Embeddings of manifolds with boundary: classification
(→Proofs of Theorem \ref{thm::embed}.b and Theorem \ref{thm::unknotting}.b) |
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+ | {{Authors|Mikhail Fedorov|Garaev Timur}} | ||
+ | {{Stub}} | ||
+ | == Introduction == | ||
+ | <wikitex>; \label{sec::intro} | ||
+ | In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. | ||
+ | In $\S$\ref{sec::linking_form} we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. | ||
+ | In $\S$\ref{sec::generalisations} which is independent from $\S$\ref{sec::example}, $\S$\ref{sec::linking_form} and $\S$\ref{sec::classification} we state generalisations of theorems from $\S$\ref{sec::general_theorems} to highly-connected manifolds. | ||
− | + | For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S$1, $\S$3]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. | |
− | + | If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$ the linking coefficient \cite[$\S$ 77]{Seifert&Threlfall1980} of two disjoint cycles. | |
− | - | + | We state the simplest results. These results can be deduced from [[Some_calculations_involving_configuration_spaces_of_distinct_points|the Haefliger-Weber deleted product criterion]] \cite[$\S$ 5]{Skopenkov2006}, see \cite[6.4]{Haefliger1963}, \cite[Theorem 1.1$\alpha\partial$]{Skopenkov2002} for the DIFF case and \cite[Theorem 1.3$\alpha\partial$]{Skopenkov2002} for the PL case. 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. | |
+ | </wikitex> | ||
− | + | == Embedding and unknotting theorems == | |
+ | <wikitex>; \label{sec::general_theorems} | ||
+ | {{beginthm|Theorem}}\label{thm::embed} | ||
+ | Assume that $N$ is a compact connected $n$-manifold. | ||
− | + | (a) Then $N$ embeds into $\mathbb R^{2n}$. | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | If | + | (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]]. | |
− | + | 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. | |
− | + | ||
− | + | {{beginthm|Theorem}}\label{thm::unknotting} | |
− | + | Assume that $N$ is a compact connected $n$-manifold and either | |
− | {{beginthm|Theorem}} | + | |
− | Assume that $N$ is a | + | (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}} | {{endthm}} | ||
− | |||
− | |||
− | |||
− | (a) $ | + | Part (a) is Whitney-Wu Unknotting Theorem, see \cite[$\S$ 2, Theorems 2.1, 2.2]{Skopenkov2016c}. |
+ | 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. | ||
− | (b) | + | The inequality in part (b) is sharp by Proposition \ref{exm::linked_boundary}. |
− | + | These basic results can be generalized to highly-connected manifolds (see $\S$\ref{sec::generalisations}). | |
− | { | + | In particular, both parts of Theorem \ref{thm::embed} are special cases of Theorem \ref{thm::k_connect_unknot}. |
− | + | ||
− | + | All theorems for manifolds with non-empty boundary stated in $\S$\ref{sec::general_theorems} and $\S$\ref{sec::generalisations} can be proved using | |
− | + | * 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. | |
− | + | ||
− | + | * handle decomposition, see e.g. the second proof of Theorem \ref{thm::embed}.b in $\S$3. | |
− | + | Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition. | |
+ | </wikitex> | ||
− | + | == Proofs of Theorem \ref{thm::embed}.b and Theorem \ref{thm::unknotting}.b == | |
+ | <wikitex>; \label{sec::pt} | ||
+ | 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. | ||
+ | |||
+ | <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}} | ||
+ | |||
+ | For the second proof we need some lemmas. | ||
+ | |||
+ | {{beginthm|Lemma}}\label{p:handle-decompose} \cite{Wall1966} | ||
+ | Assume that $N$ is a closed connected $n$-manifold. Then $N_0$ have handle decomposition with indices of attaching maps at most $n-1$. | ||
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Lemma}}\label{p:handle-add} | ||
+ | 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 $U \cup_{\phi} D^i\times D^{n-i} \to\ \mathbb R^m$. | ||
{{endthm}} | {{endthm}} | ||
− | |||
− | + | <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 $U^l=U^{l-1}\cup_{\phi_l} D^i\times D^{n-i}$. | |
− | <!-- | + | {{endproof}} |
− | + | <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. | |
− | + | ||
+ | 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 $U \cup_{\phi} D^i\times D^{n-i}$. | ||
+ | {{endproof}} | ||
+ | |||
+ | In the proof of Theorem \ref{thm::unknotting} we will use Lemma \ref{p:handle-decompose} and Lemma \ref{p:handle-add2}.(a). | ||
+ | |||
+ | {{beginthm|Lemma}}\label{p:handle-add2} | ||
+ | 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 $$F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$ respectively. Then $F\cup G$ extends to a concordance between $f_0$ and $f_1$. | ||
+ | {{endthm}} | ||
+ | |||
+ | <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 $\mathbb{R}^m$. | ||
+ | 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 $U^{l}=U^{l-1}\cup_\phi D^i\times D^{n-i}$. | ||
+ | {{endproof}} | ||
+ | |||
+ | <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: $$\bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right. $$ | ||
+ | |||
+ | 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 $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$. | ||
