Seiberg-Witten invariants and surface singularities

András Némethi Liviu I. Nicolaescu

Department of Mathematics Ohio State University Columbus, OH 43210 and Department of Mathematics University of Notre Dame Notre Dame, IN 46556

      Email: nemethi@math.ohio-state.edu and nicolaescu.1@nd.edu
URL: http://www.math.ohio-state.edu/~nemethi/ and http://www.nd.edu/~lnicolae/

Abstract

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally elliptic (including the cyclic quotient and “polygonal”) singularities, and Brieskorn-Hamm complete intersections. Some of the verifications are based on a result which describes (in terms of the plumbing graph) the Reidemeister-Turaev sign refined torsion (or, equivalently, the Seiberg-Witten invariant) of a rational homology 3-manifold M, provided that M is given by a negative definite plumbing.

These results extend previous work of Artin, Laufer and S. S.-T. Yau, respectively of Fintushel-Stern and Neumann-Wahl.

AMS Classification numbers Primary: 14B05, 14J17, 32S25, 57M27, 57R57.

Secondary: 14E15, 32S55, 57M25

Keywords: (links of) surface singularities, (ℚ-)Gorenstein singularities, rational singularities, Brieskorn-Hamm complete intersections, geometric genus, Seiberg-Witten invariants of ℚ-homology spheres, Reidemeister-Turaev torsion, Casson-Walker invariant

1 Introduction

The main goal of the present paper is to formulate a very general conjecture which relates the topological and the analytical invariants of a complex normal surface singularity whose link is a rational homology sphere.

The motivation for such a result comes from several directions. Before we present some of them, we fix some notations.

Let (X, 0) be a normal two-dimensional analytic singularity. It is well-known that from a topological point of view, it is completely characterized by its link M, which is an oriented 3-manifold. Moreover, by a result of Neumann [33], any decorated resolution graph of (X, 0) carries the same information as M. A property of (X, 0) will be called topological if it can be determined from M, or equivalently, from any resolution graph of (X, 0).

It is interesting to investigate, in which cases (some of) the analytical invariants (determined, say, from the local algebra of (X, 0)) are topological. In this article we are mainly interested in the geometric genus p_{g} of (X, 0) (for details, see section 4).

Moreover, if (X, 0) has a smoothing with Milnor fiber F, then one can ask the same question about the signature σ(F) and the topological Euler characteristic χ_{top}(F) of F as well. It is known (via some results of Laufer, Durfee, Wahl and Steenbrink) that for Gorenstein singularities, any of p_{g}, σ(F) and χ_{top}(F) determines the remaining two modulo a certain invariant K^{2} + #V of the link M. Here K is the canonical divisor, and #V is the number of irreducible components of the exceptional divisor of the resolution. We want to point out that this invariant coincides with an invariant introduced by Gompf in [14] (see Remark 4.8).

The above program has a long history. M. Artin proved in [3, 4] that the rational singularities (i.e. p_{g} = 0) can be characterized completely from the graph (and he computed even the multiplicity and embedding dimension of these singularities from the graph). In [21], H. Laufer extended these results to minimally elliptic singularities. Additionally, he noticed that the program breaks for more complicated singularities (for details, see also section 4). On the other hand, the first author noticed in [28] that Laufer’s counterexamples do not signal the end of the program. He conjectured that if we restrict ourselves to the case of those Gorenstein singularities whose links are rational homology spheres then some numerical analytical invariants (including p_{g}) are topological. This was carried out explicitly for elliptic singularities in [28].

On the other hand, in the literature there is no “good” topological candidate for p_{g} in the very general case. In fact, we are searching for a “good” topological upper bound in the following sense. We want a topological upper bound for p_{g} for any normal surface singularity, which, additionally, is optimal in the sense that for Gorenstein singularities it yields exactly p_{g}. E.g., such a “good” topological upper bound for elliptic singularities is the length of the elliptic sequence, introduced and studied by S. S.-T. Yau (see, e.g. [53]) and Laufer.

In fact, there are some other particular cases too, when a possible candidate is present in the literature. Fintushel and Stern proved in [10] that for a hypersurface Brieskorn singularity whose link is an integral homology sphere, the Casson invariant λ(M) of the link M equals σ(F)/8 (hence, by the mentioned correspondence, it determines p_{g} as well). This fact was generalized by Neumann and Wahl in [35]. They proved the same statement for all Brieskorn-Hamm complete intersections and suspensions of plane curve singularities (with the same assumption, that the link is an integral homology sphere). Moreover, they conjectured the validity of the formula for any isolated complete intersection singularity (with the same restriction about the link). For some other relevant conjectures the reader can also consult [36].

