Primary decomposition in the smooth concordance group of topologically slice knots

Abstract

We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts, each of which has infinite rank. This supports the conjectures for topologically slice knots. We also prove analogues for the associated graded groups of the bipolar filtration of topologically slice knots. Among ingredients of the proof, we use amenable $L^2$ -signatures, Ozsváth-Szabó d-invariants and Némethi’s result on Heegaard Floer homology of Seifert 3-manifolds. In an appendix, we present a general formulation of the notion of primary decomposition.

1. Introduction and main results

It is a major open problem to classify knots in 3-space modulo concordance. Our understanding is far from complete, for both topological and smooth knot concordance groups. The sophistication of the smooth case beyond topological concordance is measured by the smooth concordance group of topologically slice knots, which has been actively investigated using modern smooth techniques.

In the study of knot concordance, the notion of primary decomposition first appeared in Jerome Levine’s foundational work [35, 34]. Briefly, he constructed an algebraic concordance group of Seifert matrices and proved that it is isomorphic to the knot concordance group in high odd dimensions, whereas it gives algebraic invariants in the classical dimension [35]. He proved that a rational coefficient version of the algebraic concordance group decomposes into a direct sum of certain ‘primary parts’ indexed by irreducible factors of Alexander polynomials. This plays a crucial role in his well-known classification result that the algebraic concordance group and high odd-dimensional knot concordance groups are isomorphic to $\mathbb {Z}^{\infty }\oplus (\mathbb {Z}_2)^{\infty } \oplus (\mathbb {Z}_4)^{\infty }$ [34].

For the low-dimensional case, it is natural to ask whether the classical knot concordance groups and related objects admit analogous primary decomposition and to study the structures via primary parts. In an appendix, we formulate a general notion of primary decomposition, which specialises to several specific cases including concordance and rational homology cobordism, and discuss related questions. We hope this is useful for future study as well. The appendix also discusses known earlier results from the viewpoint of the general formulation. In particular, for topological knot concordance, there were remarkable results related to primary decomposition [36, 28, 29, 14, 13, 30].

In this article, we begin a detailed study of the smooth concordance group of topologically slice knots  ${\mathcal T}$ , via primary decomposition. Precise statements of our results are given in Subsections 1.1 and 1.2. Briefly, the conjectural primary decompositions (see Question 1.1) are direct sum decompositions, along irreducible factors of Alexander polynomials, of the quotient ${\mathcal T}/\Delta $ where $\Delta $ is the subgroup generated by knots with unit Alexander polynomials. (Taking the quotient by $\Delta $ is along the same lines of ignoring units for factorisations in a ring.) We show that the conjectures hold for a large subgroup of ${\mathcal T}$ and that there are infinitely many primary parts, each of which has infinite rank (see Theorems A and B). This provides evidence supporting primary decomposition conjectures and reveals a rich structure in ${\mathcal T}/\Delta $ , generalising a result of Hedden, Livingston and Ruberman [25] that ${\mathcal T}/\Delta $ has infinite rank.

In proving this, the essential challenge is to find irreducible polynomials $\lambda (t)$ and topologically slice knots (whose nontrivial linear combinations are) not concordant to any knot whose Alexander polynomial is relatively prime to $\lambda (t)$ . It appears to be hard, if possible at all, to do this using known smooth invariants such as those from gauge theory, Heegaard Floer homology and Khovanov homology. Our proof combines amenable $L^2$ -signatures, which are key ingredients of recent studies of topological concordance, with smooth information from Heegaard Floer homology of infinitely many branched covers of a knot. It seems intriguing to study whether more recent smooth invariants are useful in understanding primary decomposition, motivated from the results and approaches of this article.

We also prove results that support primary decomposition conjectures for the bipolar filtration of topologically slice knots (see Theorems C and D).

1.1. Primary decomposition for topologically slice knots

In what follows, we state primary decomposition conjectures and main results for ${\mathcal T}$ .

For a knot K in $S^3$ , denote the Alexander polynomial by $\Delta _K$ and regard it as an element in the Laurent polynomial ring $\mathbb {Q}[t^{\pm 1}]$ . Then $\Delta _K$ is well defined up to associates. Recall that $\lambda $ and $\mu \in \mathbb {Q}[t^{\pm 1}]$ are associates if $\lambda = at^k \mu $ for some $a\in \mathbb {Q}\smallsetminus \{0\}$ and $k\in \mathbb {Z}$ . The standard involution on $\mathbb {Q}[t^{\pm 1}]$ is defined by $\bigl (\sum a_i t^i\bigr )^* = \sum a_i t^{-i}$ . We say that $\lambda $ and $\mu \in \mathbb {Q}[t^{\pm 1}]$ are $*$ -associates if $\lambda $ is an associate of either $\mu $ or $\mu ^*$ .

Denote the smooth concordance class of a knot K by $[K]$ . Recall that $\Delta = \{[K] \in {\mathcal T} \mid \Delta _K$ is trivial $\}$ . For an irreducible $\lambda $ in $\mathbb {Q}[t^{\pm 1}]$ , let

$$ \begin{align*} {\mathcal T}_{\lambda} &=\{[K] \in {\mathcal T} \mid \Delta_K \text{ is an associate of } (\lambda\lambda^*)^k\text{ for some }k\ge 0\}, \\ {\mathcal T}^{\kern3pt\lambda} &= \{[K] \in {\mathcal T} \mid \Delta_K \text{ is relatively prime to }\lambda\}. \end{align*} $$

We remark that the product $\lambda \lambda ^*$ in the definition of ${\mathcal T}_{\lambda }$ reflects the Fox-Milnor condition that $\Delta _K$ is an associate of $ff^*$ for some $f\in \mathbb {Q}[t^{\pm 1}]$ when K is topologically slice. Note that ${\mathcal T}_{\lambda }$ and ${\mathcal T}^{\kern3pt\lambda }$ are subgroups containing $\Delta $ for every irreducible $\lambda $ . Also, ${\mathcal T}_{\lambda }={\mathcal T}_{\mu }$ and ${\mathcal T}^{\kern3pt\lambda }={\mathcal T}^{\kern3pt\mu }$ if $\lambda $ and $\mu $ are $*$ -associates.

Primary decomposition for topologically slice knots concerns natural homomorphisms

$$ \begin{align*} \Phi_L\colon \bigoplus_{[\lambda]} {\mathcal T}_{\lambda}/\Delta {\longrightarrow} {\mathcal T}/\Delta \quad\text{and}\quad \Phi_R\colon {\mathcal T}/\Delta {\longrightarrow} \bigoplus_{[\lambda]} {\mathcal T}/{\mathcal T}^{\kern3pt\lambda}. \end{align*} $$

Here the index $[\lambda ]$ of the direct sums varies over the $*$ -associate classes of irreducibles $\lambda $ that arise as the factor of an Alexander polynomial of a knot. The homomorphism $\Phi _L$ is defined to be the sum of the inclusions ${\mathcal T}_{\lambda }/\Delta \hookrightarrow {\mathcal T}/\Delta $ . Because $\Delta _K$ is a product of finitely many irreducibles, the quotient epimorphisms ${\mathcal T}/\Delta \twoheadrightarrow {\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ induce a homomorphism into the direct sum, which is our $\Phi _R$ above. This formulation is influenced by earlier work in the literature, particularly Levine [35, 34] and Cochran, Harvey and Leidy [14, 13].

An informal remark. One might regard elements in ${\mathcal T}_{\lambda }/\Delta \subset {\mathcal T}/\Delta $ as ‘ $\lambda $ -primary’ and ${\mathcal T}/\Delta \to {\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ as ‘forgetting those coprime to $\lambda $ ’ or ‘extracting the $\lambda $ -primary component’ of an element. Then, the surjectivity of $\Phi _L$ means the existence of a ‘decomposition into a sum of primary elements’, and the injectivity of $\Phi _L$ means the uniqueness of such a decomposition. Also, the injectivity of $\Phi _R$ means that ‘primary components determine an element uniquely’, and the surjectivity of $\Phi _R$ means that ‘every combination of primary components is realisable’.

Question 1.1 Primary decomposition for topologically slice knots

1. (1) Left primary decomposability: is $\Phi _L$ an isomorphism?

2. (2) Right primary decomposability: is $\Phi _R$ an isomorphism?

More generally (if they are not isomorphisms), what are their kernel and cokernel?

We conjecture an affirmative answer to (2) and that $\Phi _L$ is injective at the least. In any case, it appears to be interesting to study ${\mathcal T}_{\lambda }/\Delta $ and ${\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ , which we call left and right primary parts (or primary factors). This leads us particularly to the following.

1. (3) Nontriviality of primary parts: are ${\mathcal T}_{\lambda }/\Delta $ and ${\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ nonzero for each irreducible $\lambda $ that arises as the factor of an Alexander polynomial of a knot?

More generally (if they are nontrivial), what are the isomorphism types of the primary parts ${\mathcal T}_{\lambda }/\Delta $ and ${\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ ?

1. (4) Relationship of left and right primary parts: is the composition

   $$ \begin{align*} {\mathcal T}_{\lambda}/\Delta \hookrightarrow {\mathcal T}/\Delta \twoheadrightarrow {\mathcal T}/{\mathcal T}^{\kern3pt\lambda} \end{align*} $$
   an isomorphism?

We remark that Definition A.1 in the appendix generalises the notion of left and right primary decomposition to a broader context. Also, regarding the relationship of Question 1.1(1) and (2), see Lemma A.4 in the appendix.

The first main result of this article, which is given as Theorem A, says that there is a large subgroup of ${\mathcal T}$ for which the answers to the above questions are affirmative and many primary parts of ${\mathcal T}$ are highly nontrivial. To state the result, we use the following notation. For a subgroup ${\mathcal S}$ of ${\mathcal T}$ and an irreducible $\lambda \in \mathbb {Q}[t^{\pm 1}]$ , let ${\mathcal S}_{\lambda }$ be the subgroup of $[K]\in {\mathcal S}$ with $\Delta _K$ a power of $\lambda \lambda ^*$ and ${\mathcal S}^{\lambda }$ be the subgroup of $[K]\in {\mathcal S}$ with $\gcd (\Delta _K,\lambda )=1$ . That is, ${\mathcal S}_{\lambda } = {\mathcal S}\cap {\mathcal T}_{\lambda }$ and ${\mathcal S}^{\lambda } = {\mathcal S}\cap {\mathcal T}^{\kern3pt\lambda }$ . Then one can ask Question 1.1 for ${\mathcal S}$ in place of ${\mathcal T}$ . Let

(1.1) $$ \begin{align} \Lambda=\{(m+1)t-m \mid m \text{ is a positive integer}\}. \end{align} $$

Note that $\Lambda $ is an infinite collection of pairwise non- $*$ -associate irreducibles $\lambda $ such that $\lambda \lambda ^*$ is an Alexander polynomial of a knot.

Theorem A. There is a subgroup ${\mathcal S}$ in ${\mathcal T}$ containing $\Delta $ that satisfies the following:

1. (1) For every $\lambda \in \Lambda $ , ${\mathcal S}_{\lambda }/\Delta \cong \mathbb {Z}^{\infty }$ , ${\mathcal S}/{\mathcal S}^{\lambda } \cong \mathbb {Z}^{\infty }$ and the composition ${\mathcal S}_{\lambda }/\Delta \hookrightarrow {\mathcal S}/\Delta \twoheadrightarrow {\mathcal S}/{\mathcal S}^{\lambda }$ is an isomorphism.

2. (2) The inclusions ${\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/\Delta $ induce an isomorphism $\bigoplus _{\lambda \in \Lambda } {\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/\Delta $ .

3. (3) The surjections ${\mathcal S}/\Delta \to {\mathcal S}/{\mathcal S}^{\lambda }$ induce an isomorphism ${\mathcal S}/\Delta \to \bigoplus _{\lambda \in \Lambda } {\mathcal S}/{\mathcal S}^{\lambda }$ .

(1.2)

From Theorem A it follows that each of the primary parts ${\mathcal T}_{\lambda } /\Delta $ and ${\mathcal T}/{\mathcal T}^{\kern3pt\lambda }$ has a subgroup isomorphic to $\mathbb {Z}^{\infty }$ for all $\lambda \in \Lambda $ . An immediate consequence is the main result of [25] that ${\mathcal T}/\Delta $ has a subgroup isomorphic to $\mathbb {Z}^{\infty }$ .

Note that Theorem A(1) implies the following.

Theorem B. For each $\lambda \in \Lambda $ , there is an infinite collection of topologically slice knots $\{K_{\lambda ,i}\}_{i=1}^{\infty }$ with $\Delta _{K_{\lambda ,i}}$ a power of $\lambda \lambda ^*$ , such that any nontrivial linear combination of the $K_{\lambda ,i}$ is not smoothly concordant to any knot J with $\Delta _J$ relatively prime to $\lambda $ .

Indeed, Theorem B is equivalent to Theorem A, by an elementary formal argument.

Proof that Theorem B implies Theorem A. Suppose that Theorem B holds. Let ${\mathcal S}$ be the subgroup in ${\mathcal T}$ that is generated by $\Delta $ and the two-parameter family $\{K_{\lambda ,i}\}_{\lambda \in \Lambda , i\in {\mathbb N}}$ given by Theorem B.

We claim that, for each $\lambda \in \Lambda $ , ${\mathcal S}_{\lambda }$ is equal to the subgroup generated by $\Delta $ and the one-parameter family $\{K_{\lambda ,i}\}_{i\in {\mathbb N}}$ . To see this, first observe that for irreducibles $\lambda $ and $\mu $ that are not $*$ -associates, ${\mathcal S}_{\lambda }\subset {\mathcal S}^{\mu }$ , and so the composition

(1.3) $$ \begin{align} {\mathcal S}_{\lambda}/\Delta \hookrightarrow {\mathcal S}/\Delta \twoheadrightarrow {\mathcal S}/{\mathcal S}^{\mu} \end{align} $$

is zero. If a linear combination $[J] + \sum _{\mu } \sum _i r_{\mu ,i} [K_{\mu ,i}]$ ( $[J]\in \Delta $ , $r_{\mu ,i} \in \mathbb {Z}$ ) lies in ${\mathcal S}_{\lambda }$ , then for each $\mu \ne \lambda $ , the image of the linear combination in ${\mathcal S}/{\mathcal S}^{\mu }$ , which is represented by $\sum _{i} r_{\mu ,i} [K_{\mu ,i}]$ , should be zero by the observation. Therefore, $r_{\mu ,i}=0$ for all i and $\mu \ne \lambda $ , by the conclusion of Theorem B that the classes $[K_{\mu ,i}]$ are linearly independent in ${\mathcal S}/{\mathcal S}^{\mu }$ . This proves the claim.

