On the homology of finite abelian coverings of links

J.A.Hillman^{(1)} and M.Sakuma^{(2)}

(1) School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, AUSTRALIA

(2) Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka 560, JAPAN

ABSTRACT

Let A be a finite abelian group and M be a branched cover of an homology 3-sphere, branched over a link L, with covering group A. We show that H_{1}(M; Z[1/|A|]) is determined as a Z[1/|A|][A]-module by the Alexander ideals of L and certain ideal class invariants.

AMS Subject Classification (1991): Primary 57M25. Key words and phrases: Alexander ideal, branched covering, Dedekind domain, knot, link.

Let L : μS^{1} → Σ be a μ-component link in an homology 3-sphere Σ. The exterior of L is X(L) = Σ − N(L), where N(L) is an open regular neighbourhood of the image of L, and the group of L is πL = π_{1}(X(L)). Given an epimorphism ϕ : πL → A, we shall let X_{ϕ}(L) denote the corresponding covering space of X(L); if A is finite M_{ϕ}(L) shall denote the corresponding branched cover of Σ, branched over L. (We shall henceforth assume that A is abelian). The homology groups of finite cyclic covering spaces were among the first invariants used to distinguish knots, and computing these groups for A abelian has remained a problem of continuing interest. In [Sa95] the second author gave precise formulae for the (first) Betti numbers of X_{ϕ}(L) and M_{ϕ}(L). The formulae involved the nullities of intermediate infinite cyclic coverings of sublinks, and thus these Betti numbers are computable from the Alexander ideals of L. In the same paper there are also estimates for the order of the torsion subgroups, which are precise only for very special cases. Here we shall show that if we extend coefficients to invert |A| (the order of A) then these homology groups are determined as modules by the Alexander ideals of L, together with certain Steinitz-Fox-Smythe ideal class invariants. In particular, we may determine the part of the torsion subgroups of order coprime to |A| from the Alexander ideals of L. In the final section we shall consider cyclic branched covers of links, and we shall give examples showing that in general the Alexander polynomials alone do not determine the full torsion subgroup.

§1. Localization away from the order of A

Given a subring R of the field of rational numbers Q, let RΛ_{m} = R[t_{1}, t_{1}^{−1}, … t_{m}, t_{m}^{−1}] and let ε : RΛ_{m} → R be the augmentation homomorphism, determined by ε(t_{i}) = 1 for all i. (If R = Z we write Λ_{m} rather than ZΛ_{m}, and if m = 1 we drop the subscript). If M is a finitely presentable RΛ_{m}-module we shall let Δ_{i}(M) denote the greatest common divisor of the elements of its i^{th} elementary ideal E_{i}(M). Let ζ_{n} be a fixed primitive n^{th} root of unity, with minimal polynomial θ_{n} ∈ Λ. If χ : A → S^{1} is a character of finite order n then R_{χ} = R[ζ_{n}] ≅ RΛ/(θ_{n}) shall denote the ring of algebraic numbers generated by R and the values of χ.

If ϕ : πL → A is an epimorphism H_{∗}(X_{ϕ}(L); R) shall be considered as a module over R[A], and tensor products shall be taken over this ring, unless otherwise indicated. The Alexander module of the link L is the Λ_{μ}-module A(L) = H_{1}(X_{α}(L), ˜∗; Z), where α : πL → Z^{μ} is the abelianization and ˜∗ is the preimage of a basepoint ∗ ∈ X(L) in the maximal abelian covering space X_{α}(L). We shall also let L_{ϕ} denote the sublink of L whose components have meridians mapped nontrivially by ϕ. (Note that if ϕ is trivial then L_{ϕ} is empty and X(L_{ϕ}) = Σ).

In the next four lemmas we shall assume that A is a finite abelian group and that R = Z[1/|A|].

Lemma 1. Let P be a set of characters χ : A → S^{1} such that every subgroup B ≤ A with A/B cyclic is the kernel of exactly one character in P. Then R[A] ≅ ⨁_{χ∈P} R_{χ}.

Proof. For each χ ∈ P let e_{χ} = (|A|)^{−1}Σχ′(a)a, where the sum runs over all a ∈ A and all characters χ′ with Kerχ′ = Kerχ. Then the e_{χ} are mutually orthogonal idempotents and Σ_{χ∈P} e_{χ} = 1. //

This lemma corresponds to Lemma 9.2 of [Sa95].

