Embeddings in Euclidean space: an introduction to their classification

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* (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in `metastable range') in terms of 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 \cite{Haefliger1963}, \cite{Weber1967}, \cite{Skopenkov2002} and a survey \cite[\S5]{Skopenkov2006}. Such equivariant maps are not classified in general. The deleted product of a general manifold is less `standard object' of mathematics (if at all) than homotopy groups of spheres. Most of the explicit corollaries of the Haefliger-Weber Theorem are recovered by easier proofs, see though the [[Embeddings_just_below_the_stable_range:_classification#Classification_further_below_the_stable_range|Becker-Glover Theorem]] \cite[Theorem 6.5]{Skopenkov2016e} and [[Knotted_tori#Classification|classification of knotted tori]] \cite[Theorem 3.2]{Skopenkov2016k}. Thus I consider the Haefliger-Weber classification of embeddings not to be a readily calculable classification. This does not mean that it is completely useless: it implies the independence of the classification of embeddings ot the smooth or PL structure, and 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}.
* (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in `metastable range') in terms of 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 \cite{Haefliger1963}, \cite{Weber1967}, \cite{Skopenkov2002} and a survey \cite[\S5]{Skopenkov2006}. Such equivariant maps are not classified in general. The deleted product of a general manifold is less `standard object' of mathematics (if at all) than homotopy groups of spheres. Most of the explicit corollaries of the Haefliger-Weber Theorem are recovered by easier proofs, see though the [[Embeddings_just_below_the_stable_range:_classification#Classification_further_below_the_stable_range|Becker-Glover Theorem]] \cite[Theorem 6.5]{Skopenkov2016e} and [[Knotted_tori#Classification|classification of knotted tori]] \cite[Theorem 3.2]{Skopenkov2016k}. Thus I consider the Haefliger-Weber classification of embeddings not to be a readily calculable classification. This does not mean that it is completely useless: it implies the independence of the classification of embeddings ot the smooth or PL structure, and 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}.
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* (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) \cite{Weiss96}, \cite{Goodwillie&Weiss1999} so far did not give any new complete readily calculable
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* (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) \cite{Weiss96}, \cite{Goodwillie&Weiss1999} so far did not give any new complete readily calculable classification results. This does not mean that it is completely useless: it gives a modern abstract proof of the Haefliger-Weber Theorem; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ \cite{Weiss}.
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classification results. This does not mean that it is completely useless: it gives a modern abstract proof of the Haefliger-Weber Theorem; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ \cite{Weiss}.
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Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification.
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification.
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This is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
+
This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.
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Contents

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; see definition in [Skopenkov2016i].

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

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

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

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

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

The most interesting and very much studied case concerns embeddings S^1\to S^3 (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and \S5 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 complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold N into Euclidean space \Rr^m are known. Such classification results are the unknotting theorems in \S2, the results on the pages listed in Remark 1.4 and in \S6. Their statements, 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:

  • The complete readily calculable classification of embeddings into \Rr^m of closed connected n-manifolds is non-trivial but presently accessible only for n+3\le m\le 2n or for m=n+1\ge4;
  • For a fixed N, the more m decreases from 2n towards n+3, the more complicated classification of embeddings of N into \Rr^m becomes.

The lowest dimensional cases, i.e. all such pairs (m,n) with n\le4, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

Remark 1.2 (Readily calculable classification). The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [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.' [Wall1999, p. 119]

`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].

See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

Let me justify leaving aside some classifications of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below). For this I need the informal concept of `readily calculable (concrete) classification', which is dual to `abstract classification' or `reduction' as described by Wall. An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers). A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds. In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a `tool', necessarily more complicated and accessible only to specialists. For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1]. Here let me further illustrate the notion by describing some examples.

  • (i) 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, \S1], only the left two columns of the table are recovered by easier proofs, and homotopy groups of spheres is a `standard object' of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
  • (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in `metastable range') in terms of 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 [Haefliger1963], [Weber1967], [Skopenkov2002] and a survey [Skopenkov2006, \S5]. Such equivariant maps are not classified in general. The deleted product of a general manifold is less `standard object' of mathematics (if at all) than homotopy groups of spheres. Most of the explicit corollaries of the Haefliger-Weber Theorem are recovered by easier proofs, see though the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and classification of knotted tori [Skopenkov2016k, Theorem 3.2]. Thus I consider the Haefliger-Weber classification of embeddings not to be a readily calculable classification. This does not mean that it is completely useless: it implies the independence of the classification of embeddings ot the smooth or PL structure, and the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
  • (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. This does not mean that it is completely useless: it gives a modern abstract proof of the Haefliger-Weber Theorem; it also gives explicit results on higher homotopy groups of the space of embeddings S^1\to\Rr^n [Weiss].

Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.

Remark 1.3 (Embeddings into the sphere and Euclidean space). (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 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 6.1.a and below.

(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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], f and 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-manifold N and m\ge2n+2, any two embeddings of N into \Rr^m are isotopic.

The case m\ge2n+2 is called a `stable range' (for the classification problem; for the existence problem there is analogous result with m\ge2n+1 [Skopenkov2006, \S2]).

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

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

This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, \S5]).

