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 mathematician from other areas who want to apply or to learn theory of embeddings.

According to Zeeman [1], the classical problems of topology are the following.

• Homeomorphism Problem: Classify $n$${{Stub}} == Introduction == <wikitex>; This page is intended not only for specialists in embeddings, but also for mathematician from other areas who want to apply or to learn theory of embeddings. According to Zeeman [http://www.mccme.ru/circles/oim/foto2014/l_ht.html], the 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]]. See an [[Embedding#Definition|alternative definition of an embedding]]. 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 reviews the Knotting Problem. After making general remarks we establish notation and conventions, record some of the dimension ranges where no knotting is possible, introduce connected sum and make some comments on codimension 1 and 2 embeddings. The [[#Unknotting_theorems|unknotting theorems]] and the results on the pages [[#Links for information about embeddings|below]] record all known ''complete readily calculable'' isotopy classification results for embeddings of ''closed'' manifolds into Euclidean spaces. <!-- In particular, we do this for codimension 1 and 2 embeddings. --> The Knotting Problem is known to be hard: * There are only [[#Links for information about embeddings|a few cases]] in which there are ''complete readily calculable'' classification results. (However, the statements, if not the proofs, are simple and accessible to non-specialists.) * For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in $\Rr^3$), a complete readily calculable classification is neither known nor expected. (Note that there is an extensive study of codimension 2 embeddings not directly aiming at complete classification. Almost nothing is said here about this. See more in [[Wikipedia:Knot_theory|knot theory]].) <!--So in the title of this article we only mention embeddings of codimension greater than 2. --> * In some cases `geometric problems are reduced to algebraic problems which are even harder to solve' \cite{Wall1999}. 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}.) </wikitex> === Sphere and Euclidean space === <wikitex>; Every two embeddings $S^1\to S^2$ are isotopic (this intuitively clear assertion is Sch\"oenfliess Theorem) \cite{Rushing1973}. On the other hand, embeddings $f,g:S^1\to\Rr^2$ given by $f(x,y)=(x,y)$ and $g(x,y)=(x,-y)$ are not isotopic. This example is generalized to (locally flat) embeddings $S^{m-1}\to S^m$ and $S^{m-1}\to\Rr^m$ \cite{Rushing1973}. For $m\ge n+2$ 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. </wikitex> === Notation and conventions === <wikitex>; The following notations and conventions will be widely used for pages about embeddings. 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 $\Zz_{(k)}$ be $\Zz$ for $k$ even and $\Zz_2$ for $k$ odd. Denote by $V_{m,n}$ the Stiefel manifold of $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)$. The ''self-intersection set'' of a map $f:X\to Y$ is $\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$ For an 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$. </wikitex> === Links for information about classification of embeddings === ==== By dimensions of a manifold and a Euclidean space==== [[Embeddings just below the stable range: classification]] [[3-manifolds in 6-space]] [[4-manifolds in 7-space]] ==== By `complexity' of a manifold==== [[Knots, i.e. embeddings of spheres]] [[High codimension links]] [[Knotted tori]] [[Unknotting of projective spaces]] <!--[[Embeddings of highly-connected manifolds: classification]] [[Link maps: classification]] [[Immersions: classification]] [[Embeddings of manifolds with boundary]] [[Some open problems]] --> <wikitex>; For more information see e.g. \cite{Skopenkov2006}. </wikitex> ==Unknotting theorems == <wikitex>; {{beginthm|General Position Theorem}}\label{th1} For each $n$-manifold $N$ and $m\ge2n+2$, every two embeddings $N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$). {{endthm}} 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#General_position_and_the_Hopf_linking|Hopf linking]] $S^n\sqcup S^n\to\Rr^{2n+1}$ shows (see Figure~2.1.a of \cite{Skopenkov2006}). {{beginthm|Theorem}}\label{th2} For each connected $n$-manifold $N$, $n\ge2$ and $m\ge2n+1$, every two embeddings $N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$) \cite{Wu1958}, \cite{Wu1958a} and \cite{Wu1959}. {{endthm}} 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 linking above; *the assumption $m\ge2n+1$ because of the [[Embeddings just below the stable range#Hudson_tori|example of Hudson tori]]. {{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\frac{3n}2+2$ in the PL or DIFF