Embeddings in Euclidean space: an introduction to their classification

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1 Introduction

This page is intended not only for specialists in embeddings, but also for mathematicians from other areas who want to apply or to learn the theory of embeddings. Unless otherwise indicated, the word `isotopy' means `ambient isotopy' on this page; see definition in [Skopenkov2016i].

Remark 1.1 (some motivations). The three major classical problems of topology are the following, cf. [Zeeman1993, p. 3].

• Homeomorphism Problem: Classify $n$${{Authors|Askopenkov}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematicians from other areas who want to apply or to learn the theory of embeddings. Unless otherwise indicated, the word `isotopy' means [[Isotopy|`ambient isotopy']] on this page; see definition in \cite{Skopenkov2016i}. {{beginthm|Remark|(some motivations)}}\label{r:zee} The three major classical problems of topology are the following, cf. \cite[p. 3]{Zeeman1993}. * ''Homeomorphism Problem'': Classify $n$-manifolds. * ''Embedding Problem'': Find the least dimension $m$ such that given space admits an [[Embedding_(simple_definition)|embedding]] into $m$-dimensional Euclidean space $\Rr^m$. * ''Knotting Problem'': Classify [[Embedding_(simple_definition)|embeddings]] of a given space into another given space up to isotopy. The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others. The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark \ref{s:list} below). See also [[Wikipedia:Knot_theory|Wikipedia article on knot theory]] and \cite[Example 2.3]{Skopenkov2016t}. {{endthm}} <!-- and an [[Embedding#Definition|alternative definition of an embedding]]. {{beginthm|Remark}}\label{r:oth} Of course there are other important problems of geometric topology. Some of them are interesting because they come from applications (notably problems on vector fields on manifolds). Some of them proved to be useful for many other problems, and so gradually became interesting in themselves (notably homotopy-theoretic problems). {{endthm}} --> This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark \ref{r:zee}). After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$ are isotopic. We then establish notation and conventions. <!-- and give references to other pages on the Knotting Problem, to which this page serves as an introduction.--> We continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings. We conclude with a brief review of some important results about codimension 1 embeddings The most interesting and very much studied case concerns embeddings $S^1\to S^3$ (classical knots), or more generally, [[Knots,_i.e._embeddings_of_spheres#Codimension_2_knots|codimension 2 embeddings of spheres]]. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see [[Wikipedia:Knot_theory|Wikipedia article on knot theory]] and $\S$\ref{s:c2} for more information. The Knotting Problem is known to be hard: to the best of the author's knowledge, at the time of writing there are only a few cases in which complete readily calculable classification results (see Remark \ref{compl}) describing all isotopy classes for embeddings of a ''closed'' manifold $N$ into Euclidean space $\Rr^m$ are known. Such classification results are the [[#Unknotting_theorems|unknotting theorems]] in $\S$\ref{s:ut}, the results on the pages listed in Remark \ref{s:list} <!--[[#References to information on the classification of embeddings|]] \cite[$\S]{Skopenkov2016p}--> and in $\S$\ref{s:c1}. Their statements, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark \ref{s:list} concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2. The results and remarks given below show the following: *The complete classification of embeddings into $\Rr^m$ of closed connected $n$-manifolds is non-trivial but presently accessible only for $n+3\le m\le 2n$ or for $m=n+1\ge4$; *For a fixed $N$, the more $m$ decreases from n$ towards $n+3$, the more complicated classification of embeddings of $N$ into $\Rr^m$ becomes. <!