All the three assumptions in this result are indeed necessary:

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

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

The case 2m\ge 3n+4 is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with 2m\ge3n+3 [Skopenkov2006, \S2, \S5]).

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) 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, m\ge2n-k+1 and closed k-connected n-manifold N, any two embeddings of N into \Rr^m are isotopic.

This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, \S7] work for homologically k-connected manifolds (see \S3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness is sufficient).

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

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

For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, \S5] or [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.

For a manifold N let E^m_D(N) or E^m_{PL}(N) denote the set of smooth or piecewise-linear (PL) embeddings N\to S^m up to smooth or PL isotopy. If 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 0 for k even and 2 for k odd, so that \Zz_{\varepsilon(k)} is \Zz for k even and \Zz_2 for k odd.

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

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

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

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

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

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

  • C_f the closure of the complement in S^m\supset\Rr^m to a tight enough tubular neighborhood of f(N) and
  • \nu_f:\partial C_f\to N the restriction of the linear normal bundle of f to the subspace of unit length vectors identified with \partial C_f.
Denote by 
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 the standard embedding. The natural normal framing by vectors of length 1/2 on 
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 defines the standard embedding 
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. Denote by the same symbol 
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 the restriction of 
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 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}) and \Sigma:\pi_q(S^n)\to \pi_{q-n}^S the suspension and the stable suspension homomorphisms.

4 Embedded connected 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, \S1].

This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes [f] and [g], and is independent of the choice of the path and of the representatives f,g. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for N=S^n in [Skopenkov2015a, \S3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for X=D^0_+ a point]. The proof for arbitrary closed connected n-manifold N is analogous.

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

5 Some remarks on codimension 2 embeddings

The case of embeddings of S^n into \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 be a closed connected n-manifold. Using embedded connected sum (\S4) we can apparently produce an overwhelming multitude of embeddings N\to\Rr^{n+2} from the overwhelming multitude of embeddings S^n\to\Rr^{n+2}. (However, note that for n=2 there are embeddings f:\Rr P^2\to S^4 and g_1,g_2:S^2\to S^4 such that g_1 is not isotopic to g_2 but f\#g_1 is isotopic to f\#g_2 [Viro1973].) One can also apply Artin's spinning construction [Artin1928] E^m(N)\to E^{m+1}(S^1\times N) for m=n+2. Thus the description of E^{n+2}(N) is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

6 Codimension 1 embeddings

Theorem 6.1. (a) Any two smooth embeddings of S^n into S^{n+1} are smoothly isotopic for every n\ne3 [Smale1961], [Smale1962a], [Barden1965].

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

The analogue of part (a) holds

  • for n=1 in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, \S1.8].
  • for n=2 in the PL category (Alexander Theorem, 1923) [Rushing1973, \S1.8].
  • for every n in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].

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

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

If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between E^3(S^1\times S^1) and the union of E^3(S^1)\times\Zz and \Zz\times E^3(S^1) with `base points' i\times0 and 0\times i identified (where i is the isotopy class of the standard inclusion S^1\to\Rr^3). So the description of E^3(S^1\times S^1) would be as hopeless as that of E^3(S^1). Thus the description of E^3(N) for N a sphere with handles is apparently hopeless.

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

7 References

]{Skopenkov2016h}, only the left two columns of the table are recovered by easier proofs, and homotopy groups of spheres is a `standard object' of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification. * (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in `metastable range') in terms of 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 \cite{Haefliger1963}, \cite{Weber1967}, \cite{Skopenkov2002} and a survey \cite[\S5]{Skopenkov2006}. Such equivariant maps are not classified in general. The deleted product of a general manifold is less `standard object' of mathematics (if at all) than homotopy groups of spheres. Most of the explicit corollaries of the Haefliger-Weber Theorem are recovered by easier proofs, see though the [[Embeddings_just_below_the_stable_range:_classification#Classification_further_below_the_stable_range|Becker-Glover Theorem]] \cite[Theorem 6.5]{Skopenkov2016e} and [[Knotted_tori#Classification|classification of knotted tori]] \cite[Theorem 3.2]{Skopenkov2016k}. Thus I consider the Haefliger-Weber classification of embeddings not to be a readily calculable classification. This does not mean that it is completely useless: it implies the independence of the classification of embeddings ot the smooth or PL structure, and 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) An interesting approach of Goodwille-Weiss (calculus of embeddings) \cite{Weiss96}, \cite{Goodwillie&Weiss1999} so far did not give any new complete readily calculable classification results. This does not mean that it is completely useless: it gives a modern abstract proof of the Haefliger-Weber Theorem; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ \cite{Weiss}. Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others. {{endthm}} {{beginthm|Remark|(Embeddings into the sphere and Euclidean space)}}\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 [[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. (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]] * [[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 there is analogous result with $m\ge2n+1$ \cite[$\S]{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{Skopenkov2016h}, \cite[Figure 2.1.a]{Skopenkov2006}. {{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{Wu1958}, \cite{Wu1958a} and \cite{Wu1959} using ''the Whitney trick'' (see those references or \cite[$\S]{Rourke&Sanderson1972}). 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{Skopenkov2016h}; *the assumption $m\ge2n+1$ because of the [[Embeddings just below the stable range#Hudson_tori|example of Hudson tori]] \cite{Skopenkov2016e}. {{beginthm|Unknotting Spheres Theorem}}\label{sph} For $N=S^n$, or even for $N$ an integral homology $n$-sphere, $m\ge n+3$ or m\ge 3n+4$ in the PL or smooth category, respectively, any two embeddings of $N$ into $\Rr^m$ are isotopic. {{endthm}} This result is proved in \cite{Zeeman1960} or \cite{Haefliger1961} in the PL or smooth category, respectively. This result is also true for $m\ge n+3$ in the topological locally flat category \cite{Stallings1963}, \cite{Gluck1963}, \cite[Flattening Theorem 4.5.1]{Rushing1973}, \cite{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 was proved in \cite{Penrose&Whitehead&Zeeman1961}, \cite{Haefliger1961}, \cite{Zeeman1962}, \cite{Irwin1965}, \cite{Hudson1969}. The proofs in \cite{Haefliger&Hirsch1963}, \cite{Vrabec1977}, \cite{Weber1967}, \cite[$\S]{Adachi1993} work for ''homologically'' $k$-connected manifolds (see $\S$\ref{s:nc} for the definition; the proofs are non-trivial but the generalization is trivial, basically because the $k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$ to the image of $N$, by Alexander duality and simple connectedness of the complement, so homological $k$-connectedness is sufficient). Given Theorem \ref{haze} above, the case $m\ge2n-k+1$ can be called a `stable range for $k$-connected manifolds'. Note that if $n\le2k+1$, then every closed $k$-connected $n$-manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem \ref{haze} in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem \ref{sph}. For generalizations of the Haefliger-Zeeman Unknotting Theorem \ref{haze} [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|see]] \cite[$\S]{Skopenkov2016e} or \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}. 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 -manifolds.
  • Embedding Problem: Find the least dimension m such that given space admits an embedding into m-dimensional Euclidean space \Rr^m.
  • Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