+ | |||
+ | 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 $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$. | ||
+ | {{endproof}} | ||
− | |||
</wikitex> | </wikitex> | ||
− | == Example == | + | == Example of non-isotopic embeddings == |
− | <wikitex>; | + | <wikitex>; \label{sec::example} |
The following example is folklore. | The following example is folklore. | ||
− | {{beginthm| | + | {{beginthm|Example}} \label{exm::linked_boundary} |
− | Let $N=S^ | + | Let $N=S^k\times [0, 1]$ be the cylinder over $S^k$. |
− | Then there exist non-isotopic embeddings of $N$ | + | |
+ | (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 $k\geqslant2$. | ||
{{endthm}} | {{endthm}} | ||
+ | |||
{{beginproof}} | {{beginproof}} | ||
− | + | ''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 [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation_and_conventions|the standard embedding]].Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$. | |
− | + | ||
− | + | ''Proof of part (c).'' Clearly $l$ is well-defined. By (b) $l$ is surjective. Now take any two embeddings $f_1, f_2$ such that $l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. | |
− | + | Since $k\geqslant 2$ [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that there exists an isotopy of $f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$ and $f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$ and $f_2$ are isotopic. {{endproof}} | |
− | + | ||
− | By $ | + | |
− | + | [[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| | + | |
− | + | Denote $1_k:=(1,0,\ldots,0)\in S^k$. | |
+ | |||
+ | {{beginthm|Example}} | ||
+ | Let $N=S^k\times S^1$. Assume $k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. | ||
{{endthm}} | {{endthm}} | ||
− | |||
− | |||
− | + | The surjectivity of $l$ is given analogously to Proposition \ref{exm::linked_boundary}(b). | |
+ | The 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 \ref{thm::k_connect_classif} fails for $k=0$. | |
− | { | + | |
− | + | ||
− | + | {{beginthm|Example}} | |
− | {{beginthm| | + | 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 $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$. |
− | Let $ | + | |
− | + | ||
− | + | ||
− | Then | + | |
− | + | ||
− | + | ||
− | + | ||
{{endthm}} | {{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| | + | {{beginthm|Example}} |
− | + | (a) Let $N_0$ be the punctured 2-torus containing the meridian $x$ and the parallel $y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$ denote by $s$ the normal field of $\epsilon$-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)). | ||
+ | {{endthm}} | ||
+ | </wikitex> | ||
+ | |||
+ | == Seifert linking form == | ||
+ | <wikitex>; \label{sec::linking_form} | ||
+ | For a simpler invariant see \cite{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. | ||
+ | |||
+ | {{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 $s$. | ||
− | |||
− | |||
− | Denote by $\rho_2 \colon H_*(N | + | {{beginthm|Lemma|($L$ is well-defined)}}\label{lmm:L_well_def} |
+ | 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} 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 $2$. 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} | ||
− | + | 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}; 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}. | |
− | Let $- | + | </wikitex> |
+ | |||
+ | == Classification theorems == | ||
+ | <wikitex>; \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. | ||
+ | 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$5]{Skopenkov2016e}. | ||
+ | <!--the extension of $f$ to an embedding $\Lambda(f)$. Informally speaking $\Lambda(f)$ is obtained by adding and smoothing a cone over $f(\partial N_0)$ in $\mathbb R^{2n}$.--> | ||
+ | {{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) | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
$$ | $$ | ||
− | The | + | 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. | ||
+ | <!--Here the element $\bar L(f) \in H_1(N; \mathbb Z_2)$ is such that for every $x\in H_{n-1}(N)$ holds $\bar L(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x)\right)$.--> | ||
+ | {{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 | ||
− | The | + | (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} | ||
+ | </wikitex> | ||
+ | |||
+ | == A generalization to highly-connected manifolds == | ||
+ | <wikitex>; \label{sec::generalisations} | ||
+ | For simplicity in this paragraph we consider only punctured manifolds, see $\S$\ref{prt::arbitrary_boundary} 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. | ||
+ | {{beginthm|Theorem}}\label{thm::k_connect_embeds} | ||
+ | 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 $\mathbb R^{2n-k-1}$. | ||
+ | {{endthm}} | ||
+ | |||
+ | Part (a) is proved in \cite[Existence Theorem (a)]{Haefliger1961} for the Diff case and in \cite[Theorem 1.1]{Penrose&Whitehead&Zeeman1961}, \cite[Corollary 1.3]{Irwin1965} for PL case. | ||
+ | |||
+ | Part (b) is proved in \cite[Corollary 4.2]{Hirsch1961a} for the Diff case and in \cite[Theorem 1.2]{Penrose&Whitehead&Zeeman1961} for the PL case. | ||
