ALEXANDER DUALITY, GROPES AND LINK HOMOTOPY

VYACHESLAV S. KRUSHKAL AND PETER TEICHNER

Abstract. We prove a geometric refinement of Alexander duality for certain 2complexes, the so-called gropes, embedded into 4-space. In addition, we give new proofs and extended versions of two lemmas from [2] which are of central importance in the A-B-slice problem, the main open problem in the classification theory of topological 4-manifolds. Our methods are group theoretical, rather than using Massey products and Milnor μ-invariants as in the original proofs.

1. Introduction

Consider a finite complex X PL-embedded into the n-dimensional sphere S^{n}. Alexander duality identifies the (reduced integer) homology H_{i}(S^{n} ∖ X) with the cohomology H^{n−1−i}(X). This implies that the homology (or even the stable homotopy type) of the complement cannot distinguish between possibly different embeddings of X into S^{n}. As a consequence, there cannot be a duality for homotopy groups as one can see by considering the fundamental group of classical knot complements, i.e. the case X = S^{1} and n = 3.

However, one can still ask whether additional information about X does lead to additional information about S^{n}∖X. For example, if X is a smooth closed (n−1−i)- dimensional manifold then the cohomological fundamental class is dual to a spherical class in H_{i}(S^{n} ∖ X). Namely, it is represented by any meridinal i-sphere which by definition is the boundary of a normal disk at a point in X. This geometric picture explains the dimension shift in the Alexander duality theorem.

By reversing the roles of X and S^{n} ∖ X in this example we see that it is not true that H_{i}(X) being spherical implies that H_{n−1−i}(S^{n} ∖ X) is spherical. However, the following result shows that there is some kind of improved duality if one does not consider linking dimensions. One should think of the Gropes in our theorem as means of measuring how spherical a homology class is.

Theorem 1 (Grope Duality). If X ⊂ S^{4} is the disjoint union of closed embedded Gropes of class k then H_{2}(S^{4} ∖ X) is freely generated by r disjointly embedded closed Gropes of class k. Here r is the rank of H_{1}(X). Moreover, H_{2}(S^{4} ∖ X) cannot be generated by r disjoint maps of closed gropes of class k + 1.

As a corollary to this result we show in 4.2 that certain Milnor μ-invariants of a link in S^{3} are unchanged under a Grope concordance.

The Gropes above are framed thickenings of very simple 2-complexes, called gropes, which are inductively built out of surface stages, see Figure 1 and Section 2. For example, a grope of class 2 is just a surface with a single boundary component and gropes of bigger class contain information about the lower central series of the fundamental group. Moreover, every closed grope has a fundamental class in H_{2}(X) and one obtains a geometric definition of the Dwyer filtration

Teichner’s research at MSRI was supported by a fellowship from the Miller foundation, UC Berkeley.

π_{2}(X) ⊆ ⋯ ⊆ ϕ_{k}(X) ⊆ ⋯ ⊆ ϕ_{3}(X) ⊆ ϕ_{2}(X) = H_{2}(X)

by defining ϕ_{k}(X) to be the set of all homology classes represented by maps of closed gropes of class k into X.

Figure 1. A grope of class 2 is a surface – two closed gropes of class 4.

Figure 1 shows that each grope has a certain “type” which measures how the surface stages are attached. In Section 2 this will be made precise using certain rooted trees, compare Figure 2. In Section 4 we give a simple algorithm for obtaining the trees corresponding to the dual Gropes constructed in Theorem 1.

The simplest application of Theorem 1 (with class k = 2) is as follows. Consider the standard embedding of the 2-torus T^{2} into S^{4} (which factors through the usual picture of T^{2} in ℝ^{3}). Then the boundary of the normal bundle of T^{2} restricted to the two essential circles gives two disjointly embedded tori representing generators of H_{2}(S^{4} ∖ T^{2}) ≅ ℤ^{2}. Since both of these tori may be surgered to (embedded) spheres, H_{2}(S^{4} ∖ T^{2}) is in fact spherical. However, it cannot be generated by two maps of 2-spheres with disjoint images, since a map of a sphere may be replaced by a map of a grope of arbitrarily big class.

This issue of disjointness leads us to study the relation of gropes to classical link homotopy. We use Milnor group techniques to give new proofs and improved versions of the two central results of [2], namely the Grope Lemma and the Link Composition Lemma. Our generalization of the grope lemma reads as follows.

Theorem 2. Two n-component links in S^{3} are link homotopic if and only if they cobound disjointly immersed annulus-like gropes of class n in S^{3} × I.

This result is stronger than the version given in [2] where the authors only compare a link to the trivial link. Our new proof is considerably shorter than the original one.

Our generalization of the link composition lemma is formulated as Theorem 3 in Section 5. The reader should be cautious about the proof given in [2]. It turns out that our Milnor group approach contributes a beautiful feature to Milnor’s algebraization of link homotopy: He proved in [10] that by forgetting one component of the unlink one gets an abelian normal subgroup of the Milnor group which is the additive group of a certain ring R. We observe that the Magnus expansion of the free Milnor groups arises naturally from considering the conjugation action of the quotient group on this ring R. Moreover, we show in Lemma 5.3 that “composing” one link into another corresponds to multiplication in that particular ring R. This fact is the key in our proof of the link composition lemma.

