Embeddings in Euclidean space: an introduction to their classification
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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; the definitions of these terms are given in [Skopenkov2016i, 1].
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1960] E. Zeeman, Unknotting spheres in five dimensions, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1960] E. Zeeman, Unknotting spheres in five dimensions, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1960] E. Zeeman, Unknotting spheres in five dimensions, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1960] E. Zeeman, Unknotting spheres in five dimensions, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1960] E. Zeeman, Unknotting spheres in five dimensions, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1960] E. Zeeman, Unknotting spheres in five dimensions, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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- [Zeeman1963] E. C. Zeeman, Seminar on Combinatorial Topology, IHES, 1963 (revised 1966).
- [Zeeman1993] E.C. Zeeman, A Brief History of Topology, UC Berkeley, October 27, 1993, On the occasion of Moe Hirsch's 60th birthday, http://zakuski.utsa.edu/~gokhman/ecz/hirsch60.pdf
Remark 1.1 (Some motivations). Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- The Manifold Problem: Classify -manifolds.
- The Embedding Problem: Find the least dimension such that a given manifold admits an embedding into -dimensional Euclidean space .
- The Knotting Problem: Classify embeddings of a given manifold into another given manifold up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other areas of mathematics, most importantly, to algebraic topology (see Remark 1.4 below). See also the Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3], [Takase2006, 1].
This article gives a short guide to the problem of classifying embeddings of closed manifolds into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark 1.1 for embeddings of general manifolds into or ). After making some general remarks and giving references, in Section 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.In Section 3, we establish notation and conventions and in Section 4, we continue by introducing the connected sum operation for embeddings. We then make some remarks on codimension 2 embeddings in Section 5 and conclude with a brief review of some important results about codimension 1 embeddings in Section 6.
The most interesting and much studied case of embeddings concerns classical knots, which are embeddings , or more generally, codimension 2 embeddings of spheres. Although there have been many wonderful results in this subject in the last 100 years, these results were not directly aiming at a complete classification, which remains wide open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 5 for more information.
The Knotting Problem is known to be hard. To the best of the author's knowledge, at the time of writing there are only a few cases in which readily calculable classification results (see Remark 1.2) describing all isotopy classes for embeddings of a closed manifold into Euclidean space are known. Such classification results are presented on the pages listed in Remark 1.4, in 2 and in 6. The statements of those results, although not the proofs, are simple and accessible to non-specialists. This page and the pages listed in Remark 1.4 concern only such classification results. As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- For a fixed , the more decreases from towards , the more complicated the classification of embeddings of into becomes.
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial and presently accessible only for or for ; the lowest dimensional cases, i.e. all such pairs with , are (6,3), (4,3), (8,4), (7,4), (5,4). For information on the cases (6,3), (8,4), (7,4) see [Skopenkov2016t], [Skopenkov2016e], [Skopenkov2016f].
Remark 1.2 (Readily calculable classification). The informal concept of `readily calculable (concrete) classification' is complementary to `abstract classification' or `reduction' as described by Wall (see (c) below). See [Graham&Knuth&Patashnik89, Preface] for discussion of similar issues. To a first approximation, a classification of embeddings is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number and the manifold . This reflects the taste of the author and is intended to be used as informative but imprecise concept. Readers happy with this may skip the rest of this Remark. For other readers, we further illustrate our use of the term `readily calculable' with the following general remarks (a)-(d) and the more specific examples (i)-(iii). We feel that the advantages of the idea of `readily calculable' outweigh its imprecision. Here we describe this concept with only as much precision as sufficient to see why some classifications of embeddings are presented and some others are left aside.
Since this is a paper on the Knotting Problem of Remark 1.1, below `classification' means `classification of embeddings of a manifold into up to isotopy' (except in (d); some comments like (a) apply for more general classification problems).
(a) An important feature of a readily calculable classification is the accessibility of the statement to a general mathematical audience, which may only be familiar with basic notions of the area; this in turn may be viewed as an approximation to beauty. Another important feature is whether the classification gives an algorithm to classify the object considered, and if so, how fast the algorithm is.
(b) In the above description of a readily calculable classification, the notion of `easily' depends upon how the manifold is presented. Here we assume that the manifold is equipped with a triangulation. Then a classification involving invariants which are `easily' calculated from the (co)homology of the manifold (with integral or finite cyclic coefficients), from basic extra structures on (co)homology (like the intersection product and characteristic classes), when they are known, is readily calculable. Since we do not specify the extra structures exhaustively, this explanation is open-ended. If the manifold is presented in a different way (e.g. by a system of equations), the integral homology may not be easy to calculate.
