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; the definitions of these terms are given in [Skopenkov2016i, ${{Stub}} == 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. According to \cite[p. 3]{Zeeman93}, three major classical problems of topology are the following. * ''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|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. This article gives a short guide to the Knotting Problem of compact manifolds $N$ in Euclidean spaces and in spheres. After making general remarks 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 conclude by introducing connected sum and make some comments on codimension 1 and 2 embeddings. The most interesting and very much studied case concerns embeddings $S^1\to S^3$ (classical knots), or more generally, codimension 2 embeddings of spheres or arbitrary manifolds. 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]] for more information and $\S$\ref{s:c2} for some speculations why the classification in codimension 2 should not be accessible for manifolds, not only for spheres.  Even in general, the Knotting Problem is known to be hard: at the time of writing there are only a few cases in which ''complete readily calculable'' classification results describing all isotopy classes for embeddings of a ''closed'' manifold $N$ into Euclidean space $\Rr^m$. Cf. Remark \ref{compl}. This page and pages listed $\S$\ref{s:list} concern such results, so we have to leave out much of the other research, both for codimension $\le2$ and for codimension $\ge3$. Such results are the [[#Unknotting_theorems|unknotting theorems]] in $\S$\ref{s:ut}, the results on the [[#References to information on the classification of embeddings|pages listed below]] in $\S$\ref{s:list} and in $\S$\ref{s:c1}. Their statements, although not the proofs, are simple and accessible to non-specialists.  The results and remarks in $\S\S$\ref{s:ut},\ref{s:list},\ref{s:c2},\ref{s:c1} show that *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 $m=n+1\ge4$. *the farther we go from $m=2n$ to $m=n+3$, the more complicated classification is. 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 known information on the cases (6,3), (8,4), (7,4) see \cite{Skopenkov2016t}, \cite{Skopenkov2016e}, \cite{Skopenkov2016f}. {{beginthm|Remark|(Readily calculable classification)}}\label{compl} A `readily calculable classification' is an informal expression. In the best case this is a classification in terms of homology of a manifold (and certain structures in 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 another standard algebraic problem involving only the homology of the manifold, which is solved in particular cases, although may be unsolved in general). An important feature of a useful classification is accessibility of statement to general mathematical audience which is only familiar with basic notions of the area; this in turn is an approximation to beauty. Another important feature is whether the classification gives an algorithm and how complex the algorithm is. Many readily calculable classification results are presented in these pages. On the other hand, in some cases `geometric problems are reduced to algebraic problems which are even harder to solve' \cite{Wall1999}, and such results 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}.) {{endthm}} {{beginthm|Remark|(Embeddings into the sphere and Euclidean space)}}\label{spheu} (a) Embeddings $f,g:S^1\to\Rr^2$ given by $f(x,y)=(x,y)$ and $g(x,y)=(x,-y)$ are not isotopic. On the other hand, any two embeddings $S^1\to S^2$ are isotopic (this intuitively clear assertion is non-trivial, see $\S$\ref{s:c1}). (b) For $m\ge n+2$ the classifications of embeddings of $n$-manifolds into $S^m$ and into $\Rr^m$ are the same. Let us prove this for the smooth category. 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 (in spite of the existence of orientation-preserving diffeomorphisms $S^m\to S^m$ not isotopic to the identity). 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 ambient one \cite{Hirsch1976}, Theorem 1.3, $f$ and $f'$ are isotopic. {{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{Hirsch1976}, \cite{Rourke&Sanderson1972})}}\label{th1} For each compact $n$-manifold $N$ and $m\ge2n+2$, every two embeddings $N\to\Rr^m$ are isotopic. {{endthm}} The case $m\ge2n+2$ is called a `stable range'. The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds, as the [[Links,_i.e._embeddings_of_non-connected_manifolds#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 each compact connected $n$-manifold $N$, $n\ge2$ and $m\ge2n+1$, every two embeddings $N\to\Rr^m$ are isotopic. {{endthm}} This is proved in \cite{Wu1958}, \cite{Wu1958a} and \cite{Wu1959} using Whitney trick. 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 [[Links,_i.e._embeddings_of_non-connected_manifolds#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 $N$ a homology $n$-sphere, $m\ge n+3$ or m\ge 3n+4$ in the PL or smooth category, respectively, every two embeddings $N\to\Rr^m$ 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'. Knots in codimension 2 and the [[3-manifolds in 6-space#Examples|the Haefliger trefoil knot]] 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 each $n\ge2k+2$, $m\ge2n-k+1$ and closed $k$-connected $n$-manifold $N$, every two embeddings $N\to\Rr^m$ 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 definition). So 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 in some other pages about embeddings, including those listed in $\S$\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 a category is omitted, then the result holds (or a definition or a construction is given) in both categories. All manifolds are tacitly 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">\S$ 1].

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

The Manifold Problem: Classify $n$ -manifolds.
The Embedding Problem: Find the least dimension $m$ such that a given manifold admits an embedding into $m$ -dimensional Euclidean space $\Rr^m$ .
The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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 J. W. Alexander, H. Hopf, 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, M. Irwin, J. F. P. Hudson and others.

The Knotting Problem is related to other areas 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], [Takase2006, $\S$ 1].