Our proofs completely avoid the use of Massey products and Milnor μ-invariants and we feel that they are more geometric and elementary than the original proofs. This might be of some use in studying the still unsolved A-B-slice problem which is the main motivation behind trying to relate gropes, their duality and link homotopy. It is one form of the question whether topological surgery and s-cobordism theorems hold in dimension 4 without fundamental group restrictions. See [4] for new developments in that area.

Acknowledgement: It is a pleasure to thank Mike Freedman for many important discussions and for providing an inspiring atmosphere in his seminars. In particular, we would like to point out that the main construction of Theorem 1 is reminiscent of the methods used in the linear grope height raising procedure of [5].

2. Preliminary facts about gropes and the lower central series.

The following definitions are taken from [5].

Definition 2.1. A grope is a special pair (2-complex, circle). A grope has a class k = 1, 2, …, ∞. For k = 1 a grope is defined to be the pair (circle, circle). For k = 2 a grope is precisely a compact oriented surface Σ with a single boundary component. For k finite a k-grope is defined inductively as follow: Let {α_{i}, β_{i}, i = 1, …, genus} be a standard symplectic basis of circles for Σ. For any positive integers p_{i}, q_{i} with p_{i} + q_{i} ≥ k and p_{i_{0}} + q_{i_{0}} = k for at least one index i_{0}, a k-grope is formed by gluing p_{i}-gropes to each α_{i} and q_{i}-gropes to each β_{i}.

The important information about the “branching” of a grope can be very well captured in a rooted tree as follows: For k = 1 this tree consists of a single vertex v_{0} which is called the root. For k = 2 one adds 2·genus(Σ) edges to v_{0} and may label the new vertices by α_{i}, β_{i}. Inductively, one gets the tree for a k-grope which is obtained by attaching p_{i}-gropes to α_{i} and q_{i}-gropes to β_{i} by identifying the roots of the p_{i}- (respectively q_{i}-)gropes with the vertices labeled by α_{i}(respectively β_{i}). Figure 2 below should explain the correspondence between gropes and trees.

Note that the vertices of the tree which are above the root v_{0} come in pairs corresponding to the symplectic pairs of circles in a surface stage and that such rooted paired trees correspond bijectively to gropes. Under this bijection, the leaves (:= 1-valent vertices) of the tree correspond to circles on the grope which freely generate its fundamental group. We will sometimes refer to these circles as the tips of the

root leaves

Figure 2. A grope of class 5 and the associated tree.

grope. The boundary of the first stage surface Σ will be referred to as the bottom of the grope.

Given a group Γ, we will denote by Γ^{k} the k-th term in the lower central series of Γ, defined inductively by Γ^{1} := Γ and Γ^{k} := [Γ, Γ^{k−1}], the characteristic subgroup of k-fold commutators in Γ.

Lemma 2.2 (Lem.2.1 in [5]). For a space X, a loop γ lies in π_{1}(X)^{k}, 1 ≤ k < ω, if and only if γ bounds a map of some k-grope. Moreover, the class of a grope (G, γ) is the maximal k such that γ ∈ π_{1}(G)^{k}.

A closed k-grope is a 2-complex made by replacing a 2-cell in S^{2} with a k-grope. A closed grope is sometimes also called a sphere-like grope. Similarly, one has annuluslike k-gropes which are obtained from an annulus by replacing a 2-cell with a k-grope. Given a space X, the Dwyer’s subgroup ϕ_{k}(X) of H_{2}(X) is the set of all homology classes represented by maps of closed gropes of class k into X. Compare [5, Lem.2.3] for a translation to Dwyer’s original definition.

Theorem (Dwyer’s Theorem [1]). Let k be a positive integer and let f : X ⟶ Y be a map inducing an isomorphism on H_{1} and an epimorphism on H_{2}/ϕ_{k}. Then f induces an isomorphism on π_{1}/(π_{1})^{k}.

A Grope is a special “untwisted” 4-dimensional thickening of a grope (G, γ); it has a preferred solid torus (around the base circle γ) in its boundary. This “untwisted” thickening is obtained by first embedding G in ℝ^{3} and taking its thickening there, and then crossing it with the interval [0, 1]. The definition of a Grope is independent of the chosen embedding of G in ℝ^{3}. One can alternatively define it by a thickening of G such that all relevant relative Euler numbers vanish. Similarly, one defines sphere- and annulus-like Gropes, the capital letter indicating that one should take a 4-dimensional untwisted thickening of the corresponding 2-complex.

3. The Grope Lemma

We first recall some material from [10]. Two n-component links L and L′ in S^{3} are said to be link-homotopic if they are connected by a 1-parameter family of immersions such that distinct components stay disjoint at all times. L is said to be homotopically trivial if it is link-homotopic to the unlink. L is almost homotopically trivial if each proper sublink of L is homotopically trivial.

For a group π normally generated by g_{1}, …, g_{k} its Milnor group Mπ (with respect to g_{1}, …, g_{k}) is defined to be the quotient of π by the normal subgroup generated by the elements [g_{i}, g_{i}^{h}], where h ∈ π is arbitrary. Here we use the conventions

[g_{1}, g_{2}] := g_{1} · g_{2} · g_{1}^{−1} · g_{2}^{−1} and g^{h} := h^{−1} · g · h.

