Nonconcordant links with homology cobordant zero framed surgery manifolds
Abstract.
We use topological surgery theory to give sufficient conditions for the zero framed surgery manifold of a 3component link to be homology cobordant to the zero framed surgery on the Borromean rings (also known as the 3torus), via a topological homology cobordism preserving the free homotopy classes of the meridians.
This enables us to give examples of 3component links with unknotted components and vanishing pairwise linking numbers, such that any two of these links have homology cobordant zero surgeries in the above sense, but the zero surgery manifolds are not homeomorphic. Moreover the links are not concordant to one another, and in fact they can be chosen to be height but not height symmetric grope concordant, for each which is at least three.
1991 Mathematics Subject Classification:
57M25, 57N701. Introduction
It is well known that the study of homology cobordism of 3manifolds is essential for understanding the concordance of knots and links: homology cobordism of the exteriors of links in is equivalent to concordance in a homology , and an additional mild normal generation condition for is equivalent to topological concordance in (this also holds modulo the 4dimensional Poincaré conjecture in the smooth case).
We recall the definitions: two component links and in are said to be topologically (respectively smoothly) concordant if there exist locally flat (respectively smoothly embedded) disjoint annuli in cobounded by components of and . Two 3manifolds and bordered by a 2manifold , that is endowed with a marking , are topologically (respectively smoothly) homology cobordant if there is a topological (respectively smooth) 4manifold with , , such that the inclusions () induce isomorphisms on integral homology groups. In this paper links are oriented, and link exteriors are always bordered by under the zero framing.
In high dimensions, concordance classification results were obtained by studying homology surgery, with the aim of surgeries being to produce a homology cobordism of the exteriors (for example, see [CappellShaneson:19741, CappellShaneson:19801, LeDimet:19881]). On the other hand, for knots and links in dimension three, the zero surgery manifolds and their 4dimensional homology cobordisms have been extensively used in the literature in order to understand the structure peculiar to low dimensions, especially in the topological category. Recall that performing zero framed surgery on a link in yields a closed manifold, called the zero surgery manifold.
The classical invariants such as the knot signature and Levine’s algebraic knot concordance class [Levine:19692, Levine:19691] are obtained from the zero surgery manifold of a knot, via the Blanchfield form. Also higher order knot concordance obstructions, such as CassonGordon invariants [CassonGordon:19781, CassonGordon:19861], and CochranOrrTeichner signatures [CochranOrrTeichner:19991] are obtained from the zero surgery manifold (often together with the homology class of the meridian).
A natural interesting question is whether the homology cobordism class of a zero surgery manifold determines the concordance class of a knot or link or if it determines the homology cobordism class of the exterior.
In this paper we show, in a strong sense involving homotopy of meridians, that the answer is negative for a large class of links satisfying a certain nonvanishing condition on Milnor’s invariants, even in the framework of symmetric grope and Whitney tower generalisations of concordance and homology cobordism in the sense of [CochranOrrTeichner:19991, Cha:20121]. Also we employ topological surgery in dimension 4 to give a new construction of homology cobordisms of zero surgery manifolds. Next we state our main theorems, after which we will discuss these aspects further.
Theorem 1.1.
Suppose . Then there are infinitely many 3component links , , … with vanishing pairwise linking numbers and with unknotted components, satisfying the following for any .

The zero surgery manifolds and are not homeomorphic.

There is a topological homology cobordism between and in which the th meridians of and are freely homotopic for each .

The links and are height but not height symmetric grope concordant. In particular and are not concordant.
For a definition of height symmetric grope concordance, see Definition LABEL:Definition:gropeconcordance. Our links are obtained from the Borromean rings, which will be our for all , by performing a satellite construction along a curve lying in the kernel of the inclusion induced map .
As a counterpoint to Theorem 1.1, we show that there are infinite families of links with the same nonvanishing Milnor invariants with homeomorphic zero surgery manifolds preserving the homotopy classes of the meridians, but which are not concordant.
The Milnor invariant of an component link associated to a multiindex with , as defined in [Milnor:19571], will be denoted by . We denote its length by . Define .
Theorem 1.2.
Let be a multiindex with nonrepeating indices with length . For any there are infinitely many component links with unknotted components, satisfying the following.