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

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

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

    The most interesting and very much studied case concerns embeddings S^1\to S^3 (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and \S5 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 complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold N into Euclidean space \Rr^m are known. Such classification results are the unknotting theorems in \S2, the results on the pages listed in Remark 1.4 and in \S6. Their statements, 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:

    The lowest dimensional cases, i.e. all such pairs (m,n) with n\le4, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

    Remark 1.2 (Readily calculable classification). The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [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.' [Wall1999, p. 119]

    `The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].

    See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

    Let me justify leaving aside some classifications of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below). For this I need the informal concept of `readily calculable (concrete) classification', which is dual to `abstract classification' or `reduction' as described by Wall. An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers). A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds. In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a `tool', necessarily more complicated and accessible only to specialists. For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1]. Here let me further illustrate the notion by describing some examples.

    • (i) 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, \S1], only the left two columns of the table are recovered by easier proofs, and homotopy groups of spheres is a `standard object' of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
    • (ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in `metastable range') in terms of 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 [Haefliger1963], [Weber1967], [Skopenkov2002] and a survey [Skopenkov2006, \S5]. Such equivariant maps are not classified in general. The deleted product of a general manifold is less `standard object' of mathematics (if at all) than homotopy groups of spheres. Most of the explicit corollaries of the Haefliger-Weber Theorem are recovered by easier proofs, see though the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and classification of knotted tori [Skopenkov2016k, Theorem 3.2]. Thus I consider the Haefliger-Weber classification of embeddings not to be a readily calculable classification. This does not mean that it is completely useless: it implies the independence of the classification of embeddings ot the smooth or PL structure, and the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
    • (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. This does not mean that it is completely useless: it gives a modern abstract proof of the Haefliger-Weber Theorem; it also gives explicit results on higher homotopy groups of the space of embeddings S^1\to\Rr^n [Weiss].

    Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.

    Remark 1.3 (Embeddings into the sphere and Euclidean space). (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 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 6.1.a and below.

    (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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], f and 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-manifold N and m\ge2n+2, any two embeddings of N into \Rr^m are isotopic.

    The case m\ge2n+2 is called a `stable range' (for the classification problem; for the existence problem there is analogous result with m\ge2n+1 [Skopenkov2006, \S2]).

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

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

    This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, \S5]).

    All the three assumptions in this result are indeed necessary:

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

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

    The case 2m\ge 3n+4 is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with 2m\ge3n+3 [Skopenkov2006, \S2, \S5]).

    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) 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, m\ge2n-k+1 and closed k-connected n-manifold N, any two embeddings of N into \Rr^m are isotopic.

    This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, \S7] work for homologically k-connected manifolds (see \S3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness is sufficient).

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

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

    For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, \S5] or [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.

    For a manifold N let E^m_D(N) or E^m_{PL}(N) denote the set of smooth or piecewise-linear (PL) embeddings N\to S^m up to smooth or PL isotopy. If 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 0 for k even and 2 for k odd, so that \Zz_{\varepsilon(k)} is \Zz for k even and \Zz_2 for k odd.

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

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

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

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

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

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

    Denote by 
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     the standard embedding. The natural normal framing by vectors of length 1/2 on 
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     defines the standard embedding 
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    . Denote by the same symbol 
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     the restriction of 
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     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}) and \Sigma:\pi_q(S^n)\to \pi_{q-n}^S the suspension and the stable suspension homomorphisms.