Mπ is nilpotent of class ≤ k + 1, i.e. it is a quotient of π/(π)^{k+1}, and is generated by the quotient images of g_{1}, …, g_{k}, see [4]. The Milnor group M(L) of a link L is defined to be Mπ_{1}(S^{3} ∖ L) with respect to its meridians m_{i}. It is the largest common quotient of the fundamental groups of all links link-homotopic to L, hence one obtains

Theorem (Invariance under link homotopy [10]). If L and L′ are link homotopic then their Milnor groups are isomorphic.

The track of a link homotopy in S^{3} × I gives disjointly immersed annuli with the additional property of being mapped in a level preserving way. However, this is not really necessary for L and L′ to be link homotopic, as the following result shows.

Lemma 3.1 (Singular concordance implies homotopy [6], [7], [9]). If L ⊂ S^{3} × {0} and L′ ⊂ S^{3} × {1} are connected in S^{3} × I by disjointly immersed annuli then L and L′ are link-homotopic.

Remark . This result was recently generalized to all dimensions, see [13].

Our Grope Lemma (Theorem 2 in the introduction) further weakens the conditions on the objects that connect L and L′.

Proof of Theorem 2. Let G_{1}, …, G_{n} be disjointly immersed annulus-like gropes of class n connecting L and L′ in S^{3} × I. To apply the above Lemma 3.1, we want to replace one G_{i} at a time by an immersed annulus A_{i} in the complement of all gropes and annuli previously constructed.

Let’s start with G_{1}. Consider the circle c_{1} which consists of the union of the first component l_{1} of L, then an arc in G_{1} leading from l_{1} to l′_{1}, then the first component l′_{1} of L′ and finally the same arc back to the base point. Then the n-grope G_{1} bounds c_{1} and thus c_{1} lies in the n-th term of the lower central series of the group π_{1}(S^{3}×I ∖G), where G denotes the union of G_{2}, …, G_{n}. As first observed by Casson, one may do finitely many finger moves on the bottom stage surfaces of G (keeping the components G_{i} disjoint) such that the natural projection induces an isomorphism

π_{1}(S^{3} × I ∖ G) ≅ Mπ_{1}(S^{3} × I ∖ G)

(see [4] for the precise argument, the key idea being that the relation [m_{i}, m_{i}^{h}] can be achieved by a self finger move on G_{i} which follows the loop h.) But the latter Milnor group is normally generated by (n − 1) meridians and is thus nilpotent of class ≤ n. In particular, c_{i} bounds a disk in S^{3} × I ∖ G which is equivalent to saying that l_{1} and l′_{1} cobound an annulus A_{1}, disjoint from G_{2}, …, G_{n}.

Since finger moves only change the immersions and not the type of a 2-complex, i.e. an immersed annulus stays an immersed annulus, the above argument can be repeated n times to get disjointly immersed annuli A_{1}, …, A_{n} connecting L and L′.

4. Grope Duality

In this section we give the proof of Theorem 1 and a refinement which explains how the trees corresponding to the dual Gropes look like. Since we now consider closed gropes, the following variation of the correspondence to trees turns out to be extremely useful. Let G be a closed grope and let G′ denote G with a small 2-cell removed from its bottom stage. We define the tree T_{G} to be the tree corresponding to G′ (as defined in Section 2) together with an edge added to the root vertex. This edge represents the deleted 2-cell and it turns out to be useful to define the root of T_{G} to be the 1-valent vertex of this new edge. See Figure 4 for an example of such a tree.

Proof of Theorem 1. Abusing notation, we denote by X the core grope of the given 4-dimensional Grope in S^{4}. Thus X is a 2-complex which has a particularly simple thickening in S^{4} which we may use as a regular neighborhood. All constructions will take place in this regular neighborhood, so we may assume that X has just one connected component. Let {α_{i,j}, β_{i,j}} denote a standard symplectic basis of curves for the i-th stage X_{i} of X; these curves correspond to vertices at a distance i + 1 from the root in the associated tree. Here X_{1} is the bottom stage and thus a closed connected surface. For i > 1, the X_{i} are disjoint unions of punctured surfaces. They are attached along some of the curves α_{i−1,j} or β_{i−1,j}.

Let A_{i,j} denote the ϵ-circle bundle of X_{i} in S^{4}, restricted to a parallel displacement of α_{i,j} in X_{i}, see Figure 3. The corresponding ϵ-disk bundle, for ϵ small enough, can be used to see that the 2-torus A_{i,j} has linking number 1 with β_{i,j} and does not link other curves in the collection {α_{s,t}, β_{s,t}}. Note that if there is a higher stage attached to β_{i,j} then it intersects A_{i,j} in a single point, while if there is no stage attached to β_{i,j} then A_{i,j} is disjoint from X, and the generator of H_{2}(S^{4} ∖ X) represented by A_{i,j} is Alexander-dual to β_{i,j}. Similarly, let B_{i,j} denote a 2-torus representative of the class dual to α_{i,j}. There are two inductive steps used in the construction of the stages of dual Gropes.

Step 1. Let γ be a curve in the collection {α_{i,j}, β_{i,j}}, and let X′ denote the subgrope of X which is attached to γ. Since X is framed and embedded, a parallel copy of γ in S^{4} bounds a parallel copy of X′ in the complement of X. If there is no higher stage attached to γ then the application of Step 1 to this curve is empty.

