Two-torsion in the grope and solvable filtrations of knots

# Two-torsion in the grope and solvable filtrations of knots

Hye Jin Jang Department of Mathematics, POSTECH, Pohang, 790–784, Republic of Korea
###### Abstract.

We study knots of order in the grope filtration and the solvable filtration of the knot concordance group. We show that, for any integer , there are knots generating a subgroup of . Considering the solvable filtration, our knots generate a subgroup of distinct from the subgroup generated by the previously known 2-torsion knots of Cochran, Harvey, and Leidy. We also present a result on the 2-torsion part in the Cochran, Harvey, and Leidy’s primary decomposition of the solvable filtration.

###### 1991 Mathematics Subject Classification:
57M27, 57M25, 57N70

## 1. Introduction

Two oriented knots and in are said to be (topologically) concordant if and cobound a locally flat annulus in , where denotes the mirror image of with orientation reversed. Concordance is an equivalence relation on the set of all oriented knots in , and concordance classes form an abelian group under connected sum. It is called the knot concordance group. A knot which represents the identity is called a slice knot. Understanding the structure of the knot concordance group has been one of the main interests in the study of knot theory.

In the celebrated paper [COT03] of Cochran, Orr, and Teichner, they introduced several types of approximations of a knot being slice. They defined the concept of knots bounding (symmetric) gropes of height in for each half integer , relaxing the geometric condition of a knot bounding a slice disk. Also, a knot whose zero framed surgery bounds a -manifold which resembles slice knot exteriors is said to be -solvable, where the half integer depends on the degree of resemblance. (See Section 7 and 8 of [COT03] for the precise definitions of a grope and -solvability.) Any slice knot bounds a grope of arbitrary height in and is -solvable for any . Also, a knot bounding a grope of height is -solvable [COT03, Theorem 8.11], while the converse is unknown.

The set of concordance classes of knots bounding gropes of height in is a subgroup of , which is denoted by . Similarly, the set of concordance classes of -solvable knots is a subgroup of , which is denoted by . They form filtrations in the knot concordance group :

 {0}≤⋯≤Gh+0.5≤Gh≤⋯,G1.5≤G1≤C,
 {0}≤⋯≤Fh+0.5≤Fh≤⋯,F0.5≤F0≤C,

which are called the grope and solvable filtration respectively.

Recently, to understand the structure of the knot concordance group, the graded quotients and have been studied extensively and successfully. (See [COT03, COT04, CT07, Cha07, CK08, KK08, CHL09, Hor10, CHL11a, CHL11c, CO12, Cha14a], for example.) Interestingly, nothing is known about the structure of or except .

In this paper, we study order 2 elements in the knot concordance group and those filtrations. There were some results on the order 2 elements in the solvable filtration. For instance, is isomorphic to with Arf invariant map as an isomorphism. Also, the study of J. Levine on the algebraic concordance group [Lev70] implies that contains as a subgroup. In [Liv99], C. Livingston found infinitely many order 2 elements in using Casson-Gordon invariants [CG78, CG86]. Using the fact that the Casson-Gordon invariants vanish for -solvable knots [COT03, Section 9], Livingston’s method also gives infinitely many examples in , which generate a subgroup of . In [CHL11b], T. Cochran, S. Harvey, and C. Leidy showed that for each integer , there are infinitely many 2-torsion elements in which generate a subgroup of .

Regarding the grope filtration, much less was known about 2-torsion. It is known that and are isomorphic to and is isomorphic to [Tei02, Theorem 5]. Hence the results on the solvable filtration imply that is isomorphic to and contains as a subgroup.

In this paper, we show that has infinitely many 2-torsion elements for any :

###### Theorem A.

For any integer , there is a subgroup of isomorphic to . Moreover, the subgroup is generated by -solvable knots with vanishing poly-torsion-free-abelian (abbreviated PTFA) -signature obstructions.

