Annals of Mathematics, 0 (1991), 1–???

Not all Links are Concordant to Boundary Links

By Tim D. Cochran and Kent E. Orr*

Introduction

A link is a smooth, oriented submanifold L = {K_{1}, …, K_{m}} of S^{n+2} which is the ordered disjoint union of m manifolds each piecewise linearly homeomorphic to the n-sphere (if m = 1 L is called a knot). Knots and links play an essential role in the classification of manifolds and, in this regard, perhaps the most important equivalence relation on links is that of link concordance. L_{0} and L_{1} are concordant if there is a smooth, oriented submanifold C = {C_{1}, …, C_{m}} of S^{n+2} × [0, 1] which meets the boundary transversely in ∂C, is piecewise linearly homeomorphic to L_{0} × [0, 1] and meets S^{n+2} × {i} in L_{i} for i = 0, 1.

In the mid-1960’s M. Kervaire and J. Levine gave an algebraic classification of the high-dimensional (n > 1) knot concordance groups [26, 22]. For even n these are the trivial group and for odd n they are infinitely generated.

Extending Levine’s knot classification to links is difficult for several reasons. Firstly, the natural operation of connected-sum is not well defined on the set of link concordance classes unless m = 1. Secondly, although it is known that the fundamental group of the exterior of the link (π_{1}(S^{n+2}−(L×D^{2}))) carries vital information, these groups are large, typically containing non-abelian free subgroups. Moreover these groups are not invariant under concordance. The inclusion of a link exterior into a concordance exterior induces an isomorphism on integral homology but not on fundamental group.

However, the knot classification techniques do extend to a special class of links called boundary links. Recall that a Seifert surface for a component K_{i} of L is a compact, oriented, connected submanifold of S^{n+2} whose boundary is K_{i}. A boundary link is one which admits a collection of pairwise disjoint Seifert surfaces, one for each of its components. Since Kervaire and Levine relied heavily on Seifert surfaces, their techniques extend. Cappell and Shaneson were the first to classify boundary links modulo a similarly restricted type of concordance. Their technique of “homology surgery” was different than Levine’s even in the case of knots, and suggested further progress, culminating in the work of J. Le Dimet (see §4) [3, 24].

*Research by Tim D. Cochran is partially supported by NSF grant #DMS-8903514. Research by Kent E. Orr is partially supported by National Science Foundation Postdoctoral Fellowship

The only heretofore realized obstructions to a links being concordant to a boundary link were Massey products in the link exterior (so called higherorder linking numbers) or, equivalently, Milnor’s \overline{μ}-invariants (see [18]). These always vanish if n > 1.

In this paper we define new concordance invariants for odd-dimensional links. We then evaluate these invariants on new classes of links to produce infinite numbers of counterexamples to the long-standing conjecture that every higher-dimensional link is concordant to a boundary link and to the analagous conjecture for links in S^{3}. These were announced in [10].

The invariants are obtained as follows (this is discussed in detail in §1). To a link L we associate various “covering links” L̃ by performing various iterated p-fold covering spaces. To L̃ we associate a pair (c, λ) where c is an integer (the complexity) which measures the deviation of L from a boundary link; and λ is the Witt class of a Blanchfield linking form of L̃, lying in the Witt group W = W(R, S) of all non-singular (±)-Hermitian linking forms on finitely-generated S-torsion R-modules. Here R is the group ring [Z]_{(p)}[Z^{m}] and S = {1 + i | i ∈ I} where I is the augmentation ideal of R. Under concordance both λ and c may change, but they do so in a compatible fashion. For each positive integer k relatively prime to p set W_{k} = W. The W_{k} form a directed system as follows. If j divides k then take f_{j,k} : W_{j} ⟶ W_{k} to be the endomorphism induced by the group endomorphism of Z^{m} given by t_{i} ⟶ t_{i}^{k/j}. Our invariant λ̂(L) is then the image of λ ∈ W_{c} in the direct limit Ŵ = \underrightarrow{lim} W_{k}. Boundary links are seen to have complexity 1 and Ŵ/(image W_{1}) is seen to be infinitely generated over Q and to have infinitely generated 2 and 4-torsion subgroups. We do not herein address the general realization problem for our invariants. However this is well understood for an important class, homology boundary links, (defined in §3) and this enables us to exhibit specific families of links (the simplest of which is shown in Figure 3.6) realizing the above algebraic phenomena.

Proposition 3.11. There are infinitely many concordance classes of links in S^{2n+1}, (with vanishing Massey products if n = 1), containing no links concordant to boundary links.

Among these are infinite families with the properties that framed surgery on the components of the link yields a manifold homeomorphic to framed surgery on a boundary link, and that (if n > 1) π_{1} of the link exterior is the free group of rank m. If arcs are chosen for each of these links, so that connected sum is well-defined, then these infinite families form linearly independent sets.

This is to say that the family comes from an infinite linearly independent set of disk links (see §4). Note that we have no results for even-dimensional links.

There presently exists only one algebraic model through which link concordance might be effectively classified, and it is therefore very important to see how our invariants fit into and contribute to our knowledge of this model. Indeed, we first formulated our invariants from within this viewpoint. This model is based on work of Cappell and Shaneson, Jean Le Dimêt and Pierre Vogel [3, 24]. Vogel’s work on localization of spaces provided the correct “universal space” EW to “carry” all possible link concordance invariants. Le Dimêt espoused the viewpoint of first classifying concordance classes of links of disks (which, like braids, have a natural group structure) and then dividing out by the action of some group to get the set of link concordance classes. He also defined a “universal homotopy invariant” associated to EW whose vanishing allows for the set-up of a problem in homology surgery. Cappell and Shaneson, of course, defined the entire machine of homology surgery to classify homology cobordisms in general and concordances in particular in terms of their Γ-groups, Witt groups of forms over (in this case) the group ring of π_{1}(EW) = F^{∧}. Moreover their outline of the problem in [3] pre-dated Le Dimet and was the context of his work. We briefly sketch the relationship of our invariants to this framework and provide more details in section four. The full proof of the equality of the two approaches is technical and will appear in [13]. The interested reader should consult chapter 4 and the primary references [3, 24, 27].

In [24], Le Dimet classifies higher-dimensional m-component disk links up to disk link concordance. An m-component n-dimensional disk link is a piecewise linear locally flat embedding {1, …, m} × D^{n} ⟶ D^{n+2}, transverse to the boundary and inducing a fixed trivial embedding on the boundary. Concordances are assumed to be constant on the boundary. Le Dimet establishes an exact sequence

… ⟶ surgery-theoretic Γgroup ⟶^{h} C_{m,n} ⟶^{θ} homotopy theory of EW ⟶

where C_{m,n} is the group of concordance classes of m-component n-dimensional disk links under the operation of concatenation. Of course every disk link may be “closed” to yield a “true” spherical link but many disk links close to the same spherical link. Thus the set of concordances classes of links is the orbit space of C_{m,n} under the action of a certain group of automorphisms. The kernel of θ is known to be essentially those links which are concordant to sublinks of homology boundary links (see §3), and in particular includes all links concordant to boundary links [32, 28]. Our invariants live in the Γ group and have the advantage that they change very little under the action of the above-mentioned group of automorphisms. Our examples are sublinks of homology boundary links whose concordances classes can be distinguished, in the Γ group, from those of all boundary links. Therefore it is a consequence of our results that this Γ group is much larger, even after considering the orbit space, than the subgroup given by boundary links, which itself is largely understood [3]. Indeed we find that the quotient of the large group by the subgroups of boundary links, loosely speaking, has infinite Q rank as well as infinitely generated 2 and 4-torsion subgroups.

Subsequent to our work, several excellent papers on these and related issues have appeared. The interested reader may consult [17, 29].

The paper is organized as follows. Section 1 defines complexity and establishes our invariants. Section 2 contains the technical theorems necessary to evaluate Ŵ/W_{1}. In §3 we relate complexity to previously known invariants, exhibit families of examples and calculate our invariants for them in order to prove our main theorems. We also describe a geometrical procedure to realize, for homology boundary links, any possible (c, λ). In Section 4, we sketch the relationship between our invariants and the classification scheme of Le Dimet-Vogel-Cappell-Shaneson. Section 5 lists important open questions.

A careful analysis of a theorem of Hillman along with the classification results of Le Dimet sparked our thinking [18, Thm. 5 p. 150]. We are indebted to work of S. Cappell, R. De Meo, J. Hillmann, A. Kawauchi, J. Le Dimet, J. Levine, J. Shaneson and P. Vogel. We would like to thank Jim Davis and Bruce Williams for their help with various aspects of L-theory. In addition we are grateful for helpful conversations with W. Richter.

We shall work in the piecewise-linear category. The symbols Z_{p} and Z_{(p)} denote Z/pZ and the integers localized at p respectively. A good general reference for links, link concordance, Milnor’s \overline{μ}-invariants, homology boundary links and Blanchfield pairings for links is [18]. References for the Γ-group connection are [3, 24, 39, 36, 32]. For discussion of algebraic closure of groups see [27, 28]. For a good discussion of the inter-relations between Γ-groups and universal Blanchfield linking forms see [15, 13, 14].

1. Complexities of links

We shall be concerned with links in Z_{(p)}-homology spheres (p prime). In particular, by a p-link L we mean an m-component ordered, oriented link of n-spheres in an (n + 2)-dimensional Z_{(p)}-homology sphere Σ. Similarly a pconcordance of p-links L_{0} and L_{1} is an embedding of m copies of S^{n} × [0, 1], denoted C, in a manifold W which is a Z_{(p)}-homology S^{n+2} × [0, 1] such that (∂W, C ∩ ∂W) is (Σ_{0}, L_{0}) ⨿ −(Σ_{1}, L_{1}).

Our interest in such links arises as follows. If L′ = {J_{0}, J_{1}, …, J_{k}} is a link in a Z_{(p)}-homology sphere Σ′, then, by Alexander Duality, H_{1}(E_{L′}, Z_{(p)}) ≅ (Z_{(p)})^{m} generated by the meridians where E_{L′} denotes the exterior of L′. Then it is well-known that the p-fold cyclic cover of Σ′ branched over J_{0} is also a Z_{(p)}- homology sphere Σ (prime power covers or more general finite p-group covers are also possible although any finite p-group is polycyclic) [6; Lemma 4.2]. Consider the pair consisting of Σ and the lifts of the components of L′ (possibly including the branch set). This covering link L is oriented but in general not canonically ordered. To canonically order L, choose one component in each non-trivial Z_{p}-orbit of L. The first of these choices will not affect any of our invariants since Z_{p} acts as an equivalence of ordered links, for a total of k − 1 arbitrary choices. Note that if L′ is a 2-component link, L can be considered ordered. In general, together with these choices, our pair is a p-link as above. Moreover, if L′_{0} is concordant to L′_{1} then, by branching over the first component of a concordance, the lifts of the components of the concordance constitute a p-concordance between the two covering links L_{0} and L_{1}. Thus any invariant of the p-concordance class of a covering link L is an invariant of the p-concordance class of L′. We shall consider in particular the case that L′ is a link in S^{n+2}, whereupon we shall, by iterated branched covers, derive sequences of covering links L each of whose p-concordance class restricts the honest concordance class of the original link L′.

Suppose L = {K_{1}, …, K_{m}} is a link in an oriented Z_{(p)}-homology (n+2)- sphere Σ. By Alexander duality, H_{1}(E_{L}; Z_{(p)}) ≅ (Z_{(p)})^{m} generated by the meridians. However, in general a set of meridians will fail to generate the torsion-free part of the integral homology. We shall define the complexity of L to be a positive integer which measures this failure and we will see that covering links of boundary links have complexity 1, whereas covering links of homology boundary links generally have greater complexity.

Proposition 1.1. Let {u_{i}|i = 1, …, m} be the elements of H_{1}(E_{L}; Z) represented by a set of meridians of L. The following describe the same positive integer c(L), relatively prime to p, which we call the complexity (or pcomplexity) of L.

(1) The complexity of L is the least positive integer c for which there exists a homomorphism Ψ : H_{1}(E_{L}; Z) → Z^{m} which sends u_{i} to ce_{i} (e_{i} = (0, …, 0, 1, 0 …, 0)) for i = 1, …, m. This is clearly independent of the ordering of the components of L. Moreover the least such integer divides all other such integers.

(2) For any epimorphism ϕ : H_{1}(E_{L}; Z) ↠ Z^{m}, if M : Z^{m} ↠ H_{1}(E_{L}; Z) is the meridian map, let C be the matrix representing ϕ ∘ M (the complexity matrix). The complexity of L is the least positive integer c such that cC^{−1} is an integral matrix. (That is [det C/d] where d is the greatest common divisor of the entries of the adjoint C^{∗}).

(3) The exponent c of the quotient of (H_{1}(E_{L})/Torsion) by the subgroup generated by the meridians of L.

Proof of 1.1. The integer c_{2} of 2) clearly exists and divides det C, so is coprime to p since the u_{i} generate with Z_{(p)} coefficients. Moreover this c_{2} is independent of C since H_{1}(E_{L}) modulo torsion is Z^{m}, so C is unique up to an element of GL(m, Z). Letting Ψ = c_{2}C^{−1} ∘ ϕ it follows that c_{1} exists and is less than the c_{2}. On the other hand, given Ψ, and letting C be Ψ ∘ M with range restricted to its image and A be the inclusion of im Ψ to Z^{m}, A is c_{1}C^{−1} so c_{2} ≤ c_{1}.

If Ψ′ is such that Ψ′(u_{i}) = c′e_{i} then, similarly, A′ = c′C^{−1} is integral so c′ is a multiple of det C/d.

The equivalence with 3) is left to the reader.

We remark that if L is a covering link then c divides the order of a longitude of the covering link in H_{n}(Σ; Z) and is usually equal to it. In addition, note that the map of 1) induces (by killing meridians) a “character” on Σ; that is, an element of Hom(H_{1}(Σ), (Z_{c})^{m}) ⊂ (H^{1}(Σ; Q/Z))^{m}. Since we will be considering signature invariants associated to such Ψ, Casson-Gordon invariants are suggested [6].

Definition 1.2. L is a boundary link if the components of L bound disjoint Seifert surfaces in Σ. Equivalently, π_{1}(E_{L}) will admit an epimorphism to the free group F ⟨x_{1}, …, x_{m}⟩ which sends meridians to generators (see [37]).

Proposition 1.3. A covering link L of a boundary link is a boundary link. The complexity of a covering link L of a boundary link is 1.

Proof of 1.3. The Seifert surfaces clearly lift to the covering space and there induce an epimorphism from π_{1}(E_{L}) to F ⟨x_{1}, …, x_{m}⟩ taking meridians to generators.

Thus complexities of covering links distinguish links from boundary links. However these integers are not invariant under concordance. Indeed, it follows from [9, Thm. 2.3] that the complexities of covering links of ribbon links achieve all possible values. More philosophically, note that the complexities of covering links of a link L are purely functions of the group of L together with a set of meridians; but for any higher-dimensional link group and set of normal generators, Kervaire has shown how to construct a null-concordant link with this group data [22].

We shall henceforth assume that n is odd (n + 2 = 2q + 1). Moreover when q = 1, we will restrict our attention to links satisfying a strengthening (possibly) of the condition that the mod p Massey products of E_{L} vanish. Let G = G(L) = π_{1}(E_{L}).

Definition. A link L in a 3-dimensional Z_{(p)}-homology sphere is said to be of type L if

(i) each component of L has a parallel which is torsion in H_{1}(E_{L}; Z) (such parallel is then unique and called a longitude). (ii) there is a finitely-generated group P and a homomorphism f : G(L) → P such that f induces a 2-connected map on Z_{p}-homology. In addition the longitudes of L lie in kernel (f).

Remarks. If L is in S^{3} and the longitudes of L lie in the intersection of the transfinite mod p lower-central series, denoted G_{δ,p}, then L has property L using P = G/G_{δ,p}. The latter has vanishing H_{2} by Stallings’ theorem [38]. The only presently known links whose longitudes lie in the intersection of the finite lower-central series, namely fusions of boundary links (see [7, §8.5] [8]), are known to satisfy this stronger condition above and hence have Property L. The groups of fusions of boundary links map onto E-groups with longitudes dying [8, Thm. 2.9].

In order to show our invariants apply to particular classes of links in S^{3} we need the following.

Theorem 1.4. A sublink of a link with Property L has Property L. A covering link of a link with Property L has Property L. Therefore any covering link of a link which is concordant to a fusion of a boundary link has Property L.

Proof of 1.4. We shall show that a covering link L̃ of a link L of type L is also of type L. Suppose f : G → P is a group homomorphism, 2-connected on H_{∗}( ; Z_{p}). Suppose ϕ : P ↠ Z_{p} is an epimorphism, P̃ = ker(ϕ), G̃ = ker(ϕ ∘ f). Then f̃: G̃ → P̃ is also 2-connected on H_{∗}( ; Z_{p}). This may be proved in our situation, where ϕ factors through Z, by replacing the groups by Eilenberg-Maclane spaces and applying the ideas of the proof of [6; Lemma 4.2] to the mapping cylinder of f. More details may appear in [11]. Now suppose L is of type L with π_{1}(E_{L}) ≅ G. Since P is finitely-generated and H_{2}(P; Z_{p}) ≅ 0, H_{1}(P; Z) is Z^{m} × T where T is torsion relatively prime to p. It follows that there is a map g : F ⟨x_{1}, …, x_{m}⟩ ⟶ P which is 2-connected on Z_{p} homology, and hence that g̃ : F̃ → P̃ is also 2-connected. Since F̃ is free, H_{2}(P̃; Z_{p}) ≅ 0. Thus f̃: G̃ → P̃ satisfies all of the conditions of ii, and G̃ = π_{1}(E_{L}). Since the longitudes of L lay in the kernel of f, the longitudes of L̃ lie in the kernel of f̃, but since f̃ induces an isomorphism on H_{1}( ; Z_{p}), these longitudes must be torsion in H_{1}(G̃; Z). Hence L̃ is of type L.

If L is a fusion of a boundary link then L is an E-link in the sense of [8; p. 643] and so its group maps to such a group P. The concordance part of the last statement follows from [28; Theorem 1].

Let X(L) or merely X be the closed 2q+1 manifold obtained by performing surgery (longitudinal surgery if q = 1) on the components of L in Σ.

Proposition 1.5. There is a unique homomorphism Ψ : H_{1}(E_{L}; Z) → H_{1}(X(L); Z) → Z^{m} which satisfies Ψ(u_{i}) = c(L)e_{i} and this is given by cC^{−1} in the notation of 1.1 2 ).

Proof. Since L is of type L the surgery does not affect H_{1} modulo torsion. Any two such homomorphisms would agree on the finite index lattice generated by the u_{i} and hence would be identical.

The homomorphism Ψ of 1.5 induces a Z^{m}-covering space X̃(Ψ) of X which we call the covering space associated to Ψ. A description of X̃(Ψ) is obtained as follows. Let Ψ′ be epimorphism associated to Ψ by restricting its range to H = image (Ψ) ≈ Z^{m}. Let Ỹ denote the connected H-cover of X associated to Ψ′. Let f : H → Z^{m} be the inclusion. Recall that for any embedding f : H → G, the pull-back f^{∗}(EG) is a disjoint union of copies of EH indexed by the cosets of H in G. Applying this in our case, we see that X̃(Ψ) is a disjoint union of copies of Ỹ, indexed by cokernel (Ψ). Choosing a basepoint in X̃ fixes an identification of the set of components with the group cokernel (Ψ) and hence fixes an action of G = Z^{m} on X̃ which extends the action of H on (Ỹ, ∗). This is unique up to covering space isomorphism.

To any compact (n+2)-manifold X and such a covering space X̃(Ψ), R.C. Blanchfield (following Reidermeister) associated a linking pairing. Let R be the group ring Z_{(p)}[Z^{m}] with involution induced by \overline{t} = t^{−1} for t ∈ Z. Let S be 1 + I (I is the augmentation ideal of R → Z_{(p)}). Consider the S-torsion submodule of the R-module H_{i}(X̃(Ψ); Z_{(p)}) and let T_{i}(X̃) be its quotient by its maximal pseudo-zero submodule [18, p. 32, 117]. Blanchfield’s pairing:

λ : T_{i}(X̃) × T_{n−1−i}(X̃, ∂X̃) → S^{−1}R/R

is sesquilinear and primitive [2, Thm. 3.2], [18, p. 120, 117]. If we let X = X(L) with n + 2 = 2q + 1 and i = q then, since X is closed, gives a non-singular (−1)^{q+1}-Hermitian linking form on T_{q}(X̃).

Consider the Witt group W = W(R, S) of all non-singular (−1)^{q+1}- Hermitian linking forms on finitely-generated S-torsion R modules (of course R and thus W depends on p and m). Moreover, if L is a covering link, Z_{p} acts on X(L). The action on the meridians of L defines a Z_{p}-module structure on Z^{m} under which the complexity map of 1.5 is equivariant. Hence T_{i}(X) is a Z_{p}-module and it is easy to see from the definition of λ that it is invariant under the induced Z_{p}-action on W. For each positive integer k which is relatively prime to p set W_{k} = W. The W_{k} form a directed system as follows. If j divides k, define the homomorphism f_{j,k} : W_{j} → W_{k} induced by the group endomorphism of Z^{m} given by e_{i} → (k/j)e_{i} (multiplicatively this is t_{i} → t_{i}^{k/j} where t_{i} = e_{i} = (0, …, 1, …, 0).)

Definition 1.6. Define λ_{L} ∈ W_{c(L)} to be the Witt class of the Blanchfield pairing associated to X(L) by 1.5. Let λ̂_{L} ∈ Ŵ be the image of λ_{L} in the direct limit Ŵ ≡ lim_{⟶} W_{k}.

Theorem 1.7. If L and L′ are p-concordant p-links (with Property L if q = 1) then λ̂_{L} ≡ λ̂_{L′}.

Proof of 1.7. Let C be the concordance in the (2q + 2)-manifold W with exterior E(C) and let Z denote the (zero) framed surgery on C. Then the statements and proofs of 1.1 and 1.5 hold for E(C) and Z so there is defined a complexity of the concordance C, c(C), and a homomorphism Ψ_{Z} as in 1.5. Let Z̃ denote the corresponding covering space and λ_{Z} the corresponding Blanchfield form on T_{q}(Z̃) × T_{q+1} (Z̃, ∂Z̃). Then ∂Z = X ⨿ (−X′). If j and j′ are the inclusions then Ψ_{Z} ∘ j(u_{i}) = c(C)e_{i} = Ψ_{Z} ∘ j′(u_{i}) so, by 1.1, c(C) = dc(L) = d′c(L′) for some integers d, d′ prime to p.

Let ∂_{0}(Z̃) and ∂_{1}(Z̃) be the boundary of Z̃ corresponding to L and L′ respectively. These are covering spaces of X and X′ respectively corresponding to the maps D ∘ Ψ_{X} and D′ ∘ Ψ_{X′} where D is multiplication by d and D′ is multiplication by d′ (see 1.1).

First we shall apply a standard argument to prove that the Blanchfield form “∂λ_{Z}” (induced on ∂Z̃ by Ψ_{Z} ∘ j, Ψ_{Z} ∘ j′) is zero in W_{c(C)}. Secondly we show that ∂λ_{Z} is equal to f_{c(L),c(C)} (λ(L)) ⊕ −f_{c(L′),c(C)} (λ(L′)).

Since X → Z is a Z_{(p)}-homology equivalence, H_{q+1}(Z̃, ∂_{0}(Z̃)) is an Storsion module by a slight generalization of [36, Proposition 2.3], which we state below without proof.

Theorem 1.8. [36, Proposition 2.3] Let C_{∗} be a finitely-generated right projective chain complex over the commutative Noetherian ring RG (we need the cases G abelian, R = Z_{(p)} or Q or Z). Let F : RG → RH be induced by a group epimorphism, I = kernel (F), S = {1 + i | i ∈ I} and Λ = (RG)[S^{−1}]. Then, if C_{∗} ⊗_{RG} RH has vanishing homology in dimensions less than k, C_{∗} ⊗_{RG} Λ has vanishing homology in dimensions less than k. Thus H_{∗}(C_{∗}) ⊗ Λ = H_{∗}(C_{∗} ⊗ Λ) = 0 in dimensions less than k.

