# Embeddings in Euclidean space: an introduction to their classification

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 This page has been accepted for publication in the Bulletin of the Manifold Atlas.

## 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. Unless otherwise indicated, the word isotopy' means ambient isotopy' on this page; the definitions of these terms are given in [Skopenkov2016i, $\S$${{Authors|Askopenkov}} == 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. Unless otherwise indicated, the word isotopy' means [[Isotopy|ambient isotopy']] on this page; the definitions of these terms are given in \cite[\S\S$1].

Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].

• The Manifold Problem: Classify $n$$n$-manifolds.
• The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$.
• The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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.

The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1].

This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.

The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information.

The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 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 given below show the following:

• For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes.
• The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and 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 information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.

Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).

(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.

(b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.

(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).

(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].

(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.

(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies

(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014].

Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it.

(b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection.

Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, it suffices to show that if the compositions with the inclusion $i:\Rr^m\to S^m$$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'$$f'$ are isotopic. For showing that assume that $i\circ f$$i\circ f$ and $i\circ f'$$i\circ f'$ are isotopic. Then by general position $f$$f$ and $f'$$f'$ are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic.

Remark 1.4 (References to information on the classification of embeddings). 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:

## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).

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, Example 2.1].

Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].

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, Example 2.1];
• the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1].

Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].

The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]).

Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] 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 every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).

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

Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'.

For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.

## 3 Notation and conventions

The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.

Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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 the category is omitted, then the result holds (or a definition or a construction is given) in both categories.

The sources of all embeddings are 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
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${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor.

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$$P$ with boundary $\partial P$$\partial 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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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$$f$ to the subspace of unit length vectors identified with $\partial C_f$$\partial C_f$.
Denote by
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${\rm i}:S^q\to S^m$ the standard embedding given by
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${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on
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${\rm i}$ defines the standard embedding
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${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol
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${\rm i}_{m,q}$ the restriction of
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${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$.

Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large.

## 4 Embedded connected sum

Suppose that $m\ge n+2$$m\ge n+2$, $N$$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, 1.3, 1.4].

This operation is well-defined, i.e. the isotopy 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, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$.

## 5 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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 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].

## 6 Codimension 1 embeddings

Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$.

(b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$.

Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.

Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$.

Clearly, only the standard embedding extends to both.

Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$).

(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). So the description of $E^3(S^1\times S^1)$$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.