Lemma 2. Let χ : A → S^{1} be a character. Then

(i) H_{1}(X_{ϕ}(L); Z) ⊗ R_{χ} ≅ H_{1}(X_{χϕ}(L); Z) ⊗ R_{χ}; (ii) H_{1}(M_{ϕ}(L); Z) ⊗ R_{χ} ≅ H_{1}(M_{χϕ}(L_{χϕ}); Z) ⊗ R_{χ}. Proof. Let C_{∗} be the singular chain complex for X_{ϕ}(L), considered as a complex of free Z[A]-modules. Then C_{∗} ⊗ Z[A/kerχ] is the singular chain complex for X_{χϕ}(L) and C_{∗} ⊗ R_{χ} = (C_{∗} ⊗ Z[A/ ker χ]) ⊗ R_{χ}. Since R_{χ} is a direct summand of a localization of Z[A] it is a flat Z[A]-module, and so (i) follows.

The second assertion follows by use of the transfer. (See Chapter III.2 of [Br]). //

This lemma corresponds to step 1 of the proof of Theorem 8.1 of [Sa95].

Lemma 3. Let χ : A → S^{1} be a character. Then

H_{1}(M_{χϕ}(L_{χϕ}); Z) ⊗ R_{χ} ≅ H_{1}(X_{χϕ}(L_{χϕ}); Z) ⊗ R_{χ}.

Proof. The homomorphism from H_{1}(X_{χϕ}(L_{χϕ}); Z)⊗R_{χ} to H_{1}(M_{χϕ}(L_{χϕ}); Z)⊗R_{χ} induced by the inclusion of X_{χϕ}(L_{χϕ}) into M_{χϕ}(L_{χϕ}) is an epimorphism, with kernel generated by lifts of multiples of meridians. Since each meridian of L_{χϕ} has nontrivial image under χϕ each such generator of this kernel is annihilated by t^{d} − 1, for some proper divisor d of the order of χ. But the images of such terms in R_{χ} are invertible, and so the kernel is 0. //

This is essentially the argument of §11 of [Sa95].

Lemma 4. Let χ : A → S^{1} be a character of order n(χ), and let χ̃ : πL → Z be an epimorphism lifting χϕ. Then

(i) H_{1}(X(L); R) ≅ A(L) ⊗ R; (ii) (H_{1}(X_{χϕ}(L); Z) ⊗ R_{χ}) ⊕ R_{χ} ≅ A(L) ⊗ R_{χ} ≅ (A(L) ⊗χ̃ RΛ)/θ_{n(χ)}(A(L) ⊗χ̃ RΛ), if

χ ≠ 1; (iii) A(L)⊗χ̃RΛ ≅ H_{1}(Xχ̃(L); R)⊕RΛ. Moreover H_{1}(Xχ̃(L); R) has a square presentation

matrix as an RΛ-module and χ̃(E_{1}(L)) is a principal ideal, generated by χ̃(Δ_{1}(L)) if μ = 1 and by (t − 1)χ̃(Δ_{1}(L)) if μ > 1.

Proof. The link exterior X(L) is homotopy equivalent to a finite 2-complex with one 0-cell and Euler characteristic 0. Hence the relative homology of covering spaces of the pair (X(L), ∗) may be computed from chain complexes concentrated in degrees 1 and 2, and so H_{1}(X_{ϕ}(L), ˆ∗; R) ≅ A(L) ⊗ R[A] and H_{1}(Xχ̃(L), ˆ∗; R) ≅ A(L) ⊗ RΛ (where ˆ∗ is the preimage of ∗ in X_{ϕ}(L) or Xχ̃(L), respectively). Therefore the first two assertions and the first part of (iii) follow from the exact sequences of homology for such covering spaces, together with the facts that R_{χ} is a flat Z[A]-module and that R ⊗ R_{χ} = 0 unless χ = 1.