--and in $\S\S$\ref{s:ut}, \ref{s:c2}, \ref{s:c1}, \ref{s:list}, listed in \cite[$\S]{Skopenkov2016p}--> The lowest dimensional cases, i.e. all such pairs $(m,n)$ with $n\le4$, are [[3-manifolds_in_6-space|(6,3)]], (4,3), [[Embeddings_just_below_the_stable_range:_classification|(8,4)]], [[4-manifolds_in_7-space|(7,4)]], (5,4). For information on the cases (6,3), (8,4), (7,4) see \cite{Skopenkov2016t}, \cite{Skopenkov2016e}, \cite{Skopenkov2016f}. {{beginthm|Remark|(Readily calculable classification)}}\label{compl} Let me informally explain what I mean by a `readily calculable classification'. (Such words are used by other people who might have a similar or a different concept.) In the best case a `readily calculable classification' is a classification in terms of (co)homology of a manifold (and certain structures on (co)homology like intersection, characteristic classes etc). A readily calculable classification is also a reduction to calculation of stable [[Wikipedia:Homotopy_groups_of_spheres|homotopy groups of spheres]] when these are known (or to some other standard algebraic problems involving only the homology of the manifold, which are solved in particular cases, although may be unsolved in general). An important feature of a useful classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. Many readily calculable classification results are presented on this page and the pages listed in Remark \ref{s:list}. On the other hand, in some cases `geometric problems are reduced to algebraic problems which are even harder to solve' \cite{Wall1999}, and such reductions do not give a readily calculable classification. E.g. so far an interesting approach of \cite{Weiss96}, \cite{Goodwillie&Weiss1999} did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ \cite{Weiss}.) See discusion of similar issues in \cite[Preface]{Graham&Knuth&Patashnik89}. {{endthm}} {{beginthm|Remark|(Embeddings into the sphere and Euclidean space)}}\label{spheu} (a) The embeddings $f,g:S^1\to\Rr^2$ given by $f(x,y)=(x,y)$ and $g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in [[Wikipedia:Winding_number#Turning_number|Wikipedia article on turning number]] can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$ into $S^2$ are isotopic, see Theorem \ref{t:sch}.a and below. (b) For $m\ge n+2$ the classifications of embeddings of compact $n$-manifolds into $S^m$ and into $\Rr^m$ are the same. More precisely, for all integers $m,n$ such that $m\ge n+2$, and for every $n$-manifold $N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$ and $N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$, after a small isotopy an embedding $N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$. Hence $i_*$ is onto. To prove that $i_*$ is injective, it suffices to show that if the compositions with the inclusion $i:\Rr^m\to S^m$ of two embeddings $f,f':N\to\Rr^m$ of a compact $n$-manifold $N$ are isotopic, then $f$ and $f'$ are isotopic. For showing that assume that $i\circ f$ and $i\circ f'$ are isotopic. Then by general position $f$ and $f'$ are [[Isotopy#Other equivalence relations|non-ambiently isotopic]]. Since every non-ambient isotopy extends to an isotopy \cite[Theorem 1.3]{Skopenkov2016i}, $f$ and $f'$ are isotopic. {{endthm}} <!-- For a [[Embeddings_in_Euclidean_space:_plan_and_convention#Notation and conventions|notation and conventions]] used below on this page, we refer to \cite[$\S]{Skopenkov2016p}.--> {{beginthm|Remark|(References to information on the classification of embeddings)}}\label{s:list} The first list is structured by the dimension of the source manifold and the target Euclidean space: * [[Embeddings just below the stable range: classification]] \cite{Skopenkov2016e} * [[3-manifolds in 6-space]] \cite{Skopenkov2016t} * [[4-manifolds in 7-space]] \cite{Skopenkov2016f} Information structured by the `complexity' of the source manifold: * [[Knots, i.e. embeddings of spheres]] * [[High codimension links]] \cite{Skopenkov2016h} * [[Knotted tori]] \cite{Skopenkov2016k} For more information see e.g. \cite{Skopenkov2006}. <!