Discussions on (PTFA) -signature obstructions and vanishing PTFA -signature obstructions property appear at the end of this section and Section 2.

Note that there is another concept of knots bounding (symmetric) gropes of height in , which can be viewed as an approximation of having slice disks (see Section 7 of [COT03]). Similarly to the grope case, the sets of concordance classes of knots bounding Whitney towers of height , for every half integer , forms a filtration on . It is known that a knot bounding a grope of height also bounds a Whitney tower of the same height (see [Sch06, Corollary 2]). Hence our examples generate subgroups in the successive quotients of Whitney tower filtration as well.

To construct knots in Theorem A, we adopt the iterated infection construction in a similar way to the Cochran, Harvey, and Leidy’s method in [CHL11b] but with different infection knots. Using the infinite set of knots bounding gropes of height 2 in constructed by P. Horn [Hor10] and the infection knots used in [Cha14a] as our basic ingredients, we can find infection knots with desired properties.

Figure 1 illustrates an order 2 knot bounding a grope of height 4, but not height 4.5, in (not necessarily with vanishing PTFA -signature obstructions). Note that it is constructed by the iterated satellite construction , where and are as in Figure 4 and 5 and is the Cochran-Teichner’s knot bounding height 2 (see [CT07, Figure 3.6]). The detailed explanation on the construction will be given later.

In the proof of Theorem A, to show certain knots are not in we actually prove that they are not -solvable. That is contained in implies that the knots in Theorem A generating a subgroup of also generate a subgroup of .

Recall that Cochran, Harvey, and Leidy found knots which generate a subgroup of . We prove that our knots are distinct from Cochran, Harvey, and Leidy’s knots in :

###### Theorem B.

For any integer , there are -solvable knots with vanishing PTFA -signature obstructions which generate a subgroup of . This subgroup has trivial intersection with the subgroup generated by Cochran, Harvey and Leidy’s knots in [CHL11b].

Cochran, Harvey, and Leidy’s knots do not have the property of vanishing PTFA -signature obstructions. (Recall that, in[CHL11b], Cochran, Harvey, and Leidy’s knots have been shown not to be -solvable by calculating a nonzero PTFA -signature obstruction.) The property of PTFA -signature obstructions of our knots is a distinct feature of our knot from Cochran, Harvey, and Leidy’s knots and this implies the second part of Theorem B.

#### Results on the primary decomposition structures

In [CHL11c] they defined the concept of -solvability of knots which is a generalization of -solvability. Here is a nonnegative half integer and is an -tuple of Laurent polynomials over . They showed that the subgroups of -solvable knots form another filtration on which is coarser than , that is, .

Hence the quotient maps induce

 FnFn.5⟶∏PFnFPn.5∩Fn,

where the product is taken over all -tuples as above. This is called the primary decomposition structures of the knot concordance group by Cochran, Harvey, and Leidy [CHL11c].

They used the primary decomposition structures crucially in their proof of the existence of subgroup of as follows. For each , with some additional conditions, they show that there is a negatively amphichiral -solvable knot which does not vanish in but vanishes in for any -tuple which is strongly coprime to . (For the definition of strong coprimeness and precise conditions required for , see Section 5 or [CHL11c, Definition 4.4].) By using an infinite set of which are pairwise strongly coprime, they showed the existence of subgroup of .

On the other hand, our proof of Theorems A and B does not involve the use of the primary decomposition structures. This might be regarded as a simpler proof to the existence of a subgroup of . We are able to do that by using the recently introduced amenable -signature obstructions in the study of the knot concordance by Cha and Orr which we explain later.

Using our method, we also find new 2-torsion elements in the primary decomposition structures. A corollary of Cochran, Harvey, and Leidy’s result [CHL11a, Theorem 5.3 and Threorem 5.5] is that for any there is a set of infinitely many -tuples of the form such that, for any , there are infinitely many -solvable knots of order which

1. generate a subgroup in , but

2. vanish in , for any , .

