Embeddings in Euclidean space: an introduction to their classification

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

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 continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings. We conclude with a brief review of some important results about codimension 1 embeddings

The most interesting and very much studied case concerns embeddings $S^1\to S^3$ (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and Remark 6 for more information.

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 $\Rr^m$ are known; cf. Remark 1.1. Such classification results are the unknotting theorems in $\S$2, the results on the pages listed below in $\S$4 and in $\S$7. Their statements, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in $\S$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 in $\S\S$2,4,6,7 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 (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].

2 Unknotting theorems

Recall that the words `ambient isotopy' are abbreviated to just `isotopy'. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

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

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] [Skopenkov2006, Figure 2.1.a].

This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the 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 the Hopf link [Skopenkov2016h];
• the assumption $m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e].

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

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 the Haefliger trefoil knot 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.

This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, $\S$7] work for homologically $k$-connected manifolds (see $\S$3 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 2.4 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 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$5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 7.1.

3 Notation and conventions

The following notations and conventions will be used in some other pages about embeddings, including those listed in $\S$4.

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 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 $0$ 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 $V_{m,n}$ the Stiefel manifold of orthonormal $n$-frames in $\Rr^m$.

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

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

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

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

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

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

4 References to information on the classification of embeddings

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 [Skopenkov2016e]

• 3-manifolds in 6-space [Skopenkov2016t]

• 4-manifolds in 7-space [Skopenkov2016f]

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

• Knots, i.e. embeddings of spheres

• High codimension links [Skopenkov2016h]

• Knotted tori [Skopenkov2016k]

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

5 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, $\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 $[f]$ 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], and an action $\#$ of $E^m(S^n)$ on $E^m(N)$.

6 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$ into $\Rr^{n+2}$ is the most extensively studied case of Knotting Problem and reveals 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$5) 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].) 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].

7 Codimension 1 embeddings

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

Every embedding $S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$ [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 $E^3(S^1\times S^1)$ 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.

Let $E^m_{PL,lf}(N)$ be the set of PL locally flat embeddings $N\to S^m$ up to PL locally flat isotopy. Note that $E^{n+1}_{PL,lf}(S^p\times S^{n-p})$ can admit complete readily calculable classification even when $E^{n+1}_{PL,lf}(S^n)$ does not [Goldstein1967].

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

8 References

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• [Zeeman93] Template:Zeeman93

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

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 continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings. We conclude with a brief review of some important results about codimension 1 embeddings

The most interesting and very much studied case concerns embeddings $S^1\to S^3$ (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and Remark 6 for more information.

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 $\Rr^m$ are known; cf. Remark 1.1. Such classification results are the unknotting theorems in $\S$2, the results on the pages listed below in $\S$4 and in $\S$7. Their statements, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in $\S$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 in $\S\S$2,4,6,7 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 (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].

2 Unknotting theorems

Recall that the words `ambient isotopy' are abbreviated to just `isotopy'. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

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

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] [Skopenkov2006, Figure 2.1.a].

This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the 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 the Hopf link [Skopenkov2016h];
• the assumption $m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e].

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

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 the Haefliger trefoil knot 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.

This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, $\S$7] work for homologically $k$-connected manifolds (see $\S$3 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 2.4 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 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$5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 7.1.

3 Notation and conventions

The following notations and conventions will be used in some other pages about embeddings, including those listed in $\S$4.

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 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 $0$ 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 $V_{m,n}$ the Stiefel manifold of orthonormal $n$-frames in $\Rr^m$.

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

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

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

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

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

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

4 References to information on the classification of embeddings

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 [Skopenkov2016e]

• 3-manifolds in 6-space [Skopenkov2016t]

• 4-manifolds in 7-space [Skopenkov2016f]

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

• Knots, i.e. embeddings of spheres

• High codimension links [Skopenkov2016h]

• Knotted tori [Skopenkov2016k]

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

5 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, $\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 $[f]$ 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], and an action $\#$ of $E^m(S^n)$ on $E^m(N)$.

6 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$ into $\Rr^{n+2}$ is the most extensively studied case of Knotting Problem and reveals 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$5) 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].) 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].

7 Codimension 1 embeddings

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

Every embedding $S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$ [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 $E^3(S^1\times S^1)$ 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.

Let $E^m_{PL,lf}(N)$ be the set of PL locally flat embeddings $N\to S^m$ up to PL locally flat isotopy. Note that $E^{n+1}_{PL,lf}(S^p\times S^{n-p})$ can admit complete readily calculable classification even when $E^{n+1}_{PL,lf}(S^n)$ does not [Goldstein1967].