Step 2. Let Σ_{i} be a connected component of the i-th stage of X, and let m_{i} denote a meridian of Σ_{i} in S^{4}, that is, m_{i} is the boundary of a small normal disk to Σ_{i} at an interior point. Suppose i > 1 and let Σ_{i−1} denote the previous stage, so that Σ_{i} is attached to Σ_{i−1} along some curve, say α_{i−1,n}. The torus B_{i−1,n} meets Σ_{i} in a point, but making a puncture into B_{i−1,n} around this intersection point and connecting it by a tube with m_{i} exhibits m_{i} as the boundary of a punctured torus in the complement of X, see Figure 3.

Σ_{i} β_{i,j}  γ = α_{i,j} A_{i,j}  α_{i,j}(displaced) X′  m_{i−1} β_{i−1,n} ⊂ Σ_{i−1}  Σ_{i} m_{i} α_{i−1,n}

Figure 3. Steps 1 and 2.

By construction, H_{1}(X) is generated by those curves {α_{i,j}, β_{i,j}} which do not have a higher stage attached to them. Fix one of these curves, say β_{i,j}. We will show that its dual torus A_{i,j} is the first stage of an embedded Grope G ⊂ S^{4} ∖ X of class k. The meridian m_{i} and a parallel copy of α_{i,j} form a symplectic basis of circles for A_{i,j}. Apply Step 1 to α_{i,j}. If i = 1, the result of Step 1 is a grope at least of class k and we are done. If i > 1, apply in addition Step 2 to m_{i}. The result of Step 2 is a grope with a new genus 1 surface stage, the tips of which are the meridian m_{i−1} and a parallel copy of one of the standard curves in the previous stage, say β_{i−1,n}. The next Step 1 - Step 2 cycle is applied to these tips. Altogether there are i cycles, forming the grope G.

The trees corresponding to dual gropes constructed above may be read off the tree associated to X, as follows. Start with the tree T_{X} for X, and pick the tip (1-valent vertex), corresponding to the curve β_{i,j}. The algorithm for drawing the tree T_{G} of the grope G, Alexander-dual to β_{i,j}, reflects Steps 1 and 2 above. Consider the path p from β_{i,j} to the root of T_{X}, and start at the vertex α_{i−1,n}, adjacent to β_{i,j}. Erase all branches in T_{X}, “growing” from α_{i−1,n}, except for the edge [β_{i,j} α_{i−1,n}] which has been previously considered, and its “partner” branch [α_{i,j} α_{i−1,n}], and then move one edge down along the path p. This step is repeated i times, until the root of T_{X} is reached. The tree T_{G} is obtained by copying the part of T_{X} which is not erased, with the tip β_{i,j} drawn as the root, see figure 4.

Note the “distinguished” path in T_{G}, starting at the root and labelled by m_{i}, m_{i−1}, …, m_{1}. Each of the vertices m_{i}, m_{i−1}, …, m_{2} is trivalent (this corresponds to the fact that all surfaces constructed by applications of Step 2 have genus 1), see figures 4, 6. In particular, the class of G may be computed as the sum of classes of the gropes attached to the the “partner” vertices of m_{i}, …, m_{1}, plus 1.

We will now prove that the dual grope G is at least of class k. The proof is by induction on the class of X. For surfaces (class = 2) the construction gives tori in the

T_{X}  T_{G} α_{i−1,n} α_{i,j} β_{i,j}  m_{1}m_{2}

Figure 4. A dual tree. The branches in T_{X} to be erased are drawn with dashed lines.

collection {A_{i,j}, B_{i,j}}. Suppose the statement holds for Gropes of class less than k, and let X be a Grope of class k. By definition, for each standard pair of dual circles α, β in the first stage Σ of X there is a p-grope X_{α} attached to α and a q-grope X_{β} attached to β with p + q ≥ k. Let γ be one of the tips of X_{α}. By the induction hypothesis, the grope G_{α} dual to γ, given by the construction above for X_{α}, is at least of class p. G is obtained from G_{α} by first attaching a genus 1 surface to m_{2}, with new tips m_{1} and a parallel copy of β (Step 2), and then attaching a parallel copy of X_{β} (Step 1). According to the computation above of the class of G in terms of its tree, it is equal to p + q ≥ k.

It remains to show that the dual gropes can be made disjoint, and that they are 0-framed. Each dual grope may be arranged to lie in the boundary of a regular ϵneighborhood of X, for some small ϵ. Figure 5 shows how Steps 1 and 2 are performed at a distance ϵ from X. Note that although tori A_{i,j} and B_{i,j} intersect, at most one of them is used in the construction of a dual grope for each index (i, j). Taking distinct values ϵ_{1}, …, ϵ_{r}, the gropes are arranged to be pairwise disjoint. The same argument shows that each grope G has a parallel copy G′ with G ∩ G′ = ∅, hence its thickening in S^{4} is standard.

A_{i,j} a parallel copy of X′  X′ α_{i,j}  β_{i,j} ⊂ Σ_{i} punctured B_{i,j}  m_{i} α_{i,j}(displaced) ϵ ϵ  ϵ ϵ

Figure 5. Steps 1 and 2 at a distance ϵ from X.