The result of Neumann-Wahl [35] was reproved and reinterpreted by Collin and Saveliev (see [7] and [8]) using equivariant Casson invariant and cyclic covering techniques. But still, a possible generalization for rational homology sphere links remained open. It is important to notice that the “obvious” generalization of the above identity for rational homology spheres, namely to expect that σ(F)/8 equals the Casson-Walker invariant of the link, completely fails.

In fact, our next conjecture states that one has to replace the Casson invariant λ(M) by a certain Seiberg-Witten invariant of the link, i.e. by the difference of a certain Reidemeister-Turaev sign-refined torsion invariant and the Casson-Walker invariant (the sign-change is motivated by some sign-conventions already used in the literature).

We recall (for details, see section 2 and 3) that the Seiberg-Witten invariants associates to any spin^{c} structure σ of M a rational number sw_{M}^{0}(σ). In order to formulate our conjecture, we need to fix a “canonical” spin^{c} structure σ_{can} of M. This can be done as follows. The (almost) complex structure on X \ {0} induces a natural spin^{c} structure on X \ {0}. Its restriction to M is, by definition, σ_{can}. The point is that this structure depends only on the topology of M alone.

In fact, the spin^{c} structures correspond in a natural way to quadratic functions associated with the linking form of M; by this correspondence σ_{can} corresponds to the quadratic function −q_{LW} constructed by Looijenga and Wahl in [25].

We are now ready to state our conjecture.

Main Conjecture. Assume that (X, 0) is a normal surface singularity whose link M is a rational homology sphere. Let σ_{can} be the canonical spin^{c} structure on M. Then, conjecturally, the following facts hold.

(1) For any (X, 0), there is a topological upper bound for p_{g} given by:

sw_{M}^{0}(σ_{can}) − \frac{K^{2} + #V}{8} ≥ p_{g}.

(2) If (X, 0) is ℚ-Gorenstein, then in (1) one has equality.

(3) In particular, if (X , 0) is a smoothing of a Gorenstein singularity (X, 0) with Milnor fiber F, then

−sw_{M}^{0}(σ_{can}) = \frac{σ(F)}{8}.

If (X, 0) is numerically Gorenstein and M is a ℤ_{2}–homology sphere then σ_{can} is the unique spin structure of M; if M is an integral homology sphere then in the above formulae −sw_{M}^{0}(σ_{can}) = λ(M), the Casson invariant of M.

In the above Conjecture, we have automatically built in the following statements as well.

(a) For any normal singularity (X, 0) the topological invariant

sw_{M}^{0}(σ_{can}) − \frac{K^{2} + #V}{8}

is non-negative. Moreover, this topological invariant is zero if and only if (X, 0) is rational. This provides a new topological characterization of the rational singularities.

(b) Assume that (X, 0) (equivalently, the link) is numerically Gorenstein. Then the above topological invariant is 1 if and only if (X, 0) is minimally elliptic (in the sense of Laufer). Again, this is a new topological characterization of minimally elliptic singularities.

In this paper we will present evidence in support of the conjecture in the form of explicit verifications. The computations are rather arithmetical, involving nontrivial identities about generalized Fourier-Dedekind sums. For the reader’s convenience, we have included a list of basic properties of the Dedekind sums in Appendix B.

In general it is not easy to compute the Seiberg-Witten invariant. In our examples we use two different approaches. First, the (modified) Seiberg-Witten invariant is the sum of the Kreck-Stolz invariant and the number of certain monopoles [6, 24, 26]. On the other hand, by a result of the second author, it can also be computed as the difference of the Reidemeister-Turaev torsion and the Casson-Walker invariant [41] (for more details, see section 3). Both methods have their advantages and difficulties. The first method is rather explicit when M is a Seifert manifold (thanks to the results of the second author in [38], cf. also with [27]), but frequently the corresponding Morse function will be degenerate. Using the second method, the computation of the Reidemeister-Turaev torsion leads very often to complicated Fourier-Dedekind sums.

In section 5 we present some formulae for the needed invariant in terms of the plumbing graph. The formula for the Casson-Walker invariant was proved by Ratiu in his thesis [45], and can be deduced from Lescop’s surgery formulae as well [23]. Moreover, we also provide a similar formula for the invariant K^{2}+#V (which generalizes the corresponding formula already known for cyclic quotient singularities by Hirzebruch, see also [18, 25]). The most important result of this section describes the Reidemeister-Turaev torsion (associated with any spin^{c} structure) in terms of the plumbing graph. The proof is partially based on Turaev’s surgery formulae [49] and the structure result [48, Thm. 4.2.1]. We have deferred it to Appendix A.