+ | {{beginthm|Theorem}}\label{thm::k_connect_unknot} | ||
+ | <!--[The Haefliger-Zeeman Unknotting Theorem]--> | ||
+ | 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. | ||
+ | {{endthm}} | ||
+ | |||
+ | Part (a) is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$ 2]{Skopenkov2016c}, and is proved in \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. | ||
+ | |||
+ | Part (b) is proved in \cite[Theorem 10.3]{Hudson1969} for the PL case, using [[Isotopy#Concordance|concordance implies isotopy theorem]]. | ||
+ | |||
+ | For $k>1$ part (b) is 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}} | ||
− | + | We may hope to get around the restrictions of Theorem \ref{thm::k_connect_is_spine} using the [[Some_calculations_involving_configuration_spaces_of_distinct_points|deleted product criterion]]. | |
− | + | {{beginthm|Theorem}}\label{thm::k_connect_classif} | |
− | + | Assume $N$ is a closed $k$-connected $n$-manifold. Then for each $k\ge1$ there exists a bijection | |
$$ | $$ | ||
− | \mathrm{ | + | 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}. 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. | ||
</wikitex> | </wikitex> | ||
− | + | == Comments on non-spherical boundary == | |
− | <wikitex> | + | <wikitex>; \label{prt::arbitrary_boundary} |
− | + | {{beginthm|Theorem}}\label{thm::arbitarty_k_connect_embeds} | |
+ | Assume that $N$ is a compact $k$-connected $n$-manifold, $\partial N\neq\emptyset$, $(N, \partial N)$ is $k$-connected and $k< n - 3$. | ||
+ | Then $N$ embeds into $\mathbb R^{2n-k-1}$. | ||
+ | {{endthm}} | ||
+ | This is \cite[Theorem on p.567]{Wall1965}. | ||
+ | {{beginproof}} By Theorem \ref{thm::k_immersible} below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$. Since $N$ is $k$-connected it follows from Theorem \ref{thm::k_connect_is_spine} that $N$ collapses to an $(n-k-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $f|_X$ is an embedding, because $2(n-k) < 2n-k-1$. Since $f$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. It is clear that $f\circ h$ is an embedding.{{endproof}} | ||
+ | {{beginthm|Theorem}}\label{thm::arbitrary_k_connect_unknotting} | ||
+ | Assume that $N$ is a $n$-manifold. If $N$ has $(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$, $m \ge 2n - k$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. | ||
+ | {{endthm}} | ||
+ | Proof is similar to the proof of theorem \ref{thm::k_connect_unknot}. | ||
− | + | For a compact connected $n$-manifold with boundary, the property of having an $(n − k − 1)$-dimensional spine is close to $k$-connectedness. Indeed, the following theorem holds. | |
− | {{beginthm|Theorem}} | + | {{beginthm|Theorem}}\label{thm::k_connect_is_spine} |
− | + | Every compact connected $n$-manifold $N$ with boundary for which $(N, \partial N)$ is $k$-connected, $\pi_1(\partial N)=0$, | |
+ | $k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$-dimensional spine. | ||
+ | {{endthm}} | ||
+ | For this result see \cite[Theorem 5.5]{Wall1964a} and \cite[Lemma 5.1 and Remark 5.2]{Horvatic1969}. See also valuable remarks in \cite{Levine&Lidman2018} and \cite{Skopenkov2019}. | ||
+ | </wikitex> | ||
− | + | == Comments on immersions == | |
+ | <wikitex>; | ||
− | + | {{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}} | {{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 $\R^{2n-1}$. | |
{{endthm}} | {{endthm}} | ||
− | The | + | <!--{{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}} | ||
+ | ''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}}\ | + | {{beginthm|Theorem}}[Whitney; \cite[Theorem 6.6]{Hirsch1961a}] |
− | + | Every $n$-manifold $N$ is immersible in $\R^{2n-1}$. | |
{{endthm}} | {{endthm}} | ||
− | + | ||
− | {{beginthm|Theorem}}\label{thm:: | + | Denote by $V_{m,n}$ is Stiefel manifold of $n$-frames in $\R^m$. |
− | + | ||
− | + | {{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}} | |
+ | 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 $N$. | ||
− | {{beginthm|Theorem}}\label{thm:: | + | 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 $r=n-k,\ldots, n$ 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. | ||
+ | {{endproof}} | ||
+ | |||
+ | {{beginthm|Theorem}}\label{thm::isotop_unknot} | ||
+ | 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}} | |
+ | 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 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 $r=n-k,\ldots, n$ 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. | ||
+ | {{endproof}} | ||
</wikitex> | </wikitex> | ||
<!--== Classification == | <!--== Classification == | ||
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\cite{Becker&Glover1971} | \cite{Becker&Glover1971} | ||
</wikitex> | </wikitex> | ||
− | + | == Acknowledgments == | |
− | == | + | |
− | + | ||
... | ... | ||
− | |||
− | |||
− | |||
− | |||
== Footnotes == | == Footnotes == | ||
<references/> --> | <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. |
1 Introduction
In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In 5 we introduce an invariant of embedding of a -manifold in -space for even . In 7 which is independent from 4, 5 and 6 we state generalisations of theorems from 2 to highly-connected manifolds.