The have identical invariants: for all . In addition , and for .

There is a homeomorphism between the zero surgery manifolds and which preserves the homotopy classes of the meridians.

The links and are height but not height symmetric grope concordant. In particular and are not concordant.
The case of should be compared with Theorem 1.1 since then the links have vanishing pairwise linking numbers. To construct such links we start with certain iterated Bing doubles constructed using T. Cochran’s algorithm which realise the Milnor invariant required. We then perform satellite operations which affect the concordance class of the link but do not change the homeomorphism type of the zero surgery manifold.
We remark that we could also phrase Theorems 1.1 and 1.2 in terms of symmetric Whitney tower concordance instead of grope concordance.
In the three subsections below, we discuss some features of Theorem 1.1, regarding: (i) the use of topological surgery in dimension 4, (ii) link concordance versus zero surgery homology cobordism, and (iii) link exteriors and the homology surgery approach.
1.1. Topological surgery for 4dimensional homology cobordism
An interesting aspect of the proof of Theorem 1.1 is that we employ topological surgery in dimension 4 to give a sufficient condition for certain zero surgery manifolds of 3component links to be homology cobordant. It is well known that topological surgery in dimension 4 is useful for obtaining homology cobordisms (and consequently concordances), although the current state of the art in terms of “good” groups, for which the null disc lemma is known, is still insufficient for the general case. M. Freedman and F. Quinn showed that knots of Alexander polynomial one are concordant to the unknot [FreedmanQuinn:19901, Theorem 11.7B]. J. Davis extended the program to show that two component links with Alexander polynomial one are concordant to the Hopf link [Davis:20061]. The above two cases use topological surgery over fundamental groups and respectively. Due to the rarity of good groups for 4dimensional topological surgery, there are not many other situations where such positive results on knot and link concordance can currently be proven. As another case, S. Friedl and P. Teichner in [FriedlTeichner:20051] found sufficient conditions for a knot to be homotopy ribbon, and in particular slice, with a certain ribbon group .
We give another instance of the utility of topological surgery for constructing homology cobordisms, using the group , which is manageable from the point of view of topological surgery in dimension 4. Indeed, our sufficient condition for zero surgery manifolds to be homology cobordant focuses on the Borromean rings as a base link. The zero surgery manifold of the Borromean rings is the 3torus , whose fundamental group is .
To state our result, we use the following notation: let . Denote the zero surgery manifold of a link by as before. For a component link with vanishing pairwise linking numbers, there is a canonical homotopy class of maps which send the homotopy class of the th meridian of to that of the Borromean rings, namely the th circle factor of . After choosing an identification of , we can use this to define the coefficient homology . We say that a map is a homology equivalence if is homotopic to and induces isomorphisms on .
Theorem 1.3.
Suppose is a component link whose components have trivial Arf invariants and there exists a homology equivalence . Then there is a homology cobordism between and for which the inclusion induced maps are such that the composition from left to right takes meridians to meridians.
1.2. Link concordance versus zero surgery homology cobordism
We review the general question of whether links with homology cobordant zero surgery manifolds are concordant. The answer to the basic question is easily seen to be no, once one knows of a result of C. Livingston that there are knots not concordant to their reverses [Livingston:19831]. Note that a knot and its reverse have the same zero surgery manifold. This leads us to consider some additional conditions on the homology cobordism, involving the meridians. In what follows meridians are always positively oriented.