(c) The Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s gives necessary and sufficient conditions for embeddability of manifolds in codimension greater than 2 (in terms of Poincaré embeddings). See [Wall1999, Corollary 11.3.1] and a simpler exposition in [Cencelj&Repovs&Skopenkov2004, pp. 263-267]. `It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]. An analogue of this result for classification is [Cencelj&Repovs&Skopenkov2004, the Browder-Wall Theorem 9]. The proof is likely to be obtained (at least for the smooth category) using surgery analogously to [Wall1999, Corollary 11.3.3] as exposed in [Cencelj&Repovs&Skopenkov2004, pp. 265-267]. However, the proof did not appear in the literature. So I regard this result as a conjecture and I apologize that it is stated as a theorem (I recognize that other mathematicians have different reliability standards and might call it a theorem). The above citation of Wall applies to this conjecture because the classifications provided by this conjecture are not readily calculable in general. On the other hand, some readily calculable classifications (see [Skopenkov2016t], [Skopenkov2016f]) have been obtained by applying Kreck's modified surgery theory, see (d).
(d) Here we consider the analogous problem of the classification of manifolds, see the Manifold Problem of Remark 1.1. Classical surgery theory as developed by Browder, Novikov, Sullivan and Wall, gives a procedure for classifying smooth manifolds. While this is a major achievement of 20th century topology, with many celebrated applications, it may or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory [Kreck1999] was to develop a surgery theory which produced readily calculable classifications of manifolds more frequently and easily than classical surgery. For references to such classifications see [Kreck1999].
(i) A reduction of a classification to calculation of standard objects of mathematics when these are known is typically a readily calculable classification. E.g. the Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of (stable) homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], and homotopy groups of spheres is a `standard object' of mathematics. Thus for the cases when these groups are known, I regard this as a readily calculable classification.
(ii) The Haefliger-Weber Theorem classifies embeddings of manifolds (in a `metastable range') in terms of equivariant homotopy classes of certain equivariant maps from the `deleted product' of the manifold (defined to be the configuration space of distinct ordered pairs of points from the manifold). See survey [Skopenkov2006, 5] and [Haefliger1963], [Weber1967], [Skopenkov2002]. Such equivariant homotopy classes have not been calculated in general. Moreover, the deleted product of a general manifold is a less `standard object' of mathematics (if at all) than the homotopy groups of spheres. Thus I regard the Haefliger-Weber classification not to be readily calculable in general. However, it is an important result and implies
- the Becker-Glover Theorem [Skopenkov2016e, Theorem 6.5] and the classification of knotted tori [Skopenkov2016k, Theorem 3.2], which I regard as readily calculable classifications;
- the independence of the classification of embeddings of the smooth or PL structure on the manifold;
- the existence of an algorithm recognizing PL isotopy of given PL embeddings (and of an algorithm recognizing PL embeddability of a PL manifold) [Cadek&Krcal&Vokrınek2013, Theorem 1.1 and text after Theorem 1.4].
(iii) There is an interesting approach of Goodwille and Weiss (the calculus of embeddings) to the Knotting Problem [Weiss96], [Weiss99], [Goodwillie&Weiss1999]. So far that approach has not lead to any new readily calculable classifications, but it gives a modern abstract proof of the Haefliger-Weber Theorem (see (ii)), and it gives explicit results on higher homotopy groups of the space of embeddings [Weiss], [Arone&Turchin2014].
Remark 1.3 (Embeddings into Euclidean space and the sphere). (a) The embeddings given by and are not isotopic (because they have distinct turning numbers; readers not familiar with turning number as defined in the Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below it.
(b) For the classifications of embeddings of compact -manifolds into and into are the same. More precisely, for all integers such that , and for every -manifold , the map between the sets of isotopy classes of embeddings and , which is induced by composition with the inclusion , is a bijection.
Let us prove part (b). Since , after a small isotopy an embedding missed the point at infinity and so lies in . Hence is onto. To prove that is injective, it suffices to show that if the compositions with the inclusion of two embeddings of a compact -manifold are isotopic, then and are isotopic. For showing that assume that and are isotopic. Then by general position and are non-ambiently isotopic. Since every non-ambient isotopy extends to an isotopy [Skopenkov2016i, Theorem 1.3], and 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 -manifold and , any two embeddings of into are isotopic.
The case is called a `stable range' (for the classification problem; for the existence problem the analogous result with is called strong Whitney embedding theorem [Skopenkov2006, Theorem 2.2.a]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h, Example 2.1].
Whitney-Wu Unknotting Theorem 2.2. For every compact connected -manifold , and , any two embeddings of into are isotopic.
This is proved in [Wu1958c] using the Whitney trick (see that reference or [Rourke&Sanderson1972, 5]). See also [Smale1961, Theorem 2.3 (H. Whitney, W.T. Wu)].
All the three assumptions in this result are indeed necessary:
- the assumption because of the existence of non-trivial knots ;
- the connectedness assumption because of the existence of the Hopf link [Skopenkov2016h, Example 2.1];
- the assumption because of the example of Hudson tori [Skopenkov2016e, Example 3.1].
Unknotting Spheres Theorem 2.3. Assume that in the PL category or in the smooth category. For , or even for an integral homology -sphere, any two embeddings of into are isotopic.
