Embeddings in Euclidean space: an introduction to their classification
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1 Introduction
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This page is intended not only for specialists in embeddings, but also for mathematicians from other areas who want to apply or to learn the theory of embeddings.
Unless otherwise indicated, the word `isotopy' means `ambient isotopy' on this page; see definition in [Skopenkov2016i].
Remark 1.1 (some motivations).
Three important classical problems in topology are the following, cf. [Zeeman1993, p. 3].
- Homeomorphism Problem: Classify -manifolds.
- Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
- Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below).
See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].
This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1).
After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.
We then establish notation and conventions.
We continue by introducing the connected sum operation for embeddings.
We then make some remarks on codimension 2 embeddings.
We conclude with a brief review of some important results about codimension 1 embeddings
The most interesting and very much studied case concerns embeddings (classical knots), or more generally,
codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 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 complete 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 the unknotting theorems in 2, the results on the pages listed in Remark 1.4
and in 6.
Their statements, although not the proofs, are simple and accessible to non-specialists.
This page and the pages listed in Remark 1.4 concern only such classification results.
As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for ;
- For a fixed , the more decreases from towards , the more complicated classification of embeddings of into becomes.
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 Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [Cencelj&Repovs&Skopenkov2004]).
`It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]
`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].
See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].
Let me justify leaving aside some classification of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below.
For this I need the informal concept of readily calculable (concrete) classification for results currently presented on `embedding' pages (and, for results on classification of manifolds, on Manifold Atlas) as dual to `abstract classification' or `reduction' as described by Wall.
An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers).
A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds.
In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a tool, necessarily more complicated and accessible to specialists.
For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1].
Here let illustrate the notion by describing some examples.
- (i) The Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of stable homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
- (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of readily calculable classification.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
Remark 1.3 (Embeddings into the sphere and Euclidean space).
(a) The embeddings given by and are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below.
(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.
The case is called a `stable range' (for the classification problem; for the existence problem there is analogous result with [Skopenkov2006, 2]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].
This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, 5]).
All the three assumptions in this result are indeed necessary:
This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].
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.
This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds (see 3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness is sufficient).
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
Note that if , then every closed -connected -manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
-
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
For a manifold 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 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 .
Denote by
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the standard embedding.
The natural normal framing by vectors of length 1/2 on
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defines the standard embedding
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.
Denote by the same symbol
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the restriction of
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to
for any
.
Denote by the suspension of a space .
Denote by and the suspension and the stable suspension homomorphisms.
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, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes 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], and an action of on .
-
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].)
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
-
The analogue of part (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 part (a) for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of into are isotopic for every
(this is equivalent to the description of ).
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between 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.
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
7 References
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- [Lucas&Neto&Saeki1996] L. A. Lucas, O. M. Neto and O. Saeki, A generalization of Alexander's torus theorem to higher dimensions and an unknotting theorem for embedded in , Kobe J. Math. 13 (1996), no.2, 145–165. MR1442202 (98e:57041) Zbl 876.57045
- [Lucas&Saeki2002] L. A. Lucas and O. Saeki, Embeddings of in , Pacific J. Math. 207 (2002), no.2, 447–462. MR1972255 (2004c:57045) Zbl 1058.57022
- [Murdoch1985] I. Murdoch, The Good Apprentice, 1985, Chatto & Windus, ISBN 0-670-80940-3.
- [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
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- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
- [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
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- [Smale1961] S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391–406. MR0137124 (25 #580) Zbl 0099.39202
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- [Stallings1963] J. Stallings, On topologically unknotted spheres., (1963). MRMR0149458 (26 #6946) Zbl 0121.18202
- [Viro1973] O. J. Viro, Local knotting of sub-manifolds, Mat. Sb. (N.S.) 90(132) (1973), 173–183, 325. MR0370606 (51 #6833)
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- [Wall1965] C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two., Proc. Camb. Philos. Soc. 61 (1965), 659-664. MR0184249 (32 #1722) Zbl 0135.41602
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
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- [Weiss] M. Weiss, private communication
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- [Wu1958] W. Wu, On the realization of complexes in euclidean spaces. I, Sci. Sinica 7 (1958), 251–297. MR0099026 (20 #5471) Zbl 0183.28302
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- [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
]{Skopenkov2016h}, only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
* (ii) The Haefliger-Weber Theorem \cite[\S5]{Skopenkov2006}
* (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) \cite{Weiss96}, \cite{Goodwillie&Weiss1999} so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings $S^1\to\Rr^n$ \cite{Weiss}.)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of ''readily calculable classification''.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
{{endthm}}
{{beginthm|Remark|(Embeddings into the sphere and Euclidean space)}}\label{spheu}
(a) The embeddings $f,g:S^1\to\Rr^2$ given by $f(x,y)=(x,y)$ and $g(x,y)=(x,-y)$ are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in [[Wikipedia:Winding_number#Turning_number|Wikipedia article on turning number]] can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of $S^1$ into $S^2$ are isotopic, see Theorem \ref{t:sch}.a and below.
(b) For $m\ge n+2$ the classifications of embeddings of compact $n$-manifolds into $S^m$ and into $\Rr^m$ are the same. More precisely, for all integers $m,n$ such that $m\ge n+2$, and for every $n$-manifold $N$, the map $i_* : E_{\Rr^m}(N)\to E_{S^m}(N)$ between the sets of isotopy classes of embeddings $N\to \Rr^m$ and $N\to S^m$,
which is induced by composition with the inclusion $i \colon \Rr^m \to S^m$, is a bijection.
