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; thes definitions of these terms are given in [Skopenkov2016i].

Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].

• The 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; thes definitions of these terms are given in \cite{Skopenkov2016i}. {{beginthm|Remark|(Some motivations)}}\label{r:zee} Three important classical problems in topology are the following, cf. \cite[p. 3]{Zeeman1993}. * ''The Homeomorphism Problem'': Classify $n$-manifolds up to homeomorphism. * ''The Embedding Problem'': Find the least dimension $m$ such that given space admits an [[Embedding_(simple_definition)|embedding]] into $m$-dimensional Euclidean space $\Rr^m$. * ''The 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 the [[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 embeddings of closed manifolds $N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark \ref{r:zee} for embeddings of general manifolds $N$ into $\R^m$ or $S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$ are isotopic.In Section 3, we establish notation and conventions <!-- and give references to other pages on the Knotting Problem, to which this page serves as an introduction.--> and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$, or more generally, [[Knots,_i.e._embeddings_of_spheres#Codimension_2_knots|codimension 2 embeddings of spheres]]. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide 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 ''readily calculable'' classification results (see Remark \ref{compl}) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$ are known. Such classification results are presented on the pages listed in Remark \ref{s:list}. <!-- (either in direct form or in the form of more general ''readily calculable'' classification, see Remark \ref{compl}). (for m<3n+4$ the information on embeddings $S^n\to\R^m$ and $S^p\times S^{n-p}\to\R^m$ has to be completed there), [[#Unknotting_theorems|in $\S$\ref{s:ut}]] and in [[#Codimension_1_embeddings|$\S$\ref{s:c1}]]. [[#References to information on the classification of embeddings|]] --> The statements of those results, 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: *For a fixed $N$, the more $m$ decreases from n$ towards $n+3$, the more complicated the classification of embeddings of $N$ into $\Rr^m$ becomes. *The complete readily calculable classification of embeddings into $\Rr^m$ of closed connected $n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$ or for $m=n+1\ge4$; 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} The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (d) below). To a first approximation, a classification of embeddings $N\to\Rr^m$ is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number $m$ and the manifold $N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept, further illustrated by the following general remarks (a)-(e) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. This concept is used here to explain why some classifications of embeddings are presented and some others are left aside. <!-- Mathematicians often discuss what is more useful answer, what is less useful, and what us being useful. This remark does not intend to give a full description of `readily calculable', but is rather an invitation to further pursue the reflections by Wall (see (d) below), Graham, Knuth, Patashnik \cite[Preface]{Graham&Knuth&Patashnik89}, and others. (including classification via Poincar\'e embeddings and (ii), (iii) below). An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (see (i) below; this includes finitely generated abelian groups and other objects `easily' characterized by integers). (given by invariant or a collection of invariants)--> (a) An important feature of a readily calculable (concrete) 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. (b) A classification involving invariants which are `easily' calculated from the (co)homology on the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. For example, the integral homology of a manifold many not be easy to calculate, depending upon how the manifold is presented. (c) If a result requires several pages of formulation, several dozen pages of a proof, and recovers only explicit results which can be obtained by much simpler methods, then we would not regard this result as readily calculable. <!-- (although it might be called such if we formally apply the above description of `readily calculable').--> E.g. I regard the [[High_codimension_links#Classification_in_the_metastable_range|the Haefliger-Zeeman Theorem]] \cite[Theorem 4.1]{Skopenkov2016h} readily calculable only because it has stronger explicit corollaries than classification of embeddings $S^n\sqcup S^n\to\Rr^m$ for $m\ge2n+1$. <!-- clasification of surfaces. A readily calculable classification is a `final result' accessible to non-specialists, while an abstract classification is a less accessible `tool' (which might or night not be useful depending on whether the tool gives explicit result).--> (d) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré complexes). See \cite[Corollary 11.3.1]{Wall1999} and a simpler exposition in \cite{Cencelj&Repovs&Skopenkov2004}. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' \cite[p. 119]{Wall1999}. An analogue of this result for classification of embeddings is \cite[the Browder-Wall Theorem 9]{Cencelj&Repovs&Skopenkov2004}. To the best of my knowledge, the proof did not appear in the literature. Thus, although the proof is likely to be obtained using surgery analogously to \cite[Corollary 11.3.3]{Wall1999} as exposed in \cite[pp. 265-267]{Cencelj&Repovs&Skopenkov2004}, this result is a conjecture, and I apologize for stating it as a theorem. The above citation of Wall applies to this conjecture: as far as I know, it was never used to obtain any explicit classification results for particular manifolds. However, some readily calculable classifications of embeddings (see \cite{Skopenkov2016t}, \cite{Skopenkov2016f}) have been obtained by applying Kreck's modified surgery theory, see (e). (e) Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds (see the Homeomorphism Problem above). While this is an important achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory \cite{Kreck1999} was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see \cite{Kreck1999}. <!-- $I(H^*, ch)$-Poincaré-manifolds, as defined in \cite[$\S$ 3]{Kreck2001} are would be ``readily classifiable'' by definition and indeed see \cite{Wall1999} AS: deleted because no reference to a page or a section is given. one knows the classification up to homotopy, the computation of the normal invariants, the action of the L-groups on the structure set and the action of the group of homotopy self-equivalences on the structure set AS: deleted as a set of technicalities not relevant here signature achievement AS: sounds confusing in this context of \sgn The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in \cite{Wall1999}.) --> (i) A reduction to calculation of standard objects of mathematics when these are known is readily calculable classification. E.g. [[High_codimension_links#Classification_in_the_metastable_range|the Haefliger-Zeeman Theorem]] \cite[Theorem 4.1]{Skopenkov2016h} classifies links in terms of (stable) [[Wikipedia:Homotopy_groups_of_spheres|homotopy groups of spheres]]. These groups are not known in general. However, these groups are described for many particular cases which give [[High_codimension_links#Introduction|the table]] \cite[$\Sn$-manifolds up to homeomorphism.
• The 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$.
• The 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 the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].