    4 Embedded connected 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, \S1].

    This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes [f] and [g], and is independent of the choice of the path and of the representatives f,g. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for N=S^n in [Skopenkov2015a, \S3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for X=D^0_+ a point]. The proof for arbitrary closed connected n-manifold N is analogous.

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

    5 Some remarks on codimension 2 embeddings

    The case of embeddings of S^n into \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 be a closed connected n-manifold. Using embedded connected sum (\S4) we can apparently produce an overwhelming multitude of embeddings N\to\Rr^{n+2} from the overwhelming multitude of embeddings S^n\to\Rr^{n+2}. (However, note that for n=2 there are embeddings f:\Rr P^2\to S^4 and g_1,g_2:S^2\to S^4 such that g_1 is not isotopic to g_2 but f\#g_1 is isotopic to f\#g_2 [Viro1973].) One can also apply Artin's spinning construction [Artin1928] E^m(N)\to E^{m+1}(S^1\times N) for m=n+2. Thus the description of E^{n+2}(N) is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

    For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

    6 Codimension 1 embeddings

    Theorem 6.1. (a) Any two smooth embeddings of S^n into S^{n+1} are smoothly isotopic for every n\ne3 [Smale1961], [Smale1962a], [Barden1965].

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

    The analogue of part (a) holds

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

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

    If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between E^3(S^1\times S^1) and the union of E^3(S^1)\times\Zz and \Zz\times E^3(S^1) with `base points' i\times0 and 0\times i identified (where i is the isotopy class of the standard inclusion S^1\to\Rr^3). So the description of E^3(S^1\times S^1) would be as hopeless as that of E^3(S^1). Thus the description of E^3(N) for N a sphere with handles is apparently hopeless.

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

    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 $V_{m,n}$ 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. 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 $\Sigma:\pi_q(S^n)\to \pi_{q+1}(S^{n+1})$ and $\Sigma:\pi_q(S^n)\to \pi_{q-n}^S$ the suspension and the stable suspension homomorphisms. == 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[Theorem 1.7]{Haefliger1966}, \cite[$\Sn-manifolds.
  • Embedding Problem: Find the least dimension m such that given space admits an embedding into m-dimensional Euclidean space \Rr^m.
  • Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

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

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

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

    The most interesting and very much studied case concerns embeddings S^1\to S^3 (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and \S5 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 complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold N into Euclidean space \Rr^m are known. Such classification results are the unknotting theorems in \S2, the results on the pages listed in Remark 1.4 and in \S6. Their statements, 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:

    The lowest dimensional cases, i.e. all such pairs (m,n) with n\le4, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

    Remark 1.2 (Readily calculable classification). The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [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.' [Wall1999, p. 119]

    `The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].

    See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

    Let me justify leaving aside some classifications of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below). For this I need the informal concept of `readily calculable (concrete) classification', which is dual to `abstract classification' or `reduction' as described by Wall. An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers). A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds. In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a `tool', necessarily more complicated and accessible only to specialists. For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1]. Here let me further illustrate the notion by describing some examples.

    Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.

    Remark 1.3 (Embeddings into the sphere and Euclidean space). (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 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 6.1.a and below.

    (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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], f and 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-manifold N and m\ge2n+2, any two embeddings of N into \Rr^m are isotopic.

    The case m\ge2n+2 is called a `stable range' (for the classification problem; for the existence problem there is analogous result with m\ge2n+1 [Skopenkov2006, \S2]).

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

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

    This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, \S5]).

    All the three assumptions in this result are indeed necessary:

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

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

    The case 2m\ge 3n+4 is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with 2m\ge3n+3 [Skopenkov2006, \S2, \S5]).

    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) 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, m\ge2n-k+1 and closed k-connected n-manifold N, any two embeddings of N into \Rr^m are isotopic.

    This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, \S7] work for homologically k-connected manifolds (see \S3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness is sufficient).

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

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

    For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, \S5] or [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.

    For a manifold N let E^m_D(N) or E^m_{PL}(N) denote the set of smooth or piecewise-linear (PL) embeddings N\to S^m up to smooth or PL isotopy. If 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 0 for k even and 2 for k odd, so that \Zz_{\varepsilon(k)} is \Zz for k even and \Zz_2 for k odd.

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

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

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

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

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

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

    Denote by 
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     the standard embedding. The natural normal framing by vectors of length 1/2 on 
    Tex syntax error
     defines the standard embedding 
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    . Denote by the same symbol 
    Tex syntax error
     the restriction of 
    Tex syntax error
     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}) and \Sigma:\pi_q(S^n)\to \pi_{q-n}^S the suspension and the stable suspension homomorphisms.

    4 Embedded connected 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, \S1].

    This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes [f] and [g], and is independent of the choice of the path and of the representatives f,g. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for N=S^n in [Skopenkov2015a, \S3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for X=D^0_+ a point]. The proof for arbitrary closed connected n-manifold N is analogous.

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

    5 Some remarks on codimension 2 embeddings

    The case of embeddings of S^n into \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 be a closed connected n-manifold. Using embedded connected sum (\S4) we can apparently produce an overwhelming multitude of embeddings N\to\Rr^{n+2} from the overwhelming multitude of embeddings S^n\to\Rr^{n+2}. (However, note that for n=2 there are embeddings f:\Rr P^2\to S^4 and g_1,g_2:S^2\to S^4 such that g_1 is not isotopic to g_2 but f\#g_1 is isotopic to f\#g_2 [Viro1973].) One can also apply Artin's spinning construction [Artin1928] E^m(N)\to E^{m+1}(S^1\times N) for m=n+2. Thus the description of E^{n+2}(N) is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

    For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

    6 Codimension 1 embeddings

    Theorem 6.1. (a) Any two smooth embeddings of S^n into S^{n+1} are smoothly isotopic for every n\ne3 [Smale1961], [Smale1962a], [Barden1965].

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

    The analogue of part (a) holds

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

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

    If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between E^3(S^1\times S^1) and the union of E^3(S^1)\times\Zz and \Zz\times E^3(S^1) with `base points' i\times0 and 0\times i identified (where i is the isotopy class of the standard inclusion S^1\to\Rr^3). So the description of E^3(S^1\times S^1) would be as hopeless as that of E^3(S^1). Thus the description of E^3(N) for N a sphere with handles is apparently hopeless.

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

    7 References

    ]{Avvakumov2016}. This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes $[f]$ and $[g]$, and is independent of the choice of the path and of the representatives $f,g$. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. \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{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{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 $n\ne3$ \cite{Smale1961}, \cite{Smale1962a}, \cite{Barden1965}. (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$ \cite{Kosinski1961}, \cite{Wall1965}, \cite{Lucas&Neto&Saeki1996}, cf. \cite{Goldstein1967}. {{endthm}} The analogue of part (a) holds * for $n=1$ in the PL or topological category (Schöenfliess Theorem, 1912) \cite[$\Sn-manifolds.
  • Embedding Problem: Find the least dimension m such that given space admits an embedding into m-dimensional Euclidean space \Rr^m.
  • Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

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

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

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

    The most interesting and very much studied case concerns embeddings S^1\to S^3 (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and \S5 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 complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold N into Euclidean space \Rr^m are known. Such classification results are the unknotting theorems in \S2, the results on the pages listed in Remark 1.4 and in \S6. Their statements, 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:

    The lowest dimensional cases, i.e. all such pairs (m,n) with n\le4, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

    Remark 1.2 (Readily calculable classification). The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [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.' [Wall1999, p. 119]

    `The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].

    See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

    Let me justify leaving aside some classifications of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below). For this I need the informal concept of `readily calculable (concrete) classification', which is dual to `abstract classification' or `reduction' as described by Wall. An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers). A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds. In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a `tool', necessarily more complicated and accessible only to specialists. For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1]. Here let me further illustrate the notion by describing some examples.

    Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.

    Remark 1.3 (Embeddings into the sphere and Euclidean space). (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 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 6.1.a and below.

    (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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], f and 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-manifold N and m\ge2n+2, any two embeddings of N into \Rr^m are isotopic.

    The case m\ge2n+2 is called a `stable range' (for the classification problem; for the existence problem there is analogous result with m\ge2n+1 [Skopenkov2006, \S2]).

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

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

    This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, \S5]).

    All the three assumptions in this result are indeed necessary:

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

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

    The case 2m\ge 3n+4 is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with 2m\ge3n+3 [Skopenkov2006, \S2, \S5]).

    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) 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, m\ge2n-k+1 and closed k-connected n-manifold N, any two embeddings of N into \Rr^m are isotopic.

    This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, \S7] work for homologically k-connected manifolds (see \S3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness is sufficient).

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

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

    For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, \S5] or [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.

    For a manifold N let E^m_D(N) or E^m_{PL}(N) denote the set of smooth or piecewise-linear (PL) embeddings N\to S^m up to smooth or PL isotopy. If 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 0 for k even and 2 for k odd, so that \Zz_{\varepsilon(k)} is \Zz for k even and \Zz_2 for k odd.

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

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

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

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

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

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

    Denote by 
    Tex syntax error
     the standard embedding. The natural normal framing by vectors of length 1/2 on 
    Tex syntax error
     defines the standard embedding 
    Tex syntax error
    . Denote by the same symbol 
    Tex syntax error
     the restriction of 
    Tex syntax error
     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}) and \Sigma:\pi_q(S^n)\to \pi_{q-n}^S the suspension and the stable suspension homomorphisms.

    4 Embedded connected 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, \S1].

    This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes [f] and [g], and is independent of the choice of the path and of the representatives f,g. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for N=S^n in [Skopenkov2015a, \S3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for X=D^0_+ a point]. The proof for arbitrary closed connected n-manifold N is analogous.

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

    5 Some remarks on codimension 2 embeddings

    The case of embeddings of S^n into \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 be a closed connected n-manifold. Using embedded connected sum (\S4) we can apparently produce an overwhelming multitude of embeddings N\to\Rr^{n+2} from the overwhelming multitude of embeddings S^n\to\Rr^{n+2}. (However, note that for n=2 there are embeddings f:\Rr P^2\to S^4 and g_1,g_2:S^2\to S^4 such that g_1 is not isotopic to g_2 but f\#g_1 is isotopic to f\#g_2 [Viro1973].) One can also apply Artin's spinning construction [Artin1928] E^m(N)\to E^{m+1}(S^1\times N) for m=n+2. Thus the description of E^{n+2}(N) is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

    For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

    6 Codimension 1 embeddings

    Theorem 6.1. (a) Any two smooth embeddings of S^n into S^{n+1} are smoothly isotopic for every n\ne3 [Smale1961], [Smale1962a], [Barden1965].