Let us prove part (b). Since $n < m$, after a small isotopy an embedding $N \to S^m$ missed the point at infinity and so lies in $i(\Rr^m) \subset S^m$.
Hence $i_*$ is onto.
To prove that $i_*$ is injective, it suffices to show that if the compositions with the inclusion $i:\Rr^m\to S^m$ of two embeddings $f,f':N\to\Rr^m$ of a compact $n$-manifold $N$ are isotopic, then $f$ and $f'$ are isotopic. For showing that assume that $i\circ f$ and $i\circ f'$ are isotopic. Then by general position $f$ and $f'$ are [[Isotopy#Other equivalence relations|non-ambiently isotopic]]. Since every non-ambient isotopy extends to an isotopy \cite[Theorem 1.3]{Skopenkov2016i}, $f$ and $f'$ are isotopic.
{{endthm}}
{{beginthm|Remark|(References to information on the classification of embeddings)}}\label{s:list}
The first list is structured by the dimension of the source manifold and the target Euclidean space:
* [[Embeddings just below the stable range: classification]] \cite{Skopenkov2016e}
* [[3-manifolds in 6-space]] \cite{Skopenkov2016t}
* [[4-manifolds in 7-space]] \cite{Skopenkov2016f}
Information structured by the `complexity' of the source manifold:
* [[Knots, i.e. embeddings of spheres]]
* [[High codimension links]] \cite{Skopenkov2016h}
* [[Knotted tori]] \cite{Skopenkov2016k}
For more information see e.g. \cite{Skopenkov2006}.
{{endthm}}
==Unknotting theorems ==
; \label{s:ut}
If the category is omitted, then a result stated below holds in both the [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] and [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) category.
{{beginthm|General Position Theorem|(\cite[Theorem 3.5]{Hirsch1976}, \cite[Theorem 5.4]{Rourke&Sanderson1972})}}\label{th1}
For every compact $n$-manifold $N$ and $m\ge2n+2$, any two embeddings of $N$ into $\Rr^m$ are isotopic.
{{endthm}}
The case $m\ge2n+2$ is called a `stable range' (for the classification problem; for the existence problem there is analogous result with $m\ge2n+1$ \cite[$\S]{Skopenkov2006}).
The restriction $m\ge2n+2$ in Theorem \ref{th1} is sharp for non-connected manifolds, as the [[High_codimension_links#Examples|Hopf linking]] $S^n\sqcup S^n\to\Rr^{2n+1}$ shows \cite{Skopenkov2016h}, \cite[Figure 2.1.a]{Skopenkov2006}.
{{beginthm|Whitney-Wu Unknotting Theorem}}\label{th2}
For every compact connected $n$-manifold $N$, $n\ge2$ and $m\ge2n+1$, any two embeddings of $N$ into $\Rr^m$ are isotopic.
{{endthm}}
This is proved in \cite{Wu1958}, \cite{Wu1958a} and \cite{Wu1959} using ''the Whitney trick'' (see those references or \cite[$\S]{Rourke&Sanderson1972}).
All the three assumptions in this result are indeed necessary:
*the assumption $n\ge2$ because of the existence of non-trivial knots $S^1\to S^3$;
*the connectedness assumption because of the existence of [[High_codimension_links#Examples|the Hopf link]] \cite{Skopenkov2016h};
*the assumption $m\ge2n+1$ because of the [[Embeddings just below the stable range#Hudson_tori|example of Hudson tori]] \cite{Skopenkov2016e}.
{{beginthm|Unknotting Spheres Theorem}}\label{sph}
For $N=S^n$, or even for $N$ an integral homology $n$-sphere, $m\ge n+3$ or m\ge 3n+4$ in the PL or smooth category, respectively, any two embeddings of $N$ into $\Rr^m$ are isotopic.
{{endthm}}
This result is proved in \cite{Zeeman1960} or \cite{Haefliger1961} in the PL or smooth category, respectively. This result is also true for $m\ge n+3$ in the topological locally flat category \cite{Stallings1963}, \cite{Gluck1963}, \cite[Flattening Theorem 4.5.1]{Rushing1973}, \cite{Scharlemann1977}.
The case m\ge 3n+4$ is called a `metastable range' (for the classification problem; for the existence problem there are analogous results with m\ge3n+3$ \cite[$\S, $\S]{Skopenkov2006}).
Knots in codimension 2 and [[3-manifolds in 6-space#Examples|the Haefliger trefoil knot]] \cite[Example 2.1]{Skopenkov2016t} show that the dimension restrictions are sharp (even for $N=S^n$) in the Unknotting Spheres Theorem \ref{sph}.
Theorems \ref{th2} and \ref{sph} may be generalized as follows.
{{beginthm|The Haefliger-Zeeman Unknotting Theorem}}\label{haze}
For every $n\ge2k+2$, $m\ge2n-k+1$ and closed $k$-connected $n$-manifold $N$, any two embeddings of $N$ into $\Rr^m$ are isotopic.