--[[Unknotting of projective spaces]] [[Embeddings of highly-connected manifolds: classification]] [[Link maps: classification]] [[Immersions: classification]] [[Embeddings of manifolds with boundary]] [[Some open problems]] --> {{endthm}} </wikitex> ==Unknotting theorems == <wikitex>; \label{s:ut} If the category is omitted, then a result stated below holds in both the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] and [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) category. {{beginthm|General Position Theorem|(\cite[Theorem 3.5]{Hirsch1976}, \cite[Theorem 5.4]{Rourke&Sanderson1972})}}\label{th1} For every compact $n$-manifold $N$ and $m\ge2n+2$, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} The case $m\ge2n+2$ is called a `stable range' (for the classification problem; for the existence problem there is analogous result with $m\ge2n+1$ \cite[$\S]{Skopenkov2006}). The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds, as the [[High_codimension_links#Examples|Hopf linking]] $S^n\sqcup S^n\to\Rr^{2n+1}$ shows \cite{Skopenkov2016h}, \cite[Figure 2.1.a]{Skopenkov2006}. {{beginthm|Whitney-Wu Unknotting Theorem}}\label{th2} For every compact connected $n$-manifold $N$, $n\ge2$ and $m\ge2n+1$, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} This is proved in \cite{Wu1958}, \cite{Wu1958a} and \cite{Wu1959} using ''the Whitney trick'' (see those rferences or \cite[$\S]{Rourke&Sanderson1972}). All the three assumptions in this result are indeed necessary: *the assumption $n\ge2$ because of the existence of non-trivial knots $S^1\to S^3$; *the connectedness assumption because of the existence of [[High_codimension_links#Examples|the Hopf link]] \cite{Skopenkov2016h}; *the assumption $m\ge2n+1$ because of the [[Embeddings just below the stable range#Hudson_tori|example of Hudson tori]] \cite{Skopenkov2016e}. {{beginthm|Unknotting Spheres Theorem}}\label{sph} For $N=S^n$, or even for $N$ an integral homology $n$-sphere, $m\ge n+3$ or m\ge 3n+4$ in the PL or smooth category, respectively, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} This result is proved in \cite{Zeeman1960} or \cite{Haefliger1961} in the PL or smooth category, respectively. This result is also true for $m\ge n+3$ in the topological locally flat category \cite{Stallings1963}, \cite{Gluck1963}, \cite[Flattening Theorem 4.5.1]{Rushing1973}, \cite{Scharlemann1977}. <!--Here the local flatness assumption is indeed necessary.--> The case m\ge 3n+4$ is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with m\ge3n+3$ \cite[$\S, $\S]{Skopenkov2006}). Knots in codimension 2 and [[3-manifolds in 6-space#Examples|the Haefliger trefoil knot]] \cite[Example 2.1]{Skopenkov2016t} show that the dimension restrictions are sharp (even for $N=S^n$) in the Unknotting Spheres Theorem \ref{sph}. Theorems \ref{th2} and \ref{sph} may be generalized as follows. {{beginthm|The Haefliger-Zeeman Unknotting Theorem}}\label{haze} For every $n\ge2k+2$, $m\ge2n-k+1$ and closed $k$-connected $n$-manifold $N$, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} This was proved in \cite{Penrose&Whitehead&Zeeman1961}, \cite{Haefliger1961}, \cite{Zeeman1962}, \cite{Irwin1965}, \cite{Hudson1969}. The proofs in \cite{Haefliger&Hirsch1963}, \cite{Vrabec1977}, \cite{Weber1967}, \cite[$\S]{Adachi1993} work for ''homologically'' $k$-connected manifolds (see $\S$\ref{s:nc} for the definition; the proofs are non-trivial but the generalization is trivial, basically because the $k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$ to the image of $N$, by Alexander duality and simple connectedness of the complement, so homological $k$-connectedness is sufficient). Given Theorem \ref{haze} above, the case $m\ge2n-k+1$ can be called a `stable range for $k$-connected manifolds'. Note that if $n\le2k+1$, then every closed $k$-connected $n$-manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem \ref{haze} in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem \ref{sph}. For generalizations of the Haefliger-Zeeman Unknotting Theorem \ref{haze} [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|see]] \cite[$\S]{Skopenkov2016e} or \cite{Hudson1967}, \cite{Hacon1968}, \cite{Hudson1972}, \cite{Gordon1972}, \cite{Kearton1979}. See also Theorem \ref{t:sch}. </wikitex> == Notation and conventions == <wikitex>; \label{s:nc} The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark \ref{s:list}. For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] or [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) [[Embedding|embeddings]] $N\to S^m$ up to smooth or PL [[isotopy]]. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are assumed to be compact. Let $B^n$ be a closed $n$-ball in a closed connected $n$-manifold $N$. Denote $N_0:=Cl(N-B^n)$. Let $\varepsilon(k):=1-(-1)^k$ be <script type="math/tex">n$-manifolds.
• Embedding Problem: Find the least dimension $m$$m$ such that given space admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$.
• Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.