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category. Denote the set of all embeddings up to isotopy. We denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1. Assume that is a compact connected -manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2. Assume that is a compact connected -manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2]. Part (b) in the case is proved in [Edwards1968, 4, Corollary 5]. The case is clear. The case can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and 7 can be proved using
- analogous results for immersions of manifolds stated in 9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in 3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in 3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966] Assume that is a closed connected -manifold. Then have handle decomposition with indices of attaching maps at most .
Lemma 3.2. Assume that is a compact -manifold, is an embedding with , and is an embedding.
If , then there is an extension of to an embedding of .
Assume also that there is an embedding such that on . Suppose that on there is a field of linear independed normal vectors whose restriction to is tangent to . Then extends to a embedding .
Denote by the matrix whose rightmost block is the identity matrix, and whose other elements are zeros. Denote by the field of normal vectors on such that the -th vector has coordinates equal to the -th row in . Then the vector field is tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to an linearly independent field of vectors normal to . Then by Lemma 3.2.(b) there is an extension of to an embedding of .
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).
Lemma 3.3. Assume that is a compact -manifold, is an embedding with , are embeddings and is a concordance between and .
If , then there is an extension of to a concordance between and .
Assume also that there is a concordance between and , and on there is a field of linear independent normal vectors whose restrictions to , to , and to are tangent toTex syntax error
Denote by the matrix whose rightmost submatrix is the identity matrix, and whose other elements are zeroes. Denote by the field of normal vectors on whose -th vector has coordinates equal to the -th row in . Then is the vector field tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to a linear independent field of vectors normal to . Then by Lemma 3.3.(b) there is an extension of to a concordance .
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1. Let be the cylinder over .
(a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
(c) Then defined by the formula is well-defined and is a bijection for .
Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree . (To prove part (a) it is sufficient to take as the identity map of as a map of degree one and the constant map as a map of degree zero.) Define by the formula .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.Denote .
Example 4.2. Let . Assume . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 4.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
To prove the surjectivity of it is sufficient to take linked -spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
- is any closed orientable connected -manifold,
- is any embedding,
- if the (co)homology coefficients are omitted, then they are ,
- is even and is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).
By we denote the closure of the complement in to an closed -ball. Thus is the -sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint -cycles in with integer coefficients. Denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 5.2 ( is well-defined). The integer :
- is well-defined, i.e., does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
Denote by the reduction modulo . Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 5.3. For every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 6.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 6.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 6.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 6.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 6.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 6.4:
Theorem 6.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed -manifold. By denote the complement in to an open -ball. Thus is the -sphere.
Theorem 7.1. Assume that is a closed -connected -manifold.
(a) If , then embeds into .
(b) If and , then embeds into .
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2. Assume that is a closed -connected -manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For part (b) coincides with Theorem 2.2b.
Conjecture 7.3. Assume that is a closed -connected -manifold. Then any two embeddings of in are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4. Assume is a closed -connected -manifold. Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1. Assume that is a compact -connected -manifold, , is -connected and . Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 8.2. Assume that is a -manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an -dimensional spine is close to -connectedness. Indeed, the following theorem holds.
Theorem 8.3. Every compact connected -manifold with boundary for which is -connected, , and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from to .
Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If is immersible in with a normal -field, then is immersible in .
Theorem 9.3. Every -manifold with non-empty boundary is immersible in .
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every -manifold is immersible in .
Denote by is Stiefel manifold of -frames in .
Theorem 9.5. Suppose is a -manifold with non-empty boudary, is -connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Let us construct such a linear monomorphism by skeleta of . It is clear that a linear monomorphism exists on -skeleton of .
The obstruction to extend the linear monomorphism from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Theorem 9.6. Suppose is a connected -manifold with non-empty boudary, is -connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to extend the homotopy from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
10 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain -manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category. Denote the set of all embeddings up to isotopy. We denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1. Assume that is a compact connected -manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2. Assume that is a compact connected -manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2]. Part (b) in the case is proved in [Edwards1968, 4, Corollary 5]. The case is clear. The case can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and 7 can be proved using
- analogous results for immersions of manifolds stated in 9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in 3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in 3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966] Assume that is a closed connected -manifold. Then have handle decomposition with indices of attaching maps at most .
Lemma 3.2. Assume that is a compact -manifold, is an embedding with , and is an embedding.
If , then there is an extension of to an embedding of .
Assume also that there is an embedding such that on . Suppose that on there is a field of linear independed normal vectors whose restriction to is tangent to . Then extends to a embedding .