First, observe the following: the exteriors of two links are homology cobordant if and only if the zero framed meridians cobound framed annuli disjointly embedded in a homology cobordism of the zero surgery manifolds. (For the if direction, note that the exterior of the framed annuli is a homology cobordism of the link exteriors.) In particular it holds if two links (or knots) are concordant.
Regarding the knot case, in [CochranFranklinHeddenHorn:20111], T. Cochran, B. Franklin, M. Hedden and P. Horn considered homology cobordisms of zero surgery manifolds in which the meridians are homologous: in the smooth category, they showed that the existence of such a homology cobordism is insufficient for knots to be concordant. In the topological case this is still left unknown.
Concerning a stronger homotopy analogue, the following is unknown in both the smooth and topological cases:
Question 1.4.
If there is a homology cobordism of zero surgery manifolds of two knots in which the meridians are homotopic, are the knots concordant? Or concordant in a homology ?
For the link case, results in the literature give nonconcordant examples whose zero surgery manifolds admit a homology cobordism with homotopic meridians. As a generic example in the topological category, consider a 2component link with linking number one. The zero surgery manifold is a homology 3sphere, which bounds a contractible topological 4manifold by [FreedmanQuinn:19901, Corollary 9.3C]. Taking the connected sum of such 4manifolds, one obtains the following: the zero surgery manifolds of any two linking number one 2component links cobound a simply connected topological homology cobordism. Note that in this case the meridians are automatically homotopic. There are many linking number one 2component links which are not concordant, as can be detected, for example, by the multivariable Alexander polynomial [Kawauchi:19781, Nakagawa1978]. For related indepth study, the reader is referred to, for instance, [ChaKo:19992, FriedlPowell:20111, Cha:20121]. With our respective coauthors, we gave nonconcordant linking number one links with two unknotted components, for which abelian invariants such as the multivariable Alexander polynomial are unable to obstruct them from being concordant.
There are other examples, which have knotted components: in [CochranFranklinHeddenHorn:20111, end of Section 1], they discuss 2component linking number zero links with homeomorphic zero surgery manifolds which have nonconcordant (knotted) components. These links are obviously not concordant, and it can be seen that the homeomorphisms preserve meridians up to homotopy.
By contrast with the above examples, our links have unknotted components and vanishing pairwise linking numbers. Another feature exhibited by the links of Theorems 1.1 and 1.2 is that a great deal of the subtlety of symmetric grope concordance of links can occur within a single homology cobordism/homeomorphism class of the zero surgeries, even modulo local knot tying.
1.3. Link exteriors and the homology surgery approach
Our results serve to underline the philosophy that when investigating the relative problem of whether two links are concordant, and neither of them are the unlink, one should consider obstructions to homology cobordism of the link exteriors viewed as bordered manifolds, rather than to homology cobordism of the zero surgery manifolds, even in low dimensions. This was implemented in, for example [Kawauchi:19781, Nakagawa1978, Cha:20121] (see also [FriedlPowell:20111] for a related approach).
Although we stated our results in terms of grope concordance of links, in Theorems 1.1 and 1.2 given above, in fact we show more: the link exteriors are far from being homology cobordant, as measured in terms of Whitney towers. A more detailed discussion is given in Section LABEL:section:amenablesignature. For the purpose of distinguishing exteriors, we use the amenable CheegerGromov invariant technology for bordered 3manifolds (particularly for link exteriors) developed in [Cha:20121], generalising applications of invariants to concordance and homology cobordism in [CochranOrrTeichner:19991, CochranHarveyLeidy:20091, ChaOrr:200901].
We will now discuss our results from the viewpoint of the homology surgery approach to link concordance classification, initiated by S. Cappell and J. Shaneson [CappellShaneson:19741, CappellShaneson:19801] and implemented in high dimensions by J. Le Dimet [LeDimet:19881] using P. Vogel’s homology localisation of spaces [Vogel:19781]. The strategy consists of two parts. Consider the problem of comparing two given link exteriors. First we decide whether the exteriors have the same “Poincaré type,” which roughly means that they have homotopy equivalent Vogel homology localisations. If so, there is a common finite target space, into which the exteriors are mapped by homology equivalences rel. boundary. Once this is the case, a surgery problem is defined, and one can try to decide whether homology surgery gives a homology cobordism of the exteriors. The first step is obstructed by homotopy invariants (including Milnor invariants in the low dimension). The failure of the second step is measured by surgery obstructions, which are not yet fully formulated in the low dimension (even modulo that 4dimensional surgery might not work), since the fundamental group plays a more sophisticated central rôle; see [Powell:20121] for the beginning of an algebraic surgery approach to this problem in the context of knot slicing.
Our examples illustrate that for many Poincaré types, namely those in Theorems 1.1 and 1.2, we get a rich theory of surgery obstructions within each Poincaré type, which is invisible via zero surgery manifolds. We remark that for our links in Theorems 1.1 and 1.2, there is a homology equivalence of the exterior of each into that of a fixed one, say , since we use satellite constructions (see Section LABEL:section:constructionlinksgropeconcordance). It follows that the exteriors have the same Poincaré type in the above sense. In this paper, (parts of the notyetfullyformulated) homology surgery obstructions in dimension 4 have their incarnation in Theorem LABEL:theorem:amenablesignatureforsolvablecobordism, the Amenable Signature Theorem.
Organisation of the paper
In Section 2, we explore the implications of the hypothesis that a homology equivalence as in Theorem 1.3 exists, and we prove Theorem 1.3 in Section 3. In Section LABEL:section:constructionlinksgropeconcordance, we construct links with a given Milnor invariant with nonrepeating indices, and perform satellite operations on the links to construct the links of Theorems 1.1 and 1.2, which are height symmetric grope concordant. In Section LABEL:section:amenablesignature, we show that none of these links are height grope concordant to one another.
Acknowledgements
We would like to thank Stefan Friedl for many conversations: our first examples were motivated by joint work with him. We also thank Jim Davis for discussions relating to the proof of his theorem on Hopf links, and Prudence Heck, Charles Livingston, Kent Orr and Vladimir Turaev for helpful discussions. Finally we thank the referee for valuable comments and for being impressively expeditious.
Part of this work was completed while the first author was a visitor at Indiana University in Bloomington and the second author was a visitor at the Max Planck Institute for Mathematics in Bonn. We would like to thank these institutions for their hospitality.
The first author was partially supported by National Research Foundation of Korea grants 2013067043 and 2013053914. The second author gratefully acknowledges an AMS Simons travel grant which aided his travel to Bonn.
2. Homology type of zero surgery manifolds and the 3torus
This section discusses the hypotheses of Theorem 1.3. We begin the section by briefly reminding the reader who is familiar with Kirby calculus of a nice way to see the following well known fact.
Lemma 2.1.
The zero surgery manifold of the Borromean rings is homeomorphic to the 3torus.
Proof.
Place dots on two components of the Borromean rings and a zero near the other. Each component of the Borromean rings is a commutator in the meridians of the other two components, so this is a Kirby diagram for , whose boundary is . The 1handles (dotted circles) can be replaced with zero framed 2handles without changing the boundary. ∎
In the following proposition we expand on the meaning and implications of the condition in Theorem 1.3. Denote the exterior of a link by as before.
Proposition 2.2.
Suppose that is a 3component link. Then the following are equivalent.