In our examples we did not try to force the verification of the conjecture in the largest generality possible, but we tried to supply a rich and convincing variety of examples which cover different aspects and cases.

In order to eliminate any confusion about different notations and conventions in the literature, in most of the cases we provide our working definitions.

2 The link and its canonical spin^{c} structure

2.1 Definitions. Let (X, 0) be a normal surface singularity embedded in (ℂ^{N}, 0). Then for ϵ sufficiently small the intersection M := X ∩ S_{ϵ}^{2N−1} of a representative X of the germ with the sphere S_{ϵ}^{2N−1} (of radius ϵ) is a compact oriented 3-manifold, whose oriented C^{∞} type does not depend on the choice of the embedding and ϵ. It is called the link of (X, 0).

In this article we will assume that M is a rational homology sphere, and we write H := H_{1}(M, ℤ). By Poincaré duality H can be identified with H^{2}(M, ℤ).

It is well-known that M carries a symmetric non-singular bilinear form

b_{M} : H × H → ℚ/ℤ

called the linking form of M. If [v_{1}] and [v_{2}] ∈ H are represented by the 1cycles v_{1} and v_{2}, and for some integer n one has nv_{1} = ∂w, then b_{M}([v_{1}], [v_{2}]) = (w · v_{2})/n (mod ℤ).

2.2 The linking form as discriminant form. We briefly recall the definition of the discriminant form. Assume that L is a finitely generated free Abelian group with a symmetric bilinear form ( , ) : L × L → ℤ. Set L′ := Hom_{ℤ}(L, ℤ). Then there is a natural homomorphism i_{L} : L → L′ given by x ↦ (x, ·) and a natural extension of the form ( , ) to a rational bilinear form ( , )_{ℚ} : L′ × L′ → ℚ. If d_{1}, d_{2} ∈ L′ and nd_{j} = i_{L}(e_{j}) (j = 1, 2) for some integer n, then (d_{1}, d_{2})_{ℚ} = d_{2}(e_{1})/n = (e_{1}, e_{2})/n^{2}.

If L is non-degenerate (i.e. i_{L} is a monomorphism) then one defines the discriminant space D(L) by coker(i_{L}). In this case there is a discriminant bilinear form

b_{D(L)} : D(L) × D(L) → ℚ/ℤ

defined by b_{D(L)}([d_{1}], [d_{2}]) = (d_{1}, d_{2})_{ℚ} (mod ℤ).

Assume that M is the boundary of a oriented 4-manifold N with H_{1}(N, ℤ) = 0 and H_{2}(N, ℤ) torsion-free. Let L be the intersection lattice (H_{2}(N, ℤ), ( , )). Then L′ can be identified with H_{2}(N, ∂N, ℤ) and one has the exact sequence L → L′ → H → 0. The fact that M is a rational homology sphere implies that L is non-degenerate. Moreover (H, b_{M}) = (D(L), −b_{D(L)}). Sometimes it is also convenient to regard L′ as H^{2}(N, ℤ) (identification by Poincaré duality).

2.3 Quadratic functions and forms associated with b_{M}. A map q : H → ℚ/ℤ is called quadratic function if b(x, y) = q(x + y) − q(x) − q(y) is a bilinear form on H × H. If in addition q(nx) = n^{2}q(x) for any x ∈ H and n ∈ ℤ then q is called quadratic form. In this case we say that the quadratic function, respectively form, is associated with b. Quadratic forms are also called quadratic refinements of the bilinear form b.

In the case of the link M, we denote by Q^{c}(M) (respectively by Q(M)) the set of quadratic functions (resp. forms) associated with b_{M}. Obviously, there is a natural inclusion Q(M) ⊂ Q^{c}(M).

The set Q(M) is non-empty. It is a G := H^{1}(M, ℤ_{2}) torsor, i.e. G acts freely and transitively on Q(M). The action can be easily described if we identify G with Hom(H, ℤ_{2}) and we regard ℤ_{2} as (\frac{1}{2}ℤ)/ℤ ⊂ ℚ/ℤ. Then, the difference of any two quadratic refinements of b_{M} is an element of G, which provides a natural action G × Q(M) → Q(M) given by (χ, q) ↦ χ + q.

Similarly, the set Q^{c}(M) is non-empty and it is a Ĥ = Hom(H, ℚ/ℤ) torsor. The free and transitive action Ĥ × Q^{c}(M) → Q^{c}(M) is given by the same formula (χ, q) ↦ χ + q. In particular, the inclusion Q(M) ⊂ Q^{c}(M) is Gequivariant via the natural monomorphism G ↪ Ĥ. We prefer to replace the Ĥ action on Q^{c}(M) by an action of H. This action H × Q^{c}(M) → Q^{c}(M) