The rational classification of links
Let and be positive integers. The set of links of codimension , , is the set of smooth isotopy classes of smooth embeddings . Haefliger showed that is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. . For and for restrictions on the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group in general. In particular we determine precisely when is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras.
Keywords: smooth manifold, embedding, isotopy, link, homotopy group, Lie algebra.
2000 MSC: 57R52,57Q45; 55P62, 17B01.
Let and be positive integers. The set of links in codimension is the set
of smooth isotopy classes of smooth embeddings . This set is a finitely generated abelian group with respect to componentwise embedded connected summation . In addition to its intrinsic mathematical interest, the group of links is related to the classification of handlebodies, the mapping class groups of certain manifolds and to the set of embeddings of more general disjoint unions of manifolds: we discuss these applications in Section 3.
Up to extension problems, the computation of the group was reduced to problems in unstable homotopy theory in the seminal papers of Levine and Haefliger [16, 12, 13]. However, these problems include the determination of the unstable homotopy groups of spheres. Hence one expects that the precise computation of is in general extremely difficult.
In this paper we address the following simpler question: what is the rank of the group and in particular for which is the group finite? This question is in part motivated by analogy to rational homotopy theory and also the rational classification of link maps by Koschorke [15, 10].
The main results of this paper give an explicit formula for the rank of the group (Theorem 1.9) and also a criterion which determines precisely when the group of is finite (Corollary 1.8). We also accomplish the same tasks for the groups of framed links (Section 1.4).
1.1 Background and a useful observation
An embedding is called a link, its restrictions are called components. For one-component links, or knots, the question posed above was answered by Haefliger:
(See [12, Corollary 6.7]) Assume that . The group has rank or : it is infinite if and only if is divisible by and .
Typically one approaches multi-component links by studying the sub-links which are obtained by deleting one or more components from the original link. In particular, a link is called Brunnian if it becomes trivial after removing any one of its components. An example of a nontrivial Brunnian link is the Borromean rings; cf. . A link has unknotted components, if each of its components is a trivial knot. Denote by and respectively the subgroups of Brunnian links and links having unknotted components: observe that these two subgroups coincide for .
(See [13, Sections 2.4 and 9.3]) Assume that . Then there are isomorphisms
where the last sum is over all nonempty subsets .
Under certain dimension restrictions Haefliger and Nezhinsky have found explicit descriptions of the isotopy classes of Brunnian links in terms of homotopy groups of spheres and Stiefel manifolds [12, 13, 17, 25].
Here and are certain finitely generated abelian groups defined via homomorphisms between the homotopy groups of appropriate wedges of spheres, see Section 2.1 for their definition. The homomorphisms and are topologically defined in [13, Section 1.4]: we do not explicitly consider them in this paper. Rather we note that up to extension, the group of links is determined by the homomorphism . Moreover, since is defined using Whitehead products, the Haefliger sequence (1.1) reduces the determination of the group to a problem in unstable homotopy theory and an extension problem. In particular determining the rank of is reduced to a problem in unstable homotopy theory.
The starting point of our investigation is the following simple but crucial observation:
After tensoring with the Haefliger sequence (1.1) splits into the short exact sequences
1.2 Finiteness criteria for the group of links
In this subsection we state criteria which determine precisely when the group is finite. We begin with finiteness criteria for Brunnian links and conclude with the general case.
Assume that and . Then the group is infinite if and only if the equation has a solution in positive integers .
In the case of component links the criterion is more complicated. It involves certain finiteness-checking 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 the evident periodicity.
Assume that . Then the group is infinite if and only if there exists a point such that .
The finiteness-checking set can be considered as a nomogram: to establish infiniteness one draws the line and looks if the line intersects the set .
|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 .|
Assume that . Then the group is infinite if and only if is an integer, which is distinct from for odd, and distinct from for even.
Applying Corollary 1.6 and Theorem 1.4 we see that the group is infinite if and only if . This corrects an error in [19, Corollary 3.18] where it is stated that the group is infinite if and only if ; see Remark 2.1 for an explanation. Note that [19, Corollary 3.19] should also be modified by changing to .
Assume that . Then the group is infinite if and only if there is a subsequence satisfying one of the following conditions:
, , and ;
and there is such that ;
and the equation has a solution in positive integers.
1.3 Formulae for the rank of the group of links
In this subsection we state results on ranks of the groups of links. We give an explicit formula for the ranks of the groups and in Theorem 1.9 below. The formula for Brunnian links asserts that the rank is equal to the number of solutions of the equation from Theorem 1.4 counted with certain “multiplicities” . As a corollary to Theorem 1.9, we obtain a formula for the rank of the group of links with components having the same dimension (Theorem 1.10). Some computations are shown in Table 2 below. There is a computer application available based on Theorem 1.9 which computes these ranks in general .
To state our results we need the following notation. Denote the rank of a finitely generated abelian group by and let denote the set of positive integers. The Möbius function is defined by the formula:
Denote by the greatest common divisor of integers , …, . Denote:
Set and , if at least one of the numbers is negative. For each positive integer define the following polynomials in the indeterminate :
Set , if . Set
Set , if , and , otherwise.
Assume that and . Then
Assume that . Denote . Then
The rank of the group tends to infinity as tends to infinity.
1.4 Framed links
Let be integers such that for each . Denote by
the set of smooth isotopy classes of smooth orientation preserving embeddings (for any embedding is considered to be orientation preserving). For the set is a group called the group of partially framed links. This notion generalizes both links () and framed links () . Partially framed links play an important role in the classification of embeddings of general manifolds [3, 23, 26].
To give a formula for the rank of the group of partially framed links we first recall that the Stiefel manifold is the manifold of all -tuples of pairwise orthogonal unit vectors in ; by definition is a point. The ranks of the homotopy groups are essentially known; see Section 2.4:
Assume . Then
Assume and . Then there is an equality
Assume and . Then the set is infinite if and only if at least one of the following conditions holds:
Assume that . Then the set is infinite if and only if there is a subsequence satisfying one of the following conditions:
and either or ;
and there is such that ;
and the equation has a solution in positive integers.
1.5 Other types of links
For the group of piecewise-linear (respectively, continuous) isotopy classes of piecewise-linear (respectively, continuous) embeddings is isomorphic to the group by [13, Section 2.6] (respectively, ). Thus our results provide the rational classification of piecewise-linear and topological links as well.
1.6 The organization of the paper and its relationship to other work
We give the proofs of all the results stated above in Section 2: We start by recalling the Haefliger sequence in Section 2.1. Using this sequence at the beginning of Section 2.2 we convert the problem of the rational classification of links into a completely algebraic problem in the theory of Lie algebras. In the remainder of Section 2.2 we solve this problem and so obtain the formulae for ranks of the groups of links. In this way we convert the question of which groups of links are finite into a problem of elementary number theory. We present a solution to this latter problem in Section 2.3. Using our results for unframed links and some standard algebraic topology we obtain the rational classification of framed links in Section 2.4.
In Sections 3.1, 3.2, and further in  we apply our calculations to the classification of handlebodies, thickenings and to the computation of mapping class groups. In  the third author applies our calculations to the rational classification of embeddings of a product of two spheres; see [2, 3, 23] for particular cases. In Section 3.3 we discuss some open problems.
2.1 The Haefliger sequence for Brunnian links
By Theorem 1.2 the group of links splits as the sum of the groups of Brunnian links. There is a Haefliger sequence for Brunnian links, (2.1) below, which is analogous to the Haefliger sequence (1.1). In this subsection we recall the groups in the Haefliger sequence for Brunnian links and also the key homomorphism : we refer the reader to [13, §9.4] and [17, §1.2] for further details. We also give the definition of the groups in the Haefliger sequence (1.1) and state the Hilton–Milnor theorem which plays a key role in all computations using these sequences.