Denote by the matrix whose rightmost block is the identity matrix, and whose other elements are zeros. Denote by the field of normal vectors on such that the -th vector has coordinates equal to the -th row in . Then the vector field is tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to an linearly independent field of vectors normal to . Then by Lemma 3.2.(b) there is an extension of to an embedding of .
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).
Lemma 3.3. Assume that is a compact -manifold, is an embedding with , are embeddings and is a concordance between and .
If , then there is an extension of to a concordance between and .
Assume also that there is a concordance between and , and on there is a field of linear independent normal vectors whose restrictions to , to , and to are tangent toTex syntax error
Denote by the matrix whose rightmost submatrix is the identity matrix, and whose other elements are zeroes. Denote by the field of normal vectors on whose -th vector has coordinates equal to the -th row in . Then is the vector field tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to a linear independent field of vectors normal to . Then by Lemma 3.3.(b) there is an extension of to a concordance .
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1. Let be the cylinder over .
(a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
(c) Then defined by the formula is well-defined and is a bijection for .
Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree . (To prove part (a) it is sufficient to take as the identity map of as a map of degree one and the constant map as a map of degree zero.) Define by the formula .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.Denote .
Example 4.2. Let . Assume . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 4.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
To prove the surjectivity of it is sufficient to take linked -spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
- is any closed orientable connected -manifold,
- is any embedding,
- if the (co)homology coefficients are omitted, then they are ,
- is even and is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).
By we denote the closure of the complement in to an closed -ball. Thus is the -sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint -cycles in with integer coefficients. Denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 5.2 ( is well-defined). The integer :
- is well-defined, i.e., does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
Denote by the reduction modulo . Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 5.3. For every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 6.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 6.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 6.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 6.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 6.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 6.4:
Theorem 6.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed -manifold. By denote the complement in to an open -ball. Thus is the -sphere.
Theorem 7.1. Assume that is a closed -connected -manifold.
(a) If , then embeds into .
(b) If and , then embeds into .
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2. Assume that is a closed -connected -manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For part (b) coincides with Theorem 2.2b.
Conjecture 7.3. Assume that is a closed -connected -manifold. Then any two embeddings of in are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4. Assume is a closed -connected -manifold. Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1. Assume that is a compact -connected -manifold, , is -connected and . Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 8.2. Assume that is a -manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an -dimensional spine is close to -connectedness. Indeed, the following theorem holds.
Theorem 8.3. Every compact connected -manifold with boundary for which is -connected, , and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from to .
Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If is immersible in with a normal -field, then is immersible in .
Theorem 9.3. Every -manifold with non-empty boundary is immersible in .
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every -manifold is immersible in .
Denote by is Stiefel manifold of -frames in .
Theorem 9.5. Suppose is a -manifold with non-empty boudary, is -connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Let us construct such a linear monomorphism by skeleta of . It is clear that a linear monomorphism exists on -skeleton of .
The obstruction to extend the linear monomorphism from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Theorem 9.6. Suppose is a connected -manifold with non-empty boudary, is -connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to extend the homotopy from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
10 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain -manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category. Denote the set of all embeddings up to isotopy. We denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1. Assume that is a compact connected -manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2. Assume that is a compact connected -manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2]. Part (b) in the case is proved in [Edwards1968, 4, Corollary 5]. The case is clear. The case can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and 7 can be proved using
- analogous results for immersions of manifolds stated in 9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in 3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in 3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966] Assume that is a closed connected -manifold. Then have handle decomposition with indices of attaching maps at most .
Lemma 3.2. Assume that is a compact -manifold, is an embedding with , and is an embedding.
If , then there is an extension of to an embedding of .
Assume also that there is an embedding such that on . Suppose that on there is a field of linear independed normal vectors whose restriction to is tangent to . Then extends to a embedding .
Denote by the matrix whose rightmost block is the identity matrix, and whose other elements are zeros. Denote by the field of normal vectors on such that the -th vector has coordinates equal to the -th row in . Then the vector field is tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to an linearly independent field of vectors normal to . Then by Lemma 3.2.(b) there is an extension of to an embedding of .
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).
Lemma 3.3. Assume that is a compact -manifold, is an embedding with , are embeddings and is a concordance between and .
If , then there is an extension of to a concordance between and .
Assume also that there is a concordance between and , and on there is a field of linear independent normal vectors whose restrictions to , to , and to are tangent toTex syntax error
Denote by the matrix whose rightmost submatrix is the identity matrix, and whose other elements are zeroes. Denote by the field of normal vectors on whose -th vector has coordinates equal to the -th row in . Then is the vector field tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to a linear independent field of vectors normal to . Then by Lemma 3.3.(b) there is an extension of to a concordance .
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1. Let be the cylinder over .
(a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
(c) Then defined by the formula is well-defined and is a bijection for .
Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree . (To prove part (a) it is sufficient to take as the identity map of as a map of degree one and the constant map as a map of degree zero.) Define by the formula .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.Denote .