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

8 References

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• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

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• [Zeeman93] Template:Zeeman93

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

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 continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings. We conclude with a brief review of some important results about codimension 1 embeddings

The most interesting and very much studied case concerns embeddings $S^1\to S^3$ (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and Remark 6 for more information.

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 $\Rr^m$ are known; cf. Remark 1.1. Such classification results are the unknotting theorems in $\S$2, the results on the pages listed below in $\S$4 and in $\S$7. Their statements, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in $\S$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 in $\S\S$2,4,6,7 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 (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].

2 Unknotting theorems

Recall that the words `ambient isotopy' are abbreviated to just `isotopy'. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

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

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] [Skopenkov2006, Figure 2.1.a].

This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the 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 the Hopf link [Skopenkov2016h];
• the assumption $m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e].

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

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 the Haefliger trefoil knot 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.

This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, $\S$7] work for homologically $k$-connected manifolds (see $\S$3 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 2.4 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 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$5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 7.1.

3 Notation and conventions

The following notations and conventions will be used in some other pages about embeddings, including those listed in $\S$4.

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 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 $0$ 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 $V_{m,n}$ the Stiefel manifold of orthonormal $n$-frames in $\Rr^m$.

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

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

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

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

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

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

4 References to information on the classification of embeddings

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 [Skopenkov2016e]

• 3-manifolds in 6-space [Skopenkov2016t]

• 4-manifolds in 7-space [Skopenkov2016f]

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

• Knots, i.e. embeddings of spheres

• High codimension links [Skopenkov2016h]

• Knotted tori [Skopenkov2016k]

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

5 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, $\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 $[f]$ 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], and an action $\#$ of $E^m(S^n)$ on $E^m(N)$.

6 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$ into $\Rr^{n+2}$ is the most extensively studied case of Knotting Problem and reveals 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$5) 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].) 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].

7 Codimension 1 embeddings

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

Every embedding $S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$ [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 $E^3(S^1\times S^1)$ 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.

Let $E^m_{PL,lf}(N)$ be the set of PL locally flat embeddings $N\to S^m$ up to PL locally flat isotopy. Note that $E^{n+1}_{PL,lf}(S^p\times S^{n-p})$ can admit complete readily calculable classification even when $E^{n+1}_{PL,lf}(S^n)$ does not [Goldstein1967].

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

8 References

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• [Zeeman93] Template:Zeeman93

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

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 continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings. We conclude with a brief review of some important results about codimension 1 embeddings

The most interesting and very much studied case concerns embeddings $S^1\to S^3$ (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and Remark 6 for more information.

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 $\Rr^m$ are known; cf. Remark 1.1. Such classification results are the unknotting theorems in $\S$2, the results on the pages listed below in $\S$4 and in $\S$7. Their statements, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in $\S$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 in $\S\S$2,4,6,7 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 (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].

2 Unknotting theorems

Recall that the words `ambient isotopy' are abbreviated to just `isotopy'. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

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

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] [Skopenkov2006, Figure 2.1.a].

This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the 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 the Hopf link [Skopenkov2016h];
• the assumption $m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e].

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

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 the Haefliger trefoil knot 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.

This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, $\S$7] work for homologically $k$-connected manifolds (see $\S$3 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 2.4 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 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$5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 7.1.

3 Notation and conventions

The following notations and conventions will be used in some other pages about embeddings, including those listed in $\S$4.

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 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 $0$ 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 $V_{m,n}$ the Stiefel manifold of orthonormal $n$-frames in $\Rr^m$.

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

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

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

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

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

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

4 References to information on the classification of embeddings

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 [Skopenkov2016e]

• 3-manifolds in 6-space [Skopenkov2016t]

• 4-manifolds in 7-space [Skopenkov2016f]

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

• Knots, i.e. embeddings of spheres

• High codimension links [Skopenkov2016h]

• Knotted tori [Skopenkov2016k]

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

5 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, $\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 $[f]$ 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], and an action $\#$ of $E^m(S^n)$ on $E^m(N)$.

6 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$ into $\Rr^{n+2}$ is the most extensively studied case of Knotting Problem and reveals 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$5) 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].) 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].

7 Codimension 1 embeddings

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

Every embedding $S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$ [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 $E^3(S^1\times S^1)$ 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.

Let $E^m_{PL,lf}(N)$ be the set of PL locally flat embeddings $N\to S^m$ up to PL locally flat isotopy. Note that $E^{n+1}_{PL,lf}(S^p\times S^{n-p})$ can admit complete readily calculable classification even when $E^{n+1}_{PL,lf}(S^n)$ does not [Goldstein1967].

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

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