]{Skopenkov2016i}. {{beginthm|Remark|(Some motivations)}}\label{r:zee} Three important classical problems in topology are the following, cf. \cite[p. 3]{Zeeman1993}. * ''The Manifold Problem'': Classify $n$-manifolds. * ''The Embedding Problem'': Find the least dimension $m$ such that a given manifold admits an [[Embedding_(simple_definition)|embedding]] into $m$-dimensional Euclidean space $\Rr^m$. * ''The Knotting Problem'': Classify [[Embedding_(simple_definition)|embeddings]] of a given manifold into another given manifold 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, 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. The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark \ref{s:list} below). See also the [[Wikipedia:Knot_theory|Wikipedia article on knot theory]] and \cite[Example 2.3]{Skopenkov2016t}, \cite[$\S\S1]. Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3]. • The Manifold Problem: Classify $n$$n$-manifolds. • The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$. • The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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. The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1]. This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information. The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2. The results and remarks given below show the following: • For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes. • The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$$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$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f]. Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside. Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems). (a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. (b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate. (c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d). (d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999]. (i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies (iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014]. Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it. (b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, 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. 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 isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic. Remark 1.4 (References to information on the classification of embeddings). 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: For more information see e.g. [Skopenkov2006]. ## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]). The restriction $m\ge2n+2$$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}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h, Example 2.1]. Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)]. 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, Example 2.1]; • the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1]. Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3]. The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]). Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for $N=S^n$$N=S^n$) in the Unknotting Spheres Theorem 2.3. Theorems 2.2 and 2.3 may be generalized as follows. The Haefliger-Zeeman Unknotting Theorem 2.4. For every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification). Note that if $n\le2k+1$$n\le2k+1$, then every closed $k$$k$-connected $n$$n$-manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3. Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'. For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1. ## 3 Notation and conventions The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4. Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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$$N\to S^m$ up to smooth or PL isotopy. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are assumed to be compact. Let $B^n$$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 Tex syntax error ${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor. Denote by $V_{m,n}$$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$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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 subspace of unit length vectors identified with $\partial C_f$$\partial C_f$. Denote by Tex syntax error ${\rm i}:S^q\to S^m$ the standard embedding given by Tex syntax error ${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on Tex syntax error ${\rm i}$ defines the standard embedding Tex syntax error ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol Tex syntax error ${\rm i}_{m,q}$ the restriction of Tex syntax error ${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$. Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large. ## 4 Embedded connected sum Suppose that $m\ge n+2$$m\ge n+2$, $N$$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$$N$ is orientable), cf. [Haefliger1966, 1.3, 1.4]. This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes $[f]$$[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)$$E^m(S^n)$ [Haefliger1966, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$. ## 5 Some remarks on codimension 2 embeddings The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem. For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974]. ## 6 Codimension 1 embeddings Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$. (b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$. Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result. Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$$S^1\times D^2\to S^3$. Clearly, only the standard embedding extends to both. Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds • for $n=1$$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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$). (b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). 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. ## 7 References ]{Takase2006}. {{endthm}} This article gives a short guide to the problem of classifying embeddings of closed manifolds$N$into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark \ref{r:zee} for embeddings of general manifolds$N$into$\R^m$or$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of$N$are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings$S^1\to S^3$, or more generally, [[Knots,_i.e._embeddings_of_spheres#Codimension_2_knots|codimension 2 embeddings of spheres]]. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see [[Wikipedia:Knot_theory|Wikipedia article on knot theory]] and$\S$\ref{s:c2} for more information. The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which ''readily calculable'' classification results (see Remark \ref{compl}) describing all isotopy classes for embeddings of a closed manifold into Euclidean space$\Rr^m$are known. Such classification results are presented on the pages listed in Remark \ref{s:list}, in [[#Unknotting_theorems|$\S$\ref{s:ut}]] and in [[#Codimension_1_embeddings|$\S$\ref{s:c1}]]. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark \ref{s:list} 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 given below show the following: *For a fixed$N$, the more$m$decreases from n$ towards $n+3$, the more complicated the classification of embeddings of $N$ into $\Rr^m$ becomes. *The complete readily calculable classification of embeddings into $\Rr^m$ of closed connected $n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$ or for $m=n+1\ge4$; the lowest dimensional cases, i.e. all such pairs $(m,n)$ with $n\le4$, are [[3-manifolds_in_6-space|(6,3)]], (4,3), [[Embeddings_just_below_the_stable_range:_classification|(8,4)]], [[4-manifolds_in_7-space|(7,4)]], (5,4). For information on the cases (6,3), (8,4), (7,4) see \cite{Skopenkov2016t}, \cite{Skopenkov2016e}, \cite{Skopenkov2016f}. {{beginthm|Remark|(Readily calculable classification)}}\label{compl} The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See \cite[Preface]{Graham&Knuth&Patashnik89} for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$ and the manifold $N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside. Since this is a paper on the Knotting Problem of Remark \ref{r:zee}, below classification' means classification of embeddings of a manifold $N$ into $\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems). (a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. (b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate. (c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See \cite[Corollary 11.3.1]{Wall1999} and a simpler exposition in \cite[pp. 263-267]{Cencelj&Repovs&Skopenkov2004}. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' \cite[p. 119]{Wall1999}. An analogue of this result for classification is \cite[the Browder-Wall Theorem 9]{Cencelj&Repovs&Skopenkov2004}. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to \cite[Corollary 11.3.3]{Wall1999} as exposed in \cite[pp. 265-267]{Cencelj&Repovs&Skopenkov2004}. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see \cite{Skopenkov2016t}, \cite{Skopenkov2016f}) have been obtained by applying Kreck's modified surgery theory, see (d). (d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark \ref{r:zee}. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory \cite{Kreck1999} was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see \cite{Kreck1999}. (i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. [[High_codimension_links#Classification_in_the_metastable_range|the Haefliger-Zeeman Theorem]] \cite[Theorem 4.1]{Skopenkov2016h} classifies links in terms of (stable) [[Wikipedia:Homotopy_groups_of_spheres|homotopy groups of spheres]]. These groups are not known in general. However, these groups are described for many particular cases which give [[High_codimension_links#Introduction|the table]] \cite[$\S\S1]. Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3]. • The Manifold Problem: Classify $n$$n$-manifolds. • The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$. • The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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. The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1]. This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information. The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2. The results and remarks given below show the following: • For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes. • The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$$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$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f]. Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside. Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems). (a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. (b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate. (c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d). (d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999]. (i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies (iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014]. Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it. (b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, 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. 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 isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic. Remark 1.4 (References to information on the classification of embeddings). 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: For more information see e.g. [Skopenkov2006]. ## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]). The restriction $m\ge2n+2$$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}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h, Example 2.1]. Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)]. 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, Example 2.1]; • the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1]. Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3]. The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]). Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for $N=S^n$$N=S^n$) in the Unknotting Spheres Theorem 2.3. Theorems 2.2 and 2.3 may be generalized as follows. The Haefliger-Zeeman Unknotting Theorem 2.4. For every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification). Note that if $n\le2k+1$$n\le2k+1$, then every closed $k$$k$-connected $n$$n$-manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3. Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'. For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1. ## 3 Notation and conventions The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4. Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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$$N\to S^m$ up to smooth or PL isotopy. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are assumed to be compact. Let $B^n$$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 Tex syntax error ${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor. Denote by $V_{m,n}$$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$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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 subspace of unit length vectors identified with $\partial C_f$$\partial C_f$. Denote by Tex syntax error ${\rm i}:S^q\to S^m$ the standard embedding given by Tex syntax error ${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on Tex syntax error ${\rm i}$ defines the standard embedding Tex syntax error ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol Tex syntax error ${\rm i}_{m,q}$ the restriction of Tex syntax error ${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$. Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large. ## 4 Embedded connected sum Suppose that $m\ge n+2$$m\ge n+2$, $N$$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$$N$ is orientable), cf. [Haefliger1966, 1.3, 1.4]. This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes $[f]$$[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)$$E^m(S^n)$ [Haefliger1966, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$. ## 5 Some remarks on codimension 2 embeddings The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem. For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974]. ## 6 Codimension 1 embeddings Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$. (b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$. Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result. Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$$S^1\times D^2\to S^3$. Clearly, only the standard embedding extends to both. Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds • for $n=1$$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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$). (b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). 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. ## 7 References ]{Skopenkov2016h}, and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey \cite[$\S]{Skopenkov2006} and \cite{Haefliger1963}, \cite{Weber1967}, \cite{Skopenkov2002}. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies * the [[Embeddings_just_below_the_stable_range:_classification#Classification_further_below_the_stable_range|Becker-Glover Theorem]] \cite[Theorem 6.5]{Skopenkov2016e} and the [[Knotted_tori#Classification|classification of knotted tori]] \cite[Theorem 3.2]{Skopenkov2016k}, which I regard as readily calculable classifications; * the independence of the classification of embeddings of the smooth or PL structure on the manifold; * the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) \cite[Theorem 1.1 and text after Theorem 1.4]{Cadek&Krcal&Vokrınek2013}. (iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem \cite{Weiss96}, \cite{Weiss99}, \cite{Goodwillie&Weiss1999}. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ \cite{Weiss}, \cite{Arone&Turchin2014}. {{endthm}} {{beginthm|Remark|(Embeddings into Euclidean space and the sphere)}}\label{spheu} (a) The embeddings $f,g:S^1\to\Rr^2$ given by $f(x,y)=(x,y)$ and $g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the [[Wikipedia:Winding_number#Turning_number|Wikipedia article on turning number]] can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$ into $S^2$ are isotopic, see Theorem \ref{t:sch}.a and below it. (b) For $m\ge n+2$ the classifications of embeddings of compact $n$-manifolds into $S^m$ and into $\Rr^m$ are the same. More precisely, for all integers $m,n$ such that $m\ge n+2$, and for every $n$-manifold $N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$ and $N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$, after a small isotopy an embedding $N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$. Hence $i_*$ is onto. To prove that $i_*$ is injective, it suffices to show that if the compositions with the inclusion $i:\Rr^m\to S^m$ of two embeddings $f,f':N\to\Rr^m$ of a compact $n$-manifold $N$ are isotopic, then $f$ and $f'$ are isotopic. For showing that assume that $i\circ f$ and $i\circ f'$ are isotopic. Then by general position $f$ and $f'$ are [[Isotopy#Other equivalence relations|non-ambiently isotopic]]. Since every non-ambient isotopy extends to an isotopy \cite[Theorem 1.3]{Skopenkov2016i}, $f$ and $f'$ are isotopic. {{endthm}} {{beginthm|Remark|(References to information on the classification of embeddings)}}\label{s:list} The first list is structured by the dimension of the source manifold and the target Euclidean space: * [[Embeddings just below the stable range: classification]] \cite{Skopenkov2016e} * [[3-manifolds in 6-space]] \cite{Skopenkov2016t} * [[4-manifolds in 7-space]] \cite{Skopenkov2016f} Information structured by the complexity' of the source manifold: * [[Knots, i.e. embeddings of spheres]] \cite{Skopenkov2016s} * [[High codimension links]] \cite{Skopenkov2016h} * [[Knotted tori]] \cite{Skopenkov2016k} For more information see e.g. \cite{Skopenkov2006}. {{endthm}} ==Unknotting theorems == ; \label{s:ut} If the category is omitted, then a result stated below holds in both the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] and [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) category. {{beginthm|General Position Theorem|(\cite[Theorem 3.5]{Hirsch1976}, \cite[Theorem 5.4]{Rourke&Sanderson1972})}}\label{th1} For every compact $n$-manifold $N$ and $m\ge2n+2$, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} The case $m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$ is called [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]] \cite[Theorem 2.2.a]{Skopenkov2006}). The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds, as the [[High_codimension_links#Examples|Hopf linking]] $S^n\sqcup S^n\to\Rr^{2n+1}$ shows \cite[Example 2.1]{Skopenkov2016h}. {{beginthm|Whitney-Wu Unknotting Theorem}}\label{th2} For every compact connected $n$-manifold $N$, $n\ge2$ and $m\ge2n+1$, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} This is proved in \cite{Wu1958c} using ''the Whitney trick'' (see that reference or \cite[$\S]{Rourke&Sanderson1972}). See also \cite[Theorem 2.3 (H. Whitney, W.T. Wu)]{Smale1961}. All the three assumptions in this result are indeed necessary: *the assumption$n\ge2$because of the existence of non-trivial knots$S^1\to S^3$; *the connectedness assumption because of the existence of [[High_codimension_links#Examples|the Hopf link]] \cite[Example 2.1]{Skopenkov2016h}; *the assumption$m\ge2n+1$because of the [[Embeddings just below the stable range#Hudson_tori|example of Hudson tori]] \cite[Example 3.1]{Skopenkov2016e}. {{beginthm|Unknotting Spheres Theorem}}\label{sph} Assume that$m\ge n+3$in the PL category or m\ge 3n+4$ in the smooth category. For $N=S^n$, or even for $N$ an integral homology $n$-sphere, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} For $N=S^n$ this result is proved in \cite{Zeeman1960}, \cite[Corollary 2 of Theorem 9 in Chapter 4]{Zeeman1963} and in \cite[Existence Theorem (b) in p. 47]{Haefliger1961} in the PL and in the smooth category, respectively. For $N$ an integral homology $n$-sphere this result follows from \cite[Theorem 6.2]{Skopenkov2016e}. This result is also true for $m\ge n+3$ in the topological [[Wikipedia:Local flatness|locally flat]] category \cite[Corollary 9.3]{Stallings1963}, \cite[Theorem 1.1]{Gluck1963}; see also textbook \cite[Flattening Theorem 4.5.1]{Rushing1973} and \cite[Corollary 1.3]{Scharlemann1977}. The case m\ge 3n+4$is called a metastable range' (for the classification problem; for the existence problem there are analogous results with m\ge3n+3$ \cite[$\S,$\S]{Skopenkov2006}). Knots in codimension 2 and [[3-manifolds in 6-space#Examples|the Haefliger trefoil knot]] \cite[Example 2.1]{Skopenkov2016t} show that the dimension restrictions are sharp (even for $N=S^n$) in the Unknotting Spheres Theorem \ref{sph}. Theorems \ref{th2} and \ref{sph} may be generalized as follows. {{beginthm|The Haefliger-Zeeman Unknotting Theorem}}\label{haze} For every $n\ge2k+2$, $m\ge2n-k+1$ and closed $k$-connected $n$-manifold $N$, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} This is proved in \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963}, \cite[Existence Theorem (b) in p. 47]{Haefliger1961} in the PL or in the smooth category, respectively. The PL case uses the ideas of \cite{Penrose&Whitehead&Zeeman1961} and \cite{Irwin1965}; see also textbooks \cite[$\S]{Hudson1969}, \cite[$\S]{Rourke&Sanderson1972}. Theorem \ref{haze} remains true if ''$k$-connected'' is replaced by ''homologically $k$-connected'' (see $\S$\ref{s:nc} for the definition and \cite[Theorem 6.2]{Skopenkov2016e} for [[Embeddings_just_below_the_stable_range:_classification#A_generalization_to_highly-connected_manifolds|justification]]). Note that if $n\le2k+1$, then every closed $k$-connected $n$-manifold is topologically a sphere, so the analogue of Theorem \ref{haze} in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem \ref{sph}. Given Theorem \ref{haze} above, the case $m\ge2n-k+1$ can be called a stable range for $k$-connected manifolds'. For generalizations of Theorem \ref{haze} see [[Embeddings just below the stable range: classification|survey]] \cite{Skopenkov2016e} and \cite{Hudson1967}, \cite{Hacon1968}, \cite{Hudson1972}, \cite{Gordon1972}, \cite{Kearton1979}. See also Theorem \ref{t:sch}. == Notation and conventions == ; \label{s:nc} The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark \ref{s:list}. Unless otherwise indicated, the word isotopy' means [[Isotopy|ambient isotopy']]; see definition in \cite[$\S\S1]. Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3]. • The Manifold Problem: Classify $n$$n$-manifolds. • The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$. • The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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. The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1]. This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information. The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2. The results and remarks given below show the following: • For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes. • The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$$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$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f]. Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside. Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems). (a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. (b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate. (c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d). (d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999]. (i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies (iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014]. Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it. (b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, 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. 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 isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic. Remark 1.4 (References to information on the classification of embeddings). 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: For more information see e.g. [Skopenkov2006]. ## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]). The restriction $m\ge2n+2$$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}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h, Example 2.1]. Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)]. 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, Example 2.1]; • the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1]. Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3]. The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]). Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for $N=S^n$$N=S^n$) in the Unknotting Spheres Theorem 2.3. Theorems 2.2 and 2.3 may be generalized as follows. The Haefliger-Zeeman Unknotting Theorem 2.4. For every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification). Note that if $n\le2k+1$$n\le2k+1$, then every closed $k$$k$-connected $n$$n$-manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3. Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'. For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1. ## 3 Notation and conventions The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4. Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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$$N\to S^m$ up to smooth or PL isotopy. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are assumed to be compact. Let $B^n$$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 Tex syntax error ${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor. Denote by $V_{m,n}$$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$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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 subspace of unit length vectors identified with $\partial C_f$$\partial C_f$. Denote by Tex syntax error ${\rm i}:S^q\to S^m$ the standard embedding given by Tex syntax error ${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on Tex syntax error ${\rm i}$ defines the standard embedding Tex syntax error ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol Tex syntax error ${\rm i}_{m,q}$ the restriction of Tex syntax error ${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$. Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large. ## 4 Embedded connected sum Suppose that $m\ge n+2$$m\ge n+2$, $N$$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$$N$ is orientable), cf. [Haefliger1966, 1.3, 1.4]. This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes $[f]$$[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)$$E^m(S^n)$ [Haefliger1966, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$. ## 5 Some remarks on codimension 2 embeddings The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem. For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974]. ## 6 Codimension 1 embeddings Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$. (b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$. Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result. Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$$S^1\times D^2\to S^3$. Clearly, only the standard embedding extends to both. Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds • for $n=1$$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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$). (b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). 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. ## 7 References ]{Skopenkov2016i}. For a manifold$N$let$E^m_D(N)$or$E^m_{PL}(N)$denote the set of [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] or [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) [[Embedding|embeddings]]$N\to S^m$up to smooth or PL [[isotopy]]. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are 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 $\S$1]. Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3]. • The Manifold Problem: Classify $n$$n$-manifolds. • The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$. • The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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. The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1]. This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information. The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2. The results and remarks given below show the following: • For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes. • The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$$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$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f]. Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside. Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems). (a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. (b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate. (c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d). (d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999]. (i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies (iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014]. Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it. (b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, 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. 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 isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic. Remark 1.4 (References to information on the classification of embeddings). 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: For more information see e.g. [Skopenkov2006]. ## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]). The restriction $m\ge2n+2$$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}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h, Example 2.1]. Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)]. 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, Example 2.1]; • the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1]. Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3]. The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]). Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for $N=S^n$$N=S^n$) in the Unknotting Spheres Theorem 2.3. Theorems 2.2 and 2.3 may be generalized as follows. The Haefliger-Zeeman Unknotting Theorem 2.4. For every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification). Note that if $n\le2k+1$$n\le2k+1$, then every closed $k$$k$-connected $n$$n$-manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3. Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'. For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1. ## 3 Notation and conventions The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4. Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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$$N\to S^m$ up to smooth or PL isotopy. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are assumed to be compact. Let $B^n$$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 Tex syntax error ${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor. Denote by $V_{m,n}$$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$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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 subspace of unit length vectors identified with $\partial C_f$$\partial C_f$. Denote by Tex syntax error ${\rm i}:S^q\to S^m$ the standard embedding given by Tex syntax error ${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on Tex syntax error ${\rm i}$ defines the standard embedding Tex syntax error ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol Tex syntax error ${\rm i}_{m,q}$ the restriction of Tex syntax error ${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$. Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large. ## 4 Embedded connected sum Suppose that $m\ge n+2$$m\ge n+2$, $N$$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$$N$ is orientable), cf. [Haefliger1966, 1.3, 1.4]. This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes $[f]$$[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)$$E^m(S^n)$ [Haefliger1966, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$. ## 5 Some remarks on codimension 2 embeddings The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem. For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974]. ## 6 Codimension 1 embeddings Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$. (b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$. Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result. Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$$S^1\times D^2\to S^3$. Clearly, only the standard embedding extends to both. Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds • for $n=1$$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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$). (b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). 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. ## 7 References$ for $k$ even and $for$k$odd, so that$\Zz_{\varepsilon(k)}$is$\Zz$for$k$even and$\Zz_2$for$k$odd. Denote by${\rm pr}_k$is the projection of a Cartesian product onto the$k$th factor. Denote by$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$. Denote by${\rm i}:S^q\to S^m$the standard embedding given by${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on${\rm i}$defines the standard embedding${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol${\rm i}_{m,q}$the restriction of${\rm i}_{m,q}$to$S^p\times S^q$for any$p\le m-q-1$. Denote by$\Sigma X$the suspension of a space$X$. Denote by$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$the suspension homomorphism. Recall that$\Sigma$is an isomorphism for$q\le2n-2$. Let$\pi_k^S$be any of the groups$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$identified by the suspension isomorphism. Denote by$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$the stable suspension homomorphisms, where$M$is large. == Embedded connected sum == ; \label{s:sum} Suppose that$m\ge n+2$,$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$is orientable), cf. \cite[1.3, 1.4]{Haefliger1966}. This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes$[f]$and$[g]$, and is independent of the choice of the path and of the representatives$f,g$. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. \cite[Remark 2.3.a]{Skopenkov2015a}. The proof is written for$N=S^n$in \cite[$\S, 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]{Skopenkov2015a}. 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 $E^m(S^n)$ \cite[1.3-1.7]{Haefliger1966}, and an action $\#$ of $E^m(S^n)$ on $E^m(N)$. == Some remarks on codimension 2 embeddings == ; \label{s:c2} The case of embeddings of $S^n$ into $\Rr^{n+2}$ is the [[Knots,_i.e._embeddings_of_spheres#Codimension_2_knots|most extensively studied]] case of the Knotting Problem. In this case there is 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|embedded connected sum]] ($\S$\ref{s:sum}) 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 $f\#g_2$ \cite[22 in p. 181]{Viro1973}.) One can also apply Artin's spinning construction \cite{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 [[Isotopy|''concordance'']] see e.g. \cite{Cappell&Shaneson1974}. == Codimension 1 embeddings == ; \label{s:c1} {{beginthm|Theorem}}\label{t:sch} (a) Any two smooth embeddings of $S^n$ into $S^{n+1}$ are smoothly isotopic for every $n\ne3$. (b) Any two smooth embeddings of $S^p\times S^{n-p}$ into $S^{n+1}$ are smoothly isotopic for every \le p\le n-p$. {{endthm}} Part (a) is proved in \cite[Proposition D in$\S]{Milnor1965a} as an easy corollary of the $h$-cobordism theorem. Part (b) is proved in \cite[$\S\S1]. Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3]. • The Manifold Problem: Classify $n$$n$-manifolds. • The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$. • The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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. The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1]. This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information. The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2. The results and remarks given below show the following: • For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes. • The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$$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$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f]. Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside. Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems). (a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. (b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate. (c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d). (d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999]. (i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies (iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014]. Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it. (b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, 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. 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 isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic. Remark 1.4 (References to information on the classification of embeddings). 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: For more information see e.g. [Skopenkov2006]. ## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]). The restriction $m\ge2n+2$$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}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h, Example 2.1]. Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)]. 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, Example 2.1]; • the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1]. Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3]. The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]). Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for $N=S^n$$N=S^n$) in the Unknotting Spheres Theorem 2.3. Theorems 2.2 and 2.3 may be generalized as follows. The Haefliger-Zeeman Unknotting Theorem 2.4. For every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification). Note that if $n\le2k+1$$n\le2k+1$, then every closed $k$$k$-connected $n$$n$-manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3. Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'. For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1. ## 3 Notation and conventions The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4. Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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$$N\to S^m$ up to smooth or PL isotopy. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are assumed to be compact. Let $B^n$$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 Tex syntax error ${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor. Denote by $V_{m,n}$$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$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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 subspace of unit length vectors identified with $\partial C_f$$\partial C_f$. Denote by Tex syntax error ${\rm i}:S^q\to S^m$ the standard embedding given by Tex syntax error ${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on Tex syntax error ${\rm i}$ defines the standard embedding Tex syntax error ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol Tex syntax error ${\rm i}_{m,q}$ the restriction of Tex syntax error ${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$. Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large. ## 4 Embedded connected sum Suppose that $m\ge n+2$$m\ge n+2$, $N$$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$$N$ is orientable), cf. [Haefliger1966, 1.3, 1.4]. This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes $[f]$$[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)$$E^m(S^n)$ [Haefliger1966, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$. ## 5 Some remarks on codimension 2 embeddings The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem. For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974]. ## 6 Codimension 1 embeddings Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$. (b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$. Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result. Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$$S^1\times D^2\to S^3$. Clearly, only the standard embedding extends to both. Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds • for $n=1$$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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$). (b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). 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. ## 7 References , p. 447]{Lucas&Saeki2002} as a corollary of generalizations \cite{Kosinski1961}, \cite{Wall1965}, \cite{Goldstein1967}, \cite{Rubinstein1980}, \cite{Lucas&Neto&Saeki1996} of the following result. {{beginthm|Theorem|(Alexander torus theorem)}}\label{t:ale} \cite{Alexander1924} Every embedding$S^1\times S^1\to S^3$extends to an embedding either$D^2\times S^1\to S^3$or$S^1\times D^2\to S^3$. {{endthm}} Clearly, only the standard embedding extends to both. {{beginthm|Remark}}\label{r:pltop} (a) (on PL topological category) The analogue of Theorem \ref{t:sch}.a holds * for$n=1$in the PL or topological category (Schöenfliess Theorem, 1912) \cite[$\S\S1].

Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].

• The Manifold Problem: Classify $n$$n$-manifolds.
• The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$.
• The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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.

The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1].

This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.

The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information.

The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 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 given below show the following:

• For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes.
• The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and 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 information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.

Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).

(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.

(b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.

(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).

(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].

(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.

(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies

(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014].

Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it.

(b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection.

Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, it suffices to show that if the compositions with the inclusion $i:\Rr^m\to S^m$$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'$$f'$ are isotopic. For showing that assume that $i\circ f$$i\circ f$ and $i\circ f'$$i\circ f'$ are isotopic. Then by general position $f$$f$ and $f'$$f'$ are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic.

Remark 1.4 (References to information on the classification of embeddings). 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:

## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).

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, Example 2.1].

Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].

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, Example 2.1];
• the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1].

Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].

The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]).

Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] 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 every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).

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

Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'.

For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.

## 3 Notation and conventions

The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.

Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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 the category is omitted, then the result holds (or a definition or a construction is given) in both categories.

The sources of all embeddings are 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
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${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor.

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$$P$ with boundary $\partial P$$\partial 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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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$$f$ to the subspace of unit length vectors identified with $\partial C_f$$\partial C_f$.
Denote by
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${\rm i}:S^q\to S^m$ the standard embedding given by
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${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on
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${\rm i}$ defines the standard embedding
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${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol
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${\rm i}_{m,q}$ the restriction of
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${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$.

Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large.

## 4 Embedded connected sum

Suppose that $m\ge n+2$$m\ge n+2$, $N$$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, 1.3, 1.4].

This operation is well-defined, i.e. the isotopy 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, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$.

## 5 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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 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].

## 6 Codimension 1 embeddings

Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$.

(b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$.

Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.

Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$.

Clearly, only the standard embedding extends to both.

Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$).

(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). So the description of $E^3(S^1\times S^1)$$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.

.8]{Rushing1973}. * for $n=2$ in the PL category (Alexander Theorem, 1923) \cite[$\S\S1]. Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3]. • The Manifold Problem: Classify $n$$n$-manifolds. • The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$. • The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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. The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1]. This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6. The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information. The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2. The results and remarks given below show the following: • For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes. • The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and presently accessible only for $n+3\le m\le 2n$$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$$n\le4$, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f]. Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside. Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems). (a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is. (b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate. (c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d). (d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999]. (i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification. (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies (iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014]. Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it. (b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection. Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, 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. 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 isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic. Remark 1.4 (References to information on the classification of embeddings). 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: For more information see e.g. [Skopenkov2006]. ## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]). The restriction $m\ge2n+2$$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}$$S^n\sqcup S^n\to\Rr^{2n+1}$ shows [Skopenkov2016h, Example 2.1]. Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)]. 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, Example 2.1]; • the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1]. Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3]. The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]). Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for $N=S^n$$N=S^n$) in the Unknotting Spheres Theorem 2.3. Theorems 2.2 and 2.3 may be generalized as follows. The Haefliger-Zeeman Unknotting Theorem 2.4. For every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic. This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification). Note that if $n\le2k+1$$n\le2k+1$, then every closed $k$$k$-connected $n$$n$-manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3. Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'. For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1. ## 3 Notation and conventions The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4. Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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$$N\to S^m$ up to smooth or PL isotopy. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are assumed to be compact. Let $B^n$$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 Tex syntax error ${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor. Denote by $V_{m,n}$$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$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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 subspace of unit length vectors identified with $\partial C_f$$\partial C_f$. Denote by Tex syntax error ${\rm i}:S^q\to S^m$ the standard embedding given by Tex syntax error ${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on Tex syntax error ${\rm i}$ defines the standard embedding Tex syntax error ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol Tex syntax error ${\rm i}_{m,q}$ the restriction of Tex syntax error ${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$. Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large. ## 4 Embedded connected sum Suppose that $m\ge n+2$$m\ge n+2$, $N$$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$$N$ is orientable), cf. [Haefliger1966, 1.3, 1.4]. This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes $[f]$$[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)$$E^m(S^n)$ [Haefliger1966, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$. ## 5 Some remarks on codimension 2 embeddings The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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)$$E^{n+2}(N)$ is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem. For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974]. ## 6 Codimension 1 embeddings Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$. (b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$. Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result. Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$$S^1\times D^2\to S^3$. Clearly, only the standard embedding extends to both. Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds • for $n=1$$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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$). (b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). 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. ## 7 References .8]{Rushing1973}. * for every$n$in the topological [[Wikipedia:Local flatness|locally flat]] category (Brown-Mazur-Moise Theorem, 1960) \cite[Generalized Schöenfliess Theorem 1.8.2]{Rushing1973}. The famous counterexample to the analogue of Theorem \ref{t:sch}.a for$n=2$in the topological category is [[Wikipedia:Alexander_horned_sphere|the Alexander horned sphere]], see the corresponding [[Wikipedia:Alexander_horned_sphere|Wikipedia article]]. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of$S^n$into$S^{n+1}$are isotopic for every$n\ge3$(this is equivalent to the description of$E^{n+1}_{PL}(S^n)$). (b) (some speculations) If the extension of the Alexander torus theorem \ref{t:ale} respects isotopy, the extension 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 $\S$1].

Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].

• The Manifold Problem: Classify $n$$n$-manifolds.
• The Embedding Problem: Find the least dimension $m$$m$ such that a given manifold admits an embedding into $m$$m$-dimensional Euclidean space $\Rr^m$$\Rr^m$.
• The Knotting Problem: Classify embeddings of a given manifold into another given manifold 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, 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.

The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, $\S$$\S$1].

This article gives a short guide to the problem of classifying embeddings of closed manifolds $N$$N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds $N$$N$ into $\R^m$$\R^m$ or $S^m$$S^m$). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$$N$ are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.

The most interesting and much studied case of embeddings concerns classical knots, which are embeddings $S^1\to S^3$$S^1\to S^3$, or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and $\S$$\S$5 for more information.

The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space $\Rr^m$$\Rr^m$ are known. Such classification results are presented on the pages listed in Remark 1.4, in $\S$$\S$2 and in $\S$$\S$6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 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 given below show the following:

• For a fixed $N$$N$, the more $m$$m$ decreases from $2n$$2n$ towards $n+3$$n+3$, the more complicated the classification of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ becomes.
• The complete readily calculable classification of embeddings into $\Rr^m$$\Rr^m$ of closed connected $n$$n$-manifolds is non-trivial and 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 information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

Remark 1.2 (Readily calculable classification). The informal concept of readily calculable (concrete) classification' is complementary to abstract classification' or reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings $N\to\Rr^m$$N\to\Rr^m$ is readily calculable' if it involves a 1-1 correspondence with a set or a group which is easily' calculated from the given number $m$$m$ and the manifold $N$$N$. This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.

Since this is a paper on the Knotting Problem of Remark 1.1, below classification' means classification of embeddings of a manifold $N$$N$ into $\Rr^m$$\Rr^m$ up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).

(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.

(b) In the above description of a readily calculable classification, the notion of easily' depends upon how the manifold $N$$N$ is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.

(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).

(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].

(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, $\S$$\S$1], and homotopy groups of spheres is a standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.

(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, $\S$$\S$5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies

(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$$S^1\to\Rr^n$ [Weiss], [Arone&Turchin2014].

Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The 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)$$g(x,y)=(x,-y)$ are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$$S^1$ into $S^2$$S^2$ are isotopic, see Theorem 6.1.a and below it.

(b) For $m\ge n+2$$m\ge n+2$ the classifications of embeddings of compact $n$$n$-manifolds into $S^m$$S^m$ and into $\Rr^m$$\Rr^m$ are the same. More precisely, for all integers $m,n$$m,n$ such that $m\ge n+2$$m\ge n+2$, and for every $n$$n$-manifold $N$$N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$$i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$$N\to \Rr^m$ and $N\to S^m$$N\to S^m$, which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$$i \colon \Rr^m \to S^m$, is a bijection.

Let us prove part (b). Since $n < m$$n < m$, after a small isotopy an embedding $N \to S^m$$N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$$i(\Rr^m) \subset S^m$. Hence $i_*$$i_*$ is onto. To prove that $i_*$$i_*$ is injective, it suffices to show that if the compositions with the inclusion $i:\Rr^m\to S^m$$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'$$f'$ are isotopic. For showing that assume that $i\circ f$$i\circ f$ and $i\circ f'$$i\circ f'$ are isotopic. Then by general position $f$$f$ and $f'$$f'$ are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], $f$$f$ and $f'$$f'$ are isotopic.

Remark 1.4 (References to information on the classification of embeddings). 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:

## 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, Theorem 3.5], [Rourke&Sanderson1972, Theorem 5.4]). For every compact $n$$n$-manifold $N$$N$ and $m\ge2n+2$$m\ge2n+2$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

The case $m\ge2n+2$$m\ge2n+2$ is called a stable range' (for the classification problem; for the existence problem the analogous result with $m\ge2n$$m\ge2n$ is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).

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, Example 2.1].

Whitney-Wu Unknotting Theorem 2.2. For every compact connected $n$$n$-manifold $N$$N$, $n\ge2$$n\ge2$ and $m\ge2n+1$$m\ge2n+1$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, $\S$$\S$5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].