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

    The analogue of part (a) holds

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

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

    If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between E^3(S^1\times S^1) and the union of E^3(S^1)\times\Zz and \Zz\times E^3(S^1) with `base points' i\times0 and 0\times i identified (where i is the isotopy class of the standard inclusion S^1\to\Rr^3). So the description of E^3(S^1\times S^1) would be as hopeless as that of E^3(S^1). Thus the description of E^3(N) for N a sphere with handles is apparently hopeless.

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

    7 References

    .8]{Rushing1973}. * for $n=2$ in the PL category (Alexander Theorem, 1923) \cite[$\Sn-manifolds.
  • Embedding Problem: Find the least dimension m such that given space admits an embedding into m-dimensional Euclidean space \Rr^m.
  • Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

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

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

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

    The most interesting and very much studied case concerns embeddings S^1\to S^3 (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and \S5 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 complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold N into Euclidean space \Rr^m are known. Such classification results are the unknotting theorems in \S2, the results on the pages listed in Remark 1.4 and in \S6. Their statements, 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:

    The lowest dimensional cases, i.e. all such pairs (m,n) with n\le4, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

    Remark 1.2 (Readily calculable classification). The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [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.' [Wall1999, p. 119]

    `The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].

    See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

    Let me justify leaving aside some classifications of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below). For this I need the informal concept of `readily calculable (concrete) classification', which is dual to `abstract classification' or `reduction' as described by Wall. An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers). A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds. In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a `tool', necessarily more complicated and accessible only to specialists. For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1]. Here let me further illustrate the notion by describing some examples.

    Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.

    Remark 1.3 (Embeddings into the sphere and Euclidean space). (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 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 6.1.a and below.

    (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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], f and 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-manifold N and m\ge2n+2, any two embeddings of N into \Rr^m are isotopic.

    The case m\ge2n+2 is called a `stable range' (for the classification problem; for the existence problem there is analogous result with m\ge2n+1 [Skopenkov2006, \S2]).

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

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

    This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, \S5]).

    All the three assumptions in this result are indeed necessary:

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

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

    The case 2m\ge 3n+4 is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with 2m\ge3n+3 [Skopenkov2006, \S2, \S5]).

    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) 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, m\ge2n-k+1 and closed k-connected n-manifold N, any two embeddings of N into \Rr^m are isotopic.

    This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, \S7] work for homologically k-connected manifolds (see \S3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness is sufficient).

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

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

    For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, \S5] or [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.

    For a manifold N let E^m_D(N) or E^m_{PL}(N) denote the set of smooth or piecewise-linear (PL) embeddings N\to S^m up to smooth or PL isotopy. If 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 0 for k even and 2 for k odd, so that \Zz_{\varepsilon(k)} is \Zz for k even and \Zz_2 for k odd.

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

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

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

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

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

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

    Denote by 
    Tex syntax error
     the standard embedding. The natural normal framing by vectors of length 1/2 on 
    Tex syntax error
     defines the standard embedding 
    Tex syntax error
    . Denote by the same symbol 
    Tex syntax error
     the restriction of 
    Tex syntax error
     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}) and \Sigma:\pi_q(S^n)\to \pi_{q-n}^S the suspension and the stable suspension homomorphisms.

    4 Embedded connected 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, \S1].

    This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes [f] and [g], and is independent of the choice of the path and of the representatives f,g. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for N=S^n in [Skopenkov2015a, \S3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for X=D^0_+ a point]. The proof for arbitrary closed connected n-manifold N is analogous.

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

    5 Some remarks on codimension 2 embeddings

    The case of embeddings of S^n into \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 be a closed connected n-manifold. Using embedded connected sum (\S4) we can apparently produce an overwhelming multitude of embeddings N\to\Rr^{n+2} from the overwhelming multitude of embeddings S^n\to\Rr^{n+2}. (However, note that for n=2 there are embeddings f:\Rr P^2\to S^4 and g_1,g_2:S^2\to S^4 such that g_1 is not isotopic to g_2 but f\#g_1 is isotopic to f\#g_2 [Viro1973].) One can also apply Artin's spinning construction [Artin1928] E^m(N)\to E^{m+1}(S^1\times N) for m=n+2. Thus the description of E^{n+2}(N) is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

    For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

    6 Codimension 1 embeddings

    Theorem 6.1. (a) Any two smooth embeddings of S^n into S^{n+1} are smoothly isotopic for every n\ne3 [Smale1961], [Smale1962a], [Barden1965].

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

    The analogue of part (a) holds

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

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

    If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between E^3(S^1\times S^1) and the union of E^3(S^1)\times\Zz and \Zz\times E^3(S^1) with `base points' i\times0 and 0\times i identified (where i is the isotopy class of the standard inclusion S^1\to\Rr^3). So the description of E^3(S^1\times S^1) would be as hopeless as that of E^3(S^1). Thus the description of E^3(N) for N a sphere with handles is apparently hopeless.