Denote by the -th homotopy group of a space . For there are obvious retractions obtained by collapsing the -th sphere from the wedge. Taking the kernel of the sum over of the homomorphism induced on one obtains the finitely generated abelian group
In addition, define
and define the homomorphism
where is the obvious inclusion and is the Whitehead product. The Haefliger sequence for Brunnian links is a long exact sequence of finitely generated abelian groups which runs as follows:
The groups in the Haefliger sequence (1.1) are defined by similar formulae:
The homomorphism is also defined analogously via Whitehead products.
The homotopy groups in the sequences (1.1) and (2.1) can be expressed in terms of the homotopy groups of spheres using the Hilton–Milnor theorem which we now recall; see [17, §2.1] or [20, Section 5 of Chapter IV] for the necessary definitions. Let be the Hall basis in a free graded Lie algebra (not to be confused with a superalgebra) generated by elements of degrees . Denote by the degree of an element . The Hilton-Milnor theorem states that there is an isomorphism
The isomorphism from the the right to the left is given by the formula , where for a product of generators we denote by the analogous Whitehead product of the homotopy classes , and denotes the composition of homotopy classes.
In general the map does not extend to a multiplicative homomorphism. For instance, for odd we have . This explains the mistake in the proof [19, Corollary 3.18] where the map is assumed to be multiplicative.
2.2 Computation of ranks
A graded Lie superalgebra111 Some authors use another definition, which can be obtained from ours replacing by . Both definitions lead to the same dimensions of homogeneous components of free graded Lie superalgebras. over is a graded vector space , along with a bilinear operation satisfying the following axioms:
Respect of the grading: .
Symmetry: if , then .
Jacobi identity: if , , then
A free graded Lie superalgebra generated by a set of elements with given degrees is the quotient of the free algebra generated by the set by the ideal generated by the terms from the left parts of the axioms (2)–(3). The grading of this quotient is uniquely defined by axiom (1). Hereafter denote by the free Lie superalgebra generated by the elements of degrees .
Denote by the subalgebra of generated by the products containing each generator at least once. Denote by the subalgebra generated by the products containing each generator at least once except possibly . Denote by the subspace of formed by elements of degree . For a finitely generated abelian group identify .
Using this notation we can state a version of the Haefliger sequence (2.1) in purely algebraic terms:
For and there is an exact sequence
The linear map in the sequence is given by the formula .
Proof of Theorem 2.2.
We need only rewrite the sequence (2.1) tensored by in terms of Lie superalgebras. For a simply connected finite CW-complex the group is a graded Lie superalgebra with respect to the Whitehead product operation:
Note that the degree of a homotopy class is one less than its dimension.
A rational version of the Hilton–Milnor Theorem [8, p. 116] states that the graded Lie superalgebra
is isomorphic to the free Lie superalgebra . The isomorphism takes the homotopy class of each inclusion to the generator . Thus for each and the theorem follows. ∎
Now the rational classification of links reduces to the following purely algebraic result.
(a) The map from Theorem 2.2 is surjective.
(b) The dimension of the kernel of equals the expression on the right hand side of (1.6) .
Proof of Lemma 2.3(a).
Take . Let us prove that belongs to the image of . It suffices to consider the case when is a product of generators. Since and it follows that itself is not a generator. Thus for some products and of generators. So assertion (a) reduces to the following claim. ∎
Let and let be a product of generators (each of the generators may appear in the product several times). Then for any there are such that .
Proof of Claim 2.4.
Let us prove the claim by induction over the number of factors in the product . If there is only one factor then is a generator itself, and there is nothing to prove. Otherwise for some products containing fewer factors than . By the symmetry and the Jacobi identity we get . Apply the inductive hypothesis for and . Since contains fewer factors than it follows that for some . Analogously, since contains fewer factors than it follows that for some . Thus , which proves the claim. ∎
We say that an element has multidegree if is a product of generators in which the generator appears exactly times for each . The multidegree does not depend on the choice of representation as the product because axioms (2) and (3) above are relations between elements of the same multidegree. The element of multidegree has degree given by formula (1.2). Denote by the linear subspace of spanned by elements of multidegree .
Proof of Lemma 2.3(b).
By part (a) we have . The space is a direct sum of all the spaces