Fix $\lambda \in \Lambda $ and temporarily denote by $\mathbb {Z}^{\infty }$ the free abelian group generated by the collection $\{K_{\lambda ,i}\}_i$ . The assignment $K_{\lambda ,i} \mapsto [K_{\lambda ,i}]$ gives rise to an epimorphism $\mathbb {Z}^{\infty } \twoheadrightarrow {\mathcal S}_{\lambda }/\Delta $ by the claim. Because ${\mathcal S}$ is generated by $\{K_{\lambda ,i}\}_{\lambda ,i}$ and $[K_{\mu ,i}]=0$ in ${\mathcal S}/{\mathcal S}^{\lambda }$ for $\mu \ne \lambda $ , the composition

(1.4) $$ \begin{align} \mathbb{Z}^{\infty} \twoheadrightarrow {\mathcal S}_{\lambda}/\Delta \hookrightarrow {\mathcal S}/\Delta \twoheadrightarrow {\mathcal S}/{\mathcal S}^{\lambda} \end{align} $$

is surjective. Moreover, by the linear independence of $[K_{\lambda ,i}]$ in ${\mathcal S}/{\mathcal S}^{\lambda }$ , (1.4) is an isomorphism. From this, it follows that both $\mathbb {Z}^{\infty } \to {\mathcal S}_{\lambda }/\Delta $ and ${\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/{\mathcal S}^{\lambda }$ are isomorphisms. This shows that Theorem A(1) holds.

Because (1.3) is zero for $\mu \ne \lambda \in \Lambda $ , the composition of two horizontal arrows on the top row of (1.2) is the direct sum of the isomorphisms ${\mathcal S}_{\lambda }/\Delta \to {\mathcal S}/{\mathcal S}^{\lambda }$ . So the top row composition in (1.2) is an isomorphism. Also, the first horizontal arrow in the top row is surjective by the definition of ${\mathcal S}$ . From this, it follows that Theorem A(2) and (3) hold.

1.2. Primary decomposition for the bipolar filtration

The method of this article provides further information on the structure of ${\mathcal T}$ . To discuss this, we consider the bipolar filtration of ${\mathcal T}$ , which was defined by Cochran, Harvey and Horn [11]. It is a descending filtration

$$ \begin{align*} \{0\} \subset \cdots \subset {\mathcal T}_n \subset \cdots \subset {\mathcal T}_1 \subset {\mathcal T}_0 \subset {\mathcal T}, \end{align*} $$

where the subgroup ${\mathcal T}_n$ consists of concordance classes of certain knots called n-bipolar (see Definition 2.1). It is known that various modern smooth invariants vanish for knots in the subgroups ${\mathcal T}_0$ and ${\mathcal T}_1$ , but the associated graded groups $\mathrm {gr}_n({\mathcal T}\kern3pt):={\mathcal T}_n/{\mathcal T}_{n+1}$ are nontrivial for all n [11, 6]. Indeed, the abelian group $\mathrm {gr}_n({\mathcal T}\kern3pt)$ is known to have infinite rank for $n=0$ [15] and for all $n\ge 2$ [6].

As an attempt to understand the structure of the filtration, we formulate and study the primary decomposition of $\mathrm {gr}_n({\mathcal T}\kern3pt)$ . For an n-bipolar knot K, denote its class in $\mathrm {gr}_n({\mathcal T}\kern3pt)$ by $[K]$ . Similar to the case of ${\mathcal T}$ , for an irreducible element $\lambda $ in $\mathbb {Q}[t^{\pm 1}]$ , consider the following subgroups of $\mathrm {gr}_n({\mathcal T}\kern3pt)$ :

$$ \begin{align*} \mathrm{gr}_n({\mathcal T}\kern3pt)_{\lambda} &:=\{[K] \in \mathrm{gr}_n({\mathcal T}\kern3pt) \mid \Delta_K = (\lambda\lambda^*)^k \text{ for some } k \ge 0\}, \\ \mathrm{gr}_n({\mathcal T}\kern3pt)^{\lambda} &:= \{[K] \in \mathrm{gr}_n({\mathcal T}\kern3pt) \mid \Delta_K \text{ is relatively prime to }\lambda\}. \end{align*} $$

Also, let $\mathrm {gr}_n(\Delta ) =\{[K] \in \mathrm {gr}_n({\mathcal T}\kern3pt) \mid \Delta _K\ \text{is trivial}\}$ . The injections $\mathrm {gr}_n({\mathcal T}\kern3pt)_{\lambda }/\mathrm {gr}_n(\Delta ) \hookrightarrow \mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n(\Delta )$ and the surjections $\mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n(\Delta ) \twoheadrightarrow \mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ induce homomorphisms

$$ \begin{align*} \Phi^n_L\colon & \bigoplus\nolimits_{[\lambda]} \mathrm{gr}_n({\mathcal T}\kern3pt)_{\lambda}/\mathrm{gr}_n(\Delta) \longrightarrow \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n(\Delta), \\ \Phi^n_R\colon & \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n(\Delta) \longrightarrow \bigoplus\nolimits_{[\lambda]} \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n({\mathcal T}\kern3pt)^{\lambda}. \end{align*} $$

The following is an analogue of Question 1.1.

Question 1.2 Primary decomposition for the associated graded

1. (1) Is $\Phi ^n_L$ an isomorphism?

2. (2) Is $\Phi ^n_R$ an isomorphism?

3. (3) Are $\mathrm {gr}_n({\mathcal T}\kern3pt)_{\lambda }/\mathrm {gr}_n(\Delta )$ and $\mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ nontrivial for every irreducible $\lambda $ that arises as a factor of an Alexander polynomial of a knot?

4. (4) Is the following composition an isomorphism?

   $$ \begin{align*} \mathrm{gr}_n({\mathcal T}\kern3pt)_{\lambda}/\mathrm{gr}_n(\Delta) \hookrightarrow \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n(\Delta) \twoheadrightarrow \mathrm{gr}_n({\mathcal T}\kern3pt)/\mathrm{gr}_n({\mathcal T}\kern3pt)^{\lambda}. \end{align*} $$

The following result supports affirmative answers. Similar to the case of ${\mathcal T}$ , for a subgroup ${\mathcal S}$ in $\mathrm {gr}_n({\mathcal T}\kern3pt)$ , let ${\mathcal S}_{\lambda }={\mathcal S}\cap \mathrm {gr}_n({\mathcal T}\kern3pt)_{\lambda }$ and ${\mathcal S}^{\lambda }={\mathcal S}\cap \mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ . Recall that the collection $\Lambda $ has been defined in (1.1) above.

Theorem C. Let $n\ge 2$ . Then there is a subgroup ${\mathcal S}$ in $\mathrm {gr}_n({\mathcal T}\kern3pt)$ that contains $\mathrm {gr}_n(\Delta )$ such that

$$ \begin{align*} \bigoplus_{\lambda\in\Lambda} {\mathcal S}_{\lambda}/\mathrm{gr}_n(\Delta) \xrightarrow{\cong} {\mathcal S}/\mathrm{gr}_n(\Delta) \xrightarrow{\cong} \bigoplus_{\lambda\in\Lambda} {\mathcal S} / {\mathcal S}^{\lambda} \end{align*} $$

and ${\mathcal S}_{\lambda }/\mathrm {gr}_n(\Delta ) \cong \mathbb {Z}^{\infty } \cong {\mathcal S} / {\mathcal S}^{\lambda }$ for every $\lambda \in \Lambda $ .

By the same argument as the proof that Theorem A is equivalent to Theorem B, it is seen that Theorem C is equivalent to the following statement.

Theorem D. Let $n\ge 2$ . Then for each $\lambda \in \Lambda $ , there are infinitely many topologically slice n-bipolar knots $K_{\lambda ,i}$ ( $i=1,2,\ldots $ ) with $\Delta _{K_{\lambda ,i}}$ a power of $\lambda \lambda ^*$ that are linearly independent in $\mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ .

Also, Theorem D implies Theorem B. Indeed, if a linear combination $\#_i\, a_i K_{\lambda ,i}$ of the knots $K_{\lambda ,i}$ in Theorem D is smoothly concordant to a knot L with $\Delta _L$ relatively prime to $\lambda $ , then the class $[L]$ automatically lies in the subgroup ${\mathcal T}_n$ of ${\mathcal T}$ because so are $K_{\lambda ,i}$ , and thus $[L]$ is zero in the quotient $\mathrm {gr}_n({\mathcal T}\kern3pt)/\mathrm {gr}_n({\mathcal T}\kern3pt)^{\lambda }$ . It follows that $a_i=0$ for all i, by Theorem D. Therefore, to obtain Theorems A, B and C, it suffices to prove Theorem D.

The remaining part of this article is organised as follows. Sections 2–4 are devoted to the proof of Theorem D. In the appendix, we discuss a general formulation of the notion of primary decomposition.

1.2.1. Ingredients of the proof

The proof of the above results uses (the ideas of) several results in the literature. To extract obstructions to smooth concordance, we combine the Cheeger-Gromov $L^2 \ \rho $ -invariants or, equivalently, $L^2$ -signature defects and the Ozsváth-Szabó d-invariant defined from Heegaard Floer homology, following the approach of [6], which was motivated by earlier work of Cochran, Harvey and Horn [11]. The amenable signature theorem developed in [8, 3] and Ozsváth-Szabó’s d-invariant inequality for definite 4-manifolds [40] are among the key ingredients. We develop and use a localisation technique inspired by work of Cochran, Harvey and Leidy [14, 13], to produce representations of the fundamental group to which our $\rho $ -invariants are associated. Also, to compute and estimate d-invariants of infinitely many branched covers of the infinitely many topologically slice knots in Theorems B and D, we use Némethi’s work [38] on Heegaard Floer homology of negative definite plumbed 3-manifolds.

2. The first step of the proof of Theorem D

As a preliminary of the proof of Theorem D, we recall the definition of the bipolar filtration from [11]. Let $M(K)$ be the zero framed surgery manifold of a knot K in $S^3$ .

Definition 2.1 [11, Definition 5.1]

Let $n\ge 0$ be an integer. A compact connected 4-manifold V bounded by $M(K)$ is an n-negaton if the following are satisfied:

1. (1) The inclusion induces an isomorphism on $H_1(M(K))\to H_1(V)$ , and a meridian of K normally generates $\pi _1(V)$ .

2. (2) There are disjointly embedded closed connected surfaces $S_i$ in V that form a basis for $H_2(V)$ and have self-intersection $-1$ or, equivalently, the normal bundle of $S_i$ has Euler class $-1$ .

3. (3) For each i, the image of $\pi _1(S_i)$ lies in the nth derived subgroup $\pi _1(V)^{(n)}$ .

If there is an n-negaton bounded by $M(K)$ , then K is called n-negative. An n-positon and an n-positive knot are defined by replacing $-1$ by $+1$ in condition (2) above. A knot K is n-bipolar if K is n-positive and n-negative.

Recall that ${\mathcal T}$ is the smooth concordance group of topologically slice knots. The bipolar filtration $\{{\mathcal T}_n\}_{n\ge 0}$ of ${\mathcal T}$ is defined by

$$ \begin{align*} {\mathcal T}_n = \{[K] \in {\mathcal T} \mid K\text{ is } n\text{-bipolar} \}. \end{align*} $$

Because an $(n+1)$ -bipolar knot is n-bipolar, $\{{\mathcal T}_n\}_{n\ge 0}$ is a descending filtration. It is an open problem whether $\bigcap _{n\ge 0} {\mathcal T}_n=\{0\}$ .

2.1. Construction of a family of knots $\{K_i\}$

We start the proof of Theorem D with a construction of knots that will be shown to generate the promised infinite rank free abelian subgroup.

Fix an integer $n\ge 2$ , as in Theorem D. Also, fix another integer $m\ge 1$ . Several objects we will use below depend on $(n,m)$ , but we omit it from notation because $(m,n)$ is fixed in our arguments. Let

$$ \begin{align*} \lambda(t) = \lambda_m(t) = (m+1)t-m \in \Lambda. \end{align*} $$

For each $(n,m)$ , we will construct an infinite family of knots $\{K_i\}$ indexed by integers $i>0$ , whose Alexander polynomial is $\lambda (t)\lambda (t^{-1})$ . The construction is similar to [6, Section 2.2]. (See also [11], which influenced the construction of [6] and ours.) Let $R(J,D)$ be the knot shown in Figure 1. Here J and D are knots that will be specified later. (For now, ignore the circles $\alpha _J$ and $\alpha _D$ .) The knot $R(J,D)$ bounds an obviously seen Seifert surface of genus one, which consists of a 0-handle and two 1-handles. In Figure 1, the two 1-handles are untwisted and cross each other $2m+1$ times. So, $S=\left [\begin {smallmatrix}0 & m+1\\m & 0\end {smallmatrix}\right ]$ is a Seifert matrix. We remark that [6] and [11] use the particular case of $m=1$ .

Figure 1 The knot $R(J,D)$ .

Because $tS-S^T$ presents the Alexander module of $R(J,D)$ , a routine computation shows that $R(J,D)$ has (integral) Alexander module

(2.1) $$ \begin{align} H_1(M(R(J,D));\mathbb{Z}[t^{\pm1}]) = \mathbb{Z}[t^{\pm1}]/\langle \lambda(t)\rangle \oplus \mathbb{Z}[t^{\pm1}]/\langle \lambda(t^{-1})\rangle, \end{align} $$

and the summands $\mathbb {Z}[t^{\pm 1}]/\langle \lambda (t)\rangle $ and $\mathbb {Z}[t^{\pm 1}]/\langle \lambda (t^{-1})\rangle $ are equal to the subgroups $\langle \alpha _J\rangle $ and $\langle \alpha _D\rangle $ generated by the loops $\alpha _J$ and $\alpha _D$ shown in Figure 1. It follows that $\Delta _{R(J,D)} = \lambda (t)\lambda (t^{-1})$ . Moreover, the Blanchfield pairing

$$ \begin{align*} B\ell\colon {\mathcal A} \times {\mathcal A} \longrightarrow \mathbb{Q}(t)/\mathbb{Q}[t^{\pm1}] \end{align*} $$

on the Alexander module ${\mathcal A} := H_1(M(R(J,D));\mathbb {Q}[t^{\pm 1}])$ has exactly two metabolisers, $\langle \alpha _J\rangle $ and $\langle \alpha _D\rangle $ . Here, a submodule $P\subset {\mathcal A}$ is called a metaboliser if P is equal to $P^{\perp }:=\{x\in {\mathcal A} \mid B\ell (P,x)=0\}$ . The above observations on $R(J,D)$ hold for any choice of J and D.

As another ingredient of the construction of the promised knots $K_i$ , we will use a family of knots $\{J^i_k\}$ , which was given in [6]. For $k=0$ , an explicit construction of such $\{J^i_0\}_i$ is given in [6, Section 4]. We need that $\{J^i_0\}_i$ satisfies the following conditions (J1), (J2) and (J3) for some sequence of increasing primes $\{p_i\}$ . For a knot J and an integer p, let $\sigma _J(\omega )\in \mathbb {Z}$ be the Levine-Tristram signature function of J at $\omega \in S^1\subset \mathbb {C}$ , and let

(2.2) $$ \begin{align} \rho(J,\mathbb{Z}_d) = \frac1d \sum_{k=0}^{d-1} \sigma_J(e^{2\pi k\sqrt{-1}/d}). \end{align} $$

• (J1) For each i, $J^i_0$ is $0$ -negative.

• (J2) For each i, $|\rho (J^i_0,\mathbb {Z}_{p_i})|> 69\,713\,280\cdot (6n+8m+86)$ .

• (J3) For $i<j$ , $\rho (J^j_0,\mathbb {Z}_{p_i})=0$ .

For $k=0,\ldots ,n-2$ , $J^i_{k+1}$ is defined inductively by $J^i_{k+1} = P_k(\eta _k,J^i_k)$ , where $P_k(\eta _k,J^i_k)$ is the satellite knot shown in the right of Figure 2. The left of Figure 2 shows the pattern $P_k$ in the exterior of an unknotted circle $\eta _k$ , which is a standard solid torus. The companion is the knot $J^i_k$ .

Figure 2 The stevedore’s pattern $(P_k,\eta _k)$ and the satellite knot $P_k(\eta _k,J^i_k)$ .

Now, let $K_i = R(J_{n-1}^i,D)$ , where $D=\mathrm {Wh}^+(T)$ is the positive Whitehead double of the right-handed trefoil T. Because D is topologically slice [21], $K_i$ is topologically concordant to $R(J_{n-1}^i,U)$ , where U is the trivial knot. Because $R(J,U)$ is (smoothly) slice for any J, it follows that $K_i$ is topologically slice. By [6, Section 2.2, Lemma 2.3], property (J1) implies that each $K_i$ is n-negative and k-positive for all $k\ge 0$ . Because $\Delta _{K_i} = \lambda (t)\lambda (t^{-1})$ , it follows that the class $[K_i]$ lies in $\mathrm {gr}_n({\mathcal T}\kern3pt)_{\lambda } = ({\mathcal T}_n\cap {\mathcal T}_{\lambda }) / ({\mathcal T}_{n+1}\cap {\mathcal T}_{\lambda })$ . Therefore, to prove Theorem D, it suffices to show the following statement.

Theorem 2.2. Suppose that $K=\bigl (\#_{i=1}^r \, a_i K_i\bigr ) \# L$ ( $a_i\in \mathbb {Z}$ ) is a linear combination of the knots $K_i$ and a knot L with $\Delta _L(t)$ relatively prime to $\lambda (t)$ . If $a_i\ne 0$ for some i, then K is not $(n+1)$ -bipolar.

For the special case that L is a trivial knot and $m=1$ , the conclusion of Theorem 2.2 was shown in [6]. The general case of Theorem 2.2 requires substantially more sophisticated ideas and methods, which will be discussed in Sections 3 and 4.

2.2. Construction of a negaton

To prove Theorem 2.2 by contradiction, suppose that the knot K is $(n+1)$ -bipolar. Recall that $K_i$ is n-bipolar. The following observation will be useful. The knot L in the statement of Theorem 2.2 is automatically n-bipolar, because L is concordant to the sum of K, $-a_1K_1,\ldots$ $-a_{r-1}K_{r-1}$ , and $-a_rK_r$ and each of K and $K_i$ are n-bipolar.

We may assume $a_i\ne 0$ for all i, by removing $K_i$ when $a_i=0$ . In addition, by taking $-K$ instead of K, we may assume that $a_1>0$ . Under this assumption, we will prove that K is not $(n+1)$ -negative. To derive a contradiction, suppose that K is $(n+1)$ -negative. As done in [6, Section 2.3], we will construct a certain n-negaton for the first summand $K_1$ , which we will call $X^-$ below. Because the generality of Theorem 2.2 does not cause significant issues in this construction, we will closely follow [6], with minor additional changes related to L.

Use the n-bipolarity of L to choose an n-negaton, say, $Z^-_L$ , bounded by $-M(L)=M(-L)$ . Use the n-bipolarity of $K_i$ , choose an n-negaton $Z^-_i$ bounded by $-M(K_i)=M(-K_i)$ for each i for which $a_i>0$ and choose n-negaton $Z^-_i$ bounded by $M(K_i)$ for each i for which $a_i<0$ . Let $V^-$ be an $(n+1)$ -negaton bounded by $M(K)$ . There is a standard cobordism C bounded by the union of $\partial _-C := -M(K)$ and $\partial _+C := \bigl (\bigsqcup _i a_i M(K_i)\bigr ) \sqcup M(L)$ , which is associated with the connected sum expression $K=\bigl (\#_{i=1}^r \, a_i K_i\bigr ) \# L$ ; C is obtained by attaching to $\bigl (\bigsqcup _{i=1}^r a_i M(K_i) \times I\bigr ) \sqcup M(L)\times I$ , N 1-handles that connect the component and then attaching N 2-handles that make meridians of the involved $N+1$ knots parallel, where $N=\sum _i |a_i|$ . A detailed description of C can be found, for instance, from [17, p. 113]. Define

(2.3) $$ \begin{align} X^- := V^- \mathbin{\mathop{\cup}\limits_{\partial_-C}} C \mathbin{\mathop{\cup}\limits_{\partial_+C}} \bigg((a_1-1)Z^-_1 \sqcup \Bigl({\textstyle\bigsqcup\limits_{i>1} |a_i| Z^-_i} \Bigr) \sqcup Z^-_L \bigg). \end{align} $$

Figure 3 depicts the construction of $X^-$ . By (the argument of) [6, Lemma 2.4], $X^-$ is an n-negaton bounded by $M(K_1)$ .

Figure 3 The construction of $X^-$ . The sign of $M(K_i)$ equals that of $a_i$ .

Now, let

(2.4) $$ \begin{align} P := \operatorname{Ker} \{ H_1(M(K_1);\mathbb{Q}[t^{\pm1}]) \rightarrow H_1(X^-;\mathbb{Q}[t^{\pm1}]) \}. \end{align} $$

Because $X^-$ is an n-negaton with $n\ge 2$ , P is a metaboliser of the Blanchfield pairing on $H_1(M(K_1);\mathbb {Q}[t^{\pm 1}])$ , by [11, Theorem 5.8]. (See also the statement of [6, Lemma 2.5].) Because $K_1=R(J^1_{n-1},D)$ has exactly two metabolisers $\langle \alpha _J\rangle $ and $\langle \alpha _D\rangle $ , we have the following two cases: $P=\langle \alpha _D\rangle $ or $P=\langle \alpha _J\rangle $ . By deriving a contradiction for each case, the proof of Theorem 2.2 will be completed.

3. $L^2$ -signatures and localised mixed-type commutator series

In this section, we continue the proof of Theorem 2.2 for the case $P=\langle \alpha _D\rangle $ .

Recall that we constructed an n-negaton $X^-$ in (2.3) using the negatons $V^-$ , $Z_L^-$ and $Z_i^-$ . To obtain a hyperbolic intersection form, take the connected sum of the $(n+1)$ -negaton $V^-$ and $b_2(V^-)$ copies of $\mathbb {C} P^2$ and call the result $V^0$ . Indeed, by [11, Proposition 5.5], $V^0$ is a special type of 4-manifold called an integral $(n+1)$ -solution in [3, Definition 3.1], which particularly has a metabolic intersection form over twisted coefficients. We do not describe its definition because we do not use it directly, but we will state and use its properties later. For $Z_L^-$ and $Z_i^-$ , define integral n-solutions $Z_L^0$ and $Z_i^0$ by taking connected sum with copies of $\mathbb {C} P^2$ in the same way.

Repeat the construction of $X^-$ but now use $V^0$ , $Z_L^0$ and $Z_i^0$ in place of $V^-$ , $Z_L^-$ and $Z_i^-$ to obtain a 4-manifold $X^0$ . By the proof of [6, Lemma 2.4], $b_2(X^-)$ is equal to the sum of $b_2(V^-)$ , $b_2(Z_L^-)$ , $(a_1-1)b_2(Z_1^-)$ and $|a_i| b_2(Z_i^-)$ ( $i>1$ ). Thus, $X^0 = X^- \# (b_2(X^-) \mathbb {C} P^2)$ . Because $X^-$ is an n-negaton, it follows that $X^0$ is an integral n-solution, again by [11, Proposition 5.5].

We will attach additional pieces to $X^0$ to obtain a sequence of 4-manifolds, essentially following a technique that first appeared in [12]; see also [14, 13, 3, 6]. The notation used below is close to [3, 6]. Consider the satellite construction $J^1_{k+1} = P_k(\eta _k, J^1_k)$ . Due to [12], there is a standard cobordism, which we denote by $E_k$ , from $M(J^1_{k+1})$ to $M(J^1_k)\sqcup M(P_k)$ for $k=0,\ldots$ , $n-2$ . In Subsection 3.1, we will use an alternative description given in Figure 4, which illustrates that $E_k$ is obtained from $M(J^1_{k+1})\times [0,1]$ by attaching a 2-handle and a 3-handle: start with $M(J^1_{k+1}) = M(P_k(\eta _k, J^1_k))$ , attach a 2-handle along a zero-framed longitude of $J^1_k$ to obtain the second diagram and apply handle slide to obtain the last diagram, which is $M(J^1_k) \# M(P_k)$ . Attach a 3-handle to obtain $M(J^1_k) \sqcup M(P_k)$ .

Figure 4 A handlebody description of the standard cobordism $E_k$ .

View $K_1=R(J^1_{n-1},D)$ as the satellite knot $R(U,D)(\alpha _J,J^1_{n-1})$ and apply the same construction to obtain a standard cobordism $E_{n-1}$ from $M(K_1)$ to $M(J^1_{n-1})\sqcup M(R(U,D))$ . Now, define 4-manifolds $X_n$ , $X_{n-1},\ldots$ , $X_0=X$ as follows:

$$ \begin{align*} X_n & := X^0 \\ X_{n-1} & := X_n \mathbin{\mathop{\cup}\limits_{M(K_1)}} E_{n-1} = X^0 \mathbin{\mathop{\cup}\limits_{M(K_1)}} E_{n-1} \\ X_{n-2} & := X_{n-1} \mathbin{\mathop{\cup}\limits_{M(J^1_{n-1})}} E_{n-2} = X^0 \mathbin{\mathop{\cup}\limits_{M(K_1)}} E_{n-1} \mathbin{\mathop{\cup}\limits_{M(J^1_{n-1})}} E_{n-2} \\ &\hphantom{=}\vdots \\ X=X_0 &:= X_1 \mathbin{\mathop{\cup}\limits_{M(J^1_{1})}} E_{0} = X^0 \mathbin{\mathop{\cup}\limits_{M(K_1)}} E_{n-1} \mathbin{\mathop{\cup}\limits_{M(J^1_{n-1})}} E_{n-2} \mathbin{\mathop{\cup}\limits_{M(J^1_{n-2})}} \cdots \mathbin{\mathop{\cup}\limits_{M(J^1_{1})}} E_{0}. \end{align*} $$

See the schematic diagram in Figure 5.

Figure 5 A schematic diagram of the 4-manifold X.

We have that $H_1(X^0)=H_1(X^-)=\mathbb {Z}=\langle t \rangle $ where the generator t is represented by the meridian $\mu _K$ of K, because $X^-$ is a negaton bounded by $M(K)$ . See Definition 2.3(1). By a Mayer-Vietoris argument using this, it follows that $H_1(X_k)=\mathbb {Z}$ for all $k=0,\ldots$ , n.

3.1. A localised mixed-type commutator series associated with X

Recall from Subsection 2.1 that $p_1$ denotes the first prime used in properties (J2) and (J3), and $\lambda (t)=(m+1)t-m$ . Let

$$ \begin{align*} \Sigma := \{f(t)\in \mathbb{Q}[t^{\pm1}] \mid f(1)\ne 0, \, \gcd(f(t),\lambda(t)\lambda(t^{-1}))=1 \}. \end{align*} $$

Obviously, $\Sigma $ is a multiplicative subset. Let $\mathbb {Q}[t^{\pm }]\Sigma ^{-1}$ be the localisation.

Definition 3.1. Let G be a group endowed with a homomorphism $G\to \pi _1(X)$ that induces an epimorphism $H_1(G) \twoheadrightarrow H_1(X)$ . For $i=0,1,\ldots ,n+1$ , define subgroups ${\mathcal P}^i G$ of G inductively as follows. Let ${\mathcal P}^0 G := G$ , and let ${\mathcal P}^1 G$ be the kernel of the composition

Let ${\mathcal P}^2G$ be the kernel of the composition

Here, $\operatorname {Im} H_1(Z_L^0;\mathbb {Q}[t^{\pm 1}]\Sigma ^{-1})$ is the image of $H_1(Z_L^0;\mathbb {Q}[t^{\pm 1}]\Sigma ^{-1}) \to H_1(X;\mathbb {Q}[t^{\pm 1}]\Sigma ^{-1})$ induced by the inclusion.

For $i=2,\ldots$ , $n-1$ , let

$$ \begin{align*} {\mathcal P}^{i+1}G := \operatorname{Ker} \left\{ {\mathcal P}^i G \twoheadrightarrow \frac{{\mathcal P}^i G}{[{\mathcal P}^i G,{\mathcal P}^i G]} \rightarrow \frac{{\mathcal P}^i G}{[{\mathcal P}^i G,{\mathcal P}^i G]} {\mathbin{\mathop{\otimes}_{{\mathbb{Z}}}}} \mathbb{Q} = H_1\bigl(G;\mathbb{Q}[G/{\mathcal P}^i G]\bigr) \right\}. \end{align*} $$

Finally, define

$$ \begin{align*} {\mathcal P}^{n+1}G := \operatorname{Ker} \left\{ {\mathcal P}^n G \twoheadrightarrow \frac{{\mathcal P}^n G}{[{\mathcal P}^n G,{\mathcal P}^n G]} \twoheadrightarrow \frac{{\mathcal P}^n G}{[{\mathcal P}^n G,{\mathcal P}^n G]} {\mathbin{\mathop{\otimes}_{{\mathbb{Z}}}}} \mathbb{Z}_{p_1} = H_1\bigl(G;\mathbb{Z}_{p_1}[G/{\mathcal P}^n G]\bigr) \right\}. \end{align*} $$

It is straightforward to verify inductively that ${\mathcal P}^k G$ is a normal subgroup of G and the standard kth derived subgroup $G^{(k)}$ lies in ${\mathcal P}^k G$ for all k.

We remark that the commutator series $\{{\mathcal P}^k G\}_k$ in Definition 3.1 is ‘localised at polynomials’ in the sense of [14, Sections 3 and 4] and is of ‘mixed-coefficient’ type in the sense that we use both $\mathbb {Z}_{p_1}$ and $\mathbb {Q}$ (see [3, Section 4.1]).

In particular, define the subgroups ${\mathcal P}^i\pi _1(X_k)$ , by applying Definition 3.1 to the case $G=\pi _1(X_k) \to \pi _1(X)$ . The following properties, which we will state as Assertions A and B, are essential for our purpose. Let $\mu _k\subset M(J^1_k)$ be the meridian of $J^1_k$ . By the construction of $X_k$ , $M(J^1_k)$ is a component of $\partial X_k$ (see Figure 5), and thus $\mu _k$ represents an element in $\pi _1(X_k)$ for $k\le n-1$ . For brevity, let $J^1_n := K_1$ , so that the previous sentence holds for $k=n$ as well. Also, let $(P_{n-1},\eta _{n-1}):=(R(U,D),\alpha _D)$ , so that $J^1_k = P_{k-1}(\eta _{k-1},J^1_{k-1})$ holds for $k=n$ as well.

Assertion A. The class of $\mu _k$ lies in $\pi _1(X_k)^{(n-k)}\subset {\mathcal P}^{n-k}\pi _1(X_k)$ and is nontrivial in ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ for all $k=0$ , $1,\ldots$ , n. In particular, the class $\mu _0$ is nontrivial in ${\mathcal P}^n\pi _1(X)/{\mathcal P}^{n+1}\pi _1(X)$ .

Assertion B. For $i= 1,\ldots$ , n, the inclusion-induced map sends $\pi _1(Z_L^0)^{(i)}$ to the subgroup ${\mathcal P}^{i+1}\pi _1(X) \subset \pi _1(X)$ . In particular, $\pi _1(Z_L^0)^{(n)}$ maps to ${\mathcal P}^{n+1}\pi _1(X)$ .

Remark 3.2.

1. (1) Analogues of Assertion A for similar situations were established in earlier papers; for instance, in [12, 14, 3, 4, 6]. We will give a proof for our case, because our series ${\mathcal P}^i$ is different (notably at $i=2$ ) from those in the literature. Assertion B and its application are new, to the author’s knowledge.

2. (2) Note that both $\mu _0$ and $\pi _1(Z_L^0)^{(n)}$ map to the nth subgroup ${\mathcal P}^n \pi _1(X)$ . Due to Assertions A and B, they have opposite nature in the next stage: $\pi _1(Z_L^0)^{(n)}$ lies in ${\mathcal P}^{n+1} \pi _1(X)$ , whereas $\mu _0$ does not. This will be crucial in separating the contribution of the unknown knot L from that of $K_1$ in the linear combination $K=\bigl (\#_{i=1}^r a_i K_i\bigr ) \# L$ . See Subsection 3.2, particularly the Cheeger-Gromov $\rho $ -invariants in (3.3) and (3.8).

In the proof of Assertion A, the following fact will be useful. Let $\lambda _{k-1}$ be the zero-linking longitude of $J^1_{k-1}$ , which lies in

$$ \begin{align*} E(J^1_{k-1}) \subset E(J^1_{k-1})\cup E(P_{k-1}\sqcup \eta_{k-1}) = E(J^1_k) \subset M(J^1_k) \subset \partial X_k. \end{align*} $$

Assertion C. The inclusion $X_k \subset X_{k-1}$ induces an isomorphism ${\mathcal P}^i\pi _1(X_k)/\langle \lambda _{k-1} \rangle \cong {\mathcal P}^i\pi _1(X_{k-1})$ for all $i\le n-k+2$ . Consequently, we have

$$ \begin{align*} {\mathcal P}^{n-k+1}\pi_1(X_k)/{\mathcal P}^{n-k+2}\pi_1(X_k) \cong {\mathcal P}^{n-k+1}\pi_1(X_{k-1})/{\mathcal P}^{n-k+1}\pi_1(X_{k-1}). \end{align*} $$

Proof of Assertion C. Because $E_{k-1}$ is obtained by attaching a 2-handle to $M(J^1_k)\times [0,1]$ along $\lambda _{k-1}$ and then attaching a 3-handle (see Figure 4), $X_{k-1}=X_k\cup E_{k-1}$ is obtained from $X_k$ by the same handle attachments. It follows that $\pi _1(X_{k-1}) \cong \pi _1(X_k)/\langle \lambda _{k-1} \rangle $ . This shows the assertion for $i=0$ . Now, to proceed by induction, suppose that $i\ge 1$ and suppose that the assertion holds for $i-1$ . If $i=1$ , we have the following commutative diagram with exact rows:

Because the two rightmost horizontal arrows are to the same target $H_1(X)$ , there is an induced epimorphism ${\mathcal P}^i\pi _1(X_k) \twoheadrightarrow {\mathcal P}^i\pi _1(X_{k-1})$ and its kernel is equal to that of the epimorphism ${\mathcal P}^{i-1}\pi _1(X_k) \twoheadrightarrow {\mathcal P}^{i-1}\pi _1(X_{k-1})$ . So the assertion holds for $i=1$ . For $i=2$ , replace $H_1(X)$ in the above diagram by the quotient

$$ \begin{align*} H_1\bigl(X;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}\bigr) / \operatorname{Im} H_1\bigl(Z_L^0;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}\bigr) \end{align*} $$

and apply the same argument.

For $i\ge 3$ , we have

where $R=\mathbb {Q}$ or $\mathbb {Z}_{p_1}$ , depending on i. We claim that the rightmost vertical arrow is an isomorphism. From the claim, it follows that the assertion holds for i, once again by the argument used above. To show the claim, let $\gamma _{k-1}$ be the meridian of $J^1_{k-1}$ in the exterior $E(J^1_{k-1}) \subset M(J^1_k) \subset \partial X_k$ . Note that $\gamma _{k-1}$ is different from the meridian $\mu _{k-1}\subset M(J^1_{k-1})\subset \partial X_{k-1}$ used in the statement of Assertion A, but $\gamma _{k-1}$ and $\mu _{k-1}$ are isotopic in the cobordism $E_{k-1}$ . The meridian $\gamma _{k-1}$ is identified with the curve $\eta _{k-1}$ , which lies in the commutator subgroup $\pi _1(M(J^1_k))^{(1)}$ . Because $\pi _1(M(J^1_k))$ is normally generated by the meridian $\mu _k$ , $\gamma _{k-1}$ lies in $\langle \mu _k\rangle ^{(1)} = \langle \gamma _k\rangle ^{(1)}$ in $\pi _1(X_k)$ . By induction, it follows that $\gamma _{k-1}$ lies in $\langle \gamma _n\rangle ^{(n-k+1)}$ . Therefore, the image of $\pi _1(E(J^1_{k-1}))$ lies in $\pi _1(X_k)^{(n-k+1)}$ . Because the longitude $\lambda _{k-1}$ lies in $\pi _1(E(J^1_{k-1}))^{(1)}$ , it follows that

(3.1) $$ \begin{align} \lambda_{k-1} \in \pi_1(X_k)^{(n-k+2)}. \end{align} $$

Because $i\le n-k+2$ , (3.1) implies that $\lambda _{k-1} \in \pi _1(X_k)^{(i)} \subset ({\mathcal P}^{i-1}\pi _1(X_k))^{(1)}$ . Also, by the induction hypothesis, ${\mathcal P}^{i-1}\pi _1(X_k) \twoheadrightarrow {\mathcal P}^{i-1}\pi _1(X_{k-1})$ is an epimorphism with kernel $\langle \lambda _{k-1}\rangle $ . It follows that the rightmost vertical arrow in the above diagram is an isomorphism. This completes the proof of the assertion.

Proof of Assertion A. In the proof of Assertion C, we already showed that $\mu _k$ lies in $\pi _1(X_k)^{(n-k)}$ . This is the first part of Assertion A.

It remains to show that $\mu _k$ is nontrivial in ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ . We will use reverse induction for $k=n$ , $n-1,\ldots$ , $0$ . For $k=n$ , $\pi _1(X_n)/{\mathcal P}^1\pi _1(X_n) = H_1(X)$ by definition, and the meridian $\mu _n$ of $J^1_n = K_1$ is a generator of $H_1(X)$ . So the assertion holds.

For the case $k=n-1$ , let $\Lambda =\mathbb {Q}[t^{\pm 1}]$ for brevity, and consider the following commutative diagram:

Here, $H_1(X_n;\Lambda \Sigma ^{-1}) \to H_1(X;\Lambda \Sigma ^{-1})$ is an isomorphism, because $\pi _1(X)=\pi _1(X_0)$ is isomorphic to $\pi _1(X_n)/\langle \lambda _0,\ldots ,\lambda _{n-1}\rangle $ by Assertion C, and the longitudes $\lambda _0,\ldots ,\lambda _{n-1}$ lie in $\pi _1(X_n)^{(2)}$ by (3.1). The same argument shows that the bottom horizontal arrow is an isomorphism. (Alternatively, one may use Mayer-Vietoris arguments to show that they are isomorphisms.) Also, consider the Mayer-Vietoris sequence for $X = \overline {X\smallsetminus Z_L^0} \cup _{M(L)} Z_L^0$ :

$$ \begin{align*} H_1(M(L);\Lambda\Sigma^{-1}) \longrightarrow H_1(X\smallsetminus Z_L^0;\Lambda\Sigma^{-1}) \oplus H_1(Z_L^0;\Lambda\Sigma^{-1}) \longrightarrow H_1(X;\Lambda\Sigma^{-1}) \longrightarrow 0. \end{align*} $$

We have $H_1(M(L);\Lambda \Sigma ^{-1}) = 0$ by the hypothesis that $\Delta _L(t)$ is relatively prime to the polynomial $\lambda (t)$ . It follows that the diagonal arrow on the right-hand side of the above diagram is an isomorphism.

By our hypothesis, the kernel P of

$$ \begin{align*} \langle \alpha_J\rangle \oplus \langle\alpha_D\rangle = H_1(M(K_1);\Lambda) \longrightarrow H_1(X_n;\Lambda) \end{align*} $$

is equal to the summand $\langle \alpha _D\rangle $ . So the other summand $\langle \alpha _J\rangle $ injects into $H_1(X_n;\Lambda )$ . Because $\langle \alpha _J\rangle \cong \Lambda /\langle \lambda \rangle $ is not annihilated by $\Sigma $ , this implies that $\langle \alpha _J\rangle $ injects into $H_1(X_n;\Lambda \Sigma ^{-1}) = H_1(X_n;\Lambda )\Sigma ^{-1}$ . By the above diagram, it follows that $\langle \alpha _J\rangle $ injects into $H_1(X\smallsetminus Z_L^0;\Lambda \Sigma ^{-1})$ . Thus, $\alpha _J$ is nontrivial in $H_1(X;\Lambda \Sigma ^{-1})/\operatorname {Im} H_1(Z_L^0;\Lambda \Sigma ^{-1})$ . Therefore, by the definition of ${\mathcal P}^2\pi _1(X_n)$ , $\alpha _J$ is nontrivial in the quotient

$$ \begin{align*} {\mathcal P}^1\pi_1(X_n)/{\mathcal P}^2\pi_1(X_n) \subset H_1(X;\Lambda\Sigma^{-1})/\operatorname{Im} H_1(Z_L^0;\Lambda\Sigma^{-1}). \end{align*} $$

By Assertion C, ${\mathcal P}^1\pi _1(X_n)/{\mathcal P}^2\pi _1(X_n) \cong {\mathcal P}^1\pi _1(X_{n-1})/{\mathcal P}^2\pi _1(X_{n-1})$ . Also, $\alpha _J$ is isotopic to the meridian $\mu _{n-1}$ in $X_{n-1}$ . It follows that $\mu _{n-1}$ is nontrivial in the quotient ${\mathcal P}^1\pi _1(X_{n-1})/{\mathcal P}^2\pi _1(X_{n-1})$ . This is exactly the promised conclusion for $k=n-1$ .

Now, suppose $0\le k \le n-2$ . The induction hypothesis is that $\mu _{k+1}$ is nontrivial in the quotient ${\mathcal P}^{n-k-1}\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1})$ . To show that $\mu _k$ is nontrivial in ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ , we use an argument that is essentially the same as [3, Proof of Theorem 4.14], which was influenced by [12]. Let $R=\mathbb {Q}$ if $k\ge 1$ , and $R=\mathbb {Z}_{p_1}$ if $k=0$ . Let

$$ \begin{align*} B \colon H_1(M(J^1_{k+1});R[t^{\pm1}])\times H_1(M(J^1_{k+1});R[t^{\pm1}]) \longrightarrow R(t)/R[t^{\pm1}] \end{align*} $$

be the classical Blanchfield pairing of $J^1_{k+1}$ over R-coefficients. For brevity, let $G=\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1})$ . Consider the noncommutative Alexander module ${\mathcal A}:= H_1(M(J^1_{k+1}); RG)$ . The noncommutative Blanchfield pairing ${\mathcal B} \colon {\mathcal A}\times {\mathcal A} \to {\mathcal K}/RG$ is defined following [16, Theorem 2.13], where ${\mathcal K}$ is the skew-quotient field of $RG$ . (For $R=\mathbb {Z}_{p_1}$ , see also [3, Section 5].) In our case, because $2\le n-k\le n$ , from Definition 3.1 it follows that $G=\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1})$ is poly-torsion-free-abelian and, consequently, $RG$ is an Ore domain due to [16, Proposition 2.5] and [3, Lemma 5.2]. We will use the following known facts:

1. (1) The nontriviality of $\mu _{k+1}$ in ${\mathcal P}^{n-k-1}\pi _1(X_{k+1})/{\mathcal P}^{n-k}\pi _1(X_{k+1}) \subset G$ implies that ${\mathcal A} \cong RG {\mathbin {\mathop {\otimes }_{{R[t^{\pm 1}]}}}} H_1(M(J^1_{k+1});R[t^{\pm 1}])$ and that ${\mathcal B}(1\otimes x, 1\otimes y) = 0$ if and only if $B(x,y)=0$ . This is due to [33, Theorem 4.7], [2, Theorem 5.16] and [12, Lemma 6.5, Theorem 6.6]. See also [3, Theorem 5.4].

2. (2) The 4-manifold $X_{k+1}$ endowed with $\pi _1(X_{k+1}) \to G$ is a Blanchfield bordism in the sense of [4, Definition 4.11], due to an argument in [4, p. 3270] that uses [4, Theorem 4.13]. (The 4-manifold $W_{k+1}$ in [4] plays the role of our $X_{k+1}$ .) The only property of a Blanchfield bordism we need is the following: for all z in $\operatorname {Ker}\{{\mathcal A} \to H_1(X_{k+1};RG)\}$ , ${\mathcal B}(z,z)=0$ by [4, Theorem 4.12].

Recall that $E(P_k\sqcup \eta _k) \cup E(J^1_k) = E(J^1_{k+1}) \subset M(J^1_{k+1})\subset \partial X_{k+1}$ . Denote a zero-linking longitude of $\eta _k$ in $E(P_k\sqcup \eta _k)\subset M(J^1_{k+1})$ by $\eta _k$ , abusing notation. If $\eta _k$ is trivial in ${\mathcal P}^{n-k}\pi _1(X_{k+1})/{\mathcal P}^{n-k+1}\pi _1(X_{k+1})$ , then by the definition of ${\mathcal P}^{n-k+1}$ , $\eta _k=1\otimes \eta _k$ lies in the kernel of ${\mathcal A} \to H_1(X_{k+1};RG)$ . By (2), from this it follows that ${\mathcal B}(1\otimes \eta _k,1\otimes \eta _k) = 0$ and, consequently, by (1), $B(\eta _k,\eta _k)=0$ . Because $J^1_{k+1} = P_k(\eta _k,J^1_k)$ with $k\le n-2$ , the Alexander module $H_1(M(J^1_{k+1});R[t^{\pm 1}])$ is isomorphic to that of stevedore’s knot $P_k$ , which is a cyclic module generated by $\eta _k$ . It contradicts the nonsingularity of the classical Blanchfield pairing B. This shows that $\eta _k$ is nontrivial in ${\mathcal P}^{n-k}\pi _1(X_{k+1})/{\mathcal P}^{n-k+1}\pi _1(X_{k+1})$ , which is isomorphic to ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ by Assertion C. Because $\eta _k$ is identified with $\mu _k$ , it follows that $\mu _k$ is nontrivial in the quotient ${\mathcal P}^{n-k}\pi _1(X_k)/{\mathcal P}^{n-k+1}\pi _1(X_k)$ . This completes the proof of Assertion A.

Proof of Assertion B. Recall that Assertion B says that $\pi _1(Z_0^L)^{(i)}$ maps to ${\mathcal P}^{i+1}\pi _1(X)$ for $1\le i \le n$ . To show this for $i=1$ , observe that the composition

$$ \begin{align*} \begin{aligned} \frac{\pi_1(Z_L^0)^{(1)}}{\pi_1(Z_L^0)^{(2)}} = H_1(Z_L^0;\mathbb{Z}[t^{\pm1}]) & \longrightarrow H_1(X;\mathbb{Z}[t^{\pm1}]) \longrightarrow H_1(X;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}) \\ & \longrightarrow H_1(X;\mathbb{Q}[t^{\pm1}]\Sigma^{-1})/\operatorname{Im} H_1(Z_L^0;\mathbb{Q}[t^{\pm1}]\Sigma^{-1}) \end{aligned} \end{align*} $$

is obviously zero. By definition of ${\mathcal P}^2 \pi _1(X)$ , it follows that $\pi _1(Z_L^0)^{(1)}$ maps to ${\mathcal P}^2\pi _1(X)$ . Therefore, for all $1\le i\le n$ , $\pi _1(Z_L^0)^{(i)} = (\pi _1(Z_L^0)^{(1)})^{(i-1)}$ maps to ${\mathcal P}^2\pi _1(X)^{(i-1)}$ , which is a subgroup of ${\mathcal P}^{i+1}\pi _1(X)$ .

3.2. Obstruction from Cheeger-Gromov $\rho$ -invariants

Now, we will use the Cheeger-Gromov $\rho $ -invariant to derive a contradiction. We begin with some background. For a connected closed 3-manifold M and a homomorphism $\phi \colon \pi _1(M) \to \Gamma $ with $\Gamma $ arbitrary, the Cheeger-Gromov invariant $\rho ^{(2)}(M,\phi )\in \mathbb {R}$ is defined [10]. The value of $\rho ^{(2)}(M,\phi )$ is preserved under composition with automorphisms of $\pi _1(M)$ , so one can view $\rho ^{(2)}(M,\phi )$ as an invariant of M equipped with (the homotopy class of) a map $\phi \colon M\to B\Gamma =K(\Gamma ,1)$ , instead of $\pi _1(M) \to \Gamma $ . Even when M is not connected, $\rho ^{(2)}(M,\phi )$ is defined for $\phi \colon M\to B\Gamma $ , with additivity under disjoint union: $\rho ^{(2)}(M,\phi )=\sum _i\rho ^{(2)}(M_i,\phi |_{M_i})$ where $M=\bigcup _i M_i$ with components $M_i$ .

In this article, we do not use the definition of $\rho ^{(2)}(M,\rho ^{(2)})$ given in [10]. Instead, for our purpose, the following $L^2$ -signature defect interpretation is useful. If $M=\bigcup M_i$ bounds a 4-manifold W and $\phi \colon M\to B\Gamma $ factors through W, then $\rho ^{(2)}(M,\phi ) = \sum _i\rho ^{(2)}(M_i,\phi |_{M_i})$ is equal to the $L^2$ -signature defect $\smash {\bar \sigma ^{(2)}_{\Gamma }(W)}:=\smash {\operatorname {sign}^{(2)}_{\Gamma }(W)} -\operatorname {sign}(W)$ , where $\operatorname {sign}(W)$ is the ordinary signature and $\smash {\operatorname {sign}^{(2)}_{\Gamma }(W)}$ is the $L^2$ -signature of W over the group $\Gamma $ . We note that this approach can also be used to provide an alternative definition of $\rho ^{(2)}(M,\phi )$ for arbitrary $(M,\phi )$ . As references, see, for instance, [9], [18, Section 2], [23, Section 3], [5, Section 2.1].

For our case, let $\Gamma := \pi _1(X) / {\mathcal P}^{n+1} \pi _1(X)$ . For a connected 3-dimensional submanifold M in X, denote the composition $\pi _1(M)\to \pi _1(X)\to \Gamma $ by $\phi $ , abusing notation.

Then, by the $L^2$ -signature defect interpretation for $\rho ^{(2)}(\partial X,\phi )$ , we have

(3.2) $$ \begin{align} \rho^{(2)}(M(J^1_0),\phi) &+ \sum_{k=0}^{n-2} \rho^{(2)}(M(P_k),\phi) + \rho^{(2)}(M(R(U,D)),\phi) = \bar\sigma^{(2)}_{\Gamma}(X) \end{align} $$

$$\begin{align*} &\quad = \bar\sigma^{(2)}_{\Gamma}(V^0) + \bar\sigma^{(2)}_{\Gamma}(C) + \bar\sigma^{(2)}_{\Gamma}(Z^0_L) + \sum_{i,j} \bar\sigma^{(2)}_{\Gamma}(Z^0_{i,j}) + \sum_{k=0}^{n-1} \bar\sigma^{(2)}_{\Gamma}(E_k),\notag \end{align*}$$

where the 4-manifolds $Z^0_{i,j}$ are copies of $Z^0_i$ used in the construction of X. (See Figure 5.) The second equality is obtained by Novikov additivity of $L^2$ -signatures (for instance, see [16 Lemma 5.9]).

Recall that $\mu _0$ is the meridian of $J^{1}_{0}$ in $M(J^{1}_{0})$ . By Assertion A, $\phi (\mu _{0})$ is nontrivial in $\Gamma $ . Because $\phi (\mu _{J^{1}_{0}})$ lies in the subgroup ${\mathcal P}^n\pi _{1}(X)/{\mathcal P}^{n+1} \pi _1(X)$ of $\Gamma $ , which is a vector space over $\mathbb {Z}_{p_1}$ by Definition 3.1, it follows that $\mu _{J^1_0}$ has order $p_1$ . So the image of $\pi _1(M(J^1_0))$ in $\Gamma $ under $\phi $ is isomorphic to $\mathbb {Z}_{p_1}$ . By [8, Lemma 8.7], this implies that

(3.3) $$ \begin{align} \rho^{(2)}(M(J^1_0),\phi) = \rho(J^1_0,\mathbb{Z}_{p_1}), \end{align} $$

where $\rho (J^1_0,\mathbb {Z}_{p_1})$ is defined by (2.2).

By the explicit universal bound for the Cheeger-Gromov invariants given in [5, Theorem 1.9], we have

(3.4) $$ \begin{align} \big|\rho^{(2)}\big(M(P_k),\phi\big)\big| \le 6\cdot 69\,713\,280 \end{align} $$

because stevedore’s knot $P_k$ has six crossings. Similarly, by [5, Theorem 1.9],

(3.5) $$ \begin{align} \big|\rho^{(2)}\big(M(R(U,D)),\phi\big)\big| \le (8m+92)\cdot 69\,713\,280 \end{align} $$

because $R(U,D)$ has a diagram with $8m+92$ crossings (see Figure 1).

By [12, Lemma 2.4], the following holds for each k.

(3.6) $$ \begin{align} \bar\sigma^{(2)}_G(C)=0, \quad \bar\sigma^{(2)}_G(E_k)=0. \end{align} $$

To evaluate the terms $\bar \sigma ^{(2)}_{\Gamma }(V^0)$ and $\bar \sigma ^{(2)}_{\Gamma }(Z^0_L)$ in (3.2), we will use the following result.

Theorem 3.3 Amenable Signature Theorem [3, Theorem 3.2]

Suppose that W is an integral $(n+1)$ -solution bounded by the zero surgery manifold $M(K)$ of a knot K. Suppose that $\Gamma $ is a group that satisfies $\Gamma ^{(n+1)}=\{1\}$ and lies in Strebel’s class $D(\mathbb {Z}_{p_1})$ in the sense of [45] (or, equivalently, $\Gamma $ is a locally p-indicable group, due to [26]). If $\phi \colon \pi _1(M(K)) \to \Gamma $ is a homomorphism that factors through $\pi _1(W)$ and sends the meridian of K to an infinite-order element in $\Gamma $ , then $\rho ^{(2)}(M(K),\phi ) = \bar \sigma ^{(2)}_{\Gamma }(W) = 0$ .

In our case, $V^0$ is an integral $(n+1)$ -solution bounded by $M(K)$ . Also, the group $\Gamma $ lies in $D(\mathbb {Z}_{p_1})$ by [8, Lemma 6.8], and we have $\Gamma ^{(n+1)}=\{1\}$ because $\pi _1(X)^{(n+1)} \subset {\mathcal P}^{n+1}\pi _1(X)$ . The meridian of K has infinite order in $\Gamma $ because $\Gamma $ surjects onto $H_1(X)=\mathbb {Z}$ generated by the meridian. By Amenable Signature Theorem 3.3, it follows that

(3.7) $$ \begin{align} \bar\sigma^{(2)}_{\Gamma}(V^0) = \rho^{(2)}(M(K),\phi)=0. \end{align} $$

Now we will evaluate $\bar \sigma ^{(2)}_{\Gamma }(Z^0_L)$ , using the Amenable Signature Theorem again. Note that $\partial Z^0_L=M(L)$ . An important difference from the above paragraph is that the 4-manifold $Z^0_L$ is an integral n-solution, not $n+1$ . So, the Amenable Signature Theorem does not apply directly over $\Gamma $ , because $\Gamma ^{(n)}$ is not necessarily trivial. Instead, we proceed as follows.

Note that the map $\phi \colon \pi _1(M(L)) \to \pi _1(Z_L^0) \to \Gamma $ factors through $\pi _1(Z_L^0)/\pi _1(Z_L^0)^{(n)}$ , by Assertion B. Let G be the image of $\pi _1(Z_L^0)/\pi _1(Z_L^0)^{(n)}$ in $\Gamma $ , and let $\psi \colon \pi _1(M(L)) \to G$ be the map induced by $\phi $ . Because G injects into $\Gamma $ , we have $\rho ^{(2)}(M(L),\phi ) = \rho ^{(2)}(M(L),\psi )$ , by the $L^2$ -induction property (for instance, see [10, Eq. 2.3]). Now, because G is a subgroup of $\Gamma $ that is in $D(\mathbb {Z}_{p_1})$ , G is in Strebel’s class $D({\mathbb {Z}}_{p_1})$ , too. Also, $G^{(n)}$ is trivial because it is the image of $\pi _1(Z_L^0)/\pi _1(Z_L^0)^{(n)}$ . The meridian $\mu _L$ of L has infinite order in G, because G surjects onto $H_1(X)=\mathbb {Z}$ , which $\mu _L$ generates. Therefore, Amenable Signature Theorem 3.3 (with n in place of $n+1$ ) applies to $(M(L),\psi )$ to conclude that

(3.8) $$ \begin{align} \bar\sigma^{(2)}_{\Gamma}(Z^0_L) = \rho^{(2)}(M(L),\phi) = \rho^{(2)}(M(L),\psi) = 0. \end{align} $$

By [6, Lemma 3.3], we may assume that each $Z^0_{i,j}$ has the property that $\bar \sigma ^{(2)}_{\Gamma }(Z^0_{i,j})$ is equal to either $0$ or $\rho (J^i_0,\mathbb {Z}_{p_1})$ . (Indeed, [6, Lemma 3.3] applies when every $J^i_0$ is $0$ -negative; it is the case by property (J1) in Subsection 2.1.) Moreover, by property (J3) in Subsection 2.1, we have

(3.9) $$ \begin{align} \bar\sigma^{(2)}_{\Gamma}(Z^0_{i,j}) = \begin{cases} 0 \text{ or } \rho(J^1_0,\mathbb{Z}_{p_1}) &\text{if }i=1, \\ 0 &\text{if }i>1. \end{cases} \end{align} $$

Now, combine (3.2)–(3.9) to obtain

$$ \begin{align*} N\cdot |\rho^{(2)}(J^1_0,\mathbb{Z}_{p_1})| \le (6n+8m+86)\cdot 69\;713\;280, \end{align*} $$

where N is one plus the number of the 4-manifolds $Z^0_{1,j}$ such that $\bar \sigma ^{(2)}(Z^0_{1,j})\ne 0$ . Because $N\ge 1$ , it contradicts property (J2) in Subsection 2.1. This completes the proof that P cannot be equal to $\langle \alpha _D \rangle $ . That is, P must be $\langle \alpha _J \rangle $ .

4. Computing and estimating d-invariants

In this section, we will continue the proof of Theorem 2.2, to reach a contradiction under the hypothesis that $P=\langle \alpha _J\rangle $ .

4.1. Finite cyclic covers and their d-invariants

We begin by applying a trick introduced in [11], which we describe below. Let $K_1=K_0(\alpha ,J)$ be a satellite knot, where the pattern satisfies $\mathrm {lk}(K_0,\alpha )=0$ . Then, the identity on $E(K\sqcup \alpha )$ extends to a map $E(K_1)=E(K\sqcup \alpha )\cup E(J) \to E(K\sqcup \alpha ) \cup E(U)=E(K_0)$ that induces an isomorphism $H_1(M(K_1);\mathbb {Q}[t^{\pm 1}]) \to H_1(M(K_0);\mathbb {Q}[t^{\pm 1}])$ , under which we will identify the Alexander modules. Essentially, [11, Lemma 8.2] says the following (see also [6, Lemma 5.1]): if $K_1=K_0(\alpha ,J)$ admits a $1$ -negaton $X_1$ bounded by $M(K_1)$ and if J is unknotted by changing some positive crossings to negative, then $K_0$ has a $1$ -negaton $X_0$ bounded by $M(K_0)$ such that the two maps

$$ \begin{align*} H_1(M(K_i);\mathbb{Q}[t^{\pm1}]) \longrightarrow H_1(X_i;\mathbb{Q}[t^{\pm1}]) \quad (i=0,1) \end{align*} $$

have the identical kernel. In particular, this applies to the satellite knot $K_1 = R(U,D)(\alpha _D,J^1_{n-1})$ defined in Subsection 2.1, because $J^1_{n-1}=P_{n-2}(\eta _n, J^1_{n-2})$ is unknotted by changing a single positive crossing (see Figure 2). Note that here we use that $n\ge 2$ . Therefore, in our case, the knot $K_0 := R(U,D)$ admits a $1$ -negaton bounded by $M(K_0)$ , say, W, such that

(4.1) $$ \begin{align} \langle\alpha_J\rangle = \operatorname{Ker}\{H_1(M(K_0);\mathbb{Q}[t^{\pm1}]) \rightarrow H_1(W;\mathbb{Q}[t^{\pm1}])\}. \end{align} $$

We will derive a contradiction from the existence of this 1-negaton W for $K_0$ .

The next step is to pass to finite-degree branched cyclic covers, to which Heegaard Floer homology machinery applies, following [6, Section 5.1]. Let $\Sigma _r$ be the r-fold branched cyclic cover of $(S^3,K_0)$ . The curves $\alpha _J$ and $\alpha _D$ in Figure 1 represent homology classes in $\Sigma _r$ , say, $x_1$ and $x_2\in H_1(\Sigma _r)$ , respectively. (The classes $x_1$ and $x_2$ are defined up to covering transformation, but it will cause no problem in our argument.) Due to [37], we have $H_1(\Sigma _r) = H_1(M(K_0);\mathbb {Z}[t^{\pm 1}])/\langle t^r-1\rangle $ . Recall that $H_1(M(K_0);\mathbb {Z}[t^{\pm 1}])$ is given by (2.1). From this, by elementary computation, it follows that $H_1(\Sigma _r)=\mathbb {Z}_{(m+1)^r-m^r} \oplus \mathbb {Z}_{(m+1)^r-m^r}$ and the summands are generated by $x_1$ and $x_2$ . In particular, $\Sigma _r$ is a $\mathbb {Z}_2$ -homology sphere.

For a rational homology 3-sphere Y and a spin $^c$ structure $\mathfrak {t}$ on Y, Ozsváth and Szabó defined a correction term invariant $d(Y,\mathfrak {t})$ using the Heegaard Floer chain complex [40]. In case of a $\mathbb {Z}_2$ -homology sphere Y, the unique spin structure determines a canonical spin $^c$ structure on Y, which we denote by $\mathfrak {s}_Y$ , and all spin $^c$ structures of Y are given in the form $\mathfrak {s}_Y+c$ , where $c\in H^2(Y)$ and $+$ designates the action of $H^2(Y)$ on the set of spin $^c$ structures. For $x\in H_1(Y)$ , let $\widehat x\in H^2(Y)$ be the Poincaré dual of x. Techniques used in [6, 11] give us the following d-invariant obstruction.

Lemma 4.1. Suppose that $M(K_0)$ bounds a $1$ -negaton that satisfies (4.1). If r is sufficiently large, then $d(\Sigma _r, \mathfrak {s}_{\Sigma _r}+ k\cdot \widehat {x_1})\ge 0$ for all $k\in \mathbb {Z}$ .

Proof. Attach a 2-handle to a given $1$ -negaton W, along the zero-framed meridian of $K_0$ , to obtain a 4-manifold, which we temporarily call V. Note that $\partial V = S^3$ and the cocore of the 2-handle is a slicing disk $\Delta \subset V$ bounded by $K_0$ . Take the r-fold branched cyclic cover of $(V,\Delta )$ and call it $V_r$ . Here the r-fold branched cyclic cover is defined because, using that W is a $1$ -negaton, we see that $H_1(W\smallsetminus \Delta )=\mathbb {Z}$ generated by a meridian of $\Delta $ . Indeed, if we denote by $W_r$ the r-fold cyclic cover of W, then $V_r$ is obtained by attaching a 2-handle to $W_r$ . The 4-manifold $V_r$ is bounded by $\Sigma _r$ .

The first key step is to relate the hypothesis $P=\langle \alpha _J\rangle $ , which is associated with the infinite cyclic cover, to the kernel

$$ \begin{align*} G:=\operatorname{Ker}\{H_1(\Sigma_r)\longrightarrow H_1(V_r)\} \end{align*} $$

associated with finite covers. The following is a modified version of [6, Lemma 5.2].

Assertion. Under the hypothesis that $P=\langle \alpha _J\rangle $ , $x_1\in G$ for all large primes r.

Although we do not use it, we remark that the assertion implies that $G=\langle x_1\rangle $ , because it is known that $|G|$ is equal to $|H_1(\Sigma _m)|^{1/2} = (m+1)^r-m^r\!.$

Proof of Assertion. Let $E_r$ be the r-fold cyclic cover of the exterior $E(K_0)$ for $r \le \infty $ . Consider the following commutative diagram:

The vertical arrows are induced by coverings and inclusions. Because $\alpha _J$ lies in the kernel of the composition of the top row by the hypothesis, $\alpha _J$ is $\mathbb {Z}$ -torsion in $H_1(W;\mathbb {Z}[t^{\pm 1}])$ . That is, $a\cdot \alpha _j=0$ in $H_1(W;\mathbb {Z}[t^{\pm 1}])$ for some nonzero $a\in \mathbb {Z}$ . By the above diagram, it follows that $a\cdot x_1 \in H_1(\Sigma _r)$ lies in the kernel G of $H_1(\Sigma _r)\to H_1(V_r)$ .

Suppose that r is a prime not smaller than any prime factor of a. Under this assumption, we claim that $\gcd (a, (m+1)^r - m^r)=1$ . From this it follows that $x_1$ lies in G, because $x_1$ has order $(m+1)^r - m^r$ in $H_1(\Sigma _r)$ . This proves the assertion, modulo the proof of the claim.

To show the claim, it suffices to show that every prime factor q of a is relatively prime to $(m+1)^r - m^r\!.$ It is obviously true, if $q\mid m$ or $q\mid m+1$ . So, suppose that q divides neither m nor $m+1$ but q divides $(m+1)^r - m^r\!.$ Let $u=m^*(m+1)$ , where $m^*$ is an arithmetic inverse of $m \bmod q$ . We have $u^r \equiv 1 \bmod q$ by the hypothesis and $u^{q-1} \equiv 1 \bmod q$ by Fermat’s little theorem. Because r is a prime and $u\not \equiv 1 \bmod q$ , it follows that $r\mid q-1$ . This contradicts the assumption that $r\ge q$ . This completes the proof of the claim.

The assertion enables us to invoke a result of Cochran, Harvey and Horn [11, Theorem 6.5], which says the following: if W is a $1$ -negaton bounded by $M(K_0)$ , then $d(\Sigma _r,\widehat x)\ge 0$ for all x lying in $G=\operatorname {Ker}\{H_1(\Sigma _r) \to H_1(V_r)\}$ . Applying this to $x=k\cdot x_1$ , the proof of Lemma 4.1 is completed.

Theorem 4.2. Let $m\ge 1$ be an odd integer and $r\ge 1$ be an odd prime power. Let k be the arithmetic inverse of $2 \bmod (m+1)^r-m^r\!.$ Then $d(\Sigma _r, \mathfrak {s}_{\Sigma _r} + k\widehat x_1) \le -\frac 32$ .

On the other hand, Lemma 4.1 says that $d(\Sigma _r, \mathfrak {s}_{\Sigma _r} + k\widehat x_1)$ must be nonnegative. This contradiction implies that the kernel P defined in (2.4) cannot be equal to $\langle \alpha _J\rangle $ . This completes the proof of Theorem 2.2.

For $m=1$ , Theorem 4.2 was already shown in [6, Theorem 5.4]. (We remark that the symbol m in [6] denotes our r.) So, in the remaining part of this article, we will assume that $m>1$ .

Proof of Theorem 4.2. Let A and B be the 3-manifolds given by the surgery presentations in Figure 6. Let $Y_r$ be the connected sum of A, B and $r-3$ copies of the lens space $L_{2m+1,1}$ . (We use the orientation convention that $L_{p,1}$ is the p-framed surgery on the trivial knot U in $S^3$ .) Then, arguments in [6, Section 6.1] construct a negative definite 4-manifold W with $\partial W = Y_r \sqcup -\Sigma _r$ and $b_2(W)=(2m+1)r-4m+1$ and construct a spin $^c$ structure $\mathfrak {t}$ on W such that $c_1(\mathfrak {t})^2 = -r$ , $c_1(\mathfrak {t}|_{\Sigma _r}) = \widehat {x_1}$ and $c_1(\mathfrak {t}|_{Y_r}) = 0$ . Indeed, [6, Section 6.1] is the case of $m=1$ . We do not repeat the details here, because exactly the same method works under our assumption that $m\ge 1$ is odd. Perhaps the least obvious part is that the inverse matrix $P^{-1}$ in [6, Eq. (6.10)] should be replaced with a block matrix $P^{-1}=(R_{ij})_{1\le i,j\le r-1}$ with

$$ \begin{align*} R_{ij} = \frac{((m+1)^i-m^i)((m+1)^{r-j}-m^{r-j})}{(m+1)^r-m^r} \begin{bmatrix} 0 & (m+1)^{i-j} \\ m^{i-j} & 0 \end{bmatrix} \text{ for } i\ge j, \end{align*} $$

and $R_{ij}=R^T_{ji}$ for $i<j$ .

Figure 6 The 3-manifolds A and B. The box $2m+1$ represents $2m+1$ right-handed full twists between vertical strands.

By applying Ozsváth-Szabó’s d-invariant inequality [40, Theorem 9.6] to the negative definite 4-manifold W, and by using additivity of the d-invariant under connected sum, we have

$$ \begin{align*} \begin{aligned} d(\Sigma_r,\mathfrak{t}|_{\Sigma_r}) & \le d(Y_r,\mathfrak{t}|_{Y_r}) - \frac{c_1(\mathfrak{t})^2 + b_2(W)}{4} \\ & = d(A,\mathfrak{t}|_{A}) + d(B,\mathfrak{t}|_{B}) + (r-3)d(L_{2m+1,1},\mathfrak{t}|_{L_{2m+1,1}}) - \frac{2mr-4m+1}{4}. \end{aligned} \end{align*} $$

Because $c_1(\mathfrak {t}|_{Y_r})=0$ , we have $\mathfrak {t}|_{L_{2m+1,1}}= \mathfrak {s}_{L_{2m+1,1}}$ . By a d-invariant formula for lens spaces given in [40, Proposition 4.8], $d(L_{2m+1,1},\mathfrak {s}_{L_{2m+1,1}})=m/2$ . By Lemmas 4.3 and 4.4, which we will prove below, we have $d(A,\mathfrak {t}|_A) \le (m-7)/4$ and $d(B,\mathfrak {t}|_B) \le (m+2)/4$ . Combine them with the above inequality to obtain $d(\Sigma _r, \mathfrak {t}|_{\Sigma _r}) \le -\frac 32$ . Observe that $c_1(\mathfrak {t}|_{\Sigma _r})=\widehat {x_1} = 2k \widehat {x_1} = c_1(\mathfrak {s}_{\Sigma _r} + k \widehat {x_1})$ , because $2k\equiv 1\bmod (m+1)^r-m^r$ and $x_1$ has order $(m+1)^r-m^r\!.$ It follows that $\mathfrak {t}|_{\Sigma _r} = \mathfrak {s}_{\Sigma _r}$ , because $\Sigma _r$ is a $\mathbb {Z}_2$ -homology sphere.

Note that $|H_1(A)| = \bigl |\det \left [\begin {smallmatrix}m & -2m-1 \\ -2m-1 & m+1\end {smallmatrix}\right ]\bigr | = 3m^2+3m+1$ is odd, so there is a unique spin $^c$ structure $\mathfrak {s}$ of A such that $c_1(\mathfrak {s}) = 0$ .

Lemma 4.3. For the spin $^c$ structure $\mathfrak {s}$ on A with $c_1(\mathfrak {s}) = 0$ , $d(A,\mathfrak {s}) \le (m-7)/4$ .

For the case of B, because $|H_1(B)| = \bigl |\det \left [\begin {smallmatrix}m & -2m-1 \\ -2m-1 & m+2\end {smallmatrix}\right ] \bigr | = 3m^2+2m+1$ is even, there are exactly two spin $^c$ structures $\mathfrak {s}$ of B such that $c_1(\mathfrak {s}) = 0$ .

Lemma 4.4. If $\mathfrak {s}$ is a spin $^c$ structure of B such that $c_1(\mathfrak {s}) = 0$ , then $d(B,\mathfrak {s}) \le (m+2)/4$ .

Proof of Lemma 4.3. Let $A'$ be the 3-manifold obtained by $(m+6)$ -surgery on the knot $T\#-D$ . See Figure 7. Let Y be the 3-manifold given by the last surgery diagram in Figure 7. The four diagrams in Figure 7 describe a cobordism from $A\# A'$ to Y. More precisely, start by taking ${(A\# A') \times I}$ . Attach a 2-handle to obtain a cobordism, say, V, from $A\#A'$ to the second surgery diagram in Figure 7. Apply handle slide, to change the second surgery diagram to the third. Observe that in the third surgery diagram, the two components with framing $m+6$ and $0$ give a $S^3$ summand, whereas the two components with framing m and $2m+7$ form a link concordant to the link in the last surgery diagram, which describes Y. It follows that there is a homology cobordism, say, $V'$ , from the third surgery diagram to Y. Now $V\cup V'$ is a cobordism from $A\#A'$ to Y.

Figure 7 A cobordism from $A\# A'$ to Y.