Example 4.2. Let . Assume . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 4.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
To prove the surjectivity of it is sufficient to take linked -spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
- is any closed orientable connected -manifold,
- is any embedding,
- if the (co)homology coefficients are omitted, then they are ,
- is even and is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).
By we denote the closure of the complement in to an closed -ball. Thus is the -sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint -cycles in with integer coefficients. Denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 5.2 ( is well-defined). The integer :
- is well-defined, i.e., does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
Denote by the reduction modulo . Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 5.3. For every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 6.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 6.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 6.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 6.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 6.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 6.4:
Theorem 6.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed -manifold. By denote the complement in to an open -ball. Thus is the -sphere.
Theorem 7.1. Assume that is a closed -connected -manifold.
(a) If , then embeds into .
(b) If and , then embeds into .
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2. Assume that is a closed -connected -manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For part (b) coincides with Theorem 2.2b.
Conjecture 7.3. Assume that is a closed -connected -manifold. Then any two embeddings of in are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4. Assume is a closed -connected -manifold. Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1. Assume that is a compact -connected -manifold, , is -connected and . Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 8.2. Assume that is a -manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an -dimensional spine is close to -connectedness. Indeed, the following theorem holds.
Theorem 8.3. Every compact connected -manifold with boundary for which is -connected, , and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from to .
Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If is immersible in with a normal -field, then is immersible in .
Theorem 9.3. Every -manifold with non-empty boundary is immersible in .
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every -manifold is immersible in .
Denote by is Stiefel manifold of -frames in .
Theorem 9.5. Suppose is a -manifold with non-empty boudary, is -connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Let us construct such a linear monomorphism by skeleta of . It is clear that a linear monomorphism exists on -skeleton of .
The obstruction to extend the linear monomorphism from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Theorem 9.6. Suppose is a connected -manifold with non-empty boudary, is -connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to extend the homotopy from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
10 References
- [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
- [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
- [Fedorov2021] M. Fedorov, A description of values of Seifert form for punctured n-manifolds in (2n-1)-space. Available at the arXiv:2107.02541.
- [Haefliger&Poenaru1964] Template:Haefliger&Poenaru1964
- [Haefliger1961] A. Haefliger, Plongements différentiables de variétés dans variétés., Comment. Math. Helv.36 (1961), 47-82. MR0145538 (26 #3069) Zbl 0102.38603
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Hirsch1959] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR0119214 (22 #9980) Zbl 0118.18603
- [Hirsch1961a] M. W. Hirsch, On Imbedding Differentiable Manifolds in Euclidean Space, Annals of Mathematics, Second Series, 73(3) (1961), 566–571.
- [Horvatic1969] Template:Horvatic1969
- [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
- [Irwin1965] M. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965) 1–14. MR0182978 (32 #460) Zbl 0132.20003
- [Levine&Lidman2018] Template:Levine&Lidman2018
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
- [Saeki1999] O. Saeki, On punctured 3-manifolds in 5-sphere, Hiroshima Math. J. 29 (1999) 255--272, MR1704247 (2000h:57045)
- [Seifert&Threlfall1980] Seifert, Herbert; Threlfall, William (1980), Goldman, Michael A.; Birman, Joan S. (eds.), Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, 89, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-634850-7 MR0575168
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into , Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2019] A. Skopenkov, A short exposition of the Levine-Lidman example of spineless 4-manifolds. Available at the arXiv:1911.07330.
- [Skopenkov2022] A. Skopenkov, Invariants of embeddings of 2-surfaces in 3-space. Available at the arXiv:2201.10944.
- [Tonkonog2010] D. Tonkonog, Embedding punctured $n$-manifolds in Euclidean $(2n-1)$-space
- [Vrabec1989] J. Vrabec, Deforming a PL Submanifold of Euclidean Space into a Hyperplane., Trans. Am. Math. Soc. 312 (1989), 155-78.
- [Wall1964a] C. T. C. Wall, Differential topology, IV (theory of handle decompositions), Cambridge (1964), mimeographed notes.
- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain -manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, 1, 3]. In those pages mostly results for closed manifolds are stated.
If the category is omitted, then we assume the smooth (DIFF) category. Denote the set of all embeddings up to isotopy. We denote by the linking coefficient [Seifert&Threlfall1980, 77] of two disjoint cycles.
We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.
We do not claim the references we give are references to original proofs.
2 Embedding and unknotting theorems
Theorem 2.1. Assume that is a compact connected -manifold.
(a) Then embeds into .
(b) If has non-empty boundary, then embeds into .
Part (a) is well-known strong Whitney embedding theorem. The first proof of (b) presented below is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case, and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.
Theorem 2.2. Assume that is a compact connected -manifold and either
(a) or
(b) has non-empty boundary and .
Then any two embeddings of into are isotopic.
Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, 2, Theorems 2.1, 2.2]. Part (b) in the case is proved in [Edwards1968, 4, Corollary 5]. The case is clear. The case can be proved using the ideas presented below.
The inequality in part (b) is sharp by Proposition 4.1.
These basic results can be generalized to highly-connected manifolds (see 7). In particular, both parts of Theorem 2.1 are special cases of Theorem 7.2.
All theorems for manifolds with non-empty boundary stated in 2 and 7 can be proved using
- analogous results for immersions of manifolds stated in 9, and general position ideas, see e.g. the first proof of Theorem 2.1.b in 3.
- handle decomposition, see e.g. the second proof of Theorem 2.1.b in 3.
Observe that the `handle decomposition' proof is essentially a `straightening' of the `immersion' proof because the required results on immersions are proved using handle decomposition.
3 Proofs of Theorem 2.1.b and Theorem 2.2.b
In this section we work only in smooth category. The first proof of Theorem 2.1.b uses immersions, while the second does not.
For the second proof we need some lemmas.
Lemma 3.1. [Wall1966] Assume that is a closed connected -manifold. Then have handle decomposition with indices of attaching maps at most .
Lemma 3.2. Assume that is a compact -manifold, is an embedding with , and is an embedding.
If , then there is an extension of to an embedding of .
Assume also that there is an embedding such that on . Suppose that on there is a field of linear independed normal vectors whose restriction to is tangent to . Then extends to a embedding .
Denote by the matrix whose rightmost block is the identity matrix, and whose other elements are zeros. Denote by the field of normal vectors on such that the -th vector has coordinates equal to the -th row in . Then the vector field is tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to an linearly independent field of vectors normal to . Then by Lemma 3.2.(b) there is an extension of to an embedding of .
In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.3.(a).
Lemma 3.3. Assume that is a compact -manifold, is an embedding with , are embeddings and is a concordance between and .
If , then there is an extension of to a concordance between and .
Assume also that there is a concordance between and , and on there is a field of linear independent normal vectors whose restrictions to , to , and to are tangent toTex syntax error
Denote by the matrix whose rightmost submatrix is the identity matrix, and whose other elements are zeroes. Denote by the field of normal vectors on whose -th vector has coordinates equal to the -th row in . Then is the vector field tangent to . For denote by the projection of to the intersection of normal space to at , and tangent space to at . Since , it follows that . Hence there is an extension of to a linear independent field of vectors normal to . Then by Lemma 3.3.(b) there is an extension of to a concordance .
4 Example of non-isotopic embeddings
The following example is folklore.
Example 4.1. Let be the cylinder over .
(a) Then there exist non-isotopic embeddings of into .
(b) Then for each there exist an embedding such that .
(c) Then defined by the formula is well-defined and is a bijection for .
Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let be a map of degree . (To prove part (a) it is sufficient to take as the identity map of as a map of degree one and the constant map as a map of degree zero.) Define by the formula .
Let , where is the standard embedding.Thus .
Proof of part (c). Clearly is well-defined. By (b) is surjective. Now take any two embeddings such that . Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.
Since Unknotting Spheres Theorem implies that there exists an isotopy of and . Thus we can assume . Since it follows that normal fields on and are homotopic in class of normal fields. This implies and are isotopic.Denote .
Example 4.2. Let . Assume . Then there exists a bijection defined by the formula .
The surjectivity of is given analogously to Proposition 4.1(b). The injectivity of follows from forgetful bijection between embeddings of and a cylinder.
This example shows that Theorem 7.4 fails for .
Example 4.3. Let be the connected sum of two tori. Then there exists a surjection defined by the formula .
To prove the surjectivity of it is sufficient to take linked -spheres in and consider an embedded boundary connected sum of ribbons containing these two spheres.
Example 4.4. (a) Let be the punctured 2-torus containing the meridian and the parallel of the torus. For each embedding denote by the normal field of -length vectors to defined by orientation on (see figure (b)). Then there exists a surjection defined by the formula .
(b) Let be two embeddings shown on figure (a). Figure (c) shows that and which proves the intuitive fact that and are not isotopic. (Notice that the restrictions of and on are isotopic!) If we use the opposite normal vector field , the values of and will change but will still be different (see figure (d)).
5 Seifert linking form
For a simpler invariant see [Skopenkov2022] and references therein.
In this section assume that
- is any closed orientable connected -manifold,
- is any embedding,
- if the (co)homology coefficients are omitted, then they are ,
- is even and is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}).
By we denote the closure of the complement in to an closed -ball. Thus is the -sphere.
Lemma 5.1. There exists a nowhere vanishing normal vector field to .
This is essentially a folklore result, see an unpublished update of [Tonkonog2010] and [Fedorov2021, Lemma 5.1], cf. [Saeki1999, Lemma 4.1].
Denote by two disjoint -cycles in with integer coefficients. Denote
where is a nowhere vanishing normal field to and are the results of the shift of by .