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

    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 part (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 if each two PL embeddings of $S^n$ into $S^{n+1}$ are isotopic for every $n\ge3$ (this is equivalent to the description of $E^{n+1}_{PL}(S^n)$). Every embedding $S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$ \cite{Alexander1924}. Clearly, only the standard embedding extends to both. If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between $E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$ with `base points' $i\times0$ and -manifolds.
  • Embedding Problem: Find the least dimension m such that given space admits an embedding into m-dimensional Euclidean space \Rr^m.
  • Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

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

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

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

    The most interesting and very much studied case concerns embeddings S^1\to S^3 (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and \S5 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 complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold N into Euclidean space \Rr^m are known. Such classification results are the unknotting theorems in \S2, the results on the pages listed in Remark 1.4 and in \S6. Their statements, 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:

    The lowest dimensional cases, i.e. all such pairs (m,n) with n\le4, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

    Remark 1.2 (Readily calculable classification). The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [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.' [Wall1999, p. 119]

    `The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].

    See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

    Let me justify leaving aside some classifications of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below). For this I need the informal concept of `readily calculable (concrete) classification', which is dual to `abstract classification' or `reduction' as described by Wall. An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers). A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds. In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a `tool', necessarily more complicated and accessible only to specialists. For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1]. Here let me further illustrate the notion by describing some examples.

    Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.

    Remark 1.3 (Embeddings into the sphere and Euclidean space). (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 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 6.1.a and below.

    (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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], f and 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-manifold N and m\ge2n+2, any two embeddings of N into \Rr^m are isotopic.

    The case m\ge2n+2 is called a `stable range' (for the classification problem; for the existence problem there is analogous result with m\ge2n+1 [Skopenkov2006, \S2]).

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

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

    This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, \S5]).

    All the three assumptions in this result are indeed necessary:

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

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

    The case 2m\ge 3n+4 is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with 2m\ge3n+3 [Skopenkov2006, \S2, \S5]).

    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) 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, m\ge2n-k+1 and closed k-connected n-manifold N, any two embeddings of N into \Rr^m are isotopic.

    This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, \S7] work for homologically k-connected manifolds (see \S3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness is sufficient).

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

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

    For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, \S5] or [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.

    For a manifold N let E^m_D(N) or E^m_{PL}(N) denote the set of smooth or piecewise-linear (PL) embeddings N\to S^m up to smooth or PL isotopy. If 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 0 for k even and 2 for k odd, so that \Zz_{\varepsilon(k)} is \Zz for k even and \Zz_2 for k odd.

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

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

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

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

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

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

    Denote by 
    Tex syntax error
     the standard embedding. The natural normal framing by vectors of length 1/2 on 
    Tex syntax error
     defines the standard embedding 
    Tex syntax error
    . Denote by the same symbol 
    Tex syntax error
     the restriction of 
    Tex syntax error
     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}) and \Sigma:\pi_q(S^n)\to \pi_{q-n}^S the suspension and the stable suspension homomorphisms.

    4 Embedded connected 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, \S1].

    This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes [f] and [g], and is independent of the choice of the path and of the representatives f,g. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for N=S^n in [Skopenkov2015a, \S3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for X=D^0_+ a point]. The proof for arbitrary closed connected n-manifold N is analogous.

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

    5 Some remarks on codimension 2 embeddings

    The case of embeddings of S^n into \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 be a closed connected n-manifold. Using embedded connected sum (\S4) we can apparently produce an overwhelming multitude of embeddings N\to\Rr^{n+2} from the overwhelming multitude of embeddings S^n\to\Rr^{n+2}. (However, note that for n=2 there are embeddings f:\Rr P^2\to S^4 and g_1,g_2:S^2\to S^4 such that g_1 is not isotopic to g_2 but f\#g_1 is isotopic to f\#g_2 [Viro1973].) One can also apply Artin's spinning construction [Artin1928] E^m(N)\to E^{m+1}(S^1\times N) for m=n+2. Thus the description of E^{n+2}(N) is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

    For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

    6 Codimension 1 embeddings

    Theorem 6.1. (a) Any two smooth embeddings of S^n into S^{n+1} are smoothly isotopic for every n\ne3 [Smale1961], [Smale1962a], [Barden1965].

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

    The analogue of part (a) holds

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

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

    If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between E^3(S^1\times S^1) and the union of E^3(S^1)\times\Zz and \Zz\times E^3(S^1) with `base points' i\times0 and 0\times i identified (where i is the isotopy class of the standard inclusion S^1\to\Rr^3). So the description of E^3(S^1\times S^1) would be as hopeless as that of E^3(S^1). Thus the description of E^3(N) for N a sphere with handles is apparently hopeless.

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

    7 References

    \times i$ identified (where $i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$). So the description of $E^3(S^1\times S^1)$ would be as hopeless as that of $E^3(S^1)$. Thus the description of $E^3(N)$ for $N$ a [[2-manifolds#Orientable_surfaces|sphere with handles]] is apparently hopeless. For more on higher-dimensional codimension 1 embeddings see e.g. \cite{Lucas&Saeki2002}. == References == {{#RefList:}} [[Category:Theory]] [[Category:Manifolds]] [[Category:Embeddings of manifolds]]n-manifolds.
  • Embedding Problem: Find the least dimension m such that given space admits an embedding into m-dimensional Euclidean space \Rr^m.
  • Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.