For this result is proved in [Zeeman1960], [Zeeman1963, Corollary 2 of Theorem 9 in Chapter 4] and in [Haefliger1961, Existence Theorem (b) in p. 47] in the PL and in the smooth category, respectively. For an integral homology -sphere this result follows from [Skopenkov2016e, Theorem 6.2]. This result is also true for in the topological locally flat category [Stallings1963, Corollary 9.3], [Gluck1963, Theorem 1.1]; see also textbook [Rushing1973, Flattening Theorem 4.5.1] and [Scharlemann1977, Corollary 1.3].
The case is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with [Skopenkov2006, 2, 5]).
Knots in codimension 2 and the Haefliger trefoil knot [Skopenkov2016t, Example 2.1] show that the dimension restrictions are sharp (even for ) 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 , and closed -connected -manifold , any two embeddings of into are isotopic.
This is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8], [Haefliger1961, Existence Theorem (b) in p. 47] in the PL or in the smooth category, respectively. The PL case uses the ideas of [Penrose&Whitehead&Zeeman1961] and [Irwin1965]; see also textbooks [Hudson1969, 11], [Rourke&Sanderson1972, 5]. Theorem 2.4 remains true if -connected is replaced by homologically -connected (see 3 for the definition and [Skopenkov2016e, Theorem 6.2] for justification).
Note that if , then every closed -connected -manifold is topologically a sphere, so the analogue of Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
For generalizations of Theorem 2.4 see survey [Skopenkov2016e] and [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy'; see definition in [Skopenkov2016i, 1]. For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .
Let be for even and for odd, so that is for even and for odd.
Denote byTex syntax erroris the projection of a Cartesian product onto the th factor.
Denote by the Stiefel manifold of orthonormal -frames in .
We omit -coefficients from the notation of (co)homology groups.
For a manifold with boundary denote .
A closed manifold is called homologically -connected, if is connected and for every . This condition is equivalent to for each , where are reduced homology groups. A pair is called homologically -connected, if for every .
The self-intersection set of a map is
For a smooth embedding denote by
- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
Tex syntax errorthe standard embedding given by
Tex syntax error. The natural normal framing by vectors of length 1/2 on
Tex syntax errordefines the standard embedding
Tex syntax error. Denote by the same symbol
Tex syntax errorthe restriction of
Tex syntax errorto for any .
Denote by the suspension of a space . Denote by the suspension homomorphism. Recall that is an isomorphism for . Let be any of the groups identified by the suspension isomorphism. Denote by the stable suspension homomorphisms, where is large.
4 Embedded connected sum
Suppose that , is a closed connected -manifold and, if is orientable, an orientation of is chosen. Let us define the embedded connected sum operation of on .
Represent isotopy classes and by embeddings and whose images are contained in disjoint balls. Join the images of by an arc whose interior misses the images. Let be the isotopy class of the embedded connected sum of and along this arc (compatible with the orientation, if is orientable), cf. [Haefliger1966, 1.3, 1.4].
This operation is well-defined, i.e. the isotopy class of the embedded connected sum depends only on the the isotopy classes and , and is independent of the choice of the path and of the representatives . 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 in [Skopenkov2015a, 3, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for a point]. The proof for arbitrary closed connected -manifold is analogous.
Moreover, for embedded connected sum defines a group structure on [Haefliger1966, 1.3-1.7], and an action of on .
5 Some remarks on codimension 2 embeddings
The case of embeddings of into 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 be a closed connected -manifold. Using embedded connected sum (4) we can apparently produce an overwhelming multitude of embeddings from the overwhelming multitude of embeddings . (However, note that for there are embeddings and such that is not isotopic to but is isotopic to [Viro1973, 22 in p. 181].) One can also apply Artin's spinning construction [Artin1928] for . Thus the description of 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 into are smoothly isotopic for every .
(b) Any two smooth embeddings of into are smoothly isotopic for every .
Part (a) is proved in [Milnor1965a, Proposition D in 9] as an easy corollary of the -cobordism theorem. Part (b) is proved in [Lucas&Saeki2002, 1, p. 447] as a corollary of generalizations [Kosinski1961], [Wall1965], [Goldstein1967], [Rubinstein1980], [Lucas&Neto&Saeki1996] of the following result.
Theorem 6.2 (Alexander torus theorem). [Alexander1924] Every embedding extends to an embedding either or .
Clearly, only the standard embedding extends to both.
Remark 6.3. (a) (on PL topological category) The analogue of Theorem 6.1.a holds
- for in the PL or topological category (Schöenfliess Theorem, 1912) [Rushing1973, 1.8].
- for in the PL category (Alexander Theorem, 1923) [Rushing1973, 1.8].
- for every in the topological locally flat category (Brown-Mazur-Moise Theorem, 1960) [Rushing1973, Generalized Schöenfliess Theorem 1.8.2].
The famous counterexample to the analogue of Theorem 6.1.a for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks whether any two PL embeddings of into are isotopic for every (this is equivalent to the description of ).
(b) (some speculations) If the extension of the Alexander torus theorem 6.2 respects isotopy, the extension would give a 1-1 correspondence between and the union of and with `base points' and identified (where is the isotopy class of the standard inclusion ). So the description of would be as hopeless as that of . Thus the description of for a sphere with handles is apparently hopeless.
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