The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].

This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1). After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic. We then establish notation and conventions. We continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings. We conclude with a brief review of some important results about codimension 1 embeddings

The most interesting and very much studied case concerns embeddings $S^1\to S^3$$S^1\to S^3$ (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information.

The Knotting Problem is known to be hard: to the best of the author's knowledge, at the time of writing there are only a few cases in which complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold $N$$N$ into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are the unknotting theorems in $\S$$\S$2, the results on the pages listed in Remark 1.4 and in $\S$$\S$6. Their statements, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.

The results and remarks given below show the following:

• The complete classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial but presently accessible only for $n+3\le m\le 2n$$n+3\le m\le 2n$ or for $m=n+1\ge4$$m=n+1\ge4$;
• For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes.

The lowest dimensional cases, i.e. all such pairs $(m,n)$$(m,n)$ with $n\le4$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

Remark 1.2 (Readily calculable classification). Let me informally explain what I mean by a `readily calculable classification'. (Such words are used by other people who might have a similar or a different concept.) In the best case a `readily calculable classification' is a classification in terms of (co)homology of a manifold (and certain structures on (co)homology like intersection, characteristic classes etc). A readily calculable classification is also a reduction to calculation of stable homotopy groups of spheres when these are known (or to some other standard algebraic problems involving only the homology of the manifold, which are solved in particular cases, although may be unsolved in general). An important feature of a useful classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.

Many readily calculable classification results are presented on this page and the pages listed in Remark 1.4. On the other hand, in some cases `geometric problems are reduced to algebraic problems which are even harder to solve' [Wall1999], and such reductions do not give a readily calculable classification. E.g. so far an interesting approach of [Weiss96], [Goodwillie&Weiss1999] did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss].)

See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

Remark 1.3 (Embeddings into the sphere and Euclidean space). (a) The embeddings $f,g:S^1\to\Rr^2$$f,g:S^1\to\Rr^2$ given by $f(x,y)=(x,y)$$f(x,y)=(x,y)$ and $g(x,y)=(x,-y)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below.

(b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection.

Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, it suffices to show that if the compositions with the inclusion $i:\Rr^m\to S^m$$i:\Rr^m\to S^m$ of two embeddings $f,f':N\to\Rr^m$$f,f':N\to\Rr^m$ of a compact $n$$n$-manifold $N$$N$ are isotopic, then $f$$f$ and $f'$$f'$ are isotopic. For showing that assume that $i\circ f$$i\circ f$ and $i\circ f'$$i\circ f'$ are isotopic. Then by general position $f$$f$ and $f'$$f'$ are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic.

Remark 1.4 (References to information on the classification of embeddings). The first list is structured by the dimension of the source manifold and the target Euclidean space:

• Embeddings just below the stable range: classification [Skopenkov2016e]

• 3-manifolds in 6-space [Skopenkov2016t]

• 4-manifolds in 7-space [Skopenkov2016f]

Information structured by the `complexity' of the source manifold:

• Knots, i.e. embeddings of spheres

• High codimension links [Skopenkov2016h]

• Knotted tori [Skopenkov2016k]

For more information see e.g. [Skopenkov2006].

2 Unknotting theorems

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

General Position Theorem 2.1 ([Hirsch1976, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

The case $m\ge2n+2$$m\ge2n+2$ is called a `stable range' (for the classification problem; for the existence problem there is analogous result with $m\ge2n+1$$m\ge2n+1$ [Skopenkov2006, $\S$$\S$2]).

The restriction $m\ge2n+2$$m\ge2n+2$ in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking $S^n\sqcup S^n\to\Rr^{2n+1}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].

Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those rferences or [Rourke&Sanderson1972, $\S$$\S$5]).

All the three assumptions in this result are indeed necessary:

• the assumption $n\ge2$$n\ge2$ because of the existence of non-trivial knots $S^1\to S^3$$S^1\to S^3$;
• the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h];
• the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e].

Unknotting Spheres Theorem 2.3. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, $m\ge n+3$$m\ge n+3$ or $2m\ge 3n+4$$2m\ge 3n+4$ in the PL or smooth category, respectively, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].

The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]).

Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for $N=S^n$$N=S^n$) in the Unknotting Spheres Theorem 2.3.

Theorems 2.2 and 2.3 may be generalized as follows.

The Haefliger-Zeeman Unknotting Theorem 2.4. For every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, $\S$$\S$7] work for homologically $k$$k$-connected manifolds (see $\S$$\S$3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the $k$$k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$$\Rr^m$ to the image of $N$$N$, by Alexander duality and simple connectedness of the complement, so homological $k$$k$-connectedness is sufficient).

Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a `stable range for $k$$k$-connected manifolds'.

Note that if $n\le2k+1$$n\le2k+1$, then every closed $k$$k$-connected $n$$n$-manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.

For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, $\S$$\S$5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.