We claim that V is negative definite. To see this, view the surgery diagram of the 3-manifold $A\# A'$ as a Kirby diagram of a 4-manifold $V_0$ . That is, $V_0$ consists of one 0-handle and three 2-handles attached along the framed link diagram of $A\# A'$ in Figure 7. We have $\partial V_0=A\# A'$ . The second diagram in Figure 7 or, equivalently, the third diagram, is a Kirby diagram of the 4-manifold $V_0\cup _{A\# A'} V$ . The Kirby diagram of $V_0$ has linking matrix

$$ \begin{align*} L = \begin{bmatrix} m & -2m-1 & 0 \\ -2m-1 & m & 0 \\ 0 & 0 & m+6 \end{bmatrix}, \end{align*} $$

which has signature $1$ because $m>0$ and the top upper $2\times 2$ submatrix has negative determinant. It follows that $\operatorname {sign} V_0 = 1$ . The third diagram in Figure 7 has linking matrix

$$ \begin{align*} \begin{bmatrix} m & -2m-1 & 0 & 0\\ -2m-1 & 2m+7 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & m+6 \end{bmatrix}, \end{align*} $$

which has vanishing signature. So $\operatorname {sign} V_0\cup V = 0$ . By Novikov additivity, it follows that $\operatorname {sign} V=-1$ . Because $b_2(V)=1$ , this proves the claim that V is negative definite.

Because $V'$ is a homology cobordism, $H_*(V\cup V')=H_*(V)$ . Consequently, $V\cup V'$ is negative definite.

We will construct a generator of $H_2(V\cup V') = H_2(V)$ and use it to describe a certain spin $^c$ structure on $V\cup V'$ . Let $\sigma $ be the core of the 2-handle of $(V,A\#A')$ , and let $\alpha =\partial \sigma $ be its attaching circle, which lies in $A\#A'$ . See the second diagram in Figure 7, in which $\alpha $ is the zero-framed circle. Because the linking matrix L is a presentation for $H_1(A\#A')$ , it is seen that $H_1(A\#A')=\mathbb {Z}_{3m^2+3m+1}\oplus \mathbb {Z}_{m+6}$ , and $\alpha = (1,1)$ in $H_1(A\#A')$ . So, the order of $\alpha $ in $H_1(A\#A')$ is $(3m^2+3m+1)(m+6)/d$ , where $d:=\gcd (3m^2+3m+1, m+6)$ . (Indeed, it can be seen that d is either $91$ or $1$ .) From this, it follows that there is a 2-cycle z in $A\#A'$ such that

$$ \begin{align*} \partial z= \frac{(3m^2+3m+1)(m+6)}{d}\cdot \alpha. \end{align*} $$

Moreover, the 2-chain

$$ \begin{align*} E:= \frac{(3m^2+3m+1)(m+6)}{d} \cdot \sigma - z \end{align*} $$

is a generator of $H_2(V)=\mathbb {Z}$ , by a standard Mayer-Vietoris argument for the 2-handle attachment.

The self-intersection number $E\cdot E$ is equal to the intersection number (in $A\#A'$ ) of z and a pushoff of $\partial z$ , say, $\partial z'$ , taken along the 2-handle attachment framing (which is the zero framing in Figure 7). So $E\cdot E$ is equal to the linking number of $\partial z$ and its pushoff in the rational homology sphere $A\#A'$ . In addition, the linking number can be computed using the linking matrix L (for instance, see [7, Theorem 3.1]):

$$ \begin{align*} \begin{aligned} E\cdot E &= \mathrm{lk}_{A\#A'}(\partial z,\partial z') = \frac{(3m^2+3m+1)^2(m+6)^2}{d^2} \cdot \begin{bmatrix} 0 & -1 & 1 \end{bmatrix} L^{-1} \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \\ & = \frac{(3m^2+3m+1)(m+6)(-2m^2+3m-1)}{d^2}. \end{aligned} \end{align*} $$

Because the factor $-2m^2+3m-1$ of the numerator is even and d is odd, $E\cdot E$ is even. From this, it follows that $0\in H^2(V\cup V')$ is characteristic. Therefore, there is a spin $^c$ structure $\mathfrak {t}$ on $V\cup V'$ such that $c_1(\mathfrak {t})=0$ . By the Ozsváth-Szabó inequality [40, Theorem 9.6], we have

$$ \begin{align*} d(Y,\mathfrak{t}|_Y) - d(A,\mathfrak{t}|_A) - d(A',\mathfrak{t}|_{A'}) \ge \frac 14. \end{align*} $$

Note that $A'$ is the $(m+6)$ -surgery on the knot $T\# -D$ , and $c_1(\mathfrak {t}|_{A'})=0$ because $c_1(\mathfrak {t})=0$ . By techniques of [11, p. 2150-2151] and [24, Appendix A], $d(A',\mathfrak {t}|_{A'}) = d(L_{m+6},\mathfrak {s}_{L_{m+6}})$ . By Ozsváth-Szabó’s formula for lens spaces [40, Proposition 4.8], we have $d(L_{m+6},\mathfrak {s}_{L_{m+6}}) = (m+5)/4$ . By Lemma 4.8, which we will prove below, we have $d(Y,\mathfrak {t}|_Y) \le (2m-1)/4$ . (Note that $c_1(\mathfrak {t}|_Y)=0$ because $c_1(\mathfrak {t})=0$ .) Combining these with the above inequality, it follows that $d(A,\mathfrak {s}) \le (m-7)/4$ .

4.2. A quick summary of Némethi’s method for Seifert 3-manifolds

For the reader’s convenience, we provide a summary of Némethi’s method to compute d-invariants [38], which we will use in Subsection 4.3. We focus on the case of Seifert 3-manifolds, which is treated in [38, Section 11], and [38] provides techniques for a larger class of certain plumbed 3-manifolds.

Let Y be a Seifert 3-manifold. It is well known that Y admits a surgery presentation of a specific form, which is shown in the left side of Figure 8. The associated star-shaped plumbing graph, which is shown in the right side of Figure 8, is often used to describe the 3-manifold Y, as the boundary of a plumbed 4-manifold X: for each vertex, take a disk bundle over a 2-sphere whose Euler number is the integer decoration of the vertex. For each edge, perform $+1$ plumbing between the two disk bundles corresponding to the endpoints. The result is a 4-manifold X with $\partial X=Y$ .

Figure 8 A Seifert 3-manifold and its plumbing graph.

Let $\nu $ be the number of branches of the star-shaped graph in Figure 8. In this subsection, we assume that $\nu \ge 3$ and that X is negative definite. (We remark that not all Seifert 3-manifolds Y are described by a plumbing graph satisfying this assumption.)

We use the following notation. Let $e_0$ be the decoration of the root vertex. Let $s_{\ell }$ be the number of (non-root) vertices on the $\ell $ th branch, and let $e_{\ell ,1},\ldots$ , $e_{\ell ,s_{\ell }}$ be the decorations of those $s_{\ell }$ vertices. See Figure 8. We may assume that $e_{\ell ,j}\le -2$ for all $\ell $ , j (for instance, see [39]). Let $b_0$ and $b_{\ell ,j} \in H_2(X)$ be the classes of 2-spheres corresponding to vertices with decoration $e_0$ and $e_{\ell ,j}$ respectively ( $1\le \ell \le \nu $ , $1\le j\le s_{\ell }$ ). They form a basis for the free abelian group $H_2(X)$ . Let $b^*_0$ and $b^*_{\ell ,j}\in H^2(X)$ be basis elements (hom) dual to $b_0$ and $b_{\ell ,j}$ . With respect to these bases, the intersection form $\lambda \colon H_2(X)\times H_2(X) \to X$ or its adjoint $\lambda \colon H_2(X) \to \mathrm {Hom}(H_2(X),\mathbb {Z}) = H^2(X)$ is given as follows: $\lambda (b_0,b_0)=e_0$ , $\lambda (b_{\ell ,j}, b_{\ell ,j}) = e_{\ell ,j}$ , and for $b\ne b'$ , $\lambda (b,b')=1$ if $\{b,b'\}=\{b_0,b_{\ell ,1}\}$ or $\{b_{\ell ,j},b_{\ell ,j+1}\}$ , and $\lambda (b,b')=0$ otherwise. For $1\le \ell \le \nu $ , define a continued fraction by

$$ \begin{align*} \frac{\alpha_{\ell}}{\omega_{\ell}} := [-e_{\ell,1},\ldots,-e_{\ell,s_{\ell}}] = -e_{\ell,1} - \cfrac{1}{-e_{\ell,2} - \cfrac{1}{\,\,\,\vdots\,\,\,}}. \end{align*} $$

The condition $e_{\ell ,j}\le -2$ implies that $\alpha _{\ell }/\omega _{\ell }>1$ . So we may assume $0\le \omega _{\ell } < \alpha _{\ell }$ . Then, a standard diagonalisation process applied to the intersection form $\lambda $ gives us a diagonal matrix whose all diagonals are easily seen to be negative, possibly except one, which is equal to

$$ \begin{align*} e := e_0 + \sum_{\ell=1}^{\nu} \omega_{\ell}/\alpha_{\ell}. \end{align*} $$

So, X is negative definite if and only if $e<0$ .

The set of characteristic elements in $H^2(X)$ is defined to be

$$ \begin{align*} \operatorname{Char}(X) := \{\xi \in H^2(X) \mid \xi(b_j)\equiv \lambda(b_j,b_j) \bmod 2 \text{ for all } j\}. \end{align*} $$

The Chern class $c_1\colon \operatorname {\mathrm {Spin}}^c(X) \to \operatorname {Char}(X)$ is bijective, because $H^2(X)$ does not have any 2-torsion. Under the identification via $c_1$ , the action of an element $c\in H^2(X)$ on $\operatorname {\mathrm {Spin}}^c(X)$ is given by $\xi \mapsto \xi +2c$ for $\xi \in \operatorname {Char}(X)=\operatorname {\mathrm {Spin}}^c(X)$ . In particular, the action of $x\in H_2(X)$ on $\operatorname {Char}(X)=\operatorname {\mathrm {Spin}}^c(X)$ via $\lambda \colon H_2(X)\to H^2(X)$ is given by $\xi \mapsto \xi +2\lambda (x)$ .

The Chern class $c_1\colon \operatorname {\mathrm {Spin}}^c(Y)\to H^2(Y)$ is not injective in general, so the standard identification of spin $^c$ structures of Y is given indirectly using X: we have a bijection

$$ \begin{align*} \operatorname{\mathrm{Spin}}^c(Y) \approx \operatorname{Char}(X)/2\lambda(H_2(X)). \end{align*} $$

Here, for a spin $^c$ structure $\xi \in \operatorname {\mathrm {Spin}}^c(X)=\operatorname {Char}(X)$ , the coset $[\xi ] = \xi +2\lambda (H_2(X))$ in $\operatorname {Char}(X)/2\lambda (H_2(X))$ corresponds to the restriction of $\xi $ on Y. Essentially, the bijectivity is a consequence of the fact that $H^2(Y)$ is the cokernel of $\lambda \colon H_2(X) \to H^2(X)$ .

In [38], the notion of a distinguished representative is used to express a spin $^c$ structure of Y. Instead of the original definition (see Section 5, especially Definition 5.1 of [38]), we will use a characterisation theorem as a definition. We need the following notation. For $1\le i\le j\le s_{\ell }$ , let $n^{\ell }_{i,j}/d^{\ell }_{i,j} := [-e^{\ell }_i, \ldots , -e^{\ell }_j]$ , where $n^{\ell }_{i,j}> 0$ and $\gcd (n^{\ell }_{i,j}, d^{\ell }_{i,j}) = 1$ . Note that $\alpha _{\ell }/\omega _{\ell } = n^{\ell }_{1,s_{\ell }}/d^{\ell }_{1,s_{\ell }}$ . Define an element $K\in \operatorname {\mathrm {Spin}}^c(X) = \operatorname {Char}(X)\subset H^2(X)$ by

(4.2) $$ \begin{align} K(b_0) = -e_0-2,\quad K(b_{\ell,j}) = -e_{\ell,j}-2 \text{ for } 1\le \ell\le \nu,\, 1\le j\le s_{\ell}. \end{align} $$

The element K is characteristic because $K(b_0) \equiv \lambda (b_0,b_0)$ and $K(b_{\ell ,j}) \equiv \lambda (b_{\ell ,j},b_{\ell ,j}) \bmod 2$ . In [38], K is called the canonical spin $^c$ structure. We have $\operatorname {Char}(X) = K + 2H^2(X)$ . Consider a class $k_r\in \operatorname {Char}(X)$ of the form

(4.3) $$ \begin{align} k_r = K - 2 \bigg( a_0^{\vphantom{1}} b^*_0 + \sum_{\ell,j} a_{\ell,j}^{\vphantom{1}} b_{\ell,j}^* \bigg), \end{align} $$

where $a_0$ and $a_{\ell ,j}$ are integers. Let

(4.4) $$ \begin{align} a_{\ell} = \sum_{t=1}^{s_{\ell}-1} n^{\ell}_{t+1,s_{\ell}} a_{\ell,t}. \end{align} $$

Definition 4.5 [38, Corollary 11.7]

The class $k_r$ in (4.3) called a distinguished representative if $0\le a_{\ell } < \alpha _{\ell }$ for all $\ell $ and if

(4.5) $$ \begin{align} 0\le a_0 \le -1 -i e_0 - \sum_{\ell=1}^{\nu} \Bigl[ \frac{i\omega_{\ell} + a_{\ell}}{\alpha_{\ell}} \Bigr] \end{align} $$

for all $i>0$ . Here $[x]$ is the largest integer not greater than x.

To state Némethi’s formula for the d-invariant, we need one more notation. Let

(4.6) $$ \begin{align} \tau(i) = \sum_{t=0}^{i-1} \bigg(a_0+1-te_0 + \sum_{\ell=1}^{\nu} \Bigl[ \frac{-t\omega_{\ell} + a_{\ell}}{\alpha_{\ell}} \Bigr] \bigg). \end{align} $$

In particular, $\tau (0)=0$ .

Theorem 4.5 Némethi [38, p. 1038]

Suppose that X is negative definite and $\nu \ge 3$ . Suppose that the class $k_r$ given in (4.3) is a distinguished representative. Then, for the spin $^c$ structure $[k_r]$ of Y, the d-invariant is given by

$$ \begin{align*} d(Y,[k_r]) = \frac{k_r^2+b_2(X)}{4} - 2\cdot \min \{ \tau(i) \mid i\ge 0\}. \end{align*} $$

Remark 4.6.