This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.

The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\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 readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4. The statements of those results, 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:

• For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes.
• The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and 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$; 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). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (d) below). To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept, further illustrated by the following general remarks (a)-(e) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. This concept is used here to explain why some classifications of embeddings are presented and some others are left aside.

(a) An important feature of a readily calculable (concrete) 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.

(b) A classification involving invariants which are `easily' calculated from the (co)homology on the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. For example, the integral homology of a manifold many not be easy to calculate, depending upon how the manifold is presented.

(c) If a result requires several pages of formulation, several dozen pages of a proof, and recovers only explicit results which can be obtained by much simpler methods, then we would not regard this result as readily calculable. E.g. I regard the the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] readily calculable only because it has stronger explicit corollaries than classification of embeddings $S^n\sqcup S^n\to\Rr^m$$S^n\sqcup S^n\to\Rr^m$ for $m\ge2n+1$$m\ge2n+1$.

(d) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré complexes). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification of embeddings is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. To the best of my knowledge, the proof did not appear in the literature. Thus, although the proof is likely to be obtained using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267], this result is a conjecture, and I apologize for stating it as a theorem. The above citation of Wall applies to this conjecture: as far as I know, it was never used to obtain any explicit classification results for particular manifolds. However, some readily calculable classifications of embeddings (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (e).

(e) Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds (see the Homeomorphism Problem above). While this is an important achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].

(i) A reduction to calculation of standard objects of mathematics when these are known is readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], only the left two columns of the table are recovered by easier proofs, and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.

(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in `metastable range') in terms of equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See [Haefliger1963], [Weber1967], [Skopenkov2002] and a survey [Skopenkov2006, $\S$$\S$5]. Such equivariant maps are not classified in general. The deleted product of a general manifold is less `standard object' of mathematics (if at all) than homotopy groups of spheres. Most of the explicit corollaries of the Haefliger-Weber Theorem are recovered by easier proofs. Thus I regard the Haefliger-Weber classification of embeddings not to be readily calculable. This does not mean that it is useless: it implies

• Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable results;
• the independence of the classification of embeddings of the smooth or PL structure on the manifold;
• the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].

(iii) An interesting approach of Goodwille-Weiss (the calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. This does not mean that it is completely useless: it gives a modern abstract proof of the Haefliger-Weber Theorem; 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].

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

(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 [Skopenkov2016s]

• 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].

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 references 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.

Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i]. 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

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${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor.

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$.

Denote by

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${\rm i}:S^q\to S^m$ the standard embedding given by

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${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on

Tex syntax error

${\rm i}$ defines the standard embedding

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${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol

Tex syntax error

${\rm i}_{m,q}$ the restriction of

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${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$.

Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large.

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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]{Skopenkov2016h}, only the left two columns of the table are recovered by easier proofs, and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in `metastable range') in terms of equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See \cite{Haefliger1963}, \cite{Weber1967}, \cite{Skopenkov2002} and a survey \cite[$\S]{Skopenkov2006}. Such equivariant maps are not classified in general. The deleted product of a general manifold is less `standard object' of mathematics (if at all) than homotopy groups of spheres. Most of the explicit corollaries of the Haefliger-Weber Theorem are recovered by easier proofs. Thus I regard the Haefliger-Weber classification of embeddings not to be readily calculable. This does not mean that it is useless: it implies * [[Embeddings_just_below_the_stable_range:_classification#Classification_further_below_the_stable_range|Becker-Glover Theorem]] \cite[Theorem 6.5]{Skopenkov2016e} and [[Knotted_tori#Classification|classification of knotted tori]] \cite[Theorem 3.2]{Skopenkov2016k}, which I regard as readily calculable results; * the independence of the classification of embeddings of the smooth or PL structure on the manifold; * the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) \cite[Theorem 1.1 and text after Theorem 1.4]{Cadek&Krcal&Vokrınek2013}. (iii) An interesting approach of Goodwille-Weiss (the calculus of embeddings) \cite{Weiss96}, \cite{Goodwillie&Weiss1999} so far did not give any new complete readily calculable classification results. This does not mean that it is completely useless: it gives a modern abstract proof of the Haefliger-Weber Theorem; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ \cite{Weiss}. {{endthm}} {{beginthm|Remark|(Embeddings into Euclidean space and the sphere)}}\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 the [[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 it. (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}} {{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]] \cite{Skopenkov2016s} * [[High codimension links]] \cite{Skopenkov2016h} * [[Knotted tori]] \cite{Skopenkov2016k} For more information see e.g. \cite{Skopenkov2006}. {{endthm}} ==Unknotting theorems == ; \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}. {{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 references 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}. 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}. == Notation and conventions == ; \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}. Unless otherwise indicated, the word `isotopy' means [[Isotopy|`ambient isotopy']]; see definition in \cite{Skopenkov2016i}. 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 $n$-manifolds up to homeomorphism.

The 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$.

The 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 the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].