{{endthm}}
This was proved in \cite{Penrose&Whitehead&Zeeman1961}, \cite{Haefliger1961}, \cite{Zeeman1962}, \cite{Irwin1965}, \cite{Hudson1969}. The proofs in \cite{Haefliger&Hirsch1963}, \cite{Vrabec1977}, \cite{Weber1967}, \cite[$\S]{Adachi1993} work for ''homologically'' $k$-connected manifolds (see $\S$\ref{s:nc} for the definition; the proofs are non-trivial but the generalization is trivial, basically because the $k$-connectedness was used to ensure high enough connectedness of the complement in $\Rr^m$ to the image of $N$, by Alexander duality and simple connectedness of the complement, so homological $k$-connectedness is sufficient).
Given Theorem \ref{haze} above, the case $m\ge2n-k+1$ can be called a `stable range for $k$-connected manifolds'.
Note that if $n\le2k+1$, then every closed $k$-connected $n$-manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem \ref{haze} in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem \ref{sph}.
For generalizations of the Haefliger-Zeeman Unknotting Theorem \ref{haze} [[Embeddings just below the stable range: classification#A generalization to highly-connected manifolds|see]] \cite[$\S]{Skopenkov2016e} or \cite{Hudson1967}, \cite{Hacon1968}, \cite{Hudson1972}, \cite{Gordon1972}, \cite{Kearton1979}. See also Theorem \ref{t:sch}.
== Notation and conventions ==
; \label{s:nc}
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark \ref{s:list}.
For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of
[[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] or [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) [[Embedding|embeddings]] $N\to S^m$ up to smooth or PL [[isotopy]].
If the category is omitted, then the result holds (or a definition or a construction is given) in both categories.
The sources of all embeddings are assumed to be compact.
Let $B^n$ be a closed $n$-ball in a closed connected $n$-manifold $N$. Denote $N_0:=Cl(N-B^n)$.
Let $\varepsilon(k):=1-(-1)^k$ be -manifolds.
Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below).
See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].
This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1).
After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.
We then establish notation and conventions.
We continue by introducing the connected sum operation for embeddings.
We then make some remarks on codimension 2 embeddings.
We conclude with a brief review of some important results about codimension 1 embeddings
The most interesting and very much studied case concerns embeddings (classical knots), or more generally,
codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 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 complete 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 the unknotting theorems in 2, the results on the pages listed in Remark 1.4
and in 6.
Their statements, although not the proofs, are simple and accessible to non-specialists.
This page and the pages listed in Remark 1.4 concern only such classification results.
As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for ;
- For a fixed , the more decreases from towards , the more complicated classification of embeddings of into becomes.
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 Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [Cencelj&Repovs&Skopenkov2004]).
`It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]
`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].
See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].
Let me justify leaving aside some classification of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below.
For this I need the informal concept of readily calculable (concrete) classification for results currently presented on `embedding' pages (and, for results on classification of manifolds, on Manifold Atlas) as dual to `abstract classification' or `reduction' as described by Wall.
An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers).
A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds.
In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a tool, necessarily more complicated and accessible to specialists.
For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1].
Here let illustrate the notion by describing some examples.
- (i) The Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of stable homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
- (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of readily calculable classification.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
Remark 1.3 (Embeddings into the sphere and Euclidean space).
(a) The embeddings given by and are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below.
(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.
The case is called a `stable range' (for the classification problem; for the existence problem there is analogous result with [Skopenkov2006, 2]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].
This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, 5]).
All the three assumptions in this result are indeed necessary:
This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].
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.
This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds (see 3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness is sufficient).
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
Note that if , then every closed -connected -manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
-
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
For a manifold 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 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 .
Denote by
Tex syntax error
the standard embedding.
The natural normal framing by vectors of length 1/2 on
Tex syntax error
defines the standard embedding
Tex syntax error
.
Denote by the same symbol
Tex syntax error
the restriction of
Tex syntax error
to
for any
.
Denote by the suspension of a space .
Denote by and the suspension and the stable suspension homomorphisms.
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, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes 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], and an action of on .
-
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].)
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
-
The analogue of part (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 part (a) for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of into are isotopic for every
(this is equivalent to the description of ).
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between 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.
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
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- [Weiss96] M. Weiss, Calculus of Embeddings, Bull. Amer. Math. Soc. 33 (1996), 177-187.
- [Wu1958] W. Wu, On the realization of complexes in euclidean spaces. I, Sci. Sinica 7 (1958), 251–297. MR0099026 (20 #5471) Zbl 0183.28302
- [Wu1958a] W. Wu, On the realization of complexes in euclidean spaces. II, Sci. Sinica 7 (1958), 365–387.
- [Wu1959] W. Wu, On the realization of complexes in euclidean spaces. III, Sci. Sinica 8 (1959), 133–150. MR0098365 (20 #4825b) Zbl 0207.53104
- [Zeeman1960] E. Zeeman, Unknotting spheres in five dimensions, Bull. Amer. Math. Soc. 66 (1960) 198. MR0117737 (22 #8512a) Zbl 0117.40904
- [Zeeman1962] E. C. Zeeman, Isotopies and knots in manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall (1962), 187–193. MR0140097 (25 #3520) Zbl 1246.57069
- [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
$ for $k$ even and $ for $k$ odd, so that $\Zz_{\varepsilon(k)}$ is $\Zz$ for $k$ even and $\Zz_2$ for $k$ odd.
Denote by $V_{m,n}$ the Stiefel manifold of orthonormal $n$-frames in $\Rr^m$.
We omit $\Zz$-coefficients from the notation of (co)homology groups.