This establishes the exactness of the following sequence of S-torsion submodules

⟶ t_{q+1}(Z̃, ∂Z̃) ⟶^{∂} t_{q}(∂Z̃) ⟶^{j} t_{q}(Z̃) ⟶

for, if x ∈ H_{q+1}(Z̃, ∂Z̃) such that ∂x is S-torsion in H_{q}(∂Z̃), then ∂x is a fortiori S-torsion in H_{q}(∂Z̃, ∂_{0}Z̃). Therefore, for some s ∈ S, sx lies in the image of H_{q+1}(Z̃, ∂_{0}Z̃) → H_{q+1}(Z̃, ∂Z̃), implying that sx and hence x is S-torsion.

If P = Image ∂, then the image of P in T_{q}(∂Z̃) will be a “kernel” for the Blanchfield form ∂λ_{Z} using the standard argument [18, pg. 125]. This completes the first step.

Let X̃ be the covering space of X corresponding to Ψ, with covering group R = Z_{(p)}[τ_{1}, …, τ_{m}] and S = 1 + I. Let Ỹ be the covering space of X corresponding to f_{j,cj} ∘ Ψ with covering group R′ = Z_{(p)}[t_{1}, …, t_{m}] and S′ = 1 + I′. In general, f_{j,k}(M, λ) = (M ⊗_{R} R′, f_{∗}λ) where τ_{i} → t_{i}^{k/j}. Thus it must first be shown that T_{q}(X̃) ⊗_{R} R′ ≅ T_{q}(Ỹ). (This will fail in general. For example if m = 1, c = 3 and H_{q}(X̃) = Z[t, t^{−1}]/ < t − 1 > then H_{q}(Ỹ) will be Z[t, t^{−1}]/⟨t^{3} − 1⟩ so the S-torsion of H_{q}(X̃) is zero whereas the S′-torsion of H_{q}(Ỹ) is not!) It is clear that H_{q}(X̃) ⊗_{R} R′ ≅ H_{q}(Ỹ) as R′ modules and it is easy to see that t_{q}(X̃) ⊗_{R} R′ ⊂ t_{q}(Ỹ). The meridian map ⋁_{i=1}^{m} S^{1} → X induces a map on Z_{(p)}-homology which is at least q-connected. Consequently H_{q}(X̃, Ṽ S^{1}) is S-torsion, by 1.8. If q > 1 this implies that H_{q}(X̃) = t_{q}(X̃), establishing that t_{q}(X̃) ⊗_{R} R′ = t_{q}(Ỹ). It follows easily from the definition of λ′ (see below) that T_{q}(X̃) ⊗_{R} R′ = T_{q}(Ỹ).

If q = 1, more argument is required. First we show (by essentially rank considerations) that t_{1}(Ỹ) is contained in F ⊗_{R} R′ where F is the R-torsion submodule of H_{1}(X̃). Then, if L has type L we show that F is S-torsion. This will complete the proof that T_{q}(X̃) ⊗_{R} R′ = T_{q}(Ỹ) in all cases.

It suffices to consider the case R ⟶^{g} R′ by τ_{1} → t_{1}^{c} and τ_{i} → t_{i} if i > 1, since f above is a composition of such homomorphisms. Then H_{1}(Ỹ)

as an R-module can be identified with ⨁_{j=0}^{c−1}(H_{1}(X̃)) where x ⟶ (x_{j}) if x =

∑_{j=0}^{c−1}(x_{j} ⊗ t_{1}^{j}). Suppose zx = 0 for some z(t_{1}, …, t_{m}) ∈ S. Since t_{1} acts as an

automorphism on H_{1}(Ỹ), we can assume t_{1} does not divide z. We can express

z as ∑_{i=0}^{c−1}(g ∘ q_{i}(τ_{1}, …, τ_{m}))t_{1}^{i}. Considering z as an R-module endomorphism

of ⨁_{j=0}^{c−1}(H_{1}(X̃)) gives a c by c matrix Z over R which annihilates the column

vector (x_{j}). This matrix is obtained from the matrix whose rows are the cyclic permutations of {q_{0}, q_{1}, …, q_{c−1}} by multiplying each entry below the diagonal by τ_{1}. If det Z were zero then it would be divisible by each power of τ_{1}. It is not difficult to see that this would necessitate τ_{1} dividing each q_{i} hence t_{1}^{c} dividing z. Since this is not the case, Z is invertible in the quotient field of R, implying that each x_{j} is R-torsion, which proves the claim.

Now suppose that L and L′ have Property L. Then we have the homomorphisms F ⟶^{M} G ⟶^{f} P ↠ Z^{m} where, on Z_{(p)}-homology, M is 1-connected and f ∘M is 2-connected. Consider H_{1} of the maximal free abelian cover of P (BP) as an R′ = Z_{(p)}[t_{1}, …, t_{m}] module where T = 1 + I′. Let H_{1}(F̃) and H_{1}(G̃) be the induced R′-modules. By Smith’s results, T^{−1}H_{1}(F̃) → T^{−1}H_{1}(G̃) is onto and T^{−1}H_{1}(F̃) → T^{−1}H_{1}(P̃) is an isomorphism. It follows that T^{−1}H_{1}(G̃) is a free T^{−1}R′ module. Since H_{1}(G̃) is H_{1} of a covering space of X̃ (just like Ỹ), H_{1}(G̃) = H_{1}(X̃) ⊗_{R} R′. In particular if H_{1}(X̃) has R − S torsion then H_{1}(G̃) would have R′ − T torsion, which it does not.

Now we must show that the form λ′ on T_{q}(Ỹ) is equal to the image of λ under f_{j,jc}. It suffices to establish this for a map f sending τ_{1} → t_{1}^{n} and

τ_{i} → t_{i} for i > 1. Recall Ỹ ≅ ∐_{j=0}^{n−1}(X̃) with t_{1} permuting the copies. We identify

H_{q}(X̃) ⊗ t_{1}^{j} ⊂ H_{q}(Ỹ) with the homology of the j^{th}-component of Ỹ.

Let [·] and (·) denote the intersection form in Ỹ and X̃ respectively. Let J ≅ J′ ≅ Z^{m} where R ≡ Z_{(p)}[J] and R′ ≡ Z_{(p)}[J′]. Suppose x, y ∈ T_{q}(X̃) and α, β ∈ J′. If sx = ∂A for s ∈ S and A ∈ C_{q+1}(X̃) then ∂(A ⊗ α) = (x ⊗ α)f(s)

so λ′ on T_{q}(Ỹ) is given by λ′(x ⊗ α, y ⊗ β) = \frac{1}{f(s)} ∑_{g∈J′} [(A ⊗ α) · (y ⊗ βg)] g.

But clearly [A ⊗ α) · (y ⊗ βg)] = 0 unless βg ∈ f(J)α, so the sum becomes

= \frac{1}{f(s)} ∑_{h∈J} (A · yh) β^{−1}f(h)α

= β^{−1}f (\frac{1}{s} ∑_{h∈J}(A · yh)h) α

= β^{−1}f(λ(x, y))α.

But, by definition, (f_{∗}λ) (x ⊗ α, y ⊗ β) is β^{−1}f(λ(x, y))α as desired.

Theorem 1.7 gives us many sequences of concordance invariants, namely the Blanchfield forms of covering links (in general modulo the choices necessary to order the covering link).

Theorem 1.9. Suppose L′_{0} and L′_{1} are concordant links (with Property L if q = 1). Suppose L_{0} is a “covering link” of L′_{0} corresponding to (perhaps iterated) p-fold branched covering(s) as described above. Then, with regard to the “same” sequence of coverings to L′_{1}, there exists a set of choices of orderings of the covering links of L′_{1} (including intermediate covering links) such that λ̂_{L_{0}} = λ̂_{L_{1}} where L_{1} is the resulting covering link of L′_{1}. If L′_{0} is concordant to a boundary link then, for any covering link L_{0}, λ̂_{L_{0}} lies in the image of W_{1}.

Remark 1.10. Remember that in some important special cases the covering links may be assumed to be canonically ordered. These include the case of a single covering space of a 2-component link and the case of iterated 2-fold covers of a 2-component link, throwing away the lift of the branch set each time. In any case these orderings are irrelevant to detecting concordance to a boundary link. We shall not herein formalize these ambiguities further.

Proof. Inductively using [6, Lemma 4.2] the covering links are p-concor-dant. Ordering the intermediate covering links is necessary to speak of the “same” sequence of covering spaces. Invoke 1.4 if q = 1, then apply 1.7. Note that 1.3 is independent of any ordering.

In this paper we shall perform calculations by projecting Ŵ to a more understandable group ˆ \overline{W} as follows. First note that any epimorphism π : Z^{m} → Z extends in a unique way to π : (Z_{(p)})^{m} → Z_{(p)}. Therefore the induced homomorphism π_{∗} : W(Z_{(p)}[Z^{m}], S) → W(Z_{(p)}[Z], S) extends to a homomorphism π_{∗} : lim_{⟶} W_{j}(Z_{(p)}[Z^{m}], S) ⟶ lim_{⟶} W_{j}(Z_{(p)}[Z], S) where W_{j}(Z_{(p)}[Z], S) maps to W_{jk}(Z_{(p)}[Z], S) by t → t^{k}. Secondly, if S_{p} = {z ∈ Z[t, t^{−1}] ↪ Z_{(p)}[Z] | ϵ(z) is relatively prime to p}, then it is straightforward to see that W(Z_{(p)}[Z], S) = W(Z_{(p)}[Z], S_{p}), and the latter maps to W(Q[t, t^{−1}], S_{p}) which we call \overline{W}. Therefore any such epimorphism π induces π_{∗} : Ŵ → lim_{⟶} \overline{W} where

we denote the latter ˆ \overline{W}. Indeed it is easy to see that in fact π_{∗} maps W_{j} to \overline{W}_{j} for all j. Finally, since Q[t, t^{−1}] is a principal ideal domain, H_{q}(X̃) will have no pseudo-null submodule and T_{q}(X̃) will be homological dimension one. Thus \overline{W} is Ranicki’s (symmetric) relative L-group L(Q[t, t^{−1}], S_{p}) where S_{p} is the set of integral polynomials such that z(1) is relatively prime to p. These groups are well-understood as we shall see in Section 2. Moreover ˆ \overline{W} can be thought of as lying in L_{∗}(Q[Z_{(p)}], S_{p}).

Theorem 1.10. If L is an m-component covering link of L′ and π : Z^{m} → Z is an epimorphism then π_{∗}(λ̂_{L}) ∈ L_{∗}(Q[Z_{(p)}], S_{p}) is an invariant of the concordance class of L′ (modulo the ambiguity of ordering as in 1.9 if applicable). If L′ is a boundary link then this invariant is in the image of \overline{W}_{1} = L_{∗}(Q[t, t^{−1}], S_{p}).

2. Detecting Elements of Ŵ

In this chapter we establish that Ŵ/W_{1} is very large by showing that \frac{image(W_{nm}→_{W})}{image(W_{n}→_{W})} modulo torsion is infinitely generated if m > 1. We also describe

simple signature invariants of Ŵ which enable us to detect that the examples of §3 are not concordant to boundary links. The main theorem (2.2) is followed by several technical corollaries which will be called upon in §3 to show that the torsion free part of the algebraic knot concordance group “embeds” (in many ways) in the set of concordance classes of homology boundary links modulo boundary links. These corollaries also convey the (correct) impression that very few odd-dimensional homology boundary links are concordant to boundary links.

Let p be a fixed prime, R = Z_{(p)}[G] with involution, ϵ : Z_{(p)}[G] → Z_{(p)} the augmentation and S = 1 + ker ϵ. Let L(R, S, ±) be the Witt group of non-singular ± symmetric (i.e., ± Hermitian) linking forms over (R, S) in the sense of [35] (see also [18, p. 118]). Since these are forms defined on homological dimension one modules, it is not immediately apparent that these groups embed in the Witt groups W we have defined when G = Z^{m}. Indeed we shall not here address this. However these relative L-groups will be relevant in §4 and for all of our calculations where we pass to Q[Z]. Moreover certain results about L-groups carry over to the general Witt groups.

