# Embeddings in Euclidean space: an introduction to their 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], three major classical problems of topology are the following.

• Homeomorphism Problem: Classify $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n$-manifolds.
• Embedding Problem: Find the least dimension $m$$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.

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$$N$ in Euclidean spaces and in spheres. After making general remarks we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic. We then establish notation and conventions and give references to other pages on the Knotting Problem, to which this page serves as an introduction. We conclude by introducing connected sum and make some comments on codimension 1 and 2 embeddings.

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

Even in general, the Knotting Problem is known to be hard: at the time of writing there are only a few cases in which complete readily calculable classification results describing all isotopy classes for embeddings of a closed manifold $N$$N$ into Euclidean space $\Rr^m$$\Rr^m$. Cf. Remark 1.1. Such classification results are the unknotting theorems in $\S$$\S$2, the results on the pages listed below in $\S$$\S$4 and in $\S$$\S$7. Their statements, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in $\S$$\S$4 concern only such classification results, so we have to leave out another material, in particular in codimension $\le2$$\le2$ and also for all codimensions.

The results and remarks in $\S\S$$\S\S$2,4,6,7 show 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 farther we go from $m=2n$$m=2n$ to $m=n+3$$m=n+3$, the more complicated classification is.

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). A readily calculable classification' is an informal expression. In the best case this is a classification in terms of homology of a manifold (and certain structures in homology like intersection, characteristic classes etc). A readily calculable classification is also a reduction to calculation of stable homotopy groups of spheres when these are known (or to another standard algebraic problem involving only the homology of the manifold, which is solved in particular cases, although may be unsolved in general). An important feature of a useful classification is accessibility of statement to general mathematical audience which is only familiar with basic notions of the area; this in turn is an approximation to beauty. Another important feature is whether the classification gives an algorithm and how complex the algorithm is.

Many readily calculable classification results are presented in these pages. On the other hand, in some cases geometric problems are reduced to algebraic problems which are even harder to solve' [Wall1999], and such results do not give a readily calculable classification. 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].)

Remark 1.2 (Embeddings into the sphere and Euclidean space). (a) 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)$$g(x,y)=(x,-y)$ are not isotopic. On the other hand, any two embeddings $S^1\to S^2$$S^1\to S^2$ are isotopic (this intuitively clear assertion is non-trivial, see $\S$$\S$7).

(b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of $n$$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 Unknotting theorems

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

General Position Theorem 2.1 ([Hirsch1976], [Rourke&Sanderson1972]). For each compact $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.

The case $m\ge2n+2$$m\ge2n+2$ is called a stable range'.

The restriction $m\ge2n+2$$m\ge2n+2$ in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking $S^n\sqcup S^n\to\Rr^{2n+1}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h] [Skopenkov2006, Figure 2.1.a].

Whitney-Wu Unknotting Theorem 2.2. For each compact 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.

This is proved in [Wu1958], [Wu1958a] and [Wu1959] using Whitney trick.

All the three assumptions in this result are indeed necessary:

• the assumption $n\ge2$$n\ge2$ because of the existence of non-trivial knots $S^1\to S^3$$S^1\to S^3$;
• the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h];
• the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e].

Unknotting Spheres Theorem 2.3. For $N=S^n$$N=S^n$, or even for $N$$N$ a homology $n$$n$-sphere, $m\ge n+3$$m\ge n+3$ or $2m\ge 3n+4$$2m\ge 3n+4$ in the PL or smooth category, respectively, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic.

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

The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range'.

Knots in codimension 2 and the 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 each $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, every two embeddings $N\to\Rr^m$$N\to\Rr^m$ are isotopic.

This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, $\S$$\S$7] work for homologically $k$$k$-connected manifolds (see $\S$$\S$3 for definition).

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

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

For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, $\S$$\S$5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 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$$\S$4.

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 $\varepsilon(k):=1-(-1)^k$$\varepsilon(k):=1-(-1)^k$ be $0$$0$ for $k$$k$ even and $2$$2$ for $k$$k$ odd, so that $\Zz_{\varepsilon(k)}$$\Zz_{\varepsilon(k)}$ is $\Zz$$\Zz$ for $k$$k$ even and $\Zz_2$$\Zz_2$ for $k$$k$ odd.

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

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

For a manifold $P$$P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$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$.
• $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$$\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ and $A_f:H_s(N)\to H_{s+1}(C,\partial)$$A_f:H_s(N)\to H_{s+1}(C,\partial)$ homological Alexander duality isomorphisms, see 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:

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 embedded connected sum 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 the description of $E^{n+2}(N)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of 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

Theorem 7.1. (a) Every two smooth embeddings $S^n\to S^{n+1}$$S^n\to S^{n+1}$ are smoothly isotopic for $n\ne3$$n\ne3$ [Smale1961], [Smale1962a], [Barden1965].

(b) Every two smooth embeddings $S^p\times S^{n-p}\to S^{n+1}$$S^p\times S^{n-p}\to S^{n+1}$ are smoothly isotopic for each $2\le p\le n-p$$2\le p\le n-p$ [Kosinski1961], [Wall1965], [Lucas&Neto&Saeki1996], cf. [Goldstein1967].

The analogue of part (a) holds

• for $n=1$$n=1$ in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, $\S$$\S$1.8].
• for $n=2$$n=2$ in the PL category (Alexander Theorem, 1923) [Rushing1973, $\S$$\S$1.8].
• for every $n$$n$ in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].

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

Every embedding $S^1\times S^1\to S^3$$S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$$D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$$S^1\times D^2\to S^3$ [Alexander1924]. Clearly, only the standard embedding extends to both.

If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence $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.

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].