When is the set of embeddings finite up to isotopy?
Abstract
Given a manifold and a number , we study the following question: is the set of isotopy classes of embeddings finite? In case when the manifold is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when is a product of two spheres. In the following theorem, is a specific set depending only on the parity of and which is defined in the paper.
Theorem. Assume that and . Then the set of isotopy classes of smooth embeddings is infinite if and only if either or is divisible by , or there exists a point in the set such that .
Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings to the classification of embeddings and . The latter classification problems are reduced to homotopy ones, which are solved rationally.
Keywords: smooth manifold, embedding, isotopy, knotted torus, surgery, knot.
2000 MSC: 57R52, 57R40; 57R65.
The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 20152016 (grant No 15010092) and supported within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program. During the work on this paper the author received support also from “Dynasty” foundation and from the Simons–IUM fellowship.
1 Introduction
This paper is on the classification of embeddings of higherdimensional manifolds, see [Sko07L] for a recent survey. This generalizes the subject of classical knot theory. In general one can hope only to reduce the isotopy classification problem to problems of homotopy theory [Hae66A, Hae66C, Kos90, Le65]. Sometimes the latter can be solved but finding explicit classification is hard.
Given a manifold and a number , we study the following simpler question: is the set of isotopy classes of embeddings finite? This question is motivated by analogy to rational homotopy theory founded by J.P. Serre, D. Sullivan and D. Quillen [GM81] and rational classification of link maps by U. Koschorke [Kos90, HaKa98]. We give answers for simplest manifolds : spheres, disjoint unions of spheres (known before) and products of two spheres (new). Our main result (Theorem 1.6 below) is an exact sequence, which in some sense reduces the classification of embeddings to the classification of embeddings and . This provides much information about the set of isotopy classes of embeddings including a finiteness criterion (Theorem 1.4 below). Some results for general manifolds are available only in socalled metastable dimension [Sko07L, Kl05]. Throughout the paper we work in smooth category.
This paper concludes the series of papers [CRS07, CRS08, CFS11]. It is independent of previous ones in the sense that it uses statements from [CFS11] but neither definitions nor methodology from any of them.
Knots and links
For knots in codimension at least (i.e., ) the answer to the posed question is given by A. Haefliger:
Theorem 1.1.
[Hae66A, Corollary 6.7] Assume that . Then the set of smooth isotopy classes of smooth embeddings is infinite if and only if and is divisible by .
The classification of (partially) framed knots is closely related to the classification of knots.
Theorem 1.2.
[CFS11, Corollary 1.14] Assume that , . Then the set of smooth isotopy classes of smooth embeddings is infinite if and only if one of the following conditions holds:

and ;

and ;

and .
The classification of links is the next natural problem after the classification of knots. For there is an explicit description of the isotopy classes of links “modulo” knots and in terms of homotopy groups of spheres and Stiefel manifolds; see [Hae66C, Theorem 10.7], [Sko08P, Theorem 1.1], [Ne82]. In codimension at least there is an exact sequence involving the set of isotopy classes of links and certain homotopy groups [Hae66C, Theorem 1.3]. This sequence allows to obtain the following finiteness criterion by D. Crowley, S. Ferry and the author. The criterion involves certain finitenesschecking sets which depend only on the parity of , and which are defined in Table 1 below. A part of each set is drawn in the table; the rest of the set is obtained by obvious periodicity.
Theorem 1.3.
[CFS11, Theorem 1.5] Assume that . Then the set of smooth isotopy classes of smooth embeddings , whose components are unknotted, is infinite if and only if there exists a point such that .
is the set of pairs such that and at least one of the following conditions holds —  
for even:  for odd, even:  for odd: 