For any prime q let S_{q} = {x ∈ Z[G]|ϵ(x) is relatively prime to q} and S_{0} = {x|ϵ(x) = 1}. We shall detect elements by projecting from W(Z_{(p)}[Z^{m}], S) to W(Z_{(p)}[Z], S) (via some projection Z^{m} → Z) and then mapping into L(Q[t, t^{−1}],

S_{p}) = \overline{W}. Let ˆ \overline{W} = lim_{→}(\overline{W}_{i}) under maps f_{q} : \overline{W}_{i} → \overline{W}_{iq} induced by t → t^{q}

where (q, p) = 1. From §1 we know that the maps W_{i} ⟶^{π_{∗}} \overline{W}_{i} induce a map Ŵ → ˆ \overline{W}.

Since Q[t, t^{−1}] is a principal ideal domain, \overline{W} is well-understood [34]; [25]. Referring to [18, pp. 131–135] for example, \overline{W} ≅ ⊕L(Q[t, t^{−1}]/⟨z(t)⟩) where the sum ranges over all prime ideals generated by irreducible z(t) ∈ S_{p} such that z(t) = z(t^{−1}) or z(t) = t + 1 [34, Remark 1.3]. Fix, once and for all, such a generator z(t) for each such ideal. Moreover any form in L(Q[t, t^{−1}]/⟨z⟩) is equivalent to one defined on a direct sum of cyclic modules which have z(t) as annihilator (as opposed to (z(t))^{m}). If α is any root of z(t) then L(Q[t, t^{−1}]/⟨z⟩) ≅ L(Q[α]), the Witt group of non-singular ϵ-Hermitian forms over the field Q[α]. It follows that there are natural monomorphisms

L(Q[t, t^{−1}], S_{0}) ↪^{i_{p}} L(Q[t, t^{−1}], S_{p}) ↪ L(Q[t, t^{−1}], S).

The last group was calculated in [25].

Proposition 2.1. The kernel of the composition

L(Q[t, t^{−1}], S_{0}) ↪ \overline{W}_{i} → ˆ \overline{W}

consists of 2-torsion and 4-torsion elements if p is odd; and is trivial if p = 2.

Proof. Since L(Q[t, t^{−1}], S) has only 2 and 4-torsion [25], it suffices to show that the kernel of the endomorphism f_{q} : L(Q[t, t^{−1}], S_{0}) is torsion of order dividing q. Let G = Z and N = qZ. There is a transfer tr : L(Q[G], S_{0}) → L(Q[N], S_{q}) such that precomposing with the inclusion-induced f_{q} : L(Q[N], S_{0}) → L(Q[G], S_{0}) is the same as multiplication by q following by the inclusion i_{q}.

It should be remarked for example that the Blanchfield form of the figureeight knot is in the kernel of f_{2} [20, Proof of Example 4.2].

Returning to our discussion of L(Q[α]), if α does not have norm 1 then L(Q[α]) is a 2-group, whereas if ∥α∥ = 1 then every conjugation-preserving embedding ω_{∗} : Q[α] → C gives rise to a signature homomorphism ω_{∗} : L(Q[α]) → L(C, conjugation) ≅ Z. These signatures form a complete system of invariants for \overline{W} modulo torsion and, once having fixed the z(t) once and for all, there is a well-defined signature for every root of z(t). Therefore, if η ∈ \overline{W} and α is a complex number of norm 1 then we may speak of the α-signature of η, σ_{α}(η) corresponding to Q[t, t^{−1}]/_{⟨z_{α}⟩} → C given by 1 → 1 and t → α. Let z_{α} be fixed irreducible integral polynomial which has α as root. This is symmetric unless α = ±1. The following refines 2.1 and is our main computational tool.

Theorem 2.2. Let f_{q} : \overline{W} → \overline{W} where (q, p) = 1. Suppose γ^{q} = α. If the z_{γ}-primary component of f_{q}(η) is non-trivial, then the z_{α}-primary component of η is non-trivial. Moreover if ∥γ∥ = 1 and γ ≠ 1 then

σ_{γ}(f_{q}(η)) = \{ \begin{matrix}±σ_{α}(η) & if α ≠ 1 \\ 0 & if z_{α} ∉ S_{p}(e.g. α = 1) \end{matrix}

Proof. Assume γ ≠ −1. Consider an arbitrary primary component, say z_{β}-component of η, denoted η_{β}. Since f_{q}(η_{β}) is annihilated by z_{β}(t^{q}), it cannot contribute to the z_{γ}-primary component of f_{q}(η) unless z_{γ} divides z_{β}(t^{q}). This necessitates that γ^{q} = α is a root of z_{β} so z_{β} = z_{α}. This proves the first claim of 2.2. Obviously if z_{α} ∉ S_{p} then σ_{γ}(f(η)) = 0. Otherwise z_{α}(t^{q}) ˙=q_{1}^{n_{1}} … q_{r}^{n_{r}} has a prime decomposition with q_{1}(γ) = 0. Since \overline{γ} = γ^{−1}, γ^{−1} is also a root of q_{1}. Since z_{α}(t) = z_{α}(t^{−1}), q_{1}(t^{−1}) divides z_{α}(t^{q}) and since it shares a root with q_{1}, q_{1}(t) ˙=q_{1}(t^{−1}). Since z_{α} is irreducible, it is separable and it follows that z_{α}(t^{q}) has no roots of multiplicity greater than 1, and hence n_{1} = 1. Thus we may assume z_{α}(t^{q}) = z_{γ}(t)p(t) where (p, z_{γ}) = 1 and \overline{z}_{γ} = z_{γ}. The class η is

represented by some (⨁_{i=1}^{k} Λ/⟨z_{α}⟩, λ) where Λ = Q[t, t^{−1}] which is annihilated

by z_{α} and λ(e_{i}, e_{j}) = β_{ij}(t)/z_{α} where degree β_{ij} < degree z_{α}. Then σ_{α}(η) is the signature of the matrix (β_{ij}(α)) or (\sqrt{−1}β_{ij}(α)) (in the skew-Hermitian

case). The z_{γ}-component of f_{q}(η) is represented by (⨁_{i=1}^{k} Λ/⟨z_{γ}⟩, λ′) which

is annihilated by z_{γ} and λ′(e′_{i}, e′_{j}) = p(t^{−1})β_{ij}(t^{q})/z_{γ} where e′_{i} = pe_{i}. Then σ_{γ}(f_{q}(η)) is given by σ(p(\overline{γ})β_{ij}(α)) or σ(\sqrt{−1}p(\overline{γ})β_{ij}(α)). Since p(γ) ≠ 0, this is ±σ_{α}(η). The special case γ = −1 is handled similarly because z_{−1}(t^{q}) = t^{q} + 1 = ∏_{d|q}ϕ_{2d} if q is odd, where ϕ_{i} is the i^{th} cyclotomic polynomial.

Theorem 2.2 places extremely strong restrictions on the image of \overline{W}_{1} in ˆ\overline{W}. In particular it is easy to find elements not in the image of f_{q} as follows. Consider the set of complex numbers {γ | z_{γ} ∈ S_{p}, z_{γ^{q}} ∉ S_{p}}. By 2.2, for such γ, the z_{γ}-primary part of f_{q}(η) must be trivial. The above set is larger than one might think. Recall that, if ϕ_{d} is the d^{th} cyclotomic polynomial then ϕ_{d}(1) = 1 for d composite and q if d = q^{n} for q prime. Thus all q^{th} roots of unity except 1 are in the above set, as are q^{th} roots of −1 if p = 2 and q > 1. More interestingly, it contains elements not on the unit circle. For example, if z_{γ} is the Alexander polynomial of the figure eight knot, 3 − (t + t^{−1}), then z_{γ} ∈ S_{2} but z_{γ^{3}} ∉ S_{2}. We will use these facts in §3. For example the following shows that a link of complexity q which is concordant to a boundary link possesses a periodic signature function.

Corollary 2.3. If ∥γ∥ = 1 then σ_{γ}(f_{q}(η)) = ±σ_{γe \frac{2πik}{q}} (f_{q}(η)) for 0 ≤ k ≤ q.

Definition. Let A(η) = {γ | ∥γ∥ = 1, σ_{γ}(η) ≠ 0} (a finite set).

Corollary 2.4. A(f(η)) = q\sqrt{A(η)} the set of q^{th} roots of elements of A(η), so either f_{q}(η) is torsion or the minimum degree of an “Alexander polynomial” (annihilator) of a representative of f_{q}(η) is 2q (q if A(η) = {−1}). The cardinality of A(f_{q}(η)) is 0 mod q.

Of particular interest are the signatures associated to primitive roots of 1. Suppose q is an integer greater than 1, and 1 ≤ j ≤ q − 1, (j, q) = 1.

Definition. The q-signatures of η, denoted σ_{j/q}(η) are σ_{e^{2πij/q}}(η) as defined above.

Corollary 2.5. Consider f : \overline{W} → \overline{W} induced by t → t^{c} with (c, p) = 1 and η ∈ image f. Suppose r|c (r ≥ 1).

(a) σ_{j/rp^{k}}(η) = 0 for any k ≥ 0, that is the rp^{k}-signatures of η vanish;

(b) If (m, r) = 1 then the set of absolute values of m-signatures of η is equal to the set of absolute values of the mr-signatures of η. Precisely, σ_{j/mr}(η) = ±σ_{l/m}(η) where lr ≡ j mod m.

Proof of 2.5. Let γ = exp(2πij/rp^{k}) and γ^{c} = exp(2πijc′/p^{k}) = α. Then z_{α} ˙=ϕ_{p^{k}} the p^{k} cyclotomic polynomial. Since ϕ_{p^{k}}(1), z_{α} ∉ S_{p} so by 2.2, our first result is proved.

We can assume (m, p) = 1. Now suppose (m, r) = 1, (m, j) = 1 and 1 ≤ j < m. Since r|c we have f_{r}(β) = η for some β where f_{r} is induced by t → t^{r}. By 2.2, σ_{j/m}(η) = ±σ_{l/m}(β) where l is the residue of jr modulo m. On the other hand, if 1 ≤ j < mr and (j, mr) = 1 then σ_{j/mr}(η) = ±σ_{j/m}(β) which is ±σ_{l′/m}(β) where l′ is the residue of j modulo m. Therefore σ_{j/mr}(η) is σ_{l/m}(η) where lr ≡ j mod m.

Corollary 2.6. For any c, ˆ \overline{W}/\overline{W}_{c} is infinitely generated. (Indeed, by 2.5 \overline{W}_{nc}/\overline{W}_{n} is infinitely generated!). Moreover W_{nc}/W_{n} is infinitely generated modulo torsion.

Proof of 2.6. For example in the symmetric case p ≠ 3 let λ_{r} be the Blanchfield form of a trefoil in \overline{W}_{c3^{r}} r ≥ 0 and λ̂_{r} their images. Suppose ∑_{i=1}^{n} a_{i}λ̂_{r_{i}} is of finite order in ˆ \overline{W} where a_{i} ≠ 0 and r_{1} < ⋯ < r_{n}. By 2.2,

this element must “already be torsion” in \overline{W}_{c_{1}3^{rn}}. Specifically ∑_{i=1}^{n} a_{i}f_{3^{n_{i}}} (λ_{r_{i}})

is torsion in \overline{W} where n_{i} = r_{n} − r_{i}. Since f_{pq} = f_{q} ∘ f_{p}, this would imply that, modulo torsion, a_{n}λ_{r_{n}} lies in the image of f_{3}. By 2.5 a) with k = 0, r = c = 3 it would follow that the 3-signatures of λ_{r_{n}} = λ_{r} vanish, which is a contradiction. Since examples can easily be chosen in L(Z[t, t^{−1}], S_{0}) these will lift to W under any projection.

Corollary 2.7. If (c, p) = 1, β has a non-zero d-signature iff f_{c}(β) has a non-zero cd-signature. Therefore max{n | f_{c}(β) has a non-zero n-signature} is c times max{n | β has a non-zero n-signature}.

Proof. Suppose β has a non-zero exp(2πij/d)-signature for (j, d) = 1. By 2.2 and 2.3, f_{c}(β) has non-zero signatures corresponding to exp(2πij/dc + 2πik/c) for all 0 ≤ k < c. Suppose c has prime decomposition q_{1}^{n_{1}} … q_{r}^{n_{r}}. Letting k be the residue modulo c of Πq_{i}^{n_{i}} where (q_{i}, j) = 1 yields a primitive cd^{th} root of 1. The other implication is immediate from 2.2.