A precise description of appears in Section 6. In this setting, we show the following:

###### Theorem C.

For each , there are infinitely many -solvable knots which satisfy above (1) and (2), but generate a subgroup in having trivial intersection with Cochran, Harvey, and Leidy’s subgroup.

#### The obstructions used: amenable L2-signature defects

The obstructions we use to detect non--solvable, or non--solvable knots are amenable von Neumann -invariants, and equivalently, -signature defects.

It has been known that certain -invariants of the zero framed surgery on an -solvable knot over PTFA groups vanish ([COT03, Theorem 4.2], Theorem 2.1). They have been used as the key ingredient to detect many non--solvable knots. Cochran, Harvey, and Leidy extended the vanishing theorem of PTFA -invariants to -solvable knots and found their 2-torsion elements.

In 2009, J. Cha and K. Orr [CO12] extended the homology cobordism invariance of -invariants to amenable groups lying in Strebel’s class. Since then amenable -invariants have been used for the study of various problems on homology cobordism and knot concordance. For example, Cha and Orr [CO13] found infinitely many hyperbolic 3-manifolds which are not pairwisely homology cobordant but cannot be distinguished by any previously known methods. Also, Cha, Friedl, and Powell [Cha14b, CP14, CFP14] detected many non-concordant links which have not been detected previously. Especially, Cha [Cha14a] found a subgroup of isomorphic to , which cannot be detected by any PTFA -invariants.

In this paper, we utilize amenable -invariants to detect 2-torsion elements. The use of amenable groups as the image of group homomorphisms enables us to detect various 2-torsion elements described in Theorems A, B, and C, which are unlikely to be detected by using PTFA -signature obstructions. Also, the fact that there are a lot more independent amenable -invariants enables us to show the existence of subgroups in Theorems A, B, and C without using the primary decomposition structures. In particular, we use amenable -invariant over groups with -torsion for various choices of prime in the proof.

We also employ the recent result of Cha on Cheeger-Gromov bounds [Cha]. Note that Cochran, Harvey, and Leidy’s 2-torsion knots are not fully constructive because the explicit number of connected summands of infection knots needed was unknown. It was totally due to the absence of an explicit estimate of the universal bound for -invariants of a 3-manifold (see Theorem 2.5). Recently, Cha found an explicit universal bound in terms of topological descriptions of a 3-manifold (see Theorem 2.6). Using this result, we can present our example in a fully constructive way. We remark that not only our knots but also many previously known non--solvable knots can be modified to be fully constructive.

This article is organized as follows. In Section 2, we provide definitions and properties of von Neumann -invariants of closed 3-manifolds and -signature defects of 4-manifolds, which will be used in this paper. In Section 3, we give a specific construction of knots bounding gropes of height in which will be the prototype of 2-torsion knots in Theorems A, B, and C. In Sections 4 Theorems A and B are proven. We discuss the definition of the refined filtration related to a tuple of integral Laurent polynomials of Cochran, Harvey, and Leidy in Section 5. In Section 6, we prove Theorem B.

### Acknowledgements

The author thanks her advisor Jae Choon Cha for guidance and encouragement on this project. The author is partially supported by NRF grants 2013067043 and 2013053914.

## 2. Preliminaries on von Neumann ρ-invariants

The von Neumann -invariants on closed 3-manifolds, and equivalently, the -signature defects on 4-manifolds can be used as obstructions for a knot to being -solvable [COT03, Cha14a]. In this section, we give a brief introduction to these invariants and their properties which are useful for our purposes. For more details, we recommend [COT03, Section 5], [CT07, Section 2], and [Cha14a, Section 3].

Let be a closed, smooth, oriented 3-manifold with a Riemannian metric . Let be a countable group and a group homomorphism. Let be the -invariant of the odd signature operator of . Cheeger and Gromov defined the von Neumann -invariant by lifting the metric and the signature operator to the -cover of and using the von Neumann trace. Then the von Neumann -invariant is defined as the difference between and . It is known that is a real-valued topological invariant which does not depend on the choice of [CG85].