For a manifold $P$ with boundary $\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$.
A closed manifold $N$ is called ''homologically $k$-connected'', if $N$ is connected and $H_i(N)=0$ for every $i=1,\dots,k$.
This condition is equivalent to $\tilde H_i(N)=0$ for each $i=0,\dots,k$, where $\tilde H_i$ are reduced homology groups.
A pair $(N,\partial N)$ is called ''homologically $k$-connected'', if $H_i(N,\partial)=0$ for every $i=0,\dots,k$.
The ''self-intersection set'' of a map $f:X\to Y$ is $\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$
For a smooth embedding $f:N\to\Rr^m$ denote by
* $C_f$ the closure of the complement in $S^m\supset\Rr^m$ to a tight enough tubular neighborhood of $f(N)$ and
* $\nu_f:\partial C_f\to N$ the restriction of the linear normal bundle of $f$ to the subspace of unit length vectors identified with $\partial C_f$.
Denote by ${\rm i}:S^q\to S^m$ the standard embedding.
The natural normal framing by vectors of length 1/2 on ${\rm i}$ defines the standard embedding ${\rm i}_{m,q}:D^{m-q}\times S^q\to S^m$.
Denote by the same symbol ${\rm i}_{m,q}$ the restriction of ${\rm i}_{m,q}$ to $S^p\times S^q$ for any $p\le m-q-1$.
Denote by $\Sigma X$ the suspension of a space $X$.
Denote by $\Sigma:\pi_q(S^n)\to \pi_{q+1}(S^{n+1})$ and $\Sigma:\pi_q(S^n)\to \pi_{q-n}^S$ the suspension and the stable suspension homomorphisms.
== Embedded connected sum ==
; \label{s:sum}
Suppose that $m\ge n+2$, $N$ is a closed connected $n$-manifold and, if $N$ is orientable, an orientation of $N$ is chosen.
Let us define the ''embedded connected sum operation'' $\#$ of $E^m(S^n)$ on $E^m(N)$.
Represent isotopy classes $[f]\in E^m(N)$ and $[g]\in E^m(S^n)$ by embeddings $f:N\to\Rr^m$ and $g:S^n\to\Rr^m$ whose images are contained in disjoint balls.
Join the images of $f,g$ by an arc whose interior misses the images.
Let $[f]\#[g]$ be the isotopy class of the ''embedded connected sum'' of $f$ and $g$ along this arc (compatible with the orientation, if $N$ is orientable), cf. \cite[Theorem 1.7]{Haefliger1966}, \cite[$\Sn-manifolds.
Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below).
See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].
This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1).
After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.
We then establish notation and conventions.
We continue by introducing the connected sum operation for embeddings.
We then make some remarks on codimension 2 embeddings.
We conclude with a brief review of some important results about codimension 1 embeddings
The most interesting and very much studied case concerns embeddings (classical knots), or more generally,
codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 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 complete 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 the unknotting theorems in 2, the results on the pages listed in Remark 1.4
and in 6.
Their statements, although not the proofs, are simple and accessible to non-specialists.
This page and the pages listed in Remark 1.4 concern only such classification results.
As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for ;
- For a fixed , the more decreases from towards , the more complicated classification of embeddings of into becomes.
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 Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [Cencelj&Repovs&Skopenkov2004]).
`It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]
`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].
See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].
Let me justify leaving aside some classification of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below.
For this I need the informal concept of readily calculable (concrete) classification for results currently presented on `embedding' pages (and, for results on classification of manifolds, on Manifold Atlas) as dual to `abstract classification' or `reduction' as described by Wall.
An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers).
A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds.
In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a tool, necessarily more complicated and accessible to specialists.
For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1].
Here let illustrate the notion by describing some examples.
- (i) The Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of stable homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
- (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of readily calculable classification.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
Remark 1.3 (Embeddings into the sphere and Euclidean space).
(a) The embeddings given by and are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below.
(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.
The case is called a `stable range' (for the classification problem; for the existence problem there is analogous result with [Skopenkov2006, 2]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].
This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, 5]).
All the three assumptions in this result are indeed necessary:
This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].
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.
This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds (see 3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness is sufficient).
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
Note that if , then every closed -connected -manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
-
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
For a manifold 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 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 .
Denote by
Tex syntax error
the standard embedding.
The natural normal framing by vectors of length 1/2 on
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defines the standard embedding
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.
Denote by the same symbol
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the restriction of
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to
for any
.
Denote by the suspension of a space .
Denote by and the suspension and the stable suspension homomorphisms.
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, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes 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], and an action of on .
-
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].)
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
-
The analogue of part (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 part (a) for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of into are isotopic for every
(this is equivalent to the description of ).
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between 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.
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
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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
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- [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
]{Avvakumov2016}.
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes $[f]$ and $[g]$, and is independent of the choice of the path and of the representatives $f,g$.
The proof of this fact is based on the construction of embedded connected sum of isotopies.
Although this fact is presumably classical, a proof was not written, cf. \cite[Remark 2.3.a]{Skopenkov2015a}.
The proof is written for $N=S^n$ in \cite[$\S, proof of the Standardization Lemma 2.1.b and beginning of proof of the Group structure Lemma 2.2 for $X=D^0_+$ a point]{Skopenkov2015a}.
The proof for arbitrary closed connected $n$-manifold $N$ is analogous.