Suppose f_{c}(β) has a non-zero n-signature 1 where n is maximal. Then n = n′d, d = gcd(c, n) so β has a non-zero n′-signature (by 2.2). Thus f_{c}(β) has a non-zero n′c signature where n′c ≤ n. It follows that c = d.

Corollary 2.8. Suppose η lies in the image of f_{c} and some q-signature of η is non-zero. Let q′ = q/gcd(q, c). Then ϕ_{q′}(t^{c}) factors into distinct cyclotomic polynomials ϕ_{d_{i}} and η has a non-zero d_{i} signature for each i. In particular ϕ_{q′}(t^{c}) divides the annihilator of any representative of η so the degree of such is at least ϕ_{e}(q′)c (where here ϕ_{e} is Euler’s function).

Proof of 2.8. By the proof of 2.2 these ϕ_{d_{i}} occur with multiplicity 1. Note the ϕ_{d_{i}} component of η can only come from the ϕ_{q′} component of β if f_{c}(β) = η. By induction it suffices to show the Corollary when c is a prime. Then ϕ_{q′}(t^{c}) is ϕ_{q′c} if c|q′ and is otherwise ϕ_{q′} ϕ_{q′c}. By 2.2 β has a non-zero q′ signature and hence by 2.7 η has a non-zero q′c signature. Therefore we may assume (c, q′) = 1 and then 2.5b implies η has a non-zero q′ signature.

Remarks. 2.8 can be strengthened as in 2.5b. The number ϕ_{e}(q′)c is very large and places severe limitations on image f_{c}.

The “preservation” of signatures under the maps f_{q} allows for the definition of signatures for elements of Ŵ but this contains no more information than the foregoing.

3. Examples and Computational Methods

In this chapter we first discuss briefly the relationships between the complexities of covering links and the “pattern” of a homology boundary link (see [9]) and also the Alexander polynomials.

We then give several categories of examples. The first family of examples is striking in its simplicity and shows how to embed the rational knot concordance group in “homology boundary links modulo concordance classes of boundary links” (roughly speaking). This example is done in great detail.

Next we show that there is infinitely generated 2 and 4-torsion in “homology boundary links modulo boundary links”. These examples are also elementary.

Next we show that there are Brunnian examples (any proper sublink is trivial). This is more difficult than one might imagine. It is a sobering observation of J. Levine that a Brunnian homology boundary link is a boundary link (follows from the Freiheitsatz)!! Finally we show that given any integer d, there are examples requiring d iterated covers.

We pause to discuss the calculation of the complexities of L, calculations we will call upon in the presentation of our examples.

If ϕ : G ↠ E is any group epimorphism where H_{1}(E) has rank m, then the various complexities of L are determined by ϕ because ϕ will factor any epimorphism G ↠ Z_{p} and will lift to ϕ̃: G̃ ↠ Ẽ where G̃ is π_{1} of the branched covering space. Since ϕ̃(u_{i}) = ϕ(m_{i}) ∈ Ẽ and H_{1}(Ẽ) has rank pm, the complexity of L is determined as is the complexity map. Of course, in practice, this procedure is useful only if E is simpler than G itself. Note also that ϕ̃ recovers the properties of ϕ so we can iterate.

We shall call a link L = {K_{0}, K_{1}, …, K_{m}} in a Z_{(p)}-homology sphere a homology boundary link if its group G admits an epimorphism ϕ to the free group F (x_{0}, …, x_{m}) such that, in H_{1}( ; Z_{p}), ϕ_{∗}(m_{i}) ≡ x_{i} where m_{i} is the (unbased) i^{th} meridian. An ordinary homology boundary link in S^{n+2} satisfies the extra condition since Aut(F) → Aut(F_{ab}) is onto. An equivalent definition, parallel to the definition of a boundary link, can be obtained as follows. A generalized Seifert surface for a component K_{i} of a link L is a compact, connected, oriented submanifold V of E_{L} whose boundary is a disjoint union of parallels (longitudes) of K_{j}, 0 ≤ j ≤ m, such that [∂V] in H_{n}(∂E_{L}) is equal to the class represented by a longitude of K_{i}, that is, ∂V is only “algebrically” equal to a longitude of K_{i}. Then a homology boundary link is one which admits a system {V_{0}, V_{1}, …, V_{m}} of pair-wise disjoint generalized Seifert surfaces. If L is such then a choice of based meridians induces a map M : F → G. The r-tuple \overline{r} = (M ∘ ϕ(x_{i}))_{i} is called a pattern of L (see [9]). Note that r_{i} ≡ x_{i} in Z_{p}-homology. Thus all the complexities of a homology boundary link are determined by its pattern. Moreover if π : F ↠ Z_{p} sends x_{0} → 1 and x_{i} → 0 then π ∘ ϕ defines the p-fold cover of E_{L} corresponding to the meridian of K_{0}. Hence ϕ lifts to ϕ̃ : G̃ ↠ F̃ where F̃ is free on {x_{0}^{p}, x_{0}^{j−1}x_{i}x_{0}^{−j+1} | 1 ≤ j ≤ m 1 ≤ j ≤ p}. M lifts also to M̃ which sends x_{0}^{j−1}x_{i}x_{0}^{−j+1} to u_{ij}, a meridian of the j^{th} lift of K_{i}. Here ϕ̃(u_{0}) = r_{0}^{p} and ϕ̃(u_{ij}) = x_{0}^{j−1}r_{i}x_{0}^{−j+1} as elements of F̃. It follows immediately from 1.1 that the (first-order) complexity matrix of L is easily calculated from the r_{i} and that there exist meridians for which the complexity matrix of a boundary link is the identity. The procedure may be iterated although L̃ may not be an homology boundary link in general.

Indeed, the reader sees that the pattern is the true measure of L’s deviation from being a boundary link.

We pause to show that all of the first order complexities are often determined by a complexity associated to the infinite cyclic cover X_{∞} of E_{L} corresponding to the first meridian. For brevity we restrict to an (m+1)-component homology boundary link L with ϕ such that ϕ(m_{0}) ≡ x_{0} in integral homology. Let A = Z[t, t^{−1}]. Choose a component in X_{∞} of the inverse image of the components of L. Then we have maps

F_{ab}⟨x_{1}, …, x_{m}⟩ ⟶^{M} H_{1}(X_{∞}) ↠^{ϕ_{∞}} A^{m}

where ϕ_{∞} is an A-module map. Let C(L)_{i} 1 ≤ i ≤ m be the m-tuple of polynomials (ϕ_{∞} ∘ M(x_{i}))_{i}. Indeed, given ϕ, each such polynomial is unique up to unit of A. Since ϕ is unique up to post-composition by an element of Aut_{0}(F), C(L) is unique up to units of A and certain automorphisms of A^{m}. In particular if m = 1 this polynomial is well-defined up to unit. If X_{p} denotes the p-fold cover of E_{L} corresponding to the o^{th}-meridian and A_{p} = Z[Z_{p}] then the A_{p} module H_{1}(X_{p}) is Z ⊕ (H_{1}(X_{∞})/t^{p} = 1) where the latter is H_{1} of the branched cover \overline{X}_{p}. Therefore the complexity map

F_{p}⟨x_{ij} | 1 ≤ i ≤ m, 1 ≤ j ≤ p⟩ ⟶^{M} H_{1}(\overline{X}_{p}) ⟶^{ϕ_{p}} (A_{p})^{m}

is given by x_{i} → (C(L)_{i}/t^{p} = 1) as a map of A_{p} modules and an (integral) complexity matrix is derived from this.

Since the Alexander Module of K_{0}, H_{1}((E_{K_{0}})_{∞}), is the quotient of H_{1}(X_{∞}) by the A-module generated by {M(x_{i})1 ≤ i ≤ m}, it maps onto the quotient of A^{m} by the submodule generated by {C_{i}}. It follows that the determinant of the m × m matrix (C_{i}) divides the Alexander polynomial of K_{0}. Hence if K_{0} has Alexander polynomial 1 then all first order integer complexities c_{p} are 1. Even if L is not an homology boundary link, c_{p} will divide the exponent of H_{1} of ∑, the p-fold cover of S^{n+2} branched over K_{0}, because H_{1}(∑) is precisely the quotient of H_{1}(X_{p}) by the meridians of the covering link (see 1.1c).

Proposition 3.1. Suppose L = {K_{0}, K_{1}, …, K_{m}}. The (first-order) complexity integer c_{p}(L) divides the exponent of H_{1} of the p-fold cover of S^{n+2} branched over K_{0}. If L is a 2-component homology boundary link then the complexity polynomial c_{p} ∈ Z[Z_{p}] divides (Δ_{1}(K_{0})/t^{p} = 1).

A. The Simplest Examples

We give a family of homology boundary links, indexed by Z × θ where θ is the knot concordance group, where the link L(m, [J]) is not concordant to a boundary link if m ∉ {0, −1} and [J] is not torsion. Indeed, for each such m this embeds the rational knot concordance group in the set of link concordance classes modulo boundary links. It is also easy to show that the image of (Z − {0, −1}) × [J] is infinitely generated in the disk link concordance group modulo boundary links.

First consider the ribbon link R = {K′_{1}, K′_{2}} in S^{2k+1} shown in Figure 3.4a (the “band” is D^{1} × D^{2k−1} in general). We shall exhibit R as a homology boundary link by constructing disjoint generalized Seifert surfaces {V_{x}, V_{y}}. The surfaces {V′_{x}, V′_{y}} shown in 3.4b may be completed to {V_{x}, V_{y}} by “tubing along the band” b as shown until the boundaries of these tubes become “longitudes” of K′_{2}. If G is the group of R, F is the free group on {x, y} and {m_{1}, m_{2}} are the indicated meridians, there are homomorphisms:

(3.5)  F ⟶^{M} G ⟶^{ϕ} F

where M(x) = m_{1}, M(y) = m_{2} and ϕ is the epimorphism induced from the map to W = S^{1} ∨ S^{1} sending (S^{2k+1} − V_{x} − V_{y}) to the wedge point. Thus the pattern of this homology boundary link is (x, [x, \overline{y}]y), where we use the conventions that [x, y] = xy\overline{x}\overline{y} and \overline{x} = x^{−1}.

We will modify R to a homology boundary link L = {K_{1}, K_{2}} with identical pattern by a procedure we call “knotting a band of R”, a procedure to be discussed more later and at length in [12]. Consider the loop γ shown dashed in 3.4a. Note that ϕ(γ) = [x, \overline{y}]. Now, given any knot J in S^{2k+1}, we may “grab” the 2 strands of R enclosed by γ and tie them into the knot J. The result is shown in Figure 3.6 (m = 1). The surfaces {V_{x}, V_{y}} may also be tied in the knot J (γ meets 4 sheets) recovering 3.5 for L and its group. More

20 TIM D. COCHRAN AND KENT E. ORR Figure 3.4 Figure 3.6

generally, the number of times, −m, that the unknotted band in 3.6 “wraps around” the knotted band could be varied.

The complexity of such would be (x, [x, \overline{y}]^{m}y), and we preserve ϕ(γ) = [x, \overline{y}]. The family of homology boundary links we consider is the set of such L for varying J, and m (m ≠ 0, −1). The simplest example, namely m = 1, K = 1 and J = the trefoil knot was conjectured in [7, Fig. 8.11 p. 49, p. 66#14] to be a link with vanishing \overline{μ}-invariants but not concordant to a boundary link.

Following §1, let ∑ be the p-fold cover of S^{2k+1} branched over K_{0}, L̃ = {J_{1}, …, J_{p}} the covering p-link, μ_{i} the unbased meridian of J_{i} and X = X(L̃) the framed surgery. Then π_{1}(X) maps onto F ⟨y_{1}, …, y_{p}⟩ via ϕ̃ where ϕ̃(u_{i}) = (y_{i+1}^{−1}y_{i})^{m}y_{i} and y_{p+1} = y_{1} (here F is the subgroup of F ⟨x, y⟩ with y_{i} = x^{i−1}yx^{1−i}).