On the other hand, let be a 4-manifold and be a group homomorphism. Then there is an invariant so-called the -signature defect of , which is denoted by in this article. It is known that is equal to where is the restriction of on [Mat92, Ram93]. In this article this fact will be used frequently.

The following theorem of Cochran, Orr, and Teichner enables to use -invariants as obstructions for a knot being -solvable:

###### Theorem 2.1.

[COT03, Theorem 4.2] Let be an -solvable knot and be the zero framed surgery on . Let be a PTFA group whose -th derived subgroup vanishes, and be a group homomorphism which extends to an -solution for . Then vanishes.

Note that is assumed to be a PTFA group, that is, it has a subnormal series

 0=Gm⊲Gm−1⊲…G1⊲G0=G

such that every is torsion-free abelian.

In [CO12], Cha and Orr showed the homology cobordism invariance of -invariants over amenable groups lying in Strebel’s class for a ring , which will be called amenable -invariants in this paper. Using the result, Cha extended the vanishing property of PTFA -invariants in Theorem 2.1 for -solvable knots to amenable -invariants as follows:

###### Theorem 2.2.

[Cha14a, Theorem 1.3] Let be an -solvable knot. Let be an amenable group lying in Strebel’s class where is or , and . Let be a group homomorphism which extends to an -solution for and sends the meridian of to an infinite order element in . Then vanishes.

The definition of amenable groups lying in Strebel’s class will not be presented in this paper since the following lemma is sufficient for our purposes:

###### Lemma 2.3.

[CO12, Lemma 6.8] Suppose is a group admitting a subnormal series

 0=Gm⊲Gm−1⊲…G1⊲G0=G

whose quotient is abelian. Let be a prime. If every has no torsion coprime to , then is amenable and in . If every is torsion-free, then is amenable and in for any ring .

As seen in this lemma, the class of amenable groups in Strebel’s class is much larger than and subsumes PTFA groups as a special case.

A natural question here is whether there is a knot whose non--solvability cannot be detected by PTFA -invariants but can be detected by amenable -invariants. In this context, Cha introduced the concept of -solvable knots with vanishing PTFA -signature obstructions:

###### Definition 2.4.

[Cha14a, Definition 4.7] A knot is said to be -solvable with vanishing PTFA -signature obstructions if there is an -solution for such that for any PTFA group and for any group homomorphism which extends to , vanishes.

It turns out that the set of classes of -solvable knots with vanishing PTFA -signature obstructions is a subgroup between and (see [Cha14a, Proposition 4.8]). Cha showed that there are infinitely many classes of -solvable knots with vanishing PTFA -signature obstructions which are linearly independent in . In other words, there is a subgroup of which is isomorphic to (Theorem 1.4 of [Cha14a]). Note that all of the previously known -solvable knots which are not -solvable ([COT03, COT04, CT07, CHL09, Hor10, CHL11c, CHL11a]) are not in , since the proofs of their non--solvability are actually done by showing that they are not -solvable with vanishing PTFA -signature obstructions. Hence Cha’s knots are distinguished from any previously known nontrivial elements in .

We close this section with some useful properties about -invariants and -signature defects for our purposes.

###### Theorem 2.5.

[CG85, Cha] For any closed 3-manifold , there is a constant such that, for any homomorphism , is less than .

While Cheeger and Gromov’s proof only shows the existence of such bound , Cha found an explicit when we are given a triangulation of . Especially when is a zero framed surgery on a knot, then can be chosen as a constant multiple of the crossing number of the knot as follows:

###### Theorem 2.6.

[Cha, Theorem 1.9] Let be a knot and be the crossing number of . Suppose is a 3-manifold obtained by zero framed surgery on . Then

 |ρ(M,ϕ)|<69713280⋅c(L)

for any homomorphism into any group .