Moreover, for $m\ge n+3$ embedded connected sum defines a group structure on $E^m(S^n)$ \cite{Haefliger1966}, and an action $\#$ of $E^m(S^n)$ on $E^m(N)$.
== Some remarks on codimension 2 embeddings ==
; \label{s:c2}
The case of embeddings of $S^n$ into $\Rr^{n+2}$ is the [[Knots,_i.e._embeddings_of_spheres#Codimension_2_knots|most extensively studied]] case of the Knotting Problem.
In this case there is an overwhelming multitude of isotopy classes of embeddings.
We present some speculations as to why the classification in codimension 2 should not be accessible for manifolds, not only for spheres.
Let $N$ be a closed connected $n$-manifold.
Using [[#Embedded_connected_sum|embedded connected sum]] ($\S$\ref{s:sum}) we can apparently produce an overwhelming multitude of embeddings $N\to\Rr^{n+2}$ from the overwhelming multitude of embeddings $S^n\to\Rr^{n+2}$.
(However, note that for $n=2$ there are embeddings $f:\Rr P^2\to S^4$ and $g_1,g_2:S^2\to S^4$ such that $g_1$ is not isotopic to $g_2$ but $f\#g_1$ is isotopic to $f\#g_2$ \cite{Viro1973}.)
One can also apply Artin's spinning construction \cite{Artin1928} $E^m(N)\to E^{m+1}(S^1\times N)$ for $m=n+2$.
Thus the description of $E^{n+2}(N)$ is a very hard open problem.
It would be interesting to give a more formal (e.g. algorithmic)
illustration of the hardness of this problem.
For studies of codimension 2 embeddings of manifolds up to the weaker relation of [[Isotopy|''concordance'']] see e.g. \cite{Cappell&Shaneson1974}.
== Codimension 1 embeddings ==
; \label{s:c1}
{{beginthm|Theorem}}\label{t:sch}
(a) Any two smooth embeddings of $S^n$ into $S^{n+1}$ are smoothly isotopic for every $n\ne3$ \cite{Smale1961}, \cite{Smale1962a}, \cite{Barden1965}.
(b) Any two smooth embeddings of $S^p\times S^{n-p}$ into $S^{n+1}$ are smoothly isotopic for every \le p\le n-p$ \cite{Kosinski1961}, \cite{Wall1965}, \cite{Lucas&Neto&Saeki1996}, cf. \cite{Goldstein1967}.
{{endthm}}
The analogue of part (a) holds
* for $n=1$ in the PL or topological category (Schöenfliess Theorem, 1912) \cite[$\Sn-manifolds.
Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below).
See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].
This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1).
After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.
We then establish notation and conventions.
We continue by introducing the connected sum operation for embeddings.
We then make some remarks on codimension 2 embeddings.
We conclude with a brief review of some important results about codimension 1 embeddings
The most interesting and very much studied case concerns embeddings (classical knots), or more generally,
codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 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 complete 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 the unknotting theorems in 2, the results on the pages listed in Remark 1.4
and in 6.
Their statements, although not the proofs, are simple and accessible to non-specialists.
This page and the pages listed in Remark 1.4 concern only such classification results.
As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for ;
- For a fixed , the more decreases from towards , the more complicated classification of embeddings of into becomes.
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 Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [Cencelj&Repovs&Skopenkov2004]).
`It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]
`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].
See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].
Let me justify leaving aside some classification of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below.
For this I need the informal concept of readily calculable (concrete) classification for results currently presented on `embedding' pages (and, for results on classification of manifolds, on Manifold Atlas) as dual to `abstract classification' or `reduction' as described by Wall.
An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers).
A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds.
In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a tool, necessarily more complicated and accessible to specialists.
For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1].
Here let illustrate the notion by describing some examples.
- (i) The Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of stable homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
- (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of readily calculable classification.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
Remark 1.3 (Embeddings into the sphere and Euclidean space).
(a) The embeddings given by and are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below.
(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.
The case is called a `stable range' (for the classification problem; for the existence problem there is analogous result with [Skopenkov2006, 2]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].
This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, 5]).
All the three assumptions in this result are indeed necessary:
This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].
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.
This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds (see 3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness is sufficient).
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
Note that if , then every closed -connected -manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
-
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
For a manifold 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 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 .
Denote by Tex syntax error
the standard embedding.
The natural normal framing by vectors of length 1/2 on Tex syntax error
defines the standard embedding Tex syntax error
.
Denote by the same symbol Tex syntax error
the restriction of Tex syntax error
to for any .
Denote by the suspension of a space .
Denote by and the suspension and the stable suspension homomorphisms.
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, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes 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], and an action of on .
-
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].)
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
-
The analogue of part (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 part (a) for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of into are isotopic for every
(this is equivalent to the description of ).
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between 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.
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
7 References
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.8]{Rushing1973}.
* for $n=2$ in the PL category (Alexander Theorem, 1923) \cite[$\Sn-manifolds.
Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below).
See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].
This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1).
After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.
We then establish notation and conventions.
We continue by introducing the connected sum operation for embeddings.
We then make some remarks on codimension 2 embeddings.
We conclude with a brief review of some important results about codimension 1 embeddings
The most interesting and very much studied case concerns embeddings (classical knots), or more generally,
codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 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 complete 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 the unknotting theorems in 2, the results on the pages listed in Remark 1.4
and in 6.
Their statements, although not the proofs, are simple and accessible to non-specialists.