The composition ϕ̃ ∘ M̃ on abelianizations yields the complexity matrix (with respect to {y_{i}})

(3.7)  C = ( m + 1 −m  0 …  0  0 ; 0  m + 1 −m …  0  0 ; ⋮ ; 0  0  m + 1 −m ; −m  0  0  m + 1 ) .

whose determinant is (m + 1)^{p} − m^{p} and whose adjoint C^{∗} has as its columns the cyclic permutations of ((m + 1)^{p−1}, …, (m + 1)^{p−i}m^{i−1}, …, m^{p−1}). Since (m^{p−1}, (m + 1)^{p−1}) = 1 the complexity of L is | det C| by 1.1.

Let λ_{J}(t) ∈ L(Z[Z], S) be the ordinary Blanchfield form of J.

Proposition 3.8. For L as above, c(L̃) = c =| (m + 1)^{p} − m^{p} |= c and

λ_{L} = ⨁_{i=1}^{p} λ_{J}(C^{∗}(y_{i+1}^{−1}y_{i})) where y_{p+1} = y_{1} and λ_{J}(C^{∗}(y_{i+1}^{−1}y_{i})) is the image of

λ_{J}(t) under the map

Z[Z] ⟶^{i} Z_{(p)}[Z^{p}] ⟶^{C^{∗}} Z_{(p)}[Z^{p}]

where i(t) = y_{i+1}^{−1}y_{i}.

Proof of 3.8. To compute λ_{L}, note that K_{1} bounds an obvious plumbing of two “bands” (S^{2k−1} × [0, 1] and S^{1} × D^{2k−1}) so that the techniques of [1] may be used to give a picture of X as the boundary of D^{2k+2} with (2p − 1) handles of dimension (2k − 1) and (p − 1) handles of dimension 2 attached as shown in 3.9 for m = 1. We will alter X to X′ by handle slides. Slide all handles which “link” Q_{1} over ±U_{1} until only U_{1} links Q_{1}. Then successively slide all handles which link Q_{i} over ±U_{i} until 3.10 is achieved.

Let h be this composite homeomorphism. Let γ_{i} be the p lifts of γ to X as shown in 3.9. Note that if z_{i} is a meridian to Z_{i} as in 3.9 then h_{∗}(z_{i}) = z′_{i}

Figure 3.9

so h_{∗}(γ_{i}) = γ′_{i} as in 3.10. Thus λ_{L}, being the Blanchfield form associated to (X, C^{∗} ∘ ϕ̃) by 1.6 and 1.5, is isomorphic to that associated to (X′, C^{∗} ∘ ϕ̃ ∘

(h_{∗})^{−1}). It is easy to see that the latter is ⨁_{i=1}^{p} λ_{J}(C^{∗} ∘ ϕ̃ ∘ h_{∗}^{−1}(γ′_{i})). Since

ϕ̃(γ_{i}) = x^{i−1}([x, \overline{y}])x^{1−i} = y_{i+1}^{−1}y_{i}, 3.8 follows.

Proposition 3.11. If L(J, m) is the homology boundary link of 3.6 and J is of infinite order in the knot concordance group, then L is concordant to a boundary link iff m = 0 or −1 in which case it is a boundary link. (Neither is L concordant to a link whose first component is unknotted!).

Proof of 3.11. We assume without loss that m > 0. Note lim_{p→∞} c = ∞

unless m ∈ {0, −1}. Suppose L were concordant to a boundary or any other link each of whose first-order complexities are 1. Define π : W_{i} → \overline{W}_{i} by projection onto the first factor. It then follows from 1.10 and 2.1 that, for each prime p, π_{∗}(λ_{c}^{p}(L̃)) would, modulo torsion, lie in the image of f_{c} : \overline{W}_{1} → \overline{W}_{c}. We shall show that this would imply the vanishing of all the signatures of λ_{J}.

By 3.8 π_{∗}(λ(L̃)) equals ( ⨁_{i=1}^{p−1} λ_{J}(t^{r_{i}})) ⊕ λ_{J}(t^{−r}) where r = (m + 1)^{p−1} −

m^{p−1} and r_{i} = (m + 1)^{p−i−1}m^{i−1}. Note r > r_{i}, (r, r_{i}) = 1 and (r, c) = 1.

Figure 3.10

Suppose λ_{J} has a non-zero signature corresponding to a root of 1. Let diam(η) = max{n|η has non-zero n-signature}. Then d = diam(λ_{J}) where d > 0 and, by 2.7, if (r_{i}, p) = 1 diam λ_{J}(t^{r_{i}}) = dr_{i} < dr = diam λ_{J}(t^{−r}). Choose p so that c > d and so that p > r. By 2.7, if π_{∗}(λ_{c}^{p}(L̃)) were in the image of t → t^{c} then its diameter is a multiple of c. Since dr is not a multiple of c, we can assume λ_{J} has no such non-zero signatures.

Let Δ(t) be the Alexander polynomial of J and \{β_{j}, \overline{β}_{j}|1 ≤ j ≤ d\} its distinct roots. It is easy to see that there are at most d positive integers σ such that given σ, there exist some positive integers c′, r′ with (r′, σ) = 1 and some j satisfying β_{1}^{c′r′} = β_{j}^{c′σ}. Choose p large enough so that r is greater than the maximum such σ (compare [20, pg. 331]).

It follows that if α is a root of Δ(t^{r}) then αe^{2πik/c} is not a root of Δ(t^{r_{i}}) for 0 ≤ k < c. Let s =| A(λ_{J}) | (see 2.4 and preceeding defintion). From 2.4, | A(λ_{J}(t^{r})) |= rs and | A(f_{c}(η)) |≡ 0 modulo c for any η. Moreover if σ_{α}(λ(t^{r})) ≠ 0 and π_{∗}^{p}(λ(L)) = f_{c}(η) then σ_{α}(f_{c}(η)) ≠ 0 and by 2.3 σ_{αe^{2πik/c}}(f_{c}(η)) ≠ 0. By our choice of p above, the latter are roots only of λ_{J}(t^{r}). It follows that | A(λ_{J}(t^{r})) |≡ 0 mod c but rs ≡ 0 mod c only if s = 0 (since s ≤ d < c). Therefore all signatures of λ_{J} vanish and λ_{J} is of finite order.

Figure 3.12

B. Torsion

We shall describe an infinite family of (based) homology boundary links in S^{2n+1} each member of which is of order 2 (in the sense that L#L is nullconcordant if n > 1 and algebraically null-concordant if n = 1) but which is linearly independent over Z_{2}, even modulo concordance to boundary links. We shall describe a similar family of 4-torsion elements. As a direct consequence, Γ_{2n+2}(Z[F̂] → Z) modulo the image of Γ_{2n+2}(Z[F] → Z) contains infinitely generated abelian subgroups of 2- and 4-torsion elements.

The families are part of the family shown in Figure 3.12 (here k ∈ Z), which is identical to that of Figure 3.6 except that a “strand” of K_{2} is also “tied in the simple knot J”.

For the 2-torsion families we take m = \frac{1}{2}(3^{k} −1) for n odd and m = \frac{1}{2}(9^{k} − 1) for n even. For the 4-torsion families we set m = \frac{1}{2}(5^{k} − 1) for n odd and m = \frac{1}{2}(3^{k} −1) for n even. For the 2-torsion families, if n is odd choose J to be a simple knot with Alexander polynomial Δ_{J} = Δ_{0} = t^{2} −3t+1 (the polynomial of the figure eight knot); if n is even set Δ_{J}(t) = Δ_{0}(t)Δ_{0}(t^{3}). For the 4-torsion families, if n is odd choose J such that Δ_{J} = 9 − 5(t + t^{−1}) + (t^{2} + t^{−2}) (the polynomial of the knot 7_{7} in Alexander-Briggs notation), if n is even set

Δ_{J} ˙=Δ_{2}Δ_{3} = (5t^{2} − 11t + 5)(47t^{2} − 95t + 47).

Such J can be chosen and themselves have the desired torsion properties according to [25, Sections 1, 2, 23, 24, 25] [26].

Then each L_{k} is a homology boundary link with the same pattern as R of 3.4, namely (x, [x, \overline{y}]^{m}y) and the same complexity matrix C_{p} of 3.7. The only difference is that ϕ(γ) = ([x, \overline{y}]^{m}y)^{−1}[x, \overline{y}] = \overline{y}[\overline{y}, x]^{m−1} so ϕ̃(γ̃) = y_{1}^{−m}y_{2}^{m−1} and ϕ̃(xγ̃x^{−1}) = y_{2}^{−m}y_{1}^{m−1}. Using p = 2, c(L̃_{k}) = 2m + 1 = 3^{k} (respectively 9^{k}, 5^{k}, 3^{k}) and, as in 3.8, λ(L̃_{k}) = λ_{J}(C^{∗}(y_{1}^{−m}y_{2}^{m−1})) ⊕ λ_{J}(C^{∗}(y_{2}^{−m}y_{1}^{m−1})). This is all very easy to calculate explicitly using the Kirby calculus as we did in 3.8, if one desires. Alternatively, in the next sub-section we outline a general calculational technique. Continuing, we project y_{1} → t y_{2} → 1 to get \overline{λ}(L̃_{k}) = π_{∗}(λ(L̃_{k})) = λ_{J}(t^{−2m}) ⊕ λ_{J}(t^{−1}) in \overline{W}_{3^{k}} (respectively 9^{k}, 5^{k}, 3^{k}).

We proceed with a proof, in the case that n is odd and using the 2-torsion family, that no non-trivial sum L = #_{i=1}^{N}L_{k_{i}} is concordant to a boundary link, by applying Theorem 1.9. Let k = k_{N} = max{k_{i} | 1 ≤ i ≤ N}. We need to show that the invariants ˆ \overline{λ} = π_{∗}(λ̂) (see 1.10) is additive under connectedsum of links. We sketch a proof. First, it is well known that a 2-fold cover of S^{3} branched over a connected-sum of knots yields a connected-sum of the individual branched covers. In our case the lift of the separating 2-sphere decomposes the covering link as a connected sum of covering links of the 2^{nd} components. Directly from the definition we see, in our case, that c(L) = 3^{k} = (3^{k−k_{i}})c(L_{k_{i}}). Moreover the Blanchfield form associated to the infinite cyclic cover is well-known to be additive under connected sum. These facts yield that

\overline{λ}(L̃), as an element of \overline{W}_{3^{k}} is ∑_{i=1}^{N} f_{r_{i}}(\overline{λ}(L̃_{k_{i}})) where r_{i} = 3^{k−k_{i}}. Suppose L were

concordant to a boundary link. Then, by 1.9, there is some r ≥ 1 and η′ ∈ \overline{W}

such that f_{r3^{k}}(η′) = ∑_{i=1}^{N} f_{rr_{i}} (λ_{J}(t^{−2m}) ⊕ λ_{J}(t^{−1})). By [20, Example 4.2 and

proof] λ_{J}(t^{2n}) is trivial and {λ_{J}(t^{−n}) | n odd} is linearly independent over Z_{2}. Alternatively note that Δ_{0}(1) = −1 so λ_{J} lies in the image of L(Q[t, t^{1}], S_{0}) implying f_{n}(λ_{J}) is non-trivial by 2.1. Since Δ_{0} is not cyclotomic, the roots of Δ_{0}(t^{n}) are distinct for varying n. It now follows that there is some η ∈ \overline{W},

namely f_{3^{k−1}}(η) such that f_{3r}(η) = ∑_{i=1}^{N} λ_{J}(t^{−rr_{i}}) in \overline{W}, and (using 2.2) there is

a zero γ of Δ_{J}(t^{r}) for which the z_{γ}-primary component of f_{3r}(η) is non-zero. Setting β = γ^{3r}, the z_{β}-primary component of η is necessarily non-trivial (by 2.2). Since β is the cube of a root of Δ_{0}, z_{β} = t^{2} − 18t + 1. But z_{β}(1) is even so z_{β} ∉ S_{2} (there is no integral polynomial ρ(t) with β as zero for which ρ(1) is odd). This is a contradiction so L is not concordant to a boundary link.

If n is even the proof is the same with 9 replacing 3 except that now β is a ninth power of a root of Δ_{0} or Δ_{0}(t^{3}) so z_{β} is either t^{2} −18t+1 or t^{2} −5778t+1 neither of which lies in S_{2}.