The corresponding cellular chain complex for Xχ̃(L) (with coefficients R) is a free RΛ-chain complex C_{∗} with C_{0} ≅ RΛ, C_{1} ≅ RΛ^{a+1} and C_{2} ≅ RΛ^{a}, for some a ≥ 0. Since Im∂_{1} = Kerε = (t − 1) is free of rank 1 and projective RΛ modules are free Ker∂_{1} is free of rank a, and so H_{1}(Xχ̃(L); R) has a square presentation matrix. Since χ̃(E_{1}(L)) = E_{1}(A(L) ⊗χ̃ RΛ) = E_{0}(H_{1}(Xχ̃(L); R)) it is principal.

We may also see this directly. If μ = 1 then E_{1}(L) = (Δ_{1}(L)), and so χ̃(E_{1}(L)) = (χ̃(Δ_{1}(L))). If μ > 1 then E_{1}(L) = (t_{1} − 1, … t_{μ} − 1)(Δ_{1}(L)). (See Theorem IV.3 of [Hi]). We may assume that χ̃ maps the i^{th} meridian to d_{i} times a generator of Z. Since χ̃ is an epimorphism the highest common factor (d_{1}, … d_{μ}) must be 1 and so ((t^{d_{1}} − 1), … (t^{d_{μ}} − 1)) = (t − 1), as ideals in Λ. Hence χ̃(E_{1}(L)) = ((t^{d_{1}} − 1), … (t^{d_{μ}} − 1))(χ̃(Δ_{1}(L))) = (t − 1)(χ̃(Δ_{1}(L))). (Note also that if χ̃(Δ_{1}(L)) ≠ 0 then χ̃(E_{1}(L)) being principal implies that H_{1}(Xχ̃(L); R) has a square presentation matrix, by Theorem III.9 of [Hi]). //

It follows from these lemmas that H_{1}(X_{ϕ}(L); Z[1/|A|]) and H_{1}(M_{ϕ}(L); Z[1/|A|]) are determined as modules over Z[1/|A|][A] by the homology of cyclic covers associated with the characters χϕ and with coefficients Z[χ(A), 1/|A|], and these are essentially direct summands of quotients of the Alexander module. The coefficient rings are Dedekind domains, and the module theory of such rings is well understood.

Lemma 5. Let D be a Dedekind domain and M a finitely generated D-module, with D-torsion submodule T. Then M ≅ T ⊕ (M/T) and M/T is projective of rank r = min{j|E_{j}(M) ≠ 0}. If r > 0 then M/T ≅ D^{r−1} ⊕ ∧_{r}(M/T), and ∧_{r}(M/T) is determined by the Steinitz-Fox-Smythe row ideal class derived from any presentation matrix for M. The torsion submodule T is determined by the elementary ideals of M. Proof. Since M/T is torsion free it is projective, by the remark following Lemma 1.5 of [Mi], and so M ≅ T ⊕ (M/T). The rank of M/T is r = min{j|E_{j}(M/T) ≠ 0}. If r > 0 then (M/T) ≅ D^{r−1} ⊕ J for some ideal J of D, by Theorem 1.6 of [Mi]. It is clear that J ≅ ∧_{r}(M/T) and so the ideal class of J is just the Steinitz-Fox-Smythe row class invariant of M, by Theorem III.12 of [Hi].

If Q is a finitely generated projective complement to M/T with (M/T) ⊕ Q ≅ D^{r+s}, say, then E_{r}(M/T)E_{s}(Q) = (1), and so E_{i}(M/T) = (1) for i ≥ r. Since E_{i}(M) is the ideal generated by ∪_{0≤j≤i}E_{j}(T)E_{i−j}(M/T), it follows that E_{i}(M) = 0 for i < r and E_{r+j}(M) = E_{j}(T) for j ≥ 0. Now since T is a finitely generated torsion D-module its annihilator Ann(T) is nonzero, and so is contained in only finitely many maximal ideals {m_{i}|i ∈ I}, by Proposition I.3.6 of [Se]. The localization D_{S} with respect to the multiplicatively closed set S = D − ∪_{i∈I}m_{i} is a PID, since it is a Dedekind domain with only finitely many maximal ideals. (See the Corollary to Proposition I.3.7 of [Se]). Hence T = T_{S} is determined as an D_{S}-module by its elementary ideals E_{i}(T_{S}) = E_{i}(T)_{S}, which are generated by the images of the elementary ideals of H in D_{S}. //

Remark. Any ideal in a Dedekind domain may be generated by at most two elements. (This follows easily from Lemma 1.10 of [Mi]).

Lemma 6. Let R be a subring of Q and let H be a finitely generated RΛ-module. Then H̄ = H/θ_{n}H is determined as a module by the elementary ideals of H and an ideal class invariant. In particular, the torsion subgroup T has order |T | = |RΛ/(E_{r}(H), θ_{n})|, where r = min{j|E_{j}(H) ⊄ (θ_{n})}, and so |T | is divisible in R by |Res(Δ_{r}(H), θ_{n})|.

Proof. The ring RΛ/(θ_{n}) = R[ζ_{n}] is a Dedekind domain, by Theorem 1.4 of [Mi]. The quotient H̄ is a finitely generated R[ζ_{n}]-module, of rank r = min{j|E_{j}(H) ⊄ (θ_{n})}, and its (Z-)torsion subgroup T is also its R[ζ_{n}]-torsion submodule. Since the elementary ideals of H̄ are just the images of the elementary ideals of H in R[ζ_{n}] the first assertion follows from Lemma 5.

Let S = R[ζ_{n}] − ∪_{i∈I}m_{i} be the complement of the maximal ideals dividing Ann(T). Since T_{S} and R[ζ_{n}]_{S}/E_{0}(T_{S}) are R[ζ_{n}]_{S}-torsion modules they have finite composition series. The simple R[ζ_{n}]_{S}-modules are the quotients R[ζ_{n}]_{S}/m_{iS} ≅ R[ζ_{n}]/m_{i} (for i ∈ I) and the number of simple factors isomorphic to a given simple module R[ζ_{n}]_{S}/m_{iS} in any such composition series for R[ζ_{n}]_{S}/E_{0}(T_{S}) is the same as for T_{S}. Hence |T | = |T_{S}| = |R[ζ_{n}]_{S}/E_{0}(T_{S})| = |R[ζ_{n}]/E_{0}(T)| = |RΛ/(E_{r}(H), ϕ_{n})|.

The final observation is clear, since (E_{r}(H), θ_{n}) ⊆ (Δ_{r}(H), ϕ_{n}).//

If H is a Λ-module with a square presentation matrix P and θ_{n} does not divide Δ_{0}(H) = det P then r = 0 and H/θ_{n}H = T. In this case |H/θ_{n}H| = |Res(Δ_{0}(H), θ_{n})| = |ΠΔ_{0}(ω)|, where the product is taken over all primitive n^{th} roots of unity ω.

Theorem. Let L be a μ-component link in an homology 3-sphere Σ and let ϕ : πL → A be an epimorphism to a finite abelian group. Let R = Z[1/|A|]. Then

(i) H_{1}(X_{ϕ}(L); Z[1/|A|]) ≅ ⨁_{χ∈P} H_{1}(X_{χϕ}(L); Z) ⊗ R_{χ}; (ii) H_{1}(M_{ϕ}(L); Z[1/|A|]) ≅ ⨁_{χ∈P} H_{1}(X_{χϕ}(L_{χϕ}); Z) ⊗ R_{χ}. In particular, H_{1}(X_{ϕ}(L); Z[1/|A|]) and H_{1}(M_{ϕ}(L); Z[1/|A|]) are determined as modules over Z[1/|A|][A] by the Alexander ideals of L together with the Steinitz-Fox-Smythe row class invariants corresponding to characters χ ∈ P such that R_{χ} is not a PID. Proof. The direct sum decompositions follow from Lemmas 1-3, and the further assertions then follow from Lemmas 4 and 6. //

Let null(L; χ) = max{d | χϕ(E_{d}(L)) = 0} and let E(L; χ) = χϕ(E_{null(L;χ)+1}(L)). (Thus E(L; χ) is an ideal in the ring of cyclotomic integers Z_{χ}). Let φ(n) = [Q(ζ_{n}) : Q] be Euler’s totient function.