This page and the pages listed in Remark 1.4 concern only such classification results.
As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for ;
- For a fixed , the more decreases from towards , the more complicated classification of embeddings of into becomes.
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 Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [Cencelj&Repovs&Skopenkov2004]).
`It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]
`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].
See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].
Let me justify leaving aside some classification of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below.
For this I need the informal concept of readily calculable (concrete) classification for results currently presented on `embedding' pages (and, for results on classification of manifolds, on Manifold Atlas) as dual to `abstract classification' or `reduction' as described by Wall.
An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers).
A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds.
In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a tool, necessarily more complicated and accessible to specialists.
For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1].
Here let illustrate the notion by describing some examples.
- (i) The Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of stable homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
- (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of readily calculable classification.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
Remark 1.3 (Embeddings into the sphere and Euclidean space).
(a) The embeddings given by and are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below.
(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.
The case is called a `stable range' (for the classification problem; for the existence problem there is analogous result with [Skopenkov2006, 2]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].
This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, 5]).
All the three assumptions in this result are indeed necessary:
This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].
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.
This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds (see 3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness is sufficient).
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
Note that if , then every closed -connected -manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
-
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
For a manifold 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 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 .
Denote by Tex syntax error
the standard embedding.
The natural normal framing by vectors of length 1/2 on Tex syntax error
defines the standard embedding Tex syntax error
.
Denote by the same symbol Tex syntax error
the restriction of Tex syntax error
to for any .
Denote by the suspension of a space .
Denote by and the suspension and the stable suspension homomorphisms.
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, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes 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], and an action of on .
-
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].)
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
-
The analogue of part (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 part (a) for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of into are isotopic for every
(this is equivalent to the description of ).
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between 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.
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
7 References
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.8]{Rushing1973}.
* for every $n$ in the topological [[Wikipedia:Local flatness|locally flat]] category (Brown-Mazur-Moise Theorem, 1960) \cite[Generalized Schöenfliess Theorem 1.8.2]{Rushing1973}.
The famous counterexample to the analogue of part (a) for $n=2$ in the topological category is [[Wikipedia:Alexander_horned_sphere|the Alexander horned sphere]], see the corresponding [[Wikipedia:Alexander_horned_sphere|Wikipedia article]]. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of $S^n$ into $S^{n+1}$ are isotopic for every $n\ge3$
(this is equivalent to the description of $E^{n+1}_{PL}(S^n)$).
Every embedding $S^1\times S^1\to S^3$ extends to an embedding either $D^2\times S^1\to S^3$ or $S^1\times D^2\to S^3$ \cite{Alexander1924}. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between $E^3(S^1\times S^1)$ and the union of $E^3(S^1)\times\Zz$ and $\Zz\times E^3(S^1)$ with `base points' $i\times0$ and -manifolds.
Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below).
See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].
This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1).
After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.
We then establish notation and conventions.
We continue by introducing the connected sum operation for embeddings.
We then make some remarks on codimension 2 embeddings.
We conclude with a brief review of some important results about codimension 1 embeddings
The most interesting and very much studied case concerns embeddings (classical knots), or more generally,
codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 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 complete 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 the unknotting theorems in 2, the results on the pages listed in Remark 1.4
and in 6.
Their statements, although not the proofs, are simple and accessible to non-specialists.
This page and the pages listed in Remark 1.4 concern only such classification results.
As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for ;
- For a fixed , the more decreases from towards , the more complicated classification of embeddings of into becomes.
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 Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [Cencelj&Repovs&Skopenkov2004]).
`It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]
`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].
See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].
Let me justify leaving aside some classification of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below.
For this I need the informal concept of readily calculable (concrete) classification for results currently presented on `embedding' pages (and, for results on classification of manifolds, on Manifold Atlas) as dual to `abstract classification' or `reduction' as described by Wall.
An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers).
A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds.
In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a tool, necessarily more complicated and accessible to specialists.
For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1].
Here let illustrate the notion by describing some examples.
- (i) The Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of stable homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
- (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of readily calculable classification.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
Remark 1.3 (Embeddings into the sphere and Euclidean space).
(a) The embeddings given by and are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below.
(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.
The case is called a `stable range' (for the classification problem; for the existence problem there is analogous result with [Skopenkov2006, 2]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].
This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, 5]).
All the three assumptions in this result are indeed necessary:
This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].
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.
This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds (see 3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness is sufficient).
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
Note that if , then every closed -connected -manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
-
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
For a manifold 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 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 .
Denote by Tex syntax error
the standard embedding.
The natural normal framing by vectors of length 1/2 on Tex syntax error
defines the standard embedding Tex syntax error
.
Denote by the same symbol Tex syntax error
the restriction of Tex syntax error
to for any .
Denote by the suspension of a space .
Denote by and the suspension and the stable suspension homomorphisms.
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, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes 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], and an action of on .
-
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].)
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
-
The analogue of part (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 part (a) for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of into are isotopic for every
(this is equivalent to the description of ).
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between 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.
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
7 References
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- [Alexander1924] J. W. Alexander, On the subdivision of 3-space by polyhedron, Proc. Nat. Acad. Sci. USA, 10, (1924) 6–8. Zbl 50.0659.01
- [Artin1928] E. Artin, Zur Isotopie zwei-dimensionaler Flaechen im , Abh. Math. Sem. Hamburg Univ. 4 (1928) 174–177.