For even, odd the set is obtained from by the reflection with respect to the line . 
Knotted tori
A natural next step (after link theory and the classification of embeddings of highlyconnected manifolds) towards classification of embeddings of arbitrary manifolds is the classification of knotted tori, i.e., embeddings . The classification of knotted tori gives some insight or even precise information concerning arbitrary manifolds [Sko07F]; see also Theorem LABEL:thconsum below. Many interesting examples of embeddings are knotted tori [Hud63, MiRe71, Sko02, Sko10].
There was known an explicit description of the set of isotopy classes of knotted tori “modulo” knots in the metastable dimension , , in terms of homotopy groups of Stiefel manifolds [Sko02, Sko08]. If is a closed connected manifold then until recent results [CrSk08, Sko08Z, Sko08] no complete readily calculable descriptions of isotopy classes below the metastable dimension was known, in spite of the existence of interesting approaches of Browder–Wall and Goodwillie–Weiss [Wa65, GW99, CRS04].
The main “practical” result of the paper is an explicit criterion for the finiteness of the set of knotted tori up to isotopy below the metastable dimension:
Theorem 1.4.
Assume that and . Then the set of isotopy classes of smooth embeddings is infinite if and only if at least one of the following conditions holds:

or is divisible by ,

there exists a point such that .
Example 1.5.
[CRS08, Example 1] The set of knotted tori is finite up to isotopy.
In Theorem 1.4 the inequality is assumed by aesthetic reasons — to reduce the number of cases and thus to simplify the statement and the proof. The classification of knotted tori for is easier and is given by [Sko02, Corollary 1.5], [Sko08, Theorem 1.2], [CFS11, Lemma 1.12].
Relationship between framed knots, links and knotted tori
The main result of the paper is an exact sequence (Theorem 1.6 below), which in some sense reduces the classification of knotted tori to the classification of links and framed knots.
Let us introduce some notation and conventions. For a smooth manifold denote by the set of smooth isotopy classes of smooth embeddings . The letter “” in the notation comes from the word “embedding”. For the sets , , and are finitely generated Abelian groups with respect to “connected sum”, “framed connected sum”, and “componentwise connected sum” operation, respectively [Hae66A, Hae66C]. Denote by the subgroup of formed by all the embeddings whose second component (i.e., restriction to the sphere ) is unknotted. For the “parametric connected sum” operation gives a natural Abelian group structure on the set ; see Figure 1, Section 2, and paper [Sko15, §2.1] for details.
Theorem 1.6.
For each there is an exact sequence of finitely generated Abelian groups
For this sequence is isomorphic to the middle horizontal sequence in [Sko08, Restriction Lemma 5.2], while for general our exact sequence can be called a “desuspension” of that one; see Remark LABEL:remdesuspension below for details.
As a nontrivial corollary, we get the following formula for the rank of the group :
Corollary 1.7.
Assume that and . Then
Notice that the ranks of the groups in the righthand side are known [CFS11, Theorem 1.7 and Lemma 1.12].
Organization of the paper
In Section 2 we introduce some notation and recall some required known results. In Section 3 we prove Theorem 1.6. In Section LABEL:sect3 we deduce Theorem 1.4 from Theorem 1.6 and give an easy application (Theorem 4.1) of our approach.
The reader who wants to get a nontrivial result in a minimal time may read only subsections “Group Structure”, “Definition of ”, “Exactness at ”, and then immediately get the “only if” part of Theorem 1.4 from Theorems 1.1–1.3.
Most of the ideas of the paper can be understood from the lowdimensional examples shown in figures. Notice that the proofs may not be literally correct for the shown low dimensions. In the figures instead of the spheres we always show their images under an appropriate stereographic projection.
2 Preliminaries
Group structure
First we define of a group structure on the set of knotted tori by A. Skopenkov [Sko15, §2.1].
Let be the coordinates in space . For each identify space with the subspace of given by the equations . Denote by the reflection in the hyperplane given by the equation . Denote by and the halfspheres of the unit sphere given by the inequalities and , respectively. Then . Denote by (respectively, ) the subset of given by the inequality (respectively, the inequalities ). Denote by the scaling of the unit disc . Let be the central projection from the point . Fix an embedding and denote .
If no confusion arises we denote a map and its abbreviation by the same symbol, and also an embedding and its isotopy class by the same symbol.
For the standard embedding is given by the formula , where the number of zero coordinates equals . It does not coincide with the inclusion , , coming from the identification of space with a subspace of .
For the standard embeddings are given by one formula
Clearly, is the neighborhood of the sphere in the sphere .
Definition.
(See Figure 2 to the left.) An embedding is standardized, if

is the standard embedding;