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

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

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

    The most interesting and very much studied case concerns embeddings S^1\to S^3 (classical knots), or more generally, codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and \S5 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 complete readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold N into Euclidean space \Rr^m are known. Such classification results are the unknotting theorems in \S2, the results on the pages listed in Remark 1.4 and in \S6. Their statements, 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:

    The lowest dimensional cases, i.e. all such pairs (m,n) with n\le4, are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].

    Remark 1.2 (Readily calculable classification). The Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [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.' [Wall1999, p. 119]

    `The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].

    See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].

    Let me justify leaving aside some classifications of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below). For this I need the informal concept of `readily calculable (concrete) classification', which is dual to `abstract classification' or `reduction' as described by Wall. An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers). A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds. In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a `tool', necessarily more complicated and accessible only to specialists. For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1]. Here let me further illustrate the notion by describing some examples.

    Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description of the informal concept of readily calculable classification. This is rather an invitation to further pursue the reflections by Wall, Murdoch, Graham, Knuth, Patashnik and others.

    Remark 1.3 (Embeddings into the sphere and Euclidean space). (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 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 6.1.a and below.

    (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 non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], f and 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-manifold N and m\ge2n+2, any two embeddings of N into \Rr^m are isotopic.

    The case m\ge2n+2 is called a `stable range' (for the classification problem; for the existence problem there is analogous result with m\ge2n+1 [Skopenkov2006, \S2]).

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

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

    This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, \S5]).

    All the three assumptions in this result are indeed necessary:

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

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

    The case 2m\ge 3n+4 is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with 2m\ge3n+3 [Skopenkov2006, \S2, \S5]).

    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) 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, m\ge2n-k+1 and closed k-connected n-manifold N, any two embeddings of N into \Rr^m are isotopic.

    This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, \S7] work for homologically k-connected manifolds (see \S3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the k-connectedness was used to ensure high enough connectedness of the complement in \Rr^m to the image of N, by Alexander duality and simple connectedness of the complement, so homological k-connectedness is sufficient).

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

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

    For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, \S5] or [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.

    For a manifold N let E^m_D(N) or E^m_{PL}(N) denote the set of smooth or piecewise-linear (PL) embeddings N\to S^m up to smooth or PL isotopy. If 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 0 for k even and 2 for k odd, so that \Zz_{\varepsilon(k)} is \Zz for k even and \Zz_2 for k odd.

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

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

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

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

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

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

    Denote by 
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     the standard embedding. The natural normal framing by vectors of length 1/2 on 
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     defines the standard embedding 
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    . Denote by the same symbol 
    Tex syntax error
     the restriction of 
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     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}) and \Sigma:\pi_q(S^n)\to \pi_{q-n}^S the suspension and the stable suspension homomorphisms.

    4 Embedded connected 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. [Haefliger1966, Theorem 1.7], [Avvakumov2016, \S1].

    This operation is well-defined, i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes [f] and [g], and is independent of the choice of the path and of the representatives f,g. The proof of this fact is based on the construction of embedded connected sum of isotopies. Although this fact is presumably classical, a proof was not written, cf. [Skopenkov2015a, Remark 2.3.a]. The proof is written for N=S^n in [Skopenkov2015a, \S3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for X=D^0_+ a point]. The proof for arbitrary closed connected n-manifold N is analogous.

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

    5 Some remarks on codimension 2 embeddings

    The case of embeddings of S^n into \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 be a closed connected n-manifold. Using embedded connected sum (\S4) we can apparently produce an overwhelming multitude of embeddings N\to\Rr^{n+2} from the overwhelming multitude of embeddings S^n\to\Rr^{n+2}. (However, note that for n=2 there are embeddings f:\Rr P^2\to S^4 and g_1,g_2:S^2\to S^4 such that g_1 is not isotopic to g_2 but f\#g_1 is isotopic to f\#g_2 [Viro1973].) One can also apply Artin's spinning construction [Artin1928] E^m(N)\to E^{m+1}(S^1\times N) for m=n+2. Thus the description of E^{n+2}(N) is a very hard open problem. It would be interesting to give a more formal (e.g. algorithmic) illustration of the hardness of this problem.

    For studies of codimension 2 embeddings of manifolds up to the weaker relation of concordance see e.g. [Cappell&Shaneson1974].

    6 Codimension 1 embeddings

    Theorem 6.1. (a) Any two smooth embeddings of S^n into S^{n+1} are smoothly isotopic for every n\ne3 [Smale1961], [Smale1962a], [Barden1965].

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

    The analogue of part (a) holds

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

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

    If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between E^3(S^1\times S^1) and the union of E^3(S^1)\times\Zz and \Zz\times E^3(S^1) with `base points' i\times0 and 0\times i identified (where i is the isotopy class of the standard inclusion S^1\to\Rr^3). So the description of E^3(S^1\times S^1) would be as hopeless as that of E^3(S^1). Thus the description of E^3(N) for N a sphere with handles is apparently hopeless.

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

    7 References

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