To prove the converse part of the theorem, suppose that H_{2}(S^{4} ∖ X) is generated by r disjoint maps of closed gropes. Perturb the maps in the complement of X, so that they are immersions and their images have at most a finite number of transverse self-intersection points. The usual pushing down and twisting procedures from [3] produce closed disjoint 0-framed gropes G_{1}, …, G_{r} whose only failure to being actual Gropes lies in possible self-intersections of the bottom stage surfaces. The G_{i} still lie in the complement of X and have class k+1. The proof of the first part of Theorem 1 shows that H_{2}(Y)/ϕ_{k+1}(Y) is generated by the “Clifford tori” in the neighborhoods of self-intersection points of the G_{i}, where Y denotes the complement of all G_{i} in S^{4}. Assume X is connected (otherwise consider a connected component of X), and let X′ denote X with a 2-cell removed from its bottom stage. The relations given by the Clifford tori are among the defining relations of the Milnor group on meridians to the gropes, and Dwyer’s theorem shows (as in [5], Lem.2.6) that the inclusion map induces an isomorphism

Mπ_{1}(X′)/Mπ_{1}(X′)^{k+1} ≅ Mπ_{1}(Y)/Mπ_{1}(Y)^{k+1}.

Consider the boundary curve γ of X′. Since X is a grope of class k, by Lemma 4.1 below we get γ ∉ Mπ_{1}(X′)^{k+1}. On the other hand, γ bounds a disk in Y, hence γ = 1 ∈ Mπ_{1}(Y). This contradiction concludes the proof of Theorem 1.

T_{X}  T_{G} γ  m_{1} m_{2} m_{3}

Figure 6. Tree duality in the genus 1 case.

Lemma 4.1. Let (G, γ) be grope of class k. Then γ ∉ Mπ_{1}(G)^{k+1}.

Proof. This is best proven by an induction on k, starting with the fact that π_{1}(Σ) is freely generated by all α_{i} and β_{i}. Here Σ is the bottom surface stage of the grope (G, γ) with a standard symplectic basis of circles α_{i}, β_{i}. The Magnus expansion for the free Milnor group (see [10], [5] or the proof of Theorem 3) shows that γ = ∏ [α_{i}, β_{i}] does not lie in Mπ_{1}(Σ)^{3}. Similarly, for k > 2, π_{1}(G) is freely generated by those circles in a standard symplectic basis of a surface stage in G to which nothing else is attached. Now assume that the k-grope (G, γ) is obtained by attaching p_{i}-gropes G_{α_{i}} to α_{i} and q_{i}-gropes G_{β_{i}} to β_{i}, p_{i} + q_{i} ≥ k. By induction, α_{i} ∉ Mπ_{1}(G_{α_{i}})^{p_{i}+1} and β_{i} ∉ Mπ_{1}(G_{β_{i}})^{q_{i}+1} since p_{i}, q_{i} ≥ 1. But the free generators of π_{1}(G_{α_{i}}) and π_{1}(G_{β_{i}}) are contained in the set of free generators of π_{1}(G) and therefore γ = ∏ [α_{i}, β_{i}] ∉ Mπ_{1}(G)^{k+1}. Again, this may be seen by applying the Magnus expansion to Mπ_{1}(G).

Remark . In the case when all stages of a Grope X are tori, the correspondence between its tree T_{X} and the trees of the dual Gropes, given in the proof of theorem 1, is particularly appealing and easy to describe. Let γ be a tip of T_{X}. The tree for the Grope, Alexander-dual to γ, is obtained by redrawing T_{X}, only with γ drawn as the root, see Figure 6.

As a corollary of Theorem 1 we get the following result.

Corollary 4.2. Let L = (l_{1}, …, l_{n}) and L′ = (l′_{1}, …, l′_{n}) be two links in S^{3} × {0} and S^{3} × {1} respectively. Suppose there are disjointly embedded annulus-like Gropes A_{1}, …, A_{n} of class k in S^{3} × [0, 1] with ∂A_{i} = l_{i} ∪ l′_{i}, i = 1, …, n. Then there is an isomorphism of nilpotent quotients

π_{1}(S^{3} ∖ L)/π_{1}(S^{3} ∖ L)^{k} ≅ π_{1}(S^{3} ∖ L′)/π_{1}(S^{3} ∖ L′)^{k}.

Remark . For the readers who are familiar with Milnor’s μ̄-invariants we should mention that the above statement directly implies that for any multi-index I of length |I| ≤ k one gets μ̄_{L}(I) = μ̄_{L′}(I). For a different proof of this consequence see [8].

Proof of Corollary 4.2. The proof is a ϕ_{k}-version of Stallings’ proof of the concordance invariance of all nilpotent quotients of π_{1}(S^{3} ∖ L), see [12]. Namely, Alexander duality and Theorem 1 imply that the inclusion maps

(S^{3} × {0} ∖ L) ↪ (S^{3} × [0, 1] ∖ (A_{1} ∪ … ∪ A_{n})) ↩ (S^{3} × {1} ∖ L′)

induce isomorphisms on H_{1}( .) and on H_{2}( .)/ϕ_{k}. So by Dwyer’s Theorem they induce isomorphisms on π_{1}/(π_{1})^{k}.

5. The Link Composition Lemma

The Link Composition Lemma was originally formulated in [2]. The reader should be cautious about its proof given there; it can be made precise using Milnor’s μ̄- invariants with repeating coefficients, while this section presents an alternative proof.

Given a link L̂ = (l_{1}, …, l_{k+1}) in S^{3} and a link Q = (q_{1}, …, q_{m}) in the solid torus S^{1} × D^{2}, their “composition” is obtained by replacing the last component of L̂ with Q. More precisely, it is defined as L ∪ ϕ(Q) where L = (l_{1}, …, l_{k}) and ϕ : S^{1} × D^{2} ↪ S^{3} is a 0-framed embedding whose image is a tubular neighborhood of l_{k+1}. The meridian {1} × ∂D^{2} of the solid torus will be denoted by ∧ and we put Q̂ := Q∪∧. We sometimes think of Q or Q̂ as links in S^{3} via the standard embedding S^{1} × D^{2} ↪ S^{3}.

Theorem 3. (Link Composition Lemma)

(i) If L̂ and Q̂ are both homotopically essential in S^{3} then L ∪ ϕ(Q) is also homotopically essential. (ii) Conversely, if L∪ϕ(Q) is homotopically essential and if both L̂ and Q̂ are almost homotopically trivial, then both L̂ and Q̂ are homotopically essential in S^{3}.

L  ϕ(S^{1} × D^{2}) ϕ(∧) ∧′

Figure 7. In this example L̂ is the Borromean rings, and Q is the Bing double of the core circle of S^{1} × D^{2}.

Remark . Part (ii) does not hold without the almost triviality assumption on L̂ and Q̂. For example, let L̂ consist of just one component l_{1}, and let Q be a Hopf link contained in a small ball in S^{1}×D^{2}. Then L∪ϕ(Q) = ϕ(Q) is homotopically essential, yet L̂ is trivial.

In part (i), if either L or Q is homotopically essential, then their composition L ∪ ϕ(Q) is also essential. (Note that Q̂ and ϕ(Q̂) are homotopically equivalent, see Lemma 3.2 in [2].) If neither L nor Q is homotopically essential, then by deleting some components of L and Q if necessary, one may assume that L̂ and Q̂ are almost homotopically trivial (and still homotopically essential). In the case when L ∪ ϕ(Q) is not almost homotopically trivial part (i) follows immediately. Similarly, part (ii) can be proved in this case easily by induction on the number of components of L and Q.

Therefore, we will assume from now on that L̂, Q̂ and L ∪ ϕ(Q) are almost homotopically trivial links in S^{3}.

Lemma 5.1. If L̂ and Q̂ are both homotopically trivial in S^{3} then ϕ(∧) represents the trivial element in the Milnor group M(L ∪ ϕ(Q)).

Proof. Let ∧′ denote ϕ(S^{1} × {1}). The Milnor group M(L ∪ ϕ(Q)) is nilpotent of class k + m + 1, so it suffices to show that ϕ(∧) represents an element in π_{1}(S^{3} ∖ (L ∪ ϕ(Q)))^{k+m+1}. This will be achieved by constructing an ∞-grope G bounded by ϕ(∧) in the complement of L ∪ ϕ(Q). In fact, the construction also gives an ∞-grope G′ ⊂ S^{3} ∖ (L ∪ ϕ(Q)) bounded by ∧′.

Consider S^{1} × D^{2} as a standard unknotted solid torus in S^{3}, and let c denote the core of the complementary solid torus D^{2} × S^{1}. Since Q̂ is homotopically trivial, after changing Q by an appropriate link homotopy in S^{1}×D^{2}, ∧ bounds an immersed disk Δ ⊂ S^{3} in the complement of the new link. Denote the new link by Q again. Similarly L can be changed so that the untwisted parallel copy ∧′ of l_{k+1} bounds a disk Δ′ ⊂ S^{3} ∖ L. Recall that M(L ∪ ϕ(Q)) does not change if L ∪ ϕ(Q) is modified by a link homotopy.

The intersection number of Δ with c is trivial, since ∧ and c do not link. Replace the union of disks Δ∩(D^{2}×S^{1}) by annuli lying in ∂(D^{2}×S^{1}) to get Σ ⊂ S^{1}×D^{2}∖Q, an immersed surface bounded by ∧. Similarly the intersection number of Δ′ with the core circle of ϕ(S^{1} × D^{2}) is trivial, and ∧′ bounds Σ′ ⊂ S^{3} ∖ (L ∪ ϕ(Q)). The surfaces ϕ(Σ) and Σ′ are the first stages of the gropes G and G′ respectively.

Notice that half of the basis for H_{1}(ϕ(Σ)) is represented by parallel copies of ∧′. They bound the obvious surfaces: annuli connecting them with ∧′ union with Σ′, which provide the second stage for G. Since this construction is symmetric, it provides all higher stages for both G and G′.

Lemma 5.2. Let i : S^{3}∖ neighborhood (L̂ ∖ l_{1}) ⟶ S^{3} ∖ (L ∪ ϕ(Q) ∖ l_{1}) denote the inclusion map, and let i_{#} be the induced map on π_{1}. Then i_{#} induces a well defined map i_{∗} of Milnor groups.