3 Notation and conventions

The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.

For a manifold $N$$N$ let $E^m_D(N)$$E^m_D(N)$ or $E^m_{PL}(N)$$E^m_{PL}(N)$ denote the set of smooth or piecewise-linear (PL) embeddings $N\to S^m$$N\to S^m$ up to smooth or PL isotopy. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories.

The sources of all embeddings are assumed to be compact.

Let $B^n$$B^n$ be a closed $n$$n$-ball in a closed connected $n$$n$-manifold $N$$N$. Denote $N_0:=Cl(N-B^n)$$N_0:=Cl(N-B^n)$.

Let $\varepsilon(k):=1-(-1)^k$$\varepsilon(k):=1-(-1)^k$ be $0$$0$ for $k$$k$ even and $2$$2$ for $k$$k$ odd, so that $\Zz_{\varepsilon(k)}$$\Zz_{\varepsilon(k)}$ is $\Zz$$\Zz$ for $k$$k$ even and $\Zz_2$$\Zz_2$ for $k$$k$ odd.

Denote by $V_{m,n}$$V_{m,n}$ the Stiefel manifold of orthonormal $n$$n$-frames in $\Rr^m$$\Rr^m$.

We omit $\Zz$$\Zz$-coefficients from the notation of (co)homology groups.

For a manifold $P$$P$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$H_s(P,\partial):=H_s(P,\partial P)$.

A closed manifold $N$$N$ is called homologically $k$$k$-connected, if $N$$N$ is connected and $H_i(N)=0$$H_i(N)=0$ for every $i=1,\dots,k$$i=1,\dots,k$. This condition is equivalent to $\tilde H_i(N)=0$$\tilde H_i(N)=0$ for each $i=0,\dots,k$$i=0,\dots,k$, where $\tilde H_i$$\tilde H_i$ are reduced homology groups. A pair $(N,\partial N)$$(N,\partial N)$ is called homologically $k$$k$-connected, if $H_i(N,\partial)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$i=0,\dots,k$.

The self-intersection set of a map $f:X\to Y$$f:X\to Y$ is $\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$$\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$

For a smooth embedding $f:N\to\Rr^m$$f:N\to\Rr^m$ denote by

• $C_f$$C_f$ the closure of the complement in $S^m\supset\Rr^m$$S^m\supset\Rr^m$ to a tight enough tubular neighborhood of $f(N)$$f(N)$ and
• $\nu_f:\partial C_f\to N$$\nu_f:\partial C_f\to N$ the restriction of the linear normal bundle of $f$$f$ to the subspace of unit length vectors identified with $\partial C_f$$\partial C_f$.
• $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$$\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ and $A_f:H_s(N)\to H_{s+1}(C,\partial)$$A_f:H_s(N)\to H_{s+1}(C,\partial)$ the homological Alexander duality isomorphisms, see the well-known Alexander Duality Lemmas of [Skopenkov2008], [Skopenkov2005].

4 Embedded connected sum

Suppose that $m\ge n+2$$m\ge n+2$, $N$$N$ is a closed connected $n$$n$-manifold and, if $N$$N$ is orientable, an orientation of $N$$N$ is chosen. Let us define the embedded connected sum operation $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$.

Represent isotopy classes $[f]\in E^m(N)$$[f]\in E^m(N)$ and $[g]\in E^m(S^n)$$[g]\in E^m(S^n)$ by embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$ whose images are contained in disjoint balls. Join the images of $f,g$$f,g$ by an arc whose interior misses the images. Let $[f]\#[g]$$[f]\#[g]$ be the isotopy class of the embedded connected sum of $f$$f$ and $g$$g$ along this arc (compatible with the orientation, if $N$$N$ is orientable), cf. [Haefliger1966, Theorem 1.7], [Avvakumov2016, $\S$$\S$1].

This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes $[f]$$[f]$ and $[g]$$[g]$, and is independent of the choice of the path and of the representatives $f,g$$f,g$. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for $N=S^n$$N=S^n$ in [Skopenkov2015a, $\S$$\S$3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for $X=D^0_+$$X=D^0_+$ a point]. The proof for arbitrary closed connected $n$$n$-manifold $N$$N$ is analogous.

Moreover, for $m\ge n+3$$m\ge n+3$ embedded connected sum defines a group structure on $E^m(S^n)$$E^m(S^n)$ [Haefliger1966], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$.