Corollary. [Sa95]

(i) β_{1}(X_{ϕ}(L)) = Σ_{χ∈P} φ(n(χ))null(L; χ) and β_{1}(M_{ϕ}(L)) = Σ_{χ∈P} φ(n(χ))null(L_{χϕ}; χ); (ii) Up to powers of primes dividing |A| we have |T or(H_{1}(X_{ϕ}(L); Z)| = Π_{χ∈P} |Z_{χ}/E(L; χ)|

and |T or(H_{1}(M_{ϕ}(L); Z)| = Π_{χ∈P} |Z_{χ}/E(L_{χϕ}; χ)|. Moreover if χ has order n and χ̃ : πL → Z is an epimorphism lifting χϕ then |Z_{χ}/E(L; χ)| = |Λ/(χ̃(E(L; χ), θ_{n})| and is divisible by |Res(χ̃(Δ_{null(L;χ)+1}(L)), θ_{n})|. // Note in particular that H_{1}(M_{ϕ}(L); Z) is finite if and only if χϕ(Δ_{1}(L_{χϕ})) ≠ 0 in Q_{χ},

for all χ ∈ P.

The above argument does not work for integral homology, since Z[A] does not decompose as a direct sum of Dedekind domains. However the natural homomorphism from Z[A] to ⨁_{χ∈P} Z_{χ} is injective, and its cokernel F is a finite Z[A]-module with exponent dividing |A|. The long exact sequence of homology derived from the coefficient module sequence 0 → Z[A] → ⨁_{χ∈P} Z_{χ} → F → 0 gives rise to the sequence

… H_{2}(X(L); F) → H_{1}(X_{ϕ}(L); Z) → ⨁_{χ∈P} H_{1}(X_{χϕ}(L); Z) ⊗ Z_{χ} → H_{1}(X(L); F) → 0.

Mayberry and Murasugi give a formula for the order of H_{1}(M_{ϕ}(L); Z), when it is finite, without localization. (See Theorem 10.1 of [MM]). In our terms this formula reads approximately as follows: |H_{1}(M_{ϕ}(L); Z)| = D(ϕ)Π_{χ∈P} |Res(χ̃(Δ_{1}(L_{χϕ})), θ_{n(χ)})|, where D(ϕ) is an integer defined in [MM] which depends only on the homomorphism from Z^{μ} = (πL)^{ab} to A induced by ϕ, and n(χ) is the order of χ.

In [Sa82] it is shown that the order of the cokernel of the natural homomorphism from H_{1}(X_{α}(L); Z) to H_{1}(M_{ϕ}(L); Z) is R(ϕ) = Πn_{i}/|A|, where n_{i} is the order of the image ϕ(t_{i}) of the i^{th} meridian. How is this number related to the term D(ϕ) of [MM]?

§2. Cyclic branched coverings

In this section we shall assume that A ≅ Z/nZ and that ϕ maps each meridian of L to a generator of A. (In the terminology of [MM] the covering is meridian cyclic or ξ-cyclic).

Let ϕ̃ : πL → Z be an epimorphism lifting ϕ, and let ν = (t^{n} − 1)/(t − 1). The coefficient module sequence 0 → Z → Λ/(t^{n} − 1) → Λ/(ν) → 0 gives rise to an exact sequence H_{1}(X(L); Z) → H_{1}(X_{ϕ}(L); Z) → H_{1}(X(L); Λ/(ν)) → 0, where the first map is the transfer. Since ϕ maps each meridian of L to a generator of A the image of the transfer is the submodule generated by the lifts of the n^{th} powers of the meridians and so H_{1}(M_{ϕ}(L); Z) ≅ H_{1}(X(L); Λ/(ν)). On the other hand the latter group may be described as a quotient of the homology of Xϕ̃(L), via the short exact sequence of chain complexes 0 → C_{∗} → C_{∗} → C_{∗}/ν → 0, where C_{∗} is the singular chain complex for Xϕ̃(L). Hence H_{1}(M_{ϕ}(L); Z) ≅ H_{1}(Xϕ̃(L); Z)/νH_{1}(Xϕ̃(L); Z). It is finite if and only if |Res(ϕ̃(Δ_{1}(L)), ν)| ≠ 0, in which case it has order |Res(ϕ̃(Δ_{1}(L)), ν)| (if μ = 1) or n|Res(ϕ̃(Δ_{1}(L)), ν)| (if μ > 1). (See [Sa79], [Sa81]. Note also that these formulae are special cases of the formula of [MM], in the light of their Theorem 4.5).