The additive property of -signature defects under the union of 4-manifolds will be used several times:

Let be a boundary connected sum of two 4-dimensional manifolds and . Then for any homomorphism ,

 S(V,ϕ)=S(V1,ϕ)+S(V2,ϕ)

where the induced homomorphisms on and from are also denoted by .

Recall that the Levine-Tristram signature function of a knot is a function on defined as

 σK(ω)=sign((1−ω)A+(1−¯¯¯ω)AT),

where A is a Seifert matrix of . It is known that abelian -invariants of can be considered as the average of the Levine-Tristram signature function of :

###### Lemma 2.8.

[COT04, Proposition 5.1][Fri05, Corollary 4.3] Let be a knot with the meridian and be a homomorphism whose image is contained in an abelian subgroup of . Then

 ρ(M(K),ϕ)={∫S1σK(ω)dωif % ϕ([μ])∈G has infinite order,∑d−1r=0σK(e2πr√−1/d)if ϕ([μ])∈G has finite order d.

## 3. Construction of knots bounding gropes

In this section, we construct infinitely many negatively amphichiral knots bounding gropes of height in , with vanishing PTFA -signature obstructions. Recall that a knot is called negatively amphichiral if is isotopic to , which implies has order or in the knot concordance group. Hence knots constructed in this section can serve as prototypes for various order knots in our main theorems.

### 3.1. Infection of a knot

The basic ingredient of our construction is the infection of knots, which also has been called companion, genetic modification or satellite construction. We recall its definition briefly for completeness and arrangement of notations. More details about infection can be found in [Rol03, Section 4.D] and [COT04, Section 3].

Let be a knot and the knot exterior of . Let , , be curves whose union forms an unlink in . Let , , be knots. For each , remove a tubular neighborhood of and instead glue the knot exterior along their boundaries, in such a way that the meridian of is identified with the reverse of the longitude of and the longitude of is identified with the meridian of . The resulting 3-manifold is homeomorphic to , but the the knot type of may have been changed. It is said that is infected by ’s along ’s and the resulting knot is denoted by . We call the seed knot, ’s the axes, and ’s the infection knots.

Infection has been utilized to construct new slice knots, knots bounding gropes of arbitrary height in , or -solvable knots by the virtue of the following results:

###### Proposition 3.1.

Let and be slice knots. Then for any axis , is also slice.

###### Proposition 3.2.

[CT07, Theorem 3.8] Let be a slice knot. Suppose each bounds a grope of height in the exterior of and each is a connected sum of the Horn’s knots (see Figure 2). Then bounds a grope of height in .

Figure 2 describes the knot of P. Horn in [Hor10] by using the language of clasper surgery of K. Habiro [Hab00]. The knot is obtained by performing clasper surgery on the unknot along the tree . (For a surgery description of , see Figure 3 of [Hor10].)

###### Proposition 3.3.

[COT04, Proposition 3.1] Let be a slice knot. Suppose the homotopy class of lies in , and ’s are -solvable knots for all . Then is -solvable.

Here we present the proof of the last proposition briefly for later use.

###### Sketch of the proof.

For any -solvable knot , , there is a -solution with isomorphic to . (For its proof, see, e.g., Proposition 4.4 of [Cha14a].) Let be a slice disk exterior of . Glue to along the solid torus in attached during the zero framed surgery and the tubular neighborhood of in . Then the resulting 4-manifold is an -solution for . ∎

### 3.2. An iterated infection

Let be the negatively amphichiral knot in Figure 3. The box with label denotes full positive (negative, resp.) twists between bands but individual bands left untwisted. Let be the connected sum of two copies of with two axes as in the Figure 4. Since is negatively amphichiral, is a ribbon knot. Note that forms an unlink in and the linking numbers and are zero. For each , let be a ribbon knot and be an axis with . Let be any knot bounding a grope of height 2 in .