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 1.1 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 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, 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 $\S$ 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 $\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$ 2 and in $\S$ 6. 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 $N$ , the more $m$ decreases from $2n$ 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 (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].

The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. 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. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.

Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold $N$ into $\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).

(a) An important feature of a readily calculable 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) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold $N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of 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. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.

(c) 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 embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `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 is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).

(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major 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 of a classification to calculation of standard objects of mathematics when these are known is typically a 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, $\S$ 1], 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 a `metastable range') in terms of equivariant homotopy classes of certain 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 survey [Skopenkov2006, $\S$ 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies

the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
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) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014].

(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 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 6.1.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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], $f$ and $f'$ 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]

Embeddings of manifolds with boundary: classification

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.

For every compact $n$ -manifold $N$ and $m\ge2n+2$ , any two embeddings of $N$ into $\Rr^m$ are isotopic.

The case $m\ge2n+2$ is called a `stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).

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

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

This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$ 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].

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 the Hopf link [Skopenkov2016h, Example 2.1];
the assumption $m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1].

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

For $N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$ an integral homology $n$ -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].

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

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

Theorems 2.2 and 2.3 may be generalized as follows.

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.

This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$ 11], [Rourke&Sanderson1972, $\S$ 5]. Theorem 2.4 remains true if $k$ -connected is replaced by homologically $k$ -connected (see $\S$ 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).

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

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

For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [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, $\S$ 1]. For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of smooth or piecewise-linear (PL) 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 $0$ for $k$ even and $2$ for $k$ odd, so that $\Zz_{\varepsilon(k)}$ is $\Zz$ for $k$ even and $\Zz_2$ for $k$ 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 $V_{m,n}$ 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 $\partial C_f$ .

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

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${\rm i}$ 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

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

4 Embedded connected 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. [Haefliger1966, 1.3, 1.4].

This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes $[f]$ and $[g]$ , and is independent of the choice of the path and of the representatives $f,g$ . 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 $N=S^n$ in [Skopenkov2015a, $\S$ 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for $X=D^0_+$ a point]. The proof for arbitrary closed connected $n$ -manifold $N$ is analogous.

Moreover, for $m\ge n+3$ embedded connected sum defines a group structure on $E^m(S^n)$ [Haefliger1966, 1.3-1.7], and an action $\#$ of $E^m(S^n)$ on $E^m(N)$ .

5 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$ into $\Rr^{n+2}$ 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 $N$ be a closed connected $n$ -manifold. Using embedded connected sum ( $\S$ 4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$ . (However, note that for $n=2$ there are embeddings $f:\Rr P^2\to S^4$ and $g_1,g_2:S^2\to S^4$ such that $g_1$ is not isotopic to $g_2$ but $f\#g_1$ is isotopic to $f\#g_2$ [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] $E^m(N)\to E^{m+1}(S^1\times N)$ for $m=n+2$ . Thus the description of $E^{n+2}(N)$ 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

(a) Any two smooth embeddings of $S^n$ into $S^{n+1}$ are smoothly isotopic for every $n\ne3$ .

(b) Any two smooth embeddings of $S^p\times S^{n-p}$ into $S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$ .

Part (a) is proved in [Milnor1965a, Proposition D in $\S$ 9] as an easy corollary of the $h$ -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$ 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.

[Alexander1924] For every embedding $S^1\times S^1\to S^3$ there is an autohomeomorphism $h$ of $S^1\times S^1$ such that $f\circ h$ extends to an embedding $D^2\times S^1\to S^3$ .

Clearly, only the standard embedding extends to both.

Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds

for $n=1$ in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, $\S$ 1.8].
for $n=2$ in the PL category (Alexander Theorem, 1923) [Rushing1973, $\S$ 1.8].
for every $n$ 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 Theorem 6.1.a for $n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$ into $S^{n+1}$ are isotopic for every $n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$ ).

(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$ with `base points' $i\times0$ and $0\times i$ identified (where $i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$ ). So the description of $E^3(S^1\times S^1)$ would be as hopeless as that of $E^3(S^1)$ . Thus the description of $E^3(N)$ for $N$ a sphere with handles is apparently hopeless.

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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 $P$ 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 each $i=1,\dots,k$. 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 $H_i(N,\partial N)=0$ for each $i=0,\dots,k$. 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 space of unit length vectors identified with $\partial C_f$. * $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ and $A_f:H_s(N)\to H_{s+1}(C,\partial)$ homological Alexander duality isomorphisms, see well-known Alexander Duality Lemmas of \cite{Skopenkov2008}, \cite{Skopenkov2005}. == References to information on the classification of embeddings == ; \label{s:list} Below we list references to information about the classification of embeddings of manifolds into Euclidean space. 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}. == Embedded connected sum == ; Suppose that $N$ 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[$\S\S1].

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

The Manifold Problem: Classify $n$ -manifolds.
The Embedding Problem: Find the least dimension $m$ such that a given manifold admits an embedding into $m$ -dimensional Euclidean space $\Rr^m$ .
The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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 J. W. Alexander, H. Hopf, 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, M. Irwin, J. F. P. Hudson and others.

The Knotting Problem is related to other areas 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], [Takase2006, $\S$ 1].

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 1.1 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 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, 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 $\S$ 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 $\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$ 2 and in $\S$ 6. 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 $N$ , the more $m$ decreases from $2n$ 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 (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].