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, Example 2.1];
• the assumption $m\ge2n+1$$m\ge2n+1$ because of the example of Hudson tori [Skopenkov2016e, Example 3.1].

Unknotting Spheres Theorem 2.3. Assume that $m\ge n+3$$m\ge n+3$ in the PL category or $2m\ge 3n+4$$2m\ge 3n+4$ in the smooth category. For $N=S^n$$N=S^n$, or even for $N$$N$ an integral homology $n$$n$-sphere, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

For $N=S^n$$N=S^n$ this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For $N$$N$ an integral homology $n$$n$-sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for $m\ge n+3$$m\ge n+3$ in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].

The case $2m\ge 3n+4$$2m\ge 3n+4$ is called a metastable range' (for the classification problem; for the existence problem there are analogous results with $2m\ge3n+3$$2m\ge3n+3$ [Skopenkov2006, $\S$$\S$2, $\S$$\S$5]).

Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] 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 every $n\ge2k+2$$n\ge2k+2$, $m\ge2n-k+1$$m\ge2n-k+1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\Rr^m$$\Rr^m$ are isotopic.

This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, $\S$$\S$11], [Rourke&Sanderson1972, $\S$$\S$5]. Theorem 2.4 remains true if $k$$k$-connected is replaced by homologically $k$$k$-connected (see $\S$$\S$3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).

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

Given Theorem 2.4 above, the case $m\ge2n-k+1$$m\ge2n-k+1$ can be called a stable range for $k$$k$-connected manifolds'.

For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.

## 3 Notation and conventions

The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.

Unless otherwise indicated, the word isotopy' means ambient isotopy'; see definition in [Skopenkov2016i, $\S$$\S$1]. For a manifold $N$$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 the category is omitted, then the result holds (or a definition or a construction is given) in both categories.

The sources of all embeddings are 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
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${\rm pr}_k$ is the projection of a Cartesian product onto the $k$$k$th factor.

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$$P$ with boundary $\partial P$$\partial 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 every $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)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$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$$f$ to the subspace of unit length vectors identified with $\partial C_f$$\partial C_f$.
Denote by
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${\rm i}:S^q\to S^m$ the standard embedding given by
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${\rm i}(x_1,\ldots,x_{q+1})=(x_1,\ldots,x_{q+1},0,\ldots,0)$. The natural normal framing by vectors of length 1/2 on
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${\rm i}$ defines the standard embedding
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${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$. Denote by the same symbol
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${\rm i}_{m,q}$ the restriction of
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${\rm i}_{m,q}$ to $S^p\times S^q$$S^p\times S^q$ for any $p\le m-q-1$$p\le m-q-1$.

Denote by $\Sigma X$$\Sigma X$ the suspension of a space $X$$X$. Denote by $\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$$\Sigma:\pi_q(S^n)\to\pi_{q+1}(S^{n+1})$ the suspension homomorphism. Recall that $\Sigma$$\Sigma$ is an isomorphism for $q\le2n-2$$q\le2n-2$. Let $\pi_k^S$$\pi_k^S$ be any of the groups $\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$$\pi_{2k+2}(S^{k+2})\cong \pi_{2k+3}(S^{2k+3})\cong\ldots$ identified by the suspension isomorphism. Denote by $\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$$\Sigma^\infty:\pi_q(S^n)\to \pi_{q+M}(S^{n+M})=\pi_{q-n}^S$ the stable suspension homomorphisms, where $M$$M$ is large.

## 4 Embedded connected sum

Suppose that $m\ge n+2$$m\ge n+2$, $N$$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, 1.3, 1.4].

This operation is well-defined, i.e. the isotopy 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, 1.3-1.7], and an action $\#$$\#$ of $E^m(S^n)$$E^m(S^n)$ on $E^m(N)$$E^m(N)$.

## 5 Some remarks on codimension 2 embeddings

The case of embeddings of $S^n$$S^n$ into $\Rr^{n+2}$$\Rr^{n+2}$ is the most extensively studied case of the Knotting Problem. In this case there is 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$$N$ be a closed connected $n$$n$-manifold. Using embedded connected sum ($\S$$\S$4) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$$N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$$S^n\to\Rr^{n+2}$. (However, note that for $n=2$$n=2$ 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, 22 in p. 181].) 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 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].

## 6 Codimension 1 embeddings

Theorem 6.1. (a) Any two smooth embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $n\ne3$$n\ne3$.

(b) Any two smooth embeddings of $S^p\times S^{n-p}$$S^p\times S^{n-p}$ into $S^{n+1}$$S^{n+1}$ are smoothly isotopic for every $2\le p\le n-p$$2\le p\le n-p$.

Part (a) is proved in [Milnor1965a, Proposition D in $\S$$\S$9] as an easy corollary of the $h$$h$-cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, $\S$$\S$1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.

Theorem 6.2 (Alexander torus theorem). [Alexander1924] 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$.

Clearly, only the standard embedding extends to both.

Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.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 Theorem 6.1.a for $n=2$$n=2$ in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of $S^n$$S^n$ into $S^{n+1}$$S^{n+1}$ are isotopic for every $n\ge3$$n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$$E^{n+1}_{PL}(S^n)$).

(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between $E^3(S^1\times S^1)$$E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$$E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$$\Zz\times E^3(S^1)$ with base points' $i\times0$$i\times0$ and $0\times i$$0\times i$ identified (where $i$$i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$$S^1\to\Rr^3$). So the description of $E^3(S^1\times S^1)$$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.