Define inductively

 Jk=Kkηk(Jk−1)

for , and define

 Jn=Kα,β(Jn−1,¯¯¯¯¯¯¯¯¯¯¯Jn−1),

where is the mirror image of . It is shown that is negatively amphichiral in Lemma 2.1 of [CHL11a]. We show that bounds a grope of height in .

Let be an unknot, and define inductively

 Uk=Kkηk(Uk−1)

for , and

 Un=Kα,β(Un−1,¯¯¯¯¯¯¯¯¯¯¯¯Un−1).

Suggested by Cha [Cha14a, Section 4.2], can be decomposed as

 X(U0)⊔X(¯¯¯¯¯¯¯U0)⊔X(K1⊔η1)⊔X(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯K1⊔η1)⊔…X(Kn−1⊔ηn−1)⊔X(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Kn−1⊔ηn−1)⊔X(K⊔α⊔β),

where is the mirror image of , considered as a 2-component link. Note that is a slice knot by Proposition 3.1. It can be shown that and , considered as curves in , represent elements in by following the same argument as in the proof of Lemma 4.9 of [Cha14a].

On the other hand, it is easy to see that is isotopic to . By Proposition 3.2, bounds a grope of height in .

### 3.3. Infection knots with a height 2 grope and vanishing signature integral

In this subsection we construct a special set of infection knots :

###### Proposition 3.4.

For an arbitrary constant , there is a family of knots and an increasing sequence of odd primes satisfying the following properties:

1. bounds a grope of height 2 in ,

2. and for (mod ),

3. for any , whenever , and

4. and ,

where is a primitive -th root of unity.

Its proof resembles that of [Cha14a, Proposition 4.12], which shows the existence of knots satisfying the condition (2), (3), and (4).

###### Proof.

The Levine-Tristram signature function of Horn’s knots in Figure 2 is calculated by Horn:

###### Proposition 3.5.

[Hor10, Proposition 3.1, Lemma 6.1] Each bounds a grope of height 2 in whose Levine-Tristram signature function is as follows:

 σPm(θ)={2if θm<∣θ∣≤π,0if 0≤∣θ∣<θm,

where is a real number in such that

 cos(θm)=23√m−123√m.

Choose an increasing sequence of positive integers such that there is at least one prime number, say , between and . Then the knot has the bump Levine-Tristram signature function supported by neighborhoods of and . Note that for all , the supports are disjoint each other.

Let be the -cable of , and let . By the property of the Levine-Tristram signature function of the cable of a knot (Lemma 4.13 of [Cha14a]), we have . Therefore, for , we have . Thus, if and only if mod . Also, for , we have since vanishes for any .

Also we have . It follows that . Now the desired knot is obtained by taking the connected sum of sufficiently large number of copies of . ∎

We use in place of in the construction of negatively amphichiral knots bounding gropes of height in in Subsection 3.2 and call the resulting knot for simplicity. These knots are the prototype of knots in our main theorems (Theorems A, B and C). The constant in Proposition 3.4 will be specified explicitly in the following sections.

### 3.4. Vanishing higher-order PTFA L2-signature obstructions

Here we show that each in Subsection 3.3 are -solvable with vanishing PTFA -signature obstructions (Definition 2.4), that is, there is an -solution for such that for any group homomorphism with a PTFA group which extends to , is equal to zero.

According to the proof of Proposition 3.3, is -solvable with a special -solution which is the union of a -solution for , the orientation-reversed , and a slice disk exterior of . For any group homomorphism with a PTFA group which extends to , by the Novikov additivity of -signature defects (Theorem 2.7), is equal to

 S(V0,ϕ)+S(V1,ϕ)+S(−V1,ϕ).

Since and is a slice knot, by Theorem 1.2 of [Cha14a], is equal to zero. On the other hand, since , the image of restricted to lies in a cyclic subgroup of . Since is torsion-free, this cyclic subgroup must be trivial or isomorphic to . By Theorem 2.8, is equal to zero or the integral of the Levine-Tristram signature function of over , which is also equal to zero by Proposition 3.4 (4). Similarly, is also zero. Hence vanishes, and this finishes the proof.