Now suppose n is odd and use the 4-torsion family. Consider L = #_{i=1}^{N}b_{i}L_{k_{i}} where b_{i} ∈ {1, 2, 3}. Proceed as above with 5 replacing 3. It follows that there

is some r and η ∈ \overline{W} such that f_{5r}(η) = ∑_{i=1}^{N} b_{i} (λ_{J}(t^{−2mrr_{i}}) ⊕ λ_{J}(t^{−rr_{i}})) where

r_{i} = 5^{k−k_{i}}. By 2.1, b_{N}λ_{J}(t^{−r}) is non-trivial so it has a non-trivial z_{γ}-primary part for some γ, root of Δ_{J}(t^{r}). However, since Δ_{J} is not cyclotomic, the roots of Δ_{J}(t^{n}) and Δ_{J}(t^{n′}) are distinct if n ≠ n′ so f_{5r}(η) has a non-trivial z_{γ}-primary component. Thus the z_{β}-primary component of η is non-zero for some β a fifth power of a root α of Δ_{J}. But one can calculate that z_{β}(1) is even by showing that the degree of z_{β} is 4 and that z_{β}(1) modulo 2 is a function of β + β^{−1} which is in turn a function of α + α^{−1} where β = α^{5}.

If n is even, proceed as immediately above with 3 replacing 5. We claim that the roots of Δ_{2}(t^{m}) and Δ_{3}(t^{n}) are distinct if n, m ∈ Z^{+}. Suppose not. Then we would have (11 ± \sqrt{21})^{n} · (94)^{m} = (95 ± \sqrt{21})^{m} · (10)^{n} for n, m ∈ Z^{+}. Note that (11 ± \sqrt{21})^{2} = 2(5x + 11 ± \sqrt{21}) for some x ∈ Z[\sqrt{21}], implying that (11 ± \sqrt{21})^{n} = 5y + 2^{n−1}(11 ± \sqrt{21}) for some y in Z[\sqrt{21}]. Therefore the left-hand side of the above equation is not in 5Z[\sqrt{21}] which is a contradiction. Moreover the roots of Δ_{2}(t^{n}) are distinct for varying n (similarly Δ_{3}(t^{n})) since they are not cyclotomic. Therefore the above analysis holds for some β a cube of a root of Δ_{2}Δ_{3}. One calculates z_{β} explicitly to see that z_{β}(1) is even in any case.

This completes the proof that these families are linearly independent modulo concordance to boundary links. It remains to show that they are torsion. For this we appeal to [12]. As shown there, the set of based homology boundary links with a fixed pattern is closed under connected sum. Just as for boundary links, the system of (generalized) Seifert surfaces yield a Seifert-type form. Loosely speaking, if q > 1, the set of such links modulo “homology boundary link cobordism”, is isomorphic to the group of “cobordism classes” of such forms. If q = 1, the map from links to forms is at least additive on the class of links under consideration here [12]. Moreover, if the Seifert surfaces are highly-connected (“simple”) then the Seifert form is equivalent to a Blanchfield form associated to the free covering space of the link exterior (as detailed for boundary links in [15]). In our present situation the “free” Blanchfield forms are λ_{J}(ϕ(γ)) (see 3.16 of this paper, proofs in [12]) which are clearly of order 2 or 4 as needed. Therefore by the remarks above, the links themselves are of order 2 or 4 if q > 1.

C. Brunnian Examples, etcetera

We shall now give a classes of examples including Brunnian links, for which all first-order complexities are 1, so that they cannot be distinguished from boundary links (up to concordance) by our invariants corresponding to a single covering space nor by the invariants of Gilmer and Livingston [17].

These will be distinguished however in iterated covers. For any n, there are such examples which require n branched covers as we shall show at the end of this section.

In Figure 3.13 is shown a 3-component fusion of a boundary link in which K_{y}, K_{z} are unknotted. K_{x} is obtained from the unknot by knotting a band.

In the classical dimension this yields an Alexander polynomial 1 knot which is topologically (locally flatly) null-concordant and has the same Seifert form as an unknot. In higher odd dimensions, if J is a simple knot then it is not hard to see that the resulting K_{x} will be a simple knot which has the Seifert form of the unknot. It follows that K_{x} is unknotted [31]. Therefore all of the first-order complexities of this link are 1. However the 2-fold cover branched over K_{y} gives a 4-component covering link which has the link of Figure 3.6 as a sublink. Therefore the link of 3.13 will not in general be concordant to a boundary link and, of course, this would be detected by our invariants by taking another covering space as in 3.6. To get K_{x} unknotted in the classical dimension and to get a Brunnian link, we need merely make the band through K_{z} more exotic as shown in Figure 3.14.

Figure 3.14 shows a 2-component Brunnian link L = {K_{x}, K_{y}} which is a fusion of a boundary link. The 3-component link H = {K_{x}, K_{y}, K_{z}} is a homology boundary link by the proof of Theorem 2.3 of [8]. H has unknotted components but is not Brunnian. We shall show that neither L nor H is concordant to a boundary link for suitable J. We will use the fact that L is a sublink of H and that our invariants are easy to compute for homology boundary links, especially those which are obtained from a ribbon link R by “tying a knot in a band” as H patently is.

We need a technique to realize desired invariants without a lot of work. To that end we will outline various actions of the group of boundary cobordism classes of boundary disk links on the group of “homology cobordism classes” of homology boundary disk links with fixed pattern. As a special case we recover a proof that the above groups are abstractly isomorphic [14]. In this paper we use these actions merely to calculate our invariants for links obtained by “knotting a band” in ribbon homology boundary links and specifically only for the following two examples. Therefore details of the technique are left to [12].

Suppose L is an m-component homology boundary link in S^{2n+1} with fixed generalized Seifert surfaces V = {V_{i}}. Suppose also we are given a “generalized basing” of L in the following sense. We take our hand and grab a handful of strands of L, like stalks of spaghetti, and we group these strands into m collections of vertical strands an example of which is shown in Figure 3.15a for the case m = 3. If X is the result of framed surgery on L, let γ_{i}, 1 ≤ i ≤ m be the classes in π_{1}(X, ∗) which enclose the m subcollections as shown. Given this data (L, γ_{i}, …, γ_{m}), we wish to modify L by acting on it with

Figure 3.13

a simple boundary disk link J representing a class, say α, in the group of such modulo boundary link cobordism. To do so we first take J, as shown in Figure 3.15b and push-off parallel copies of each of its m components in such a way that the resultant boundary link of 3.15c achieves the identical number of strands, including orientations, as in 3.15a. The resulting boundary disk link is denoted γ^{#}(J). We then may modify L by deleting the trivial disk link Γ of 3.15a and inserting the non-trivial disk link γ^{#}(J). It is not difficult to show that the resulting link (L, γ, α) is an homology boundary link with the same pattern as that of L. Its invariant is not difficult to compute either, in terms of the cobordism classes of Seifert type forms [23] or, equivalently, in terms of Γ-groups or in terms of the universal Blanchfield form. Some of these are discussed in section 4. We briefly outline the language in which we can most easily state our result about the invariant of (L, γ, α). Since L is an homology boundary link with basepoint ∗, this induces θ : π_{1}(X) ↠ F ⟨x_{1}, …, x_{m}⟩ and hence a “free” cover X_{ω}. Let A = Z[F ⟨x_{1}, …, x_{m}⟩] and let H_{∗}(X, A) denote the right A-module H_{∗}(X_{ω}; Z).

Figure 3.14

Let Λ denote the Cohn localization of A with respect to the augmentation ϵ : A → R (R = Z or Z_{(p)}). Then, just as in §5 of [15] there is a (−1)^{n+1}- Hermitian Blanchfield form (assuming n > 1)

λ : \overline{H_{n}(X, A)} ⟶ Hom_{Λ}(H_{n}(X, A), Λ/A)

(— denotes conjugate module). If n = 1, and if we have a degree one map E_{L} → E_{R} for a ribbon link R, which factors θ, then we can replace H_{n}(X, A) by the kernel of H_{n}(X, A) → H_{n}(X_{R}, A) and get the existence of such a λ.

Theorem 3.16. Let (L, V, γ_{1}, …, γ_{m}) be a homology boundary link as above. Let α represent a simple boundary disk link as above. Then

λ((L, γ, α)) = λ(L, V) ⊕ γ_{∗}(λ(α))

where γ_{∗} : L(A, S) → L(A, S) is induced by γ : F → F defined by γ(x_{i}) = θ(γ_{i}).

The invariant λ is related to our invariants of §1 as follows. Again this is discussed in §4.

Suppose p is prime and ϵ : Z[F] → Z_{(p)}, and N is an index p normal subgroup of F. Then there is a transfer tr : L_{q}(A, S_{ϵ}) → L_{q}(Z[N], S′_{ϵ′}).

Figure 3.15

Theorem 3.17. For a homology boundary link (L, V), our invariant, of §1, corresponding to certain iterated p-fold covers of E_{L} is the image of the “free” Blanchfield pairing λ(L, V) under specific transfers as above followed by the projection L_{q}(A, S) → L_{q}(Z[F_{ab}], S′′_{ϵ}).

We now return to our analysis of L and H of 3.14.

Figure 3.18 shows partial generalized Seifert surfaces {V_{x}, V_{y}, V_{z}} for R which may be completed by tubing along the band b in the labeled direction. With respect to the indicated meridians, the pattern of R (and H) is {x, y, [[y, x], z^{−1}xz]z}. Also note that H is obtained from R by “tying the knot J in a band” corresponding the loop γ (γ = (γ ∘m_{x}^{−1})∘m_{x}) whose image under ϕ is [x, z^{−1}]. The map ϕ : G_{H} ↠ F induces an epimorphism Ψ : G_{L} ↠ E where

NOT ALL LINKS ARE CONCORDANT TO BOUNDARY LINKS 31

Figure 3.18

p : F ↠ E defines E as the quotient of F by the normal subgroup generated by r = [[y, x], z^{−1}xz]z.

We shall now take a 2-fold cover branched over K_{y}. Labelling the lifts K_{x_{1}}, K_{x_{2}}, K_{z_{1}}, K_{z_{2}} we then take a 2-fold cover branched over K_{x_{1}}. Note that {K_{x_{1}}, K_{z_{1}}} is very similar to 3.4a and 3.6 so that a proof along more concrete lines would not be so difficult. Let w = yxy^{−1}, W = xwx^{−1}, z_{1} = z, z_{2} = yzy^{−1}, Z_{1} = xz_{1}x^{−1}, Z_{2} = xz_{2}x^{−1}. Then the meridian map of H and ϕ can be lifted to give a homomorphism ˜ ϕ̃∘˜M̃: F ⟨w, W, z_{i}, Z_{i}⟩ ⟲ whose abelianization is the complexity map C ∈ GL( , Z) given by C(w) = w, C(W) = W, C(z_{i}) = r̃_{i}, C(Z_{i}) = R̃_{i} where r̃_{1} = 3z_{1} − 2Z_{1} + w − W, R̃_{1} = 3Z_{1} − 2z_{1} + W − w, r̃_{2} = z_{2} + W − w, R̃_{2} = Z_{2} + w − W. Refer to diagram below for clarification.

F ⟨w, W, z_{i}, Z_{i}⟩ \underrightarrow{˜_{M}} ˜ G̃_{H} \underrightarrow{˜_{ϕ}} F ⟨w, W, z_{i}, Z_{i}⟩
↓  ↓˜p̃
F ⟨w, W ⟩ ⟶ ˜ G̃_{L} ⟶  ˜  Ẽ

Note that the R̃_{i}, r̃_{i} are merely the lifts of r and hence the images of the four meridians ˜ M̃(z_{i}), ˜ M̃(Z_{i}) corresponding to the four lifts of K_{z}. Now, using the “same” meridians for L, we have induced a complexity map C_{L} : Z⟨w, W ⟩ → E′ where E′ is the abelianization of E presented by ⟨w, W, z_{i}, Z_{i} | r̃_{i}, R̃_{i} i = 1, 2⟩ and C_{L}(w) = w, C_{L}(W) = W.

One sees that E′ ≈ Z⟨w, z_{1}⟩ and that with respect to this basis C_{L}(w) = w

and C_{L}(W) = w + 5z_{1}. Therefore the complexity of L is 5 and C_{L} = ( \begin{matrix}1 & 1 \\ 0 & 5 \end{matrix} )

with respect to these bases.