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- [Cencelj&Repovs&Skopenkov2004] M. Cencelj, D. Repov\v s and A. Skopenkov, On the Browder-Levine-Novikov embedding theorems, Proc. of the Steklov Math. Inst. 247 (2004) 280-290.
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- [Gordon1972] C. Gordon, Embedding piecewise linear manifolds with boundary., Proc. Camb. Philos. Soc. 72 (1972), 21-25. MR0295359 (45 #4425) Zbl 0236.57009
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- [Haefliger&Hirsch1963] A. Haefliger and M. W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. see also MR0149494 (26 #6981) Zbl 0113.38607
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- [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
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- [Lucas&Neto&Saeki1996] L. A. Lucas, O. M. Neto and O. Saeki, A generalization of Alexander's torus theorem to higher dimensions and an unknotting theorem for embedded in , Kobe J. Math. 13 (1996), no.2, 145–165. MR1442202 (98e:57041) Zbl 876.57045
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- [Rushing1973] T. Rushing, Topological embeddings., Pure and Applied Mathematics, 52. New York-London: Academic Press. XIII, 1973. MRMR0348752 (50 #1247) Zbl 0295.57003
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- [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
- [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
- [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
- [Skopenkov2016i] A. Skopenkov, Isotopy, submitted to Bull. Man. Atl.
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- [Viro1973] O. J. Viro, Local knotting of sub-manifolds, Mat. Sb. (N.S.) 90(132) (1973), 173–183, 325. MR0370606 (51 #6833)
- [Vrabec1977] J. Vrabec, Knotting a -connected closed -manifold in , Trans. Amer. Math. Soc. 233 (1977), 137–165. MR0645405 (58 #31097) Zbl 386.57013
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\times i$ identified (where $i$ is the isotopy class of the standard inclusion $S^1\to\Rr^3$). So the description of $E^3(S^1\times S^1)$ would be as hopeless as that of $E^3(S^1)$. Thus the description of $E^3(N)$ for $N$ a [[2-manifolds#Orientable_surfaces|sphere with handles]] is apparently hopeless.
For more on higher-dimensional codimension 1 embeddings see e.g. \cite{Lucas&Saeki2002}.
== References ==
{{#RefList:}}
[[Category:Theory]]
[[Category:Manifolds]]
[[Category:Embeddings of manifolds]]n-manifolds.
Embedding Problem: Find the least dimension such that given space admits an embedding into -dimensional Euclidean space .
Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
The Embedding and Knotting Problems have played an outstanding role in the development of topology. Various methods for the investigation of these problems were created by such classical figures as G. Alexander, H. Hopf, E. van Kampen, K. Kuratowski, S. MacLane, L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Zeeman, W. Browder, J. Levine, S. P. Novikov, A. Haefliger, M. Hirsch, J. F. P. Hudson, M. Irwin and others.
The Knotting Problem is related to other branches of mathematics, most importantly, to algebraic topology (see Remark 1.4 below).
See also Wikipedia article on knot theory and [Skopenkov2016t, Example 2.3].
This article gives a short guide to the problem of classifying, up to isotopy, embeddings of closed manifolds into Euclidean space or into spheres (i.e., to the Knotting Problem of Remark 1.1).
After making general remarks and giving some references we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of are isotopic.
We then establish notation and conventions.
We continue by introducing the connected sum operation for embeddings.
We then make some remarks on codimension 2 embeddings.
We conclude with a brief review of some important results about codimension 1 embeddings
The most interesting and very much studied case concerns embeddings (classical knots), or more generally,
codimension 2 embeddings of spheres. Although there have been great results in the last 100 years, these results were not directly aiming at a complete classification which remains widely open. Almost nothing is said here about this, but see Wikipedia article on knot theory and 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 complete 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 the unknotting theorems in 2, the results on the pages listed in Remark 1.4
and in 6.
Their statements, although not the proofs, are simple and accessible to non-specialists.
This page and the pages listed in Remark 1.4 concern only such classification results.
As a consequence, we leave aside a large body of work, especially but not only in codimension 2.
The results and remarks given below show the following:
- The complete readily calculable classification of embeddings into of closed connected -manifolds is non-trivial but presently accessible only for or for ;
- For a fixed , the more decreases from towards , the more complicated classification of embeddings of into becomes.
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 Browder-Novikov-Wall theorem of 1960s classifies higher-dimensional manifolds. (The statement in terms of Poincar\'e complexes is given e.g. in [Wall1999].) Analogously, the Browder-Casson-Haefliger-Sullivan-Wall theorem of 1960s classifies embeddings of manifolds in codimension greater than 2. (The statement in terms of Poincar\'e complexes is given in [Wall1999, Corollaries 11.3.1 and 11.3.3], see a simpler exposition in [Cencelj&Repovs&Skopenkov2004]).
`It is misleading to regard this as a complete solution to the problem of embeddings: the problems raised seem to the author in some respects to be harder than the original geometrical problems.' [Wall1999, p. 119]
`The modern world is full of theories which are proliferating at a wrong level of generality, we're so good at theorizing, and one theory spawns another, there's a whole industry of abstract activity which people mistake for thinking.' [Murdoch1985].
See discusion of similar issues in [Graham&Knuth&Patashnik89, Preface].
Let me justify leaving aside some classification of embeddings (including classification via Poincar\'e embeddings and (ii), (iii) below.