## 4. Proof of Theorems A and B

### 4.1. Proof of Theorem A modulo infection axis analysis

We construct knots using the iterated infection construction in Section 3 with the following choice of , , and ’s:

1. Let be the connected sum of two copies of as in Figure 4. Take curves and in Figure 4 as axes.

2. For any , let be the knot in Figure 5. Note that is a ribbon knot with cyclic Alexander module , which is generated by in Figure 5. We take as axes.

3. By Theorem 2.5, there is a constant which is greater than

 ∣∣∣ρ(M(K),ϕK)+n−1∑k=1ρ(M(Kk),ϕk)+n−1∑k=1ρ(M(¯¯¯¯¯¯¯Kk),¯¯¯¯¯¯ϕk)∣∣∣,

for any choice of homomorphisms , and , . Note that has the crossing number at most 16 and and has at most 12 for all . By Theorem 2.6, can be an explicit bound of the above equation, regardless of the choices of , and . Choose this constant as in the construction of in Proposition 3.4.

###### Remark.

In general, there is no harm to choose as and any slice knot with cyclic Alexander module as . We choose specific and to find a particular in the construction of .

We prove that knots generate a subgroup of . It is enough to show that no nontrivial finite connected sum of knots is -solvable.

Suppose not. We may assume there is such that has an -solution for , just by deleting a finite number of ’s.

We are going to construct a -manifold and a group homomorphism on , and then compute the -signature defect of in two different ways. By showing two values are not equal, we get a contradiction and finish the proof of the claim.

In general, for a seed knot , its axes and infection knots , there is a standard cobordism from to the disjoint union . It is

 M(˜K)×[0,1]⊔⨿ri=1(−M(˜Ji)×[0,1])/∼,

where the tubular neighborhood of in is identified with the solid torus attached during the surgery process. See the proof of Lemma 2.3 of [CHL09] for more detail.

Since a connected sum can be viewed as an infection, there is a cobordism from to . Let be the cobordism from to . Also Let , , be a cobordism from to . Note that , the orientation reversed , is a cobordism from to . Finally let , , be the -solution for constructed in the proof of Proposition 3.3. That is, is the union of , a slice exterior of , , a -solution for , and .

Define

 Wn=(r⨿j=2−Vj)⨿⨿rj=2M(Jj)En⨿M(J)V

and

 Wn−1=En−1⨿M(J1)Wn,

where 4-manifolds are glued along homeomorphic boundary components. For , define inductively

 Wi=Ei⨿M(Ji+11)Wi+1⨿M(¯Ji+11)−Ei

Figure 6 will be helpful to understand the construction.

Now we define a homomorphism on . Denote by for simplicity. We construct a normal series , , of .

Let for , and , be the -th rational derived subgroup of . Recall that the -th rational derived subgroup of a group is defined inductively as the kernel of the map

 G(k)⟶G(k)[G(k),G(k)]⊗ZQ,

with the initial condition .

By abuse of notation, we denote the element in represented by the curve as . Observe that for any element , lies in the first rational derived subgroup of . Let be the subset of multiplicatively generated by

 {p(μjβμ−j)|j∈Z},

where is the Alexander polynomial of and is the meridian of . We want to localize with . To do that, we have to check whether is a right divisor set since may not be commutative.

A multiplicative subset of a (noncommutative) domain is a right divisor set of if is not in and for any and , the intersection is nonempty. It is well-known that the right localized ring exists if and only if is a right divisor set of [Pas85, p.427], [CHL11c, Section 3.1]. The following result is useful to show that is a right divisor set:

###### Proposition 4.1.

[CHL11c, Proposition 4.1] Let be a ring and be a group. Suppose is a normal subgroup of such that the group ring is a domain. If is a right divisor set of that is -invariant ( for all ), then is a right divisor set of .

Note that is a right divisor set of since is a commutative ring. Also is