Since H was obtained from the slice link R by tying the knot J using a loop γ where ϕ(γ) = [x, z^{−1}], the Blanchfield pairing of H corresponding to the free cover defined by ϕ (the surfaces {V_{x}, V_{y}, V_{z}}) is λ_{J}([x, z^{−1}]) ∈ L_{q}(F ⟨x, y, z⟩, S), where λ_{J} is the ordinary Blanchfield form of J and the above is its image under t ↦ [x, z^{−1}] (see 3.16). Upon transfer corresponding to the first branched cover we obtain λ_{J}([x, z^{−1}]) ⊕ λ_{J}(y[x, z^{−1}]y^{−1}) and after the second transfer we get λ_{J}(Z_{1}^{−1}z_{1}) ⊕ λ_{J}(z_{1}^{−1}Z_{1}) ⊕ λ_{J}([w, z_{2}^{−1}]) ⊕ λ_{J}([W, Z_{2}^{−1}]). Upon abelianization and applying ˜ p̃ the result is λ_{J}(z_{1}^{2}) ⊕ λ_{J}(z_{1}^{−2}) (since z_{1} = −Z_{1} in E′). After composing with C^{∗}, our invariant of L is λ_{J}(w^{−2}z_{1}^{2}) ⊕ λ_{J}(w^{2}z_{1}^{−2}) in L_{q}(Z_{(3)}[Z × Z], S). Projecting onto z_{1} = t we get λ_{J}(z_{1}^{2}) ⊕ λ_{J}(z_{1}^{−2}) and the map Z⟨w, W ⟩ → Z⟨w, z_{1}⟩ projects to W ↦ t^{5}. If L were concordant to a boundary link then, λ_{J}(t^{2}) ⊕ λ_{J}(t^{−2}) would lie in the image of t → t^{5} (at least modulo torsion). It is then easy to see that, for links in S^{2n+1} where n is odd (so that λ_{J}(t^{−1}) = λ_{J}(t)), this necessitates, for example, that J have zero signatures corresponding to a primitive 10-th root of 1. This completes our analysis of 3.14.

For any k, there exist 2-component homology boundary links L for which precisely k covering spaces are necessary to distinguish L from a boundary link (up to concordance). There is a ribbon link R in any dimension which has pattern (x, y[y, [y, … [y, x]]] …]) (here this is a k-fold commutator) [9, Prop. 2.3]. Choose a 2-disk in that sphere whose boundary γ projects under ϕ to y. Then “tie R in a knot J via γ” (assume 2n + 1 when n is odd and use Seifert form of trefoil knot). The result is L (with above pattern) whose “free Blanchfield form” with respect to ϕ is λ_{J}(y). Now take iterated 2-fold branched covers over the first component. The resulting links are not homology boundary links but the j^{th} covering link group possesses an epimorphism ϕ̃ : G ↠ ⟨y, z| relations in [,] ⟩ where

\begin{aligned}ϕ̃(μ_{2}) & = y[y, [y, … [y, y^{j}z^{−1}y^{1−j}]] …] and \\ ϕ̃(μ_{1}) & = z[z, [z, … [z, z^{j}y^{−1}z^{1−j}]] …] j ≥ 1, \end{aligned}

and the length of these commutators is k − j. So L will have complexity 1

until the k^{th} level when it will have complexity matrix ( \begin{matrix}−2 & 1 \\ 1 & −2 \end{matrix} ). Its form

will be the image of λ_{J}(y) under k compositions of transferring and killing the meridian ϕ(μ_{1}^{2}) of the branching component. It can be inductively shown that the result at each stage is λ_{J}(ϕ̃(μ_{2})) ⊕ λ_{J}(ϕ̃(μ_{1})) giving λ_{J}(y) ⊕ λ_{J}(z) at the final stage after abelianizing. Composing with C^{∗} and the map y → t, z → t^{−1} we get λ_{J}(t) ⊕ λ_{J}(\overline{t}) which is not in the image of t ↦ t^{3}. Details are left to the reader.

4. Relation to Γ-groups

In this section we explain how the invariants of §1 are identical to invariants in certain Γ-groups which appear in the work of Le Dimet and Cappell and Shaneson.

Suppose that W is the wedge of m circles. Pierre Vogel has defined an homology localization W ⟶^{θ} EW which is initial among CW complexes with the property that, for any finite CW pair (X, A) such that X/A is contractible, [X, EW] → [A, EW] is a bijection (see [24; §1] for precise definition). In particular if h : W → E(L) is a meridian map for a higher-dimensional link complement, then h factors θ. In [24], Le Dimet classifies higher-dimensional m-component disk links in D^{n+2} up to disk link concordance by exhibiting an exact sequence (see §0) involving a homotopy invariant associated to the space EW and a Γ-group associated to π_{1}(EW) = F̂. This group F̂, is also the algebraic closure of the free group F, in the sense of [28], and has a convenient description as a direct limit of high-dimensional link groups. Consider the diagrams of group rings and augmentations below:

\begin{matrix}Z & F ⟶ ZF \\ ↓  Φ  ↓ϵ \\ Z & F ⟶^{ϵ} Z \end{matrix}  \begin{matrix}Z & F ⟶ ZF \\ ↓  Ψ  ↓ϵ \\ Z & F̂ ⟶^{ϵ} Z. \end{matrix}

Le Dimet shows that a quotient of the Cappell-Shaneson homology surgery group Γ_{n+3}(Ψ) classifies concordance classes of disk links with vanishing homotopy invariant. By Cappell-Shaneson [3], boundary links (up to boundary link concordance) are classified by Γ_{n+3}(Φ). Thus, to find disk links not concordant to boundary links, it suffices to show that φ : Γ_{n+3}(ZF ⟶^{ϵ} Z) ⟶ Γ̃_{n+3}(ZF̂ ⟶^{ϵ} Z) is not onto where the target is the appropriate quotient of Γ_{n+3}(ZF̂ ⟶ Z) (the nature of this quotient is a difficult technical matter which we will not discuss here). To achieve the same for true (spherical links) it is necessary that this cokernel survive after the quotient by the action of a group of automorphisms A of F̂. Before proceeding we should remark that, since F̂ is a direct limit of link groups and Γ commutes with direct limits, the above remarks about cokernel (φ) were already sketched in [3, 14]. Also, all of the examples we considered in §3 have vanishing homotopy invariant. Indeed this homotopy invariant may always vanish in higher dimensions because every link may be concordant to a sublink of a homology boundary link (see [32]). It is interesting to note that the invariants of §1 are defined for an arbitrary link, not just one which is the closure of a disk link with vanishing homotopy invariant. Therefore the invariants of §1 are a priori more general than their Γ-theoretic analogues which we describe below.

Perhaps our main result from this point of view is the following.

Theorem 4.1. Let F be the free group on {x_{1}, …, x_{m}} m > 1 and φ : F ⟶ F̂ be the algebraic closure. If n is odd then the cokernel of φ_{∗} : Γ_{n+3}(ZF ⟶ Z) ⟶ Γ(ZF̂ ⟶ Z) contains a subgroup of infinite Q rank as well as infinite linearly independent sets of 2 and 4-torsion elements. Moreover this linear independence is unaffected by varying elements of these sets by the action induced by the group of automorphisms A of F̂ (this consists of automorphisms which send φ(x_{i}) to λ_{i}φ(x_{i})λ_{i}^{−1} 1 ≤ i ≤ m for some λ_{i} ∈ F̂).

Elements in the cokernel of φ_{∗} may be detected as follows. The epimorphism F ⟨x_{1}, …, x_{m}⟩ ⟶ Z_{p} given by x_{1} → 1, x_{i} → 0 i > 1, induces index p subgroups F′ and F̂′ of F and F̂ respectively and φ′ : F′ ⟶ F̂′. There are corresponding transfer maps tr : Γ_{n+3}(ZF ⟶ Z) ⟶ Γ_{n+3}(Z_{(p)}F′ ⟶ Z_{(p)}) and tr^{∧} : Γ_{n+3}(ZF̂ ⟶ Z) ⟶ Γ_{n+3}(Z_{(p)}F̂′ ⟶ Z_{(p)}) such that tr^{∧} ∘ φ_{∗} = φ′_{∗} ∘ tr. In general we can iterate this transfer construction. Then we abelianize. The abelianization of F′ is Z^{p(m−1)+1}. Letting N = p(m − 1) + 1, it can be shown (similarly to 1.1) that the abelianization of F̂′ maps to Z_{(p)}^{N}. Thus φ′_{∗} induces φ_{∗} : Γ_{n+3}(Z_{(p)}[Z^{N}] ⟶ Z_{(p)}) ⟶ Γ_{n+3}(Z_{(p)}[Z_{(p)}^{N}] ⟶ Z_{(p)}). Suppose π̂ is the map F̂′ ⟶ Z_{(p)}^{N}.

Theorem 4.2. If L is an m-component disk link in S^{n+2} (n odd) with vanishing homotopy invariant (of Le Dimet) and α_{L} ∈ Γ_{n+3}(ZF̂ ⟶ Z) represents L, then the invariant λ̂_{L} of 1.6 is equal to the image of α_{L} under (perhaps iterated) transfer(s) tr^{∧} followed by π̂_{∗} then taking the associated Blanchfield linking pairing in L^{n+3}(Z_{(p)}[Z_{(p)}^{N}], S) (see [35]).

Theorem 4.2 will be proved in [13]. Note that [10; §3] outlines a slightly different approach.

Theorem 4.1 is proved independently of 4.2 by describing elements α ∈ Γ_{n+3}(ZF̂ ⟶ Z) which descend under π̂ ∘ tr^{∧} to elements ᾱ in the cokernel of φ_{∗}. In fact it suffices to project to a single Z and seek elements in the cokernel of L^{n+3}(Q[Z], S_{p}) ⟶ L^{n+3}(Q[Z_{(p)}], S_{p}) which is precisely the map

\overline{W} ⟶ ˆ \overline{W} which we have considered (see 1.10–1.11). Thus in a sense we have already proved 4.1. By the techniques of §2 there are many such ᾱ (see Corollary 2.6). It is not immediate that such ᾱ lie in the image of tr^{∧} but this is not a serious problem. In fact choosing α in the image of the inclusion induced î_{∗} : Γ_{n+3}(ZF̂′ ⟶ Z) ⟶ Γ_{n+3}(ZF̂ ⟶ Z) provides a rich source of examples that are very understandable under tr^{∧}. The torsion examples are provided by the homology boundary links L of 3.12. More precisely, when viewed as an homology boundary disk link, such L is represented by a class β ∈ Γ_{n+3}(ZF ⟶ Z). The pattern (γ_{1}, γ_{2}) of such L induces a map F ⟶^{γ} F by x_{i} → γ_{i} which factorizes φ as F ⟶^{γ} F ⟶^{ψ} F. Then the classes ψ_{∗}(β) ranging over all L in Chapter 3 section B form the claimed set of torsion elements.

The action of an element of A on Γ_{n+3}(ZF̂ ⟶ Z) can be shown to descend via π_{∗} ∘ tr^{∧} to an action on Γ_{n+3}(Z_{(p)}[Z_{(p)}^{N}] ⟶ Z_{(p)}) which is a block sum of cyclic permutations of order 1 or p. This corresponds precisely to ordering the components of the covering link (consistent with the Z_{p} action). Since the image of tr^{∧} can be seen to be invariant under the natural Z_{p} action, this descended action is trivial. Therefore, linear independence of elements in the cokernel of ϕ_{∗} suffices to prove 4.1.

5. Open Problems

(1) Is every even-dimensional link concordant to a trivial link?

(2) Is every link in S^{n+2} (n > 1) concordant to an homology boundary link? (Conjecture: No)

(3) Is every link in S^{n+2} (n > 1) concordant to a sublink of an homology boundary link?

(4) If two boundary links are concordant are they boundary concordant [3]? Is the map Γ_{2n}(ZF ⟶ Z) ⟶ Γ_{2n}(ZF̂ ⟶ Z) injective?

(5) If a link in S^{3} has vanishing Milnor’s invariants, is it concordant to an homology boundary link? A sublink of an homology boundary link?

(6) Can every element of Γ_{2n}(ZF̂ ⟶ Z) be detected by our methods?

Rice University, Houston, TX Indiana University, Bloomington, IN

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(Received ??) (Revised ??)