Suppose that ϕ̃(Δ_{1}(L)) ≠ 0 and let H be the image of H_{1}(X_{α}(L); Z) in H_{1}(Xϕ̃(L); Z). The cokernel of the inclusion of H into H_{1}(Xϕ̃(L); Z) is Z^{μ−1} = (Λ/(t − 1))^{μ−1} (by an iterated Wang sequence argument). Hence H_{1}(M_{ϕ}(L); Z) is an extension of (Z/nZ)^{μ−1} = (Λ/(t − 1, ν))^{μ−1} by H/νH. Since H_{1}(Xϕ̃(L); Z) has a square presentation matrix with nonzero determinant it is a torsion Λ-module of projective dimension ≤ 1. Hence H is also a torsion Λ-module of projective dimension ≤ 1 and so also has a square presentation matrix with nonzero determinant, by Theorem III.9 of [Hi]. If ∇ϕ̃(L) is the latter determinant then ∇ϕ̃(L) = ϕ̃(Δ_{1}(L))/(t − 1)^{μ−1}, by Lemma III.5 of [Hi]. If n = p^{r} is a prime power then Res(∇ϕ̃(L), ν) ≡ ε(∇ϕ̃(L))^{p^{r}−p} modulo (p), and so H/νH is finite of order prime to p if (ε(∇ϕ̃(L)), p) = 1. Moreover if s_{1} < s_{2} then (θ_{p^{s_{1}}} , θ_{p^{s_{2}}} ) = (θ_{p^{s_{1}}} , p). Hence H/νH ≅ ⨁_{s=1}^{r}(H/θ_{p^{s}}H) and Lemma 5 applies to give the structure of these summands.

If ϕ maps all meridians to the same generator of A then ϕ̃ is (up to sign) the total linking number homomorphism and ∇(L) is the Hosokawa polynomial of L. (In the terminology of [MM] the covering is strictly cyclic). In this case the formula for the order of H_{1}(M_{ϕ}(L); Z) was first given in [Fo] for knots in S^{3}, and in [HK] for links in S^{3}.

If L is a knot (i.e., μ = 1) then H = A(L) is the Alexander module of L, ∇(L) = Δ_{1}(L) and ε(∇(L)) = ±1, and H_{1}(X_{ϕ}(L); Z) ≅ H_{1}(M_{ϕ}(L); Z)⊕Z. If we specialize further to the case when n = p is a prime then H_{1}(M_{ϕ}(L); Z) is a finite Z[ζ_{p}]-module of order prime to p. When p = 2 it is easily seen that any finite group of odd order may be realised. (Cyclic groups of odd order may be realized by knots with cyclic knot module; the general case follows on taking connected sums). The modules that arise from knots in this way when p is odd have been determined in [Da]. His description involves consideration of the linking pairing on the torsion of H_{1}(M_{ϕ}(L); Z), on which the covering group acts isometrically, and triviality of an ideal class invariant of the torsion, deriving from the fact that M_{ϕ}(L) is homotopy equivalent to a finite complex on which A acts cellularly. He gives also criteria for the case μ > 1 and n = p an odd prime, modulo p-primary torsion; here the ideal class invariant for the torsion may be nontrivial, but is determined by the Steinitz-Fox-Smythe invariant for the torsion-free part.

If the first homology of such a cyclic branched cover of S^{3} is finite then its order is determined by the Alexander polynomial of the branch set. However the polynomials need not determine the structure. For example, the Alexander ideals for the knots 6_{1} and 9_{46} are E_{1}(6_{1}) = E_{1}(9_{46}) = (2t^{2} − 5t + 2), E_{2}(6_{1}) = (1), E_{2}(9_{46}) = (3, t + 1) and E_{j}(6_{1}) = E_{j}(9_{46}) = (1) for all j ≥ 3. Hence these knots each have first Alexander polynomial 2t^{2} −5t+2 and higher Alexander polynomials 1, but the first homology groups of the 2-fold branched covers are Z/9Z and (Z/3Z)^{2}, respectively.