category, respectively, every two embeddings $N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$) \cite{Zeeman1960}, \cite{Stallings1963}, \cite{Gluck1963}, \cite{Haefliger1961}, \cite{Adachi1993}. {{endthm}} This result is also true for $m\ge n+3$ in the TOP locally flat category \cite{Rushing1973}, \cite{Scharlemann1977}. Here the local flatness assumption is indeed necessary. Knots in codimension 2 and the higher-dimensional analogue of [[3-manifolds in 6-space#The Haefliger trefoil knot|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 $n\ge2k+2$, $m\ge2n-k+1$ and each closed homologically $k$-connected $n$-manifold $N$, every two embeddings $N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$). {{endthm}} So $m\ge2n-k+1$ can be called a `stable range for $k$-connected manifolds'. The Haefliger-Zeeman Unknotting Theorem \ref{haze} was proved directly in \cite{Penrose&Whitehead&Zeeman1961}, \cite{Haefliger1961}, \cite{Zeeman1962}, \cite{Irwin1965}, \cite{Hudson1969} for ''homotopically'' $k$-connected manifolds. The proofs in \cite{Haefliger&Hirsch1963}, \cite{Vrabec1977}, \cite{Weber1967}, \cite{Adachi1993}, \S7 work for ''homologically'' $k$-connected manifolds. Note that if $n\le2k+1$, then every closed homologically $k$-connected $n$-manifold is a homology 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} see [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|here]] or \cite{Hudson1967}, \cite{Hacon1968}, \cite{Hudson1972}, \cite{Gordon1972}, \cite{Kearton1979}. </wikitex> == Embedded connected sum == <wikitex>; 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 action'' $\#$ of $E^m(S^n)$ on $E^m(N)$. Represent elements of $E^m(N)$ and of $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{Haefliger1966}, Theorem 1.7, \cite{Avvakumov2016}, \S1. If $m\le n+1$, this operation is not well-defined. If $m\ge n+2$, 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]$ and $[g]$, but is independent of the choice of the path and of the representatives $f,g$. Although this fact is presumably classical, a proof was not written, cf. \cite{Skopenkov2015a}, Remark 2.3.a. This fact proved analogously to the case $X=D^0_+$ of \cite{Skopenkov2015a}, Standardization Lemma 2.1.b, because the construction of $\#$ has an analogue for isotopy. Moreover, for $m\ge n+3$ (embedded) connected sum defines a group structure on $E^m(S^n)$ \cite{Haefliger1966}, and an action $\#$ of $E^m(S^n)$ on $E^m(N)$. </wikitex> == Codimension 2 embeddings == <wikitex>; A description of $E^3(S^1)$ and, more generally, of $E^{n+2}(S^n)$ is a well-known very hard open problem. Let $N$ be a closed connected $n$-manifold. Using connected sums we can 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 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$ \cite{Viro1973}. One can also apply Artin's spinning construction \cite{Artin1928} $E^{n+2}(N)\to E^{n+3}(S^1\times N)$. Thus 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 hardness of this problem. On the other hand, if one studies embeddings up to the weaker relation of ''concordance'', then much is known. See e.g. \cite{Levine1969a}, \cite{Cappell&Shaneson1974} and \cite{Ranicki1998}. </wikitex> <!-- See open problems on classification `modulo knots' to appear below. --> == Codimension 1 embeddings == <wikitex>; Every embedding $S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$ \cite{Alexander1924}. Clearly, only the standard embedding extends to both. When one proves that this extension respects isotopy, this would give a 1-1 correspondence $E^3(S^1\times S^1)\to (E^3(S^1)\times\Zz)\vee(E^3(S^1)\times\Zz)$. 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. It is known that * $\#E^2(S^1)=1$ (Schoenfliess theorem) \cite{Rushing1973}. * $\#E^3(S^2)=1$ (Alexander theorem); a counterexample to the topological version is the Alexander Horned Sphere \cite{Rushing1973}. * The description of $E^{n+1}_{PL}(S^n)$ is equivalent to the PL Schoenfliess problem and therefore is very hard for $n\ge3$ \cite{Rushing1973}. * $\#E^{n+1}_D(S^n)=1$ for $n\ge6$ \cite{Smale1962a}. * $\#E^{n+1}_D(S^p\times S^{n-p})=1$ for \le p\le n-p$ \cite{Kosinski1961}, \cite{Wall1965}, \cite{Lucas&Neto&Saeki1996}, cf. \cite{Goldstein1967}. Note that $E^{n+1}_{PL,lf}(S^p\times S^{n-p})$ can be known even when $E^{n+1}_{PL,lf}(S^n)$ is unknown \cite{Goldstein1967} (here $lf$ stands for locally flat). For more on higher-dimensional codimension 1 embeddings see e.g. \cite{Lucas&Saeki2002}. </wikitex> == Conclusion == <wikitex>; Thus 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 lowest dimensional cases are $(m,n)=$ [[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). </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Embeddings of manifolds]]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.