Lemma 5.2 ( is well-defined). The integer :
- is well-defined, i.e., does not change when is replaced by ,
- does not change when or are changed to homologous cycles and,
- does not change when is changed to an isotopic embedding.
The first bullet was stated and proved in unpublished update of [Tonkonog2010] and in [Fedorov2021, Lemma 5.3], other two bullets are simple.
Lemma 5.2 implies that generates a bilinear form
denoted by the same letter and called Seifert linking form.
Denote by the reduction modulo . Define the dual to Stiefel-Whitney class to be the class of the cycle on which two general position normal fields to are linearly dependent.
Lemma 5.3. For every the following equality holds:
This Lemma was stated in a unpublished update of [Tonkonog2010]; a proof is presented in [Fedorov2021, Lemma 6.1] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].
6 Classification theorems
Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.
Let be a closed orientable connected -manifold. By we denote the complement in to an open -ball. Thus is the -sphere. For a free Abelian group , let be the group of bilinear forms such that and is even for each (the second condition automatically holds for n odd).
Definition 6.1. For each even define an invariant . For each embedding construct any PL embedding by adding a cone over . Now let , where is Whitney invariant, [Skopenkov2016e, 5].
Lemma 6.2. The invariant is well-defined for .
Proof. Note that Unknotting Spheres Theorem implies that unknots in . Thus can be extended to embedding of an -ball into . Unknotting Spheres Theorem implies that -sphere unknots in . Thus all extensions of are isotopic in PL category. Note also that if and are isotopic then their extensions are isotopic as well. And Whitney invariant is invariant for PL embeddings.
Definition 6.3 of if is even and is torsion-free. Take a collection such that . For each such that define
where .
Note also that depends on choice of collection . The following Theorems hold for any choice of .
Theorem 6.4. Let be a closed connected orientable -manifold with torsion-free, , even. The map
is one-to-one.
Lemma 6.5. For each even and each the following equality holds: .
An equivalemt statement of Theorem 6.4:
Theorem 6.6. Let be a closed connected orientable -manifold with torsion-free, , even. Then
(a) The map is an injection.
(b) The image of consists of all symmetric bilinear forms such that . Here is the normal Stiefel-Whitney class.
This is the main Theorem of [Tonkonog2010]
7 A generalization to highly-connected manifolds
For simplicity in this paragraph we consider only punctured manifolds, see 8 for a generalization.
Denote by a closed -manifold. By denote the complement in to an open -ball. Thus is the -sphere.
Theorem 7.1. Assume that is a closed -connected -manifold.
(a) If , then embeds into .
(b) If and , then embeds into .
Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.
Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.
Theorem 7.2. Assume that is a closed -connected -manifold.
(a) If and , then any two embeddings of into are isotopic.
(b) If and and then any two embeddings of into are isotopic.
Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].
Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.
For part (b) is a corollary of Theorem 7.4 below. For part (b) coincides with Theorem 2.2b.
Conjecture 7.3. Assume that is a closed -connected -manifold. Then any two embeddings of in are isotopic.
We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.
Theorem 7.4. Assume is a closed -connected -manifold. Then for each there exists a bijection
where denote for even and for odd.
For definition of and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2()]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes and differs from the general case.
8 Comments on non-spherical boundary
Theorem 8.1. Assume that is a compact -connected -manifold, , is -connected and . Then embeds into .
This is [Wall1965, Theorem on p.567].
Theorem 8.2. Assume that is a -manifold. If has -dimensional spine, , , then any two embeddings of into are isotopic.
Proof is similar to the proof of theorem 7.2.
For a compact connected -manifold with boundary, the property of having an -dimensional spine is close to -connectedness. Indeed, the following theorem holds.
Theorem 8.3. Every compact connected -manifold with boundary for which is -connected, , and , has an -dimensional spine.
For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].
9 Comments on immersions
Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in is homotopy equivalent to the space of linear monomorphisms from to .
Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If is immersible in with a normal -field, then is immersible in .
Theorem 9.3. Every -manifold with non-empty boundary is immersible in .
Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every -manifold is immersible in .
Denote by is Stiefel manifold of -frames in .
Theorem 9.5. Suppose is a -manifold with non-empty boudary, is -connected. Then is immersible in for each .
Proof. It suffices to show that exists an immersion of in . It suffices to show that exists a linear monomorphism from to . Let us construct such a linear monomorphism by skeleta of . It is clear that a linear monomorphism exists on -skeleton of .
The obstruction to extend the linear monomorphism from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such linear monomorphism exists.
Theorem 9.6. Suppose is a connected -manifold with non-empty boudary, is -connected and . Then every two immersions of in are regulary homotopic.
Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from to . Lets cunstruct such homotopy on each -skeleton of . It is clear that homotopy exists on -skeleton of .
The obstruction to extend the homotopy from -skeleton to -skeleton lies in .
For we know . For we have since is -connected and has non-empty boundary.
Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.
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