.
An isotopy is standardized, if for each the embedding is standardized.
Lemma 2.1.
[Sko15, Standardization Lemma 2.1] Assume that . Then
(a) any embedding is isotopic to a standardized embedding; and
(b) if two standardized embeddings are isotopic then there is a standardized isotopy between them.
Lemma 2.2.
[Sko15, Group Structure Lemma 2.2] Assume that . Then an Abelian group structure on the set is welldefined by the following construction.

Let be two embeddings. Take standardized embeddings isotopic to them. By definition, set to be the isotopy class of the embdedding

Set to be the isotopy class of the embedding .

Set to be the isotopy class of the standard embedding .
Lemma 2.3.
[Sko15, Lemma 3.1] Assume that . Then an embedding is isotopic to the standard embedding if and only if it extends to an embedding .
These lemmas are the only results of [Sko15] used in the present paper.
The next two subsections give some insight for the proof of Theorem 1.6 although they are not used formally in that proof.
Action of knots
Let us define an action of the group of knots on the set of embeddings and prove a particular case of Theorem 1.4 (see the paragraph after Lemma 2.4).
Define a map as follows. Represent an element of by an embedding such that the images and are separated by a hyperplane. Join these images by an arc whose interior misses the images. Let be the embedded connected sum of the knot and the standard embedding along this arc. For or the manifold is disconnected, thus we need to specify that the endpoints of the arc belong to and . Clearly, for the map is welldefined by this construction; cf. Lemma 3.2 below.
Description of a similar action for a general manifold is a hard open problem [CrSk08, Sko08Z]. Fortunately, in our situation enough information can be obtained:
Lemma 2.4.
[CRS08, Proposition 8] For the map is injective.
This lemma immediately implies the case “ divisible by 4” of Theorem 1.4, by Theorem 1.1 above. In fact is a direct summand in [Sko15, Theorem 1.1].
Let us define the standard surgery used in the proof of Lemma 2.4. Informally, it is a surgery over the “torus” along a “meridian” , ; see Figure 2.
To give a formal definition, fix a diffeomorphism . Let the embedding be given by the formula
Define the result of the standard surgery over the standard embedding to be the smooth unknotted embedding obtained by gluing and together. Define the result of the standard surgery over a standardized embedding to be the embedding obtained by gluing and together:
Proof of Lemma 2.4.
It suffices to construct a left inverse of . Take an element of . By Standardization Lemma 2.1.a it can be realized by a standardized embedding . Set to be the result of the standard surgery over .
Let us prove that the isotopy class of is welldefined, i.e., does not depend on the choice of within an isotopy class. Take two isotopic standardized embeddings . By Standardization Lemma 2.1.b there is a standardized isotopy between and . Then is an isotopy between and . That is, the isotopy classes of and are the same.
Let us prove that . Take an element of . Represent it by an embedding such that and are separated by a hyperplane. Then . ∎
Framed knots
Let us recall an approach to the classification of (partially) framed knots.
By a framing of an embedded manifold we mean a system of ordered orthogonal normal unit vector fields on the manifold. Denote by the Stiefel manifold of framings of the origin of . Clearly, the group is isomorphic to the group of framed embeddings up to framed isotopy. The group structure on the set is constructed literally as on above (with replaced by in Lemmas 2.1–2.2) or equivalently as on framed embeddings up to framed isotopy in [Hae66A].
The following result in some sense reduces the classification of framed knots to the classification of knots and computations of homotopy groups.
Theorem 2.5.
[CRS08, Theorem 9,4)] For there is an exact sequence
The theorem is proved analogously to its particular case [Hae66A, Corollary 5.9]. We sketch the proof here for convenience of the reader.
Sketch of the proof.
Definition of homomorphisms. The map is the composition of the restrictioninduced map and the map induced by the standard embedding .
The map is the obstruction to the existence of a framing on an embedding defined as follows. Take a (unique up to homotopy) framing of the disc . Take a (unique up to homotopy) framing of the disc . Thus the sphere is equipped both with the framing and the framing. Using the framing identify each fiber of the normal bundle to with the space . To each point assign the framing at the point . This leads to a map . By definition is the homotopy class of this map.
The map is defined as follows. Represent as a smooth map linear in each fiber , . Define to be the composition of the embedding and the standard embedding , i.e., for each , .
The exactness at the terms and is checked directly.
Proof of the exactness at the term . Let be an embedding. Then is isotopic to a standardized embedding , i.e., satisfying the conditions:

