) at x. Since i<m+1-(n-i), it follows that \pi_{i}(V_{m+1, n-i})=0. Hence there is an extension of v' to a linear independent field of vectors normal to G_1(D^i\times 0\times [0, 1]). Then by Lemma 3.3.(b) there is an extension of F\cup G_1 to a concordance (U \cup_{\phi} D^i\times D^{n-i})\times [0, 1].

\square


4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. 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}.

(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 well-defined and is a bijection for k\geqslant2.

Proof. 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).

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 the standard embedding.Thus \mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a.

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 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.
\square
(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.

Denote 1_k:=(1,0,\ldots,0)\in S^k.

Example 4.2. 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).

The surjectivity of l is given analogously to Proposition 4.1(b). The injectivity of l follows from forgetful bijection \mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1] between embeddings of N_0 and a cylinder.

This example shows that Theorem 7.4 fails for k=0.

Example 4.3. Let N=S^k_a\times S^1 \# S^k_b\times S^1 be the connected sum of two tori. Then there exists a surjection l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z defined by the formula l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1)).

To prove the surjectivity of l it is sufficient to take linked k-spheres in \mathbb R^{2k+1} and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (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). Figure (c) shows that l(f_1)=1 and l(f_2)=0 which proves the intuitive fact that f_1 and f_2 are not isotopic. (Notice that the restrictions of f_1 and f_2 on x\cup y are isotopic!) If we use the opposite normal vector field s'=-s, the values of l(f_1) and l(f_2) will change but will still be different (see figure (d)).

5 Seifert linking form

For a simpler invariant see [Skopenkov2022] and references therein.

In this section assume that

  • N is any closed orientable connected n-manifold,
  • f\colon N_0 \to \mathbb R^{2n-1} is any embedding,
  • if the (co)homology coefficients are omitted, then they are \mathbb Z,
  • n is even and H_1(N) is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).

By N_0 we denote the closure of the complement in N to an closed n-ball. Thus \partial N_0 is the (n-1)-sphere.

Lemma 5.1. There exists a nowhere vanishing normal vector field to f(N_0).

This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].

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

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

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

Lemma 5.2 (L is well-defined). The integer L(f)(x, y):

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

The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.

Lemma 5.2 implies that L(f) generates a bilinear form

\displaystyle L(f):H_{n-1}(N_0)\times H_{n-1}(N_0)\to\Z

denoted by the same letter and called Seifert linking form.

Denote by \rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2) the reduction modulo 2. Define the dual to Stiefel-Whitney class \mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2) to be the class of the cycle on which two general position normal fields to f(N_0) are linearly dependent.

Lemma 5.3. For every X, Y \in H_{n-1}(N_0) the following equality holds:

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

This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let 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. 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).

Definition 6.1. For each even n define an invariant W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2). For each embedding f\colon N_0\to\mathbb R^{2n-1} construct any PL embedding g\colon N\to\mathbb R^{2n} by adding a cone over f(\partial N_0). Now let W\Lambda([f]) = W(g), where W is Whitney invariant, [Skopenkov2016e, \S5].

Lemma 6.2. The invariant W\Lambda is well-defined for n\ge4.

Proof. Note that Unknotting Spheres Theorem implies that \partial N_0 unknots in \mathbb R^{2n}. Thus f|_{\partial N_0} can be extended to embedding of an n-ball B^n into \mathbb R^{2n}. Unknotting Spheres Theorem implies that n-sphere unknots in \mathbb R^{2n}. Thus all extensions of f are isotopic in PL category. Note also that if f and g are isotopic then their extensions are isotopic as well. And Whitney invariant W is invariant for PL embeddings.

\square

Definition 6.3 of G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N) if n is even and H_1(N) is torsion-free. Take a collection \{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0 such that W\Lambda(f_z)=z. For each f such that W\Lambda(f)=z define

\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)

where x,y\in H_{n-1}(N_0).

Note also that G depends on choice of collection \{f_z\}. The following Theorems hold for any choice of \{f_z\}.

Theorem 6.4. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n 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}),

is one-to-one.

Lemma 6.5. For each even n\in H_{n-1}(N) and each x the following equality holds: W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right).

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let N be a closed connected orientable n-manifold with H_1(N) torsion-free, n\ge 4, n even. Then

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

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

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see \S8 for a generalization.

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.

Theorem 7.1. Assume that N is a closed k-connected n-manifold.

(a) If n\geq 2k+3, then N embeds into \mathbb R^{2n-k}.

(b) If n\geq 2k+2 and k\geq0, then N_0 embeds into \mathbb R^{2n-k-1}.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that N is a closed k-connected n-manifold.

(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 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.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, \S 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 k>1 part (b) is a corollary of Theorem 7.4 below. For k=0 part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for k=1. By Theorem 9.6 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 8.3 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 concordance implies isotopy Theorem f and g are isotopic.
\square

Conjecture 7.3. Assume that N is a closed 1-connected 4-manifold. Then any two embeddings of N_0 in \mathbb R^7 are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume N is a closed k-connected n-manifold. Then for each k\ge1 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)}),

where \mathbb Z_{(s)} denote \mathbb Z for s even and \mathbb Z_2 for s odd.

For definition of W_0' and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2(W_0')]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes \mathrm{Emb}^{2n-1}(N_0) and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. 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}.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion f\colon N\to\mathbb R^{2n-k-1}. Since N is k-connected it follows from Theorem 8.3 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.
\square

Theorem 8.2. 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.

Proof is similar to the proof of theorem 7.2.

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.

Theorem 8.3. 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.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in \R^m is homotopy equivalent to the space of linear monomorphisms from TM to \R^m.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]]

If N is immersible in \R^{m+r} with a normal 
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-field, then N is immersible in \R^m.

Theorem 9.3. Every n-manifold N with non-empty boundary is immersible in \R^{2n-1}.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every n-manifold N is immersible in \R^{2n-1}.

Denote by V_{m,n} is Stiefel manifold of n-frames in \R^m.

Theorem 9.5. Suppose N is a n-manifold with non-empty boudary, (N,\partial N) is k-connected. Then N is immersible in \R^m for each m\geq2n-k-1.

Proof. It suffices to show that exists an immersion of N in \mathbb R^{2n-k-1}. It suffices to show that exists a linear monomorphism from TM to \mathbb R^{2n-k-1}. Let us construct such a linear monomorphism by skeleta of N. It is clear that a linear monomorphism exists on 0-skeleton of N.

The obstruction to extend the linear monomorphism from (r-1)-skeleton to 
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-skeleton lies in H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n})).

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, \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.

\square

Theorem 9.6. Suppose N is a connected n-manifold with non-empty boudary, (N, \partial N) is k-connected and m\geq2n-k. Then every two immersions of N in \R^m are regulary homotopic.

Proof.

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 
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-skeleton of N. It is clear that homotopy exists on 0-skeleton of N. The obstruction to extend the homotopy from (r-1)-skeleton to 
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-skeleton lies in H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n})).

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, \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.

\square

10 References

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