# High codimension embeddings: classification

## 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.

According to [Zeeman93, p. 3], the classical problems of topology are the following.

• Embedding Problem: Find the least dimension $m$$m$ such that given space admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$.
• Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

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

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. At the time of writing, the unknotting theorems and the results on the pages listed below in Section 4 record all known complete readily calculable isotopy classification results for embeddings of closed manifolds into Euclidean spaces which are known to the author.

The Knotting Problem is known to be hard:

• There are only a few cases, see $\S$$\S$4, in which there are complete readily calculable classification results. (However, the statements, although not the proofs, are simple and accessible to non-specialists.)
• For the best known specific case, i.e. for codimension 2 embeddings (in particular, for the classical theory of knots in $\Rr^3$$\Rr^3$), a complete readily calculable classification is neither known nor expected at the time of writing. (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 $\S$$\S$6. See more in Wikipedia article on knot theory.)
• In some cases geometric problems are reduced to algebraic problems which are even harder to solve' [Wall1999]. E.g. so far an interesting approach of [Weiss96], [Goodwillie&Weiss1999] did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss].)

The list of known results in the rest of this page shows that the complete classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial but presently accessible only for $n+3\le m\le 2n$$n+3\le m\le 2n$ or for $m=n+1\ge4$$m=n+1\ge4$. The lowest dimensional cases, i.e. all such pairs $(m,n)$$(m,n)$ with $n\le4$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For known information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

Remark 1.1 (Readily calculable classification). By readily calculable classification I mean 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 homotopy groups of spheres (or to another standard algebraic problem involving only homology of our manifold, which problem is solved in particular cases, although could be unsolved generally). 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.

Remark 1.2 (Sphere and Euclidean space). (a) Every two embeddings $S^1\to S^2$$S^1\to S^2$ are isotopic (this intuitively clear assertion is Schöenfliess Theorem) [Rushing1973]. On the other hand, embeddings $f,g:S^1\to\Rr^2$$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].

(b) 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.

## 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)$.

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

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

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

• $C_f$$C_f$ the closure of the complement in $S^m\supset\Rr^m$$S^m\supset\Rr^m$ to a tight enough tubular neighborhood of $f(N)$$f(N)$ and
• $\nu_f:\partial C_f\to N$$\nu_f:\partial C_f\to N$ the restriction of the linear normal bundle of $f$$f$ to the space of unit length vectors identified with $\partial C_f$$\partial C_f$.

## 3 Unknotting theorems

General Position Theorem 3.1 ([Hirsch1976], [Rourke&Sanderson1972]). 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$$m\ge2n+2$ in Theorem 3.1 is sharp for non-connected manifolds, as the Hopf linking $S^n\sqcup S^n\to\Rr^{2n+1}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h] [Skopenkov2006, Figure 2.1.a].

Theorem 3.2 ([Wu1958], [Wu1958a] and [Wu1959]). 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$).

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 3.3 ([Zeeman1960], [Stallings1963], [Gluck1963], [Haefliger1961], [Adachi1993]). 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$$m\ge\frac{3n}2+2$ in the PL or smooth category, respectively, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic (i.e. $\#E^m(N)=1$$\#E^m(N)=1$).

This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Rushing1973], [Scharlemann1977].

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 3.3.

Theorems 3.2 and 3.3 may be generalized as follows.

The Haefliger-Zeeman Unknotting Theorem 3.4. 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$).

The Haefliger-Zeeman Unknotting Theorem 3.4 was proved directly in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969] for homotopically $k$$k$-connected manifolds. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, $\S$$\S$7] work for homologically $k$$k$-connected manifolds.

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

Note that if $n\le2k+1$$n\le2k+1$, then every closed homologically $k$$k$-connected $n$$n$-manifold is a homology sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 3.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 3.3.

For generalizations of the Haefliger-Zeeman Unknotting Theorem 3.4 see [Skopenkov2016e, $\S$$\S$5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979].

## 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:

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

## 5 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 operation $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$.

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

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

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

## 6 Codimension 2 embeddings

In this subsection 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^m(N)\to E^{m+1}(S^1\times N)$$E^m(N)\to E^{m+1}(S^1\times N)$ for $m=n+2$$m=n+2$.

Thus 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].

## 7 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.

If it could be proven 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].

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

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