5 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is an overwhelming multitude of isotopy classes of embeddings. We present some speculations as to why the classification in codimension 2 should not be accessible for manifolds, not only for spheres.

Let $N$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ there are embeddings $f:\Rr P^2\to S^4$$f:\Rr P^2\to S^4$ and $g_1,g_2:S^2\to S^4$$g_1,g_2:S^2\to S^4$ such that $g_1$$g_1$ is not isotopic to $g_2$$g_2$ but $f\#g_1$$f\#g_1$ is isotopic to $f\#g_2$$f\#g_2$ [Viro1973].) One can also apply Artin's spinning construction [Artin1928] $E^m(N)\to E^{m+1}(S^1\times N)$$E^m(N)\to E^{m+1}(S^1\times N)$ for $m=n+2$$m=n+2$. Thus the description of $E^{n+2}(N)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

6 Codimension 1 embeddings

Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$ [Smale1961], [Smale1962a], [Barden1965].

(b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$ [Kosinski1961], [Wall1965], [Lucas&Neto&Saeki1996], cf. [Goldstein1967].

The analogue of part (a) holds

• for $n=1$$n=1$ in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, $\S$$\S$1.8].
• for $n=2$$n=2$ in the PL category (Alexander Theorem, 1923) [Rushing1973, $\S$$\S$1.8].
• for every $n$$n$ in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].

The famous counterexample to the analogue of part (a) for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$).

Every embedding $S^1\times S^1\to S^3$$S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$$D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$$S^1\times D^2\to S^3$ [Alexander1924]. Clearly, only the standard embedding extends to both.

If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with `base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). So the description of $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ would be as hopeless as that of $E^3(S^1)$$E^3(S^1)$. Thus the description of $E^3(N)$$E^3(N)$ for $N$$N$ a sphere with handles is apparently hopeless.

For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].

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$ for $k$ even and $ for $k$ odd, so that $\Zz_{\varepsilon(k)}$ is $\Zz$ for $k$ even and $\Zz_2$ for $k$ odd. Denote by $V_{m,n}$ the Stiefel manifold of orthonormal $n$-frames in $\Rr^m$. We omit $\Zz$-coefficients from the notation of (co)homology groups. For a manifold $P$ with boundary $\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$. A closed manifold $N$ is called ''homologically $k$-connected'', if $N$ is connected and $H_i(N)=0$ for every $i=1,\dots,k$. This condition is equivalent to $\tilde H_i(N)=0$ for each $i=0,\dots,k$, where $\tilde H_i$ are reduced homology groups. A pair $(N,\partial N)$ is called ''homologically $k$-connected'', if $H_i(N,\partial)=0$ for every $i=0,\dots,k$. The ''self-intersection set'' of a map $f:X\to Y$ is $\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$ For a smooth embedding $f:N\to\Rr^m$ denote by * $C_f$ the closure of the complement in $S^m\supset\Rr^m$ to a tight enough tubular neighborhood of $f(N)$ and * $\nu_f:\partial C_f\to N$ the restriction of the linear normal bundle of $f$ to the subspace of unit length vectors identified with $\partial C_f$. * $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ and $A_f:H_s(N)\to H_{s+1}(C,\partial)$ the homological Alexander duality isomorphisms, see the well-known Alexander Duality Lemmas of \cite{Skopenkov2008}, \cite{Skopenkov2005}. == Embedded connected sum == ; \label{s:sum} Suppose that $m\ge n+2$, $N$ is a closed connected $n$-manifold and, if $N$ is orientable, an orientation of $N$ is chosen. Let us define the ''embedded connected sum operation'' $\#$ of $E^m(S^n)$ on $E^m(N)$. Represent isotopy classes $[f]\in E^m(N)$ and $[g]\in E^m(S^n)$ by embeddings $f:N\to\Rr^m$ and $g:S^n\to\Rr^m$ whose images are contained in disjoint balls. Join the images of $f,g$ by an arc whose interior misses the images. Let $[f]\#[g]$ be the isotopy class of the ''embedded connected sum'' of $f$ and $g$ along this arc (compatible with the orientation, if $N$ is orientable), cf. \cite[Theorem 1.7]{Haefliger1966}, \cite[$\Sn-manifolds.

Embedding Problem: Find the least dimension $m$$m$ such that given space admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$.

Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.

The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].