Remark . Given two groups G and H normally generated by g_{i} respectively h_{j}, let MG and MH be their Milnor groups defined with respect to the given sets of normal generators. If a homomorphism ϕ : G ⟶ H maps each g_{i} to one of the h_{j} then it induces a homomorphism Mϕ : MG ⟶ MH. In general, ϕ : G ⟶ H induces a homomorphism of the Milnor groups if and only if ϕ(g_{i}) commutes with ϕ(g_{i})^{ϕ(g)} in MH for all i and all g ∈ G.

Proof of Lemma 5.2. The Milnor groups M(L̂∖l_{1}) and M(L∪ϕ(Q)∖l_{1}) are generated by meridians. Moreover, i_{#}(m_{i}) = m_{i} for i = 2, …, k and i_{#}(m_{k+1}) = ϕ(∧) where m_{1}, …, m_{k+1} are meridians to the components of L̂. Hence to show that i_{∗} is welldefined it suffices to prove that all the commutators

[ϕ(∧), (ϕ∧)^{i_{#}(g)}], g ∈ π_{1}(S^{3} ∖ (L̂ ∖ l_{1})),

are trivial in M(L ∪ ϕ(Q) ∖ l_{1})). Consider the following exact sequence, obtained from deleting the component q_{1} of Q.

ker(ψ) ⟶ M(L ∪ ϕ(Q) ∖ l_{1}) ⟶^{ψ} M(L ∪ ϕ(Q) ∖ (l_{1} ∪ ϕ(q_{1}))) ⟶ 0

An application of Lemma 5.1 to (L̂ ∖ l_{1}) and to (Q̂ ∖ q_{1}) shows that ψ(ϕ(∧)) = 1 and hence ϕ(∧), ϕ(∧)^{g} ∈ ker(ψ). The observation that ker(ψ) is generated by the meridians to ϕ(q_{1}) and hence is commutative finishes the proof of Lemma 5.2.

Proof of Theorem 3. Let M(F_{m_{1},... ,m_{s+1}}) be the Milnor group of a free group, i.e. the Milnor group of the trivial link on s + 1 components with meridians m_{i}. Let R(y_{1}, …, y_{s}) be the quotient of the free associative ring on generators y_{1}, …, y_{s} by the ideal generated by the monomials y_{i_{1}} ⋯ y_{i_{r}} with one index occurring at least twice. The additive group (R(y_{1}, …, y_{s}), +) of this ring is free abelian on generators y_{i_{1}} ⋯ y_{i_{r}} where all indices are distinct. Milnor [10] showed that setting m_{s+1} = 1 induces a short exact sequence of groups

1 ⟶ (R(y_{1}, …, y_{s}), +) ⟶^{r} M(F_{m_{1},... ,m_{s+1}}) ⟶^{i} M(F_{m_{1},... ,m_{s}}) ⟶ 1

where r is defined on the above free generators by left-iterated commutators with m_{s+1}:

r(y_{j_{1}} ⋯ y_{j_{k}}) := [m_{j_{1}}, [m_{j_{2}}, …, [m_{j_{k}}, m_{s+1}] …]].

In particular, r(0) = 1 and r(1) = m_{s+1}. Obviously, the above extension of groups splits by sending m_{i} to m_{i}. This splitting induces the following conjugation action of M(F_{m_{1},... ,m_{s}}) on R(y_{1}, …, y_{s}). Let Y := y_{j_{1}} ⋯ y_{j_{k}}, then

m_{i} · r(Y) · m_{i}^{−1} = [m_{i}, r(Y)] · r(Y) = [m_{i}, [m_{j_{1}}, [m_{j_{2}}, …, [m_{j_{k}}, m_{s+1}] …]] · r(Y)
= r((y_{i} + 1) · Y)

which implies that m_{i} acts on R(y_{1}, …, y_{s}) by ring multiplication with y_{i} + 1 on the left. Since m_{i} generate the group M(F_{m_{1},... ,m_{s}}) this defines a well defined homomorphism of M(F_{m_{1},... ,m_{s}}) into the units of the ring R(y_{1}, …, y_{s}). In fact, this is the Magnus expansion, well known in the context of free groups (rather than free Milnor groups). We conclude in particular, that the abelian group (R(y_{1}, …, y_{s}), +) is generated by y_{i} as a module over the group M(F_{m_{1},... ,m_{s}}).

Returning to the notation of Theorem 3, we have the following commutative diagram of group extensions. We use the fact that the links L ∪ ϕ(Q) ∖ l_{1} and L̂ ∖ l_{1} are homotopically trivial. Here y_{i} are the variables corresponding to the link L and z_{j} are the variables corresponding to ϕ(Q).

R(y_{2}, ..., y_{k}, z_{2}, ..., z_{m}) ⟶^{r} M(L ∪ ϕ(Q) ∖ l_{1}) ⟶^{i} M(L ∪ ϕ(Q) ∖ (l_{1} ∪ ϕ(q_{1})))
↑^{σ}  ↑^{lc}  ↑^{lc}
R(y_{2}, …, y_{k})  ⟶^{r̄}  M(L̂ ∖ l_{1})  ⟶^{j}  M(L ∖ l_{1})

Recall that by definition lc(m_{i}) = m_{i} for all meridians m_{2}, …, m_{k} of L∖l_{1}. Moreover, the link composition map lc sends the meridian m_{k+1} to the ∧-curve of ϕ(Q).