is the abbreviation of the standard embedding ;

.
Take a framing of the disc . Represent this framing by an embedding linear in each fiber , . Clearly, . Thus the embedding extends to the embedding . So is isotopic to the standard embedding , cf. Lemma 2.3 above. Thus . Analogously . ∎
3 The exact sequence
First let us define the homomorphisms in Theorem 1.6. These maps are welldefined for and are homomorphisms for .
Throughout this section we replace the group in Theorem 1.6 by the group . The former and the latter groups are identified, say, by the isomorphism induced by the central projection from the point .
The map is restrictioninduced. It follows directly by the construction of the group structure that the map is a homomorphism for .
Definition of .
The map is defined as follows; see Figure 3. Represent an element of by an embedding such that the restriction is standard and . The latter condition can be always achieved because can be moved aside a neighbourhood of by an appropriate isotopy. Join the images and by an arc, whose interior misses these images. Let be the embedded connected sum of and along this arc. For or the manifold is disconnected, thus we need to specify that the endpoints of the above arc belong to and .
To show that the map is welldefined, we need the following result due to A. Haefliger [Hae66C, Proof of Theorem 7.1]. For convenience of the reader we present the proof obtained in a discussion with A. Skopenkov and A. Zhubr.
Lemma 3.1.
If two embeddings with standard restrictions to are isotopic, then there is an isotopy between them fixed on .
Proof.
Take two isotopic embeddings whose restrictions to are standard. Move the images aside along the great halfcircles passing through the points and having no other common points with . This isotopy allows to assume that .
Since and are isotopic it follows that there is an orientationpreserving diffeomorphism fixed on such that on . Then both the standard embedding and the composition are tubular neighborhoods of in . By the uniqueness of tubular neighborhoods [Hir76, Theorem 5.5 in Chapter 4] it follows that there is an isotopy fixed on such that , , and is an automorphism of the trivial bundle .
The latter automorphism can be made identical on the contractible subset by an appropriate isotopy because automorphisms of the bundle up to isotopy are in bijection with smooth maps up to smooth homotopy. Thus we may assume that on .
Then is the required isotopy between and . Indeed, the assumption implies that is welldefined, and on . Finally, since is fixed on it follows that on . ∎
Lemma 3.2.
Assume that . Then the map is welldefined by the above construction.
Proof.
Let us show that the isotopy class depends neither on the choice of a particular representative within an isotopy class nor on the choice of arc joining and . Take two isotopic embeddings whose restrictions to are standard such that . Take two arcs and joining with and respectively.
By Lemma 3.1 there is an isotopy between and fixed on . We may assume that by uniformly moving aside a neighbourhood of .
Join the two arcs and by a general position family of arcs with the endpoints at and . Since , by general position it follows that the interior of each arc misses and . Let