# High codimension embeddings: classification

## 1 Introduction

According to Zeeman [1], the 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 embeds 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 the dimension ranges where no knotting is possible and make some comments on codimension 1 and 2 embeddings. The unknotting results 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$$\Rr^3$), a complete readily calculable classification is neither known nor expected. (Note that there is an extensive study of codimension 2 embeddings not directly aiming at complete classification. Almost nothing is said here about this. See more in knot theory.)
• In some cases geometric problems are reduced to algebraic problems which are even harder to solve' [Wall1999]. E.g. so far an interesting approach of [Goodwillie&Weiss1999] did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss].)

### 1.1 Sphere and Euclidean space

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. (For $m$m such classifications are of course different.)

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$$self$-$intersection$$intersection$ $set$$set$ of a map $f:X\to Y$$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 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 spherical normal bundle of $f$$f$.

## 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 an embedding $D^n\to N$$D^n\to N$ (or an orientation of $N$$N$, if $N$$N$ is orientable) is chosen. If the images of embeddings $f:N\to\Rr^m$$f:N\to\Rr^m$ and $g:S^n\to\Rr^m$$g:S^n\to\Rr^m$ are contained in disjoint cubes, then we can define (unlinked) embedded connected sum $f\#g:N\to\Rr^m$$f\#g:N\to\Rr^m$. We make connected summation along $f(D^n)$$f(D^n)$ and a path in $\Rr^m$$\Rr^m$ joining $f(Int D^n)$$f(Int D^n)$ to $g(S^n)$$g(S^n)$. If $m\le n+2$$m\le n+2$, this operation is not well-defined, i.e. depends on the choice of the path. If $m\ge n+3$$m\ge n+3$, this operation is well-defined, i.e. is independent on the choice of the path. 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 $\#:E^m(S^n)\to E^m(N)$$\#:E^m(S^n)\to 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}$. 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)$ seems to be very hard open problem. For $N\ne S^n$$N\ne S^n$ see e.g. [Cappell&Shaneson], [Levine], [Ranicki].

It would be interesting to give a more formal explanation of why the description of $E^{n+2}(N)$$E^{n+2}(N)$ is hard, using known information that the description of $E^{n+2}(S^n)$$E^{n+2}(S^n)$ is hard. 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].
• $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).

## 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$$S^1\times D^2\to S^3$ [Alexander1924]. Clearly, only the standard embedding extends to both. When one proves that this extension respects isotopy, this gives a 1-1 correspondence $E^3(S^1\times S^1)\to (E^3(S^1)\times\Zz)\vee(E^3(S^1)\times\Zz)$$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)$$E^3(S^1\times S^1)$ is as hopeless as that of $E^3(S^1)$$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$$\#E^2(S^1)=1$ (Schoenfliess theorem).
• $\#E^3(S^2)=1$$\#E^3(S^2)=1$ (Alexander theorem); a counterexample to the topological version is the Alexander Horned Sphere.
• 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$.
• $\#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].

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$.