1. (1) For each $k_r$ given by (4.3), the minimum in Theorem 4.6 can be found in finite steps. To see this, let $\Delta _i = \tau (i+1)-\tau (i)$ . Then we have

   $$ \begin{align*} \Delta_i & \ge a_0 + 1 - ie_0 + \sum_{\ell=1}^{\nu} \frac{- i \omega_{\ell} + a_{\ell} + \alpha_{\ell} - 1}{\alpha_{\ell}} \\ & = -e\cdot i + \bigg( 1+a_0 + \sum_{\ell=1}^{\nu} \frac{a_{\ell}-\alpha_{\ell}+1}{\alpha_{\ell}} \bigg) \ge 0 \end{align*} $$
   if i is not smaller than
   $$ \begin{align*} R := \bigg( 1+a_0 + \sum_{\ell=1}^{\nu} \frac{a_{\ell}-\alpha_{\ell}+1}{\alpha_{\ell}} \bigg) \Big/ (-e). \end{align*} $$
   Here we use that e is negative because X is negative definite. So, the minimum in Theorem 4.6 can be taken over $0\le i \le \max \{0,R\}$ .

2. (2) A similar argument shows that it can be determined in finite steps whether a class $k_r$ given by (4.3) satisfies Definition 4.5. Indeed, the right- hand side of (4.5) is bounded from below by $-(1+\sum a_{\ell }/\alpha _{\ell })-ei$ . Because $e<0$ , (4.5) is satisfied for all large i, and thus it suffices to check (4.5) for only finitely many i.

Using (1) and (2), it is straightforward to write a practically efficient algorithm (and computer code) to find distinguished representatives of all spin $^c$ structures of Y and compute the associated d-invariants.

4.3. d-invariants of the 3-manifolds Y and B

Let Y be the 3-manifold given by the last surgery diagram in Figure 7 or, equivalently, by the first surgery diagram in Figure 9. The following lemma gives an estimate of the d-invariant of Y, which is used to complete the proof of Lemma 4.3. Note that there are two spin $^c$ structures on Y satisfying $c_1(\mathfrak {s})=0$ , because $|H_1(Y)| = 2m^2-3m+1$ is even.

Figure 9 Surgery diagram calculus that gives a plumbing tree for Y. The symbol $(*)$ means handle slides and elimination of components with 0-framed meridians.

Lemma 4.8. If $\mathfrak {s}$ is a spin $^c$ structure on Y such that $c_1(\mathfrak {s})=0$ , then $d(Y,\mathfrak {s}) \le (2m-1)/4$ .

Proof. The surgery diagram calculus in Figure 9 shows that Y is a Seifert 3-manifold. The last plumbing graph in Figure 9 describes a plumbed 4-manifold X with $\partial X=Y$ .

To compute the d-invariant, we will apply the method discussed in Subsection 4.2. Using the notation in Subsection 4.2, denote the basis of $H_2(X)$ by $\{b_0$ , $b_{1,1},\ldots$ , $b_{1,2m}$ , $b_{2,1}$ , $b_{3,1}$ , $b_{4,1}\}$ and the dual basis of $H^2(X)$ by $\{b^*_0$ , $b^*_{1,1},\ldots$ , $b^*_{1,2m}$ , $b^*_{2,1}$ , $b^*_{3,1}$ , $b^*_{4,1}\}$ . The intersection form $\lambda \colon H_2(X)\times H_2(X)\to \mathbb {Z}$ is computed straightforwardly from the plumbing graph:

$$ \begin{align*} \lambda = \left[ \begin{array}{c|ccccc|c|c|c} -2 & 1 & & & & & 1 & 1 & 1 \\ \hline 1 & -2 & 1 & & & & & & \\ & 1 & -2 & 1 & & & & & \\ & & \ddots & \ddots & \ddots & & & \\ & & & 1 & -2 & 1 & & & \\ & & & & 1 & -2 & & & \\ \hline 1 & & & & & & -2 & & \\ \hline 1 & & & & & & & -3 & \\ \hline 1 & & & & & & & & -m-1 \end{array} \right]_{(2m+4)\times(2m+4)}. \end{align*} $$

Also, using the definition in Subsection 4.2, it is routine to compute the following:

$$ \begin{align*} \Bigl( \frac{\alpha_1}{\omega_1},\, \frac{\alpha_2}{\omega_2},\, \frac{\alpha_3}{\omega_3},\, \frac{\alpha_4}{\omega_4} \Bigr) = \Bigl( \frac{2m+1}{2m},\,\frac21,\,\frac31,\,\frac{m+1}{1} \Bigr). \end{align*} $$

So, the orbifold Euler number is given by

$$ \begin{align*} e = -2 + \frac{2m}{2m+1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{m+1} = -\frac{(m-1)(2m-1)}{6(m+1)(2m+1)}. \end{align*} $$

For $m>1$ , we have $e<0$ , so X is negative definite.

We will describe two spin $^c$ structures $[k_1]$ and $[k_2]\in \operatorname {Char}(X)/2\lambda (H_2(X))$ . Let

$$ \begin{align*} k_1 &:= -2 b^*_{1,2m} - b^*_{3,1} + 2b^*_{4,1}, \\ k_2 &:= -2 b^*_{1,2m-1} + b^*_{3,1} - 2b^*_{4,1}. \end{align*} $$

It is straightforward to show that $k_1$ and $k_2$ are distinguished representatives in the sense of Definition 4.5. Indeed, in our case, the canonical spin $^c$ structure described in (4.2) is given by $K= b^*_{3,1} -(m+1) b^*_{4,1}$ , and $k_1$ is of the form (4.3), where

$$ \begin{align*} a_0 = 0,\quad (a_{1,1},\ldots,a_{1,2m}) = (0,\ldots,0,1), \quad a_{2,1}=0,\quad a_{3,1}=1,\quad a_{4,1}=1. \end{align*} $$

By (4.4), we have $(a_0,a_1,a_2,a_3,a_4) = (0,1,0,1,\tfrac {m-3}{2})$ . This satisfies the conditions in Definition 4.5, so $k_1$ is a distinguished representative. The class $k_2$ is shown to be a distinguished representative, too, by similar computation. In this case, we have

$$ \begin{align*} a_0 = 0,\quad (a_{1,1},\ldots,a_{1,2m}) = (0,\ldots,0,1,0), \quad a_{2,1}=0,\quad a_{3,1}=0,\quad a_{4,1}=m-2 \end{align*} $$

and $(a_0,a_1,a_2,a_3,a_4) = (0,2,0,0,m-2)$ .

Under the adjoint $\lambda \colon H_2(X) \to H^2(X)=\mathrm {Hom}(H_2(X),\mathbb {Z})$ of the intersection form of X, $k_1$ and $k_2$ are respectively the images of

$$ \begin{align*} x_1 &= 2b_0 + 2b_{1,1}+\cdots+2b_{1,2m} + b_{2,1} + b_{3,1}, \\ x_2 &= 4b_1 + 4b_{1,1}+\cdots+4b_{1,2m-1}+2b_{1,2m} + 2b_{2,1} + 2b_{3,1} + 2b_{4,1}. \end{align*} $$

Recall that $\operatorname {Char}(X)\subset H^2(X)$ is identified with $\operatorname {\mathrm {Spin}}^c(X)$ via $c_1$ , and thus for a spin $^c$ structure $[k] \in \operatorname {Char}(X)/2\lambda (H_2(X))$ of Y, we have $c_1([k]) = k|_Y$ . So, $c_1([k]) = 0$ if and only if k lies in the kernel of $H^2(X) \to H^2(Y)$ or, equivalently, k lies in the image of $\lambda \colon H_2(X) \to H^2(X)$ . From this observation, it follows that $c_1([k_i]) = 0$ for $i=1,2$ , because $k_i = \lambda (x_i)$ . Also, $[k_1] \ne [k_2]$ because $x_1-x_2\notin 2H_2(X)$ . Therefore, to complete the proof, it suffices to show $d(Y,[k_i]) \le (2m-1)/4$ for $i=1,2$ .

We have

$$ \begin{align*} k_1^2 = \lambda(x_1,x_1) = -3, \quad k_2^2 = \lambda(x_2,x_2) = -m-4. \end{align*} $$

The last thing we need is the minimum of the values of $\tau (i)$ defined in (4.6). Recall the notation $\Delta _i = \tau (i+1)-\tau (i)$ from Remark 4.7.

Assertion. For both $k_1$ and $k_2$ and for all $i\ge 0$ , $\Delta _i \ge 0$ .

We will provide a proof of the assertion for $m\ge 23$ . For $m<23$ , the assertion is verified by direct inspection using Remark 4.7 (indeed, the author used a computer program), so we omit details for $m<23$ . Note that it suffices to use $m\ge 23$ to prove the main results of this article, Theorems A, B, C and D.

For the case of $k_1$ , by Remark 4.7, we have $\Delta _i\ge 0$ for $i\ge R$ , where

$$ \begin{align*} R=8+(48m-6)/(2m-1)(m-1). \end{align*} $$

Because $m\ge 23$ , $R\le 10$ . So, $\Delta _i\ge 0$ for $i\ge 10$ . For $i\le 10$ , using (4.6), we have

$$ \begin{align*} \Delta_i &= 1 + 2i + \bigg[ \frac{-2mi+1}{2m+1} \bigg] + \bigg[ \frac{-i}{2} \bigg] + \bigg[ \frac{-i+1}{3} \bigg] + \bigg[ \frac{-i+(m-3)/2}{m+1} \bigg] \\ & \ge 1+2i-i+\frac{-i-1}{2} + \frac{-i-1}{3} = \frac{i+1}{6} \ge 0. \end{align*} $$

This shows the claim for $k_1$ . For the case of $k_2$ , we proceed in the same way. We have

$$ \begin{align*} R=7+(48m-6)/(2m-1)(m-1). \end{align*} $$

Because $m\ge 23$ , $R\ge 10$ , and thus $\Delta _i\ge 0$ for $i\ge 10$ by Remark 4.7. For $1\le i\le 10$ , using (4.6), we have

$$ \begin{align*} \Delta_i &= 1 + 2i + \bigg[ \frac{-2mi+1}{2m+1} \bigg] + \bigg[ \frac{-i}{2} \bigg] + \bigg[ \frac{-i}{3} \bigg] + \bigg[ \frac{-i+m-2}{m+1} \bigg] \\ & \ge 1+2i-i+\frac{-i-1}{2} + \frac{-i-2}{3} = \frac{i-1}{6} \ge 0. \end{align*} $$

For $i=0$ , a direct computation using (4.6) gives $\Delta _0=1$ . This completes the proof of the assertion.

From the assertion, it follows that $\min \{\tau (i)\mid i\ge 0\}=0$ because $\tau (0)=0$ . So, by Némethi’s Theorem 4.6,

$$ \begin{align*} d(Y,[k_i]) = \frac{k_i^2 + 2m+4}{4} = \begin{cases} (2m-1)/4 &\text{for } i=1, \\ m/4 &\text{for } i=2. \end{cases} \end{align*} $$

Therefore $d(Y,[k_i]) \le (2m-1)/4$ holds for $i=1,2$ .

Now, to complete the proof of Theorem 2.2, it only remains to prove of Lemma 4.4, which estimates d-invariants of the 3-manifold B in Figure 6. In the proof below, we will use that B is a Seifert 3-manifold as well.

Proof of Lemma 4.4. Recall that B is the 3-manifold described in Figure 6. Let $\mathfrak {s}$ be a spin $^c$ structure of B satisfying $c_1(\mathfrak {s})=0$ . The goal is to show that $d(B,\mathfrak {s})\le (m+2)/4$ .

As we will see in what follows, it will be useful to consider $-B$ instead of B. That is, we will show $d(-B,\mathfrak {s})\ge -(m+2)/4$ . Figure 10 shows that $-B$ is a Seifert 3-manifold. We use the notation of Subsection 4.2. The associated plumbed 4-manifold X has $H_2(X)=\mathbb {Z}^{2m}$ , with basis elements $b_0$ , $b_{1,1},\ldots$ , $b_{1,m}$ , $b_{2,1},\ldots$ , $b_{2,m-2}$ , $b_{3,1}$ . The associated decorations are $e_0=-2$ , $e_{1,j} = -2$ for all j, $e_{2,j}=-2$ for all j and $e_{3,1}=-2m-1$ . Let $b^*_0$ , $b^*_{\ell ,j}$ be the dual basis elements in $H^2(X)$ . Because

$$ \begin{align*} e = -2 + \frac{m}{m+1} + \frac{m-2}{m-1} + \frac{1}{2m+1} = \frac{-3m^2 -2m -1}{(m^2-1)(2m+1)} < 0, \end{align*} $$

the 4-manifold X is negative definite. This is why we use $-B$ instead of B.

Figure 10 Surgery diagram calculus showing that $-B$ is a Seifert 3-manifold. The symbol $(*)$ means handle slides (or Rolfsen twist [41, p. 264]) followed by elimination of a component together with a 0-framed meridian.

The canonical spin $^c$ structure K of X is given by $K=(2m-1)b^*_{3,1}$ . Let

$$ \begin{align*} x_1 &= b_{1,1} + b_{1,3} + \cdots + b_{1,m} + b_{3,1}, \\ x_2 &= b_{2,1} + b_{2,3} + \cdots + b_{2,m-2} + b_{3,1}, \end{align*} $$

and let $k_i = \lambda (x_i) \in H^2(X)$ for $i=1$ , $2$ . Then $k_i \in \operatorname {Char}(X)=K+2H^2(X)$ , so that $[k_i] \in \operatorname {Char}(X)/2\lambda (H_2(X))=\operatorname {\mathrm {Spin}}^c(-B)$ is a spin $^c$ structure of $-B$ . Similar to the proof of Lemma 4.8, $c_1([k_i]) = k_i|_{-B} = 0$ . Also, $[k_1] \ne [k_2]$ in $\operatorname {\mathrm {Spin}}^c(-B)$ because $x_1-x_2 \notin 2H_2(X)$ . It follows that $[k_1]$ and $[k_2]$ are the two spin $^c$ structures of $-B$ with $c_1=0$ . So, it suffices to show that $d(-B,[k_i])\ge -(m+2)/4$ for $i=1$ , $2$ . Instead of determining the values exactly, we will present a simpler argument that gives the promised estimate. We have $k_1^2 = \lambda (x_1,x_1) = -3m-2$ . So, by Ozsváth-Szabó’s inequality [40, Theorem 9.6],

(4.7) $$ \begin{align} d(-B,[k_1]) \ge \frac{k_1^2 + b_2(X)}{4} = \frac{-m-2}{4}. \end{align} $$

Similarly, because $k_2^2 = \lambda (x_2,x_2) = -3m$ , we have

(4.8) $$ \begin{align} d(-B,[k_2]) \ge \frac{k_2^2 + b_2(X)}{4} = \frac{-m}{4}. \end{align} $$

So, we have $d(-B,[k_i]) \ge -(m+2)/4$ for $i=1$ , $2$ . (Indeed, it can be shown that the equality holds in (4.7) and (4.8), by using the technique described in Subsection 4.2, as we did in the proof of Lemma 4.8.)