There is a homology equivalence .

The preferred longitudes generate the link module .

The pairwise linking numbers of vanish and .
Furthermore, (any of) the above conditions imply that has multivariable Alexander polynomial , and implies that the Milnor invariant is equal to .
Proof.
First we will observe (2) and (3) are equivalent. Longitudes of represent elements in if and only if they are zero in ; that is, the pairwise linking numbers are zero. If this is the case, is isomorphic to , since is obtained by attaching three 2handles to along the longitudes and then attaching three 3handles along the boundary. It follows that longitudes generate if and only if .
Suppose (1) holds. Denote the meridians of by () and the linking number of the th and th components by . The th longitude , which is homologous to , is zero in . Since forms a basis of , it follows by linear independence that for any and . Also, . This shows that (3) holds.
Suppose (3) holds. Start with a map that sends to the th factor and to a point. Observe that factors through the inclusion induced map and the identifications ; this follows from the fact that is generated by the and and that both and are quotient maps, with their kernels generated by the . Since is a , elementary obstruction theory shows that extends to a map .
Consider the universal coefficient spectral sequence (see e.g. [Levine:19771, Theorem 2.3]) . We have since , and . It follows that . By duality, . Also, since the cover of is noncompact. Since and for , it follows that is a homology equivalence. This completes the proof of the equivalence of (1), (2) and (3).
Suppose (1), (2) and (3) hold. Recall that the scalar multiplication of a loop by in the module is defined to be conjugation by . Since and commute as elements of the fundamental group, we have in . From this and (2), it follows that there is an epimorphism of onto . Since the zeroth elementary ideal of is the principal ideal generated by , it follows that is a factor of . We now invoke the Torres condition (see e.g. [Kawauchi:19961, Theorem 7.4.1]): where is the sublink of with the first component deleted and is the pairwise linking number. Since by (3), we have . It follows that is a factor of . Similarly and are factors. Therefore .
To show the last part, suppose that . By [Kawauchi:19961, Proposition 7.3.14], the singlevariable Alexander polynomial of is given by
It follows that has Conway polynomial , by the standard substitution . In [Cochran:19851, Theorem 5.1], Cochran identified the coefficient of in with for 3component links with pairwise linking number zero. Applying this to our case, it follows that . ∎
3. Construction of homology cobordisms using topological surgery
This section gives the proof of Theorem 1.3. The proof will use surgery theory, and will parallel the proof given by Davis in [Davis:20061] (see also [Hillman:20021, Section 7.6]). We will provide some details in order to fill in where the treatment in [Davis:20061] was terse, and to convince ourselves that the analogous arguments work in the case of interest.
For the convenience of the reader we restate Theorem 1.3 here.
Theorem 1.3.
Suppose is a component link whose components have trivial Arf invariants and there exists a homology equivalence . Then there is a homology cobordism between and for which the inclusion induced maps are such that the composition from left to right takes meridians to meridians.
Remark 3.1.
It is an interesting question to determine whether there are extra conditions which can be imposed in order to see that the Arf invariants of the components are forced to vanish by the homological assumptions. In the cases of knots and two component links with Alexander polynomial one, the Arf invariants of the components are automatically trivial. For knots computes the Arf invariant by [Levine:19661]. For two component links one observes that and give the Alexander polynomials of the components, by the Torres condition, and then applies Levine’s theorem. These arguments do not seem to extend to the three component case of current interest.
The proof of Theorem 1.3 will occupy the rest of this section. In order to produce a homology cobordism, we will first show that there is a normal cobordism between normal maps and . Interestingly, we can work with smooth manifolds in order to establish the existence of a normal cobordism. This will make arguments which invoke tangent bundles and transversality easier to digest. Only at the end of the proof of Theorem 1.3, where we take connected sums with the manifold, and where we claim that the vanishing of a surgery obstruction implies that surgery can be done, do we need to leave the realm of smooth manifolds.
Definition 3.2.
Let be an dimensional manifold with a vector bundle . A degree one normal map over is an manifold with a map which induces an isomorphism , together with a stable trivialisation .
A degree one normal cobordism between normal maps and is an –dimensional cobordism between and with a map extending and , which induces an isomorphism
together with a stable trivialisation .
For us, let , and let be its tangent bundle. We fix a framing on the stable tangent bundle of the target torus once and for all. Note that this canonically determines a trivialisation of the tangent bundle of , for any map , by the following diagram, in which the bottom composition is the constant map, denoted , and the top composition is the pull back . The middle composition is the induced framing.