Weber has given examples to show that in general the Alexander polynomials alone do not even determine the prime divisors of the order of the torsion of H_{1}(M_{n}(K); Z[1/n]), if this group is infinite [We]. It remains open to what extent the Alexander ideals determine the torsion. Acknowledgement. This work was done while the second author was visiting the Universities of Melbourne and Sydney. He would like to thank both universities for their kind hospitality.

References

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[Da] Davis, J.F. The homology of cyclic branched covers, Math. Ann. 301 (1995), 507-518.

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[Hi] Hillman, J.A. Alexander Ideals of Links, Lecture Notes in Mathematics 895, Springer-Verlag, Berlin - Heidelberg - New York (1981).

[HK] Hosokawa, F. and Kinoshita, S. On the homology group of branched cyclic covering spaces of links, Osaka J. Math. 12 (1960), 331-355.

[MM] Mayberry, J.P. and Murasugi, K. Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982), 143-173.

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Annals of Mathematics Study 72, Princeton University Press, Princeton (1971).

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[Sa81] Sakuma, M. On the polynomials of periodic links, Math. Ann 257 (1981), 487-494.

[Sa82] Sakuma, M. On regular coverings of links, Math. Ann. 260 (1982), 303-315.

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Canad. J. Math. 47 (1995), 201-224.

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In general if the first homology is infinite then even the order of the torsion is not determined by the Alexander polynomials alone. (The first such examples were given in [We]). The knots 3_{1}♯6_{1} and 8_{11} each have first Alexander polynomial (t^{2} −t+1)(2t^{2} −5t+ 2) = θ_{6}(2t^{2}−5t+2) and higher polynomials 1. But the first homology groups of their 6-fold cyclic branched covers are Z^{2} ⊕ (Z/9Z) ⊕ (Z/7Z)^{2} ⊕ (Z/37Z) and Z^{2} ⊕ (Z/9Z) ⊕ (Z/7Z)^{2}, respectively. The first elementary ideals of these knots are both (θ_{6}(2t^{2} − 5t + 2)) but E_{2}(3_{1}♯6_{1}) = (θ_{6}, 2t^{2} − 5t + 2) and E_{2}(8_{11}) = (1). By our main theorem, we can determine the part of the torsion subgroups of order prime to 6 from the Alexander ideals.

In fact even the Alexander ideals do not determine the order of the torsion subgroup in general. However the only counterexamples we have found thus far lie beyond the standard tables. Let ψ = 13t^{2} − 25t + 13, Δ = θ_{6}ψ and S = Λ/Δ. The maximal ideal (3, t + 1) < Λ contains Δ. The image J of this ideal in S is a maximal ideal and S/J ≅ Z/3Z. The Λ-modules S and J are finitely generated, and t − 1 acts invertibly on each, since ε(Δ) = 1. Since these modules also admit Blanchfield pairings (for instance those defined by b_{S}([f], [g]) = fḡ/Δ ∈ Q(t)/Λ for all f, g ∈ Λ and b_{J}([f], [g]) = fḡ/3Δ ∈ Q(t)/Λ for all f, g ∈ (3, t + 1), respectively) they may be realized by classical knots K_{S} and K_{J}, say, by Theorem 8.3 of [Ke]. The Alexander ideals of these knots are the same, since E_{0}(S) = E_{0}(J) = (Δ) and E_{i}(S) = E_{i}(J) = (1) for all i > 0.

Let ν = (t^{6} −1)/(t−1) = θ_{2}θ_{3}θ_{6}. Applying the functor −⊗_{S} (S/νS) to the short exact sequence 0 → J → S → S/J → 0 we obtain the exact sequence 0 → T or_{1}^{S}(S/J, S/νS) → J/νJ → S/νS → S/(J, νS) → 0, in which the first and last terms have order 3. Now the torsion subgroup of S/νS is θ_{6}S/νS (≅ S/(ψ, θ_{2}θ_{3})), which maps trivially to S/(J, νS). Hence |T ors(J/νJ)| = 3|T ors(S/νS)|. Thus the torsion subgroups of the homology of the 6-fold cyclic branched covers of these knots are distinct, despite the Alexander ideals being the same.

(Note also that J Mod ψ is a nonprincipal maximal ideal in S/ψS ≅ Z[1/13, \sqrt{−51}]. This was essentially the original example of a nontrivial Steinitz-Fox-Smythe invariant in knot theory [FS]).