For this I need the informal concept of readily calculable (concrete) classification for results currently presented on `embedding' pages (and, for results on classification of manifolds, on Manifold Atlas) as dual to `abstract classification' or `reduction' as described by Wall.
An `explicit' classification of embeddings for a particular manifold is a clasification in terms of integers (in particular, in terms of finitely generated abelian groups and other objects `easily' characterized by integers).
A readily calculable classification of embeddings for general manifolds is a classification which `easily' (`readily') admits classifications of embeddings for some particular manifolds, and which `cannot' be recovered by `easier' means for those particular manifolds.
In other words, a readily calculable classification is more like a `final result' accessible to non-specialists, while an abstract classification is more like a tool, necessarily more complicated and accessible to specialists.
For more specific (and so less perfect) description of this notion see [Skopenkov2005, footnote 1].
Here let illustrate the notion by describing some examples.
- (i) The Haefliger-Zeeman Theorem [Skopenkov2016h, Theorem 4.1] classifies links in terms of stable homotopy groups of spheres. These groups are not known in general. However, these groups are described for many particular cases which give the table [Skopenkov2016h, 1], only the left two columns of the table are recovered by easier proofs, and stable homotopy groups of spheres is a standard object of mathematics. Thus I consider the Haefliger-Zeeman classification of links to be a readily calculable classification.
- (iii) An interesting approach of Goodwille-Weiss (calculus of embeddings) [Weiss96], [Goodwillie&Weiss1999] so far did not give any new complete readily calculable classification results. (However, it gives a modern abstract proof of certain earlier known results; it also gives explicit results on higher homotopy groups of the space of embeddings [Weiss].)
Mathematicians often discuss what is more useful answer, what is less useful, and in which sense. The above does not intend to give a full description to informal concept of readily calculable classification.
It is rather an invitation to further pursue the line of thought began by Wall, Murdoch, Graham, Knuth, Patashnik and others.
Remark 1.3 (Embeddings into the sphere and Euclidean space).
(a) The embeddings given by and are not isotopic
(because they have distinct turning numbers; readers not familiar with turning number as defined in Wikipedia article on turning number can accept this intuitively clear assertion without proof). On the other hand, any two embeddings of into are isotopic, see Theorem 6.1.a and below.
(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.
The case is called a `stable range' (for the classification problem; for the existence problem there is analogous result with [Skopenkov2006, 2]).
The restriction in Theorem 2.1 is sharp for non-connected manifolds, as the Hopf linking shows [Skopenkov2016h], [Skopenkov2006, Figure 2.1.a].
This is proved in [Wu1958], [Wu1958a] and [Wu1959] using the Whitney trick (see those references or [Rourke&Sanderson1972, 5]).
All the three assumptions in this result are indeed necessary:
This result is proved in [Zeeman1960] or [Haefliger1961] in the PL or smooth category, respectively. This result is also true for in the topological locally flat category [Stallings1963], [Gluck1963], [Rushing1973, Flattening Theorem 4.5.1], [Scharlemann1977].
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.
This was proved in [Penrose&Whitehead&Zeeman1961], [Haefliger1961], [Zeeman1962], [Irwin1965], [Hudson1969]. The proofs in [Haefliger&Hirsch1963], [Vrabec1977], [Weber1967], [Adachi1993, 7] work for homologically -connected manifolds (see 3 for the definition; the proofs are non-trivial but the generalization is trivial, basically because the -connectedness was used to ensure high enough connectedness of the complement in to the image of , by Alexander duality and simple connectedness of the complement, so homological -connectedness is sufficient).
Given Theorem 2.4 above, the case can be called a `stable range for -connected manifolds'.
Note that if , then every closed -connected -manifold is a sphere, so the analogue of the Haefliger-Zeeman Unknotting Theorem 2.4 in the smooth category is wrong, and in the PL category gives nothing more than the Unknotting Spheres Theorem 2.3.
For generalizations of the Haefliger-Zeeman Unknotting Theorem 2.4 see [Skopenkov2016e, 5] or [Hudson1967], [Hacon1968], [Hudson1972], [Gordon1972], [Kearton1979]. See also Theorem 6.1.
3 Notation and conventions
-
The following notations and conventions will be used below in this page and also in some other pages about embeddings, including those listed in Remark 1.4.
For a manifold 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 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 .
Denote by Tex syntax error
the standard embedding.
The natural normal framing by vectors of length 1/2 on Tex syntax error
defines the standard embedding Tex syntax error
.
Denote by the same symbol Tex syntax error
the restriction of Tex syntax error
to for any .
Denote by the suspension of a space .
Denote by and the suspension and the stable suspension homomorphisms.
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, Theorem 1.7], [Avvakumov2016, 1].
This operation is well-defined,
i.e. the isotopy class of class of the embedded connected sum depends only on the the isotopy classes 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], and an action of on .
-
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].)
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
-
The analogue of part (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 part (a) for in the topological category is the Alexander horned sphere, see the corresponding Wikipedia article. The well-known very hard Schöenfliess Problem asks if each two PL embeddings of into are isotopic for every
(this is equivalent to the description of ).
Every embedding extends to an embedding either or [Alexander1924]. Clearly, only the standard embedding extends to both.
If it could be proven that this extension respects isotopy, this would give a 1-1 correspondence between 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.
For more on higher-dimensional codimension 1 embeddings see e.g. [Lucas&Saeki2002].
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