The existence of the homomorphism lc on the Milnor group level already implies our claim (ii) in Theorem 3: By assumption, l_{1} represents the trivial element in M(L̂ ∖ l_{1}) since L̂ is homotopically trivial. Consequently, lc(l_{1}) = l_{1} is also trivial in M(L ∪ ϕ(Q) ∖ l_{1}) and hence by [10] the link L ∪ ϕ(Q) is homotopically trivial.

The key fact in our approach to part (i) of Theorem 3 is the following result which says that link composition corresponds to ring multiplication.

Lemma 5.3. The homomorphism σ : R(y_{2}, …, y_{k}) ⟶ R(y_{2}, …, y_{k}, z_{2}, … z_{m}) is given by ring multiplication with r^{−1}(∧) on the right.

Note that by Lemma 5.1 ∧ is trivial in M(L ∪ ϕ(Q) ∖ (l_{1} ∪ ϕ(q_{1}))), so that it makes sense to consider r^{−1}(∧). We will abbreviate this important element by ∧_{R}.

Proof of Lemma 5.3. Since the above diagram commutes and (R(y_{2}, …, y_{k}), +) is generated by y_{i} as a module over the group M(F_{m_{2},... ,m_{k}}) it suffices to check our claim for these generators y_{i}. We get by definition

lc(r̄(y_{i})) = lc([m_{i}, m_{k+1}]) = [m_{i}, ∧] = (m_{i} · ∧ · m_{i}^{−1}) · ∧^{−1} = r((y_{i} + 1) · ∧_{R}) · ∧^{−1}
= r(y_{i} · ∧_{R}).

We are using the fact that conjugation by m_{i} corresponds to left multiplication by (y_{i} + 1).

Since L is homotopically trivial and L̂ is homotopically essential, it follows that 0 ≠ l_{1} ∈ ker(j). After possibly reordering the y_{i} this implies in addition that for some integer a ≠ 0 we have

r̄^{−1}(l_{1}) = a · (y_{2} ⋯ y_{k}) + terms obtained by permutations from y_{2} ⋯ y_{k}.

Setting all the meridians m_{i} of L to 1 (which implies setting the variables y_{i} to 0), we get a commutative diagram of group extensions

R(z_{2}, …, z_{m})  ⟶^{r}  M(ϕ(Q))  ⟶^{i}  M(ϕ(Q) ∖ ϕ(q_{1}))
↑^{p}  ↑  ↑
R(y_{2}, ..., y_{k}, z_{2}, ..., z_{m}) ⟶^{r} M(L ∪ ϕ(Q) ∖ l_{1}) ⟶^{i} M(L ∪ ϕ(Q) ∖ (l_{1} ∪ ϕ(q_{1})))

Since Q̂ (and equivalently ϕ(Q̂)) is homotopically essential we have 0 ≠ ∧ ∈ ker(i). This shows that p(∧_{R}) ≠ 0. The almost triviality of Q̂ implies in addition that after possibly reordering the z_{j} we have for some integer b ≠ 0

p(∧_{R}) = b · (z_{2} ⋯ z_{m}) + terms obtained by permutations from z_{2} ⋯ z_{m}.

It follows from Lemma 5.3 that r^{−1}(l_{1}) = r̄^{−1}(l_{1}) · ∧_{R}. This product contains the term

ab · (y_{2} ⋯ y_{k} · z_{2} ⋯ z_{m}),

the coefficient ab of which is non-zero. This completes the proof of Theorem 3.

Remark . Those readers who are familiar with Milnor’s μ̄-invariants will have recognized that the above proof in fact shows that the first non-vanishing μ̄-invariants are multiplicative under link composition.

References

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[2] M. H. Freedman and X. S. Lin, On the (A, B)-slice problem, Topology Vol. 28 (1989), No. 1, 91-110.

[3] M. H. Freedman and F. Quinn, The topology of 4-manifolds, Princeton Math. Series 39, Princeton, NJ, 1990.

[4] M. H. Freedman and P. Teichner, 4-Manifold Topology I: Subexponential Groups., Invent. Math., 122 (1995) no.3, 509-529.

[5] M. H. Freedman and P. Teichner, 4-Manifold Topology II: Dwyer’s filtration and surgery kernels, Invent. Math., 122 (1995) no.3, 531-557.

[6] C. Giffen, Link concordance implies link homotopy, Math. Scand., 45 (1979), 243-254.

[7] D. Goldsmith, Concordance implies homotopy for classical links in S^{3}, Comment. Math. Helvitici, 54 (1979), 347-355.

[8] V. S. Krushkal, Additivity properties of Milnor’s μ̄-invariants, to appear.

[9] X. S. Lin, On equivalence relations of links in 3-manifolds, Preprint (1985).

[10] J. Milnor, Link Groups, Ann. Math 59 (1954), 177-195.

[11] J. Milnor, Isotopy of links, Algebraic geometry and topology, Princeton Univ. Press, 1957, pp. 280-306.

[12] J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 1970-1981.

[13] P. Teichner, Stratified Morse Theory and Link Homotopy, Habilitationsschrift, in preparation.

Vyacheslav S. Krushkal, Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027

  E-mail address: krushkal@math.msu.edu

Peter Teichner, Department of Mathematics, University of California in San Diego, La Jolla, CA, 92093-0112

Current address: Department of Mathematics, University of California in Berkeley, CA, 94720-5190

  E-mail address: teichner@euclid.ucsd.edu