See an alternative definition of an embedding.

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 reviews the Knotting Problem. After making general remarks we establish notation and conventions, record some of the dimension ranges where no knotting is possible, introduce connected sum and make some comments on codimension 1 and 2 embeddings. The unknotting theorems and the results on the pages below record all known complete readily calculable isotopy classification results for embeddings of closed manifolds into Euclidean spaces.

The Knotting Problem is known to be hard:

• There are only a few cases in which there are complete readily calculable classification results. (However, the statements, if not the proofs, are simple and accessible to non-specialists.)
• For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in $\Rr^3$$\Rr^3$), a complete readily calculable classification is neither known nor expected. (Note that there is an extensive study of codimension 2 embeddings not directly aiming at complete classification. Almost nothing is said here about this. See more in knot theory.)
• In some cases `geometric problems are reduced to algebraic problems which are even harder to solve' [Wall1999]. E.g. so far an interesting approach of [Weiss96], [Goodwillie&Weiss1999] did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss].)

1.1 Sphere and Euclidean space

Every two embeddings $S^1\to S^2$$S^1\to S^2$ are isotopic (this intuitively clear assertion is Sch\"oenfliess Theorem) [Rushing1973]. On the other hand, embeddings $f,g:S^1\to\Rr^2$$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. This example is generalized to (locally flat) embeddings $S^{m-1}\to S^m$$S^{m-1}\to S^m$ and $S^{m-1}\to\Rr^m$$S^{m-1}\to\Rr^m$ [Rushing1973].

For $m\ge n+2$$m\ge n+2$ classifications of embeddings of $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\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$$i:\Rr^m\to S^m$ of two embeddings $f,f':N\to\Rr^m$$f,f':N\to\Rr^m$ of a compact $n$$n$-manifold $N$$N$ are isotopic, then $f$$f$ and $f'$$f'$ are isotopic (in spite of the existence of orientation-preserving diffeomorphisms $S^m\to S^m$$S^m\to S^m$ not isotopic to the identity). For showing that assume that $i\circ f$$i\circ f$ and $i\circ f'$$i\circ f'$ are isotopic. Then by general position $f$$f$ and $f'$$f'$ are non-ambiently isotopic. Since every non-ambient isotopy extends to an ambient one [Hirsch1976], Theorem 1.3, $f$$f$ and $f'$$f'$ are isotopic.

1.2 Notation and conventions

The following notations and conventions will be widely used for pages about embeddings.

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 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$$B^n$ be a closed $n$$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 $\Zz_{(k)}$$\Zz_{(k)}$ be $\Zz$$\Zz$ for $k$$k$ even and $\Zz_2$$\Zz_2$ for $k$$k$ odd.

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

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

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

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 an 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$$f$ to the space of unit length vectors identified with $\partial C_f$$\partial C_f$.

1.3 Links for information about classification of embeddings

1.3.1 By dimensions of a manifold and a Euclidean space

Embeddings just below the stable range: classification

3-manifolds in 6-space

4-manifolds in 7-space

1.3.2 By `complexity' of a manifold

Knots, i.e. embeddings of spheres

High codimension links

Knotted tori

Unknotting of projective spaces

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

2 Unknotting theorems

General Position Theorem 2.1. For each $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$$\#E^m(N)=1$).

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 (see Figure~2.1.a of [Skopenkov2006]).

Theorem 2.2. For each connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$$\#E^m(N)=1$) [Wu1958], [Wu1958a] and [Wu1959].

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 linking above;
• the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori.

Unknotting Spheres Theorem 2.3. For $N=S^n$$N=S^n$, or even for $N$$N$ a homology $n$$n$-sphere, $m\ge n+3$$m\ge n+3$ or $m\ge\frac{3n}2+2$$m\ge\frac{3n}2+2$ in the PL or DIFF category, respectively, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$$\#E^m(N)=1$) [Zeeman1960], [Stallings1963], [Gluck1963], [Haefliger1961], [Adachi1993].

This result is also true for $m\ge n+3$$m\ge n+3$ in the TOP locally flat category [Rushing1973], [Scharlemann1977]. Here the local flatness assumption is indeed necessary.

Knots in codimension 2 and the higher-dimensional analogue of the Haefliger trefoil knot 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 $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and each closed homologically $k$$k$-connected $n$$n$-manifold $N$$N$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$$\#E^m(N)=1$).

So $m\ge2n-k+1$$m\ge2n-k+1$ can be called a `stable range for $k$$k$-connected manifolds'.

The Haefliger-Zeeman Unknotting Theorem 2.4 was proved directly in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969] for homotopically $k$$k$-connected manifolds. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993], \S7 work for homologically $k$$k$-connected manifolds.

Note that if $n\le2k+1$$n\le2k+1$, then every closed homologically $k$$k$-connected $n$$n$-manifold is a homology 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 here or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979].

3 Embedded connected sum

Suppose that $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 action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$.

Represent elements of $E^m(N)$$E^m(N)$ and of $E^m(S^n)$$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], \S1.

If $m\le n+1$$m\le n+1$, this operation is not well-defined. If $m\ge n+2$$m\ge n+2$, 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]$, but is independent of the choice of the path and of the representatives $f,g$$f,g$. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a], Remark 2.3.a. This fact proved analogously to the case $X=D^0_+$$X=D^0_+$ of [Skopenkov2015a], Standardization Lemma 2.1.b, because the construction of $\#$$\#$ has an analogue for isotopy.

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)$.

4 Codimension 2 embeddings

A description of $E^3(S^1)$$E^3(S^1)$ and, more generally, of $E^{n+2}(S^n)$$E^{n+2}(S^n)$ is a well-known very hard open problem.

Let $N$$N$ be a closed connected $n$$n$-manifold.

Using connected sums we can 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 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^{n+2}(N)\to E^{n+3}(S^1\times N)$$E^{n+2}(N)\to E^{n+3}(S^1\times N)$.

Thus 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 hardness of this problem.

On the other hand, if one studies embeddings up to the weaker relation of concordance, then much is known. See e.g. [Levine1969a], [Cappell&Shaneson1974] and [Ranicki1998].

5 Codimension 1 embeddings

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.

When one proves that this extension respects isotopy, this would give a 1-1 correspondence $E^3(S^1\times S^1)\to (E^3(S^1)\times\Zz)\vee(E^3(S^1)\times\Zz)$$E^3(S^1\times S^1)\to (E^3(S^1)\times\Zz)\vee(E^3(S^1)\times\Zz)$. So the description of $E^3(S^1\times S^1)$$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.

It is known that

• $\#E^2(S^1)=1$$\#E^2(S^1)=1$ (Schoenfliess theorem) [Rushing1973].
• $\#E^3(S^2)=1$$\#E^3(S^2)=1$ (Alexander theorem); a counterexample to the topological version is the Alexander Horned Sphere [Rushing1973].
• The description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$ is equivalent to the PL Schoenfliess problem and therefore is very hard for $n\ge3$$n\ge3$ [Rushing1973].
• $\#E^{n+1}_D(S^n)=1$$\#E^{n+1}_D(S^n)=1$ for $n\ge6$$n\ge6$ [Smale1962a].
• $\#E^{n+1}_D(S^p\times S^{n-p})=1$$\#E^{n+1}_D(S^p\times S^{n-p})=1$ for $2\le p\le n-p$$2\le p\le n-p$ [Kosinski1961], [Wall1965], [Lucas&Neto&Saeki1996], cf. [Goldstein1967].

Note that $E^{n+1}_{PL,lf}(S^p\times S^{n-p})$$E^{n+1}_{PL,lf}(S^p\times S^{n-p})$ can be known even when $E^{n+1}_{PL,lf}(S^n)$$E^{n+1}_{PL,lf}(S^n)$ is unknown [Goldstein1967] (here $lf$$lf$ stands for locally flat).

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

6 Conclusion

Thus 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$. The lowest dimensional cases are $(m,n)=$$(m,n)=$ (6,3), (4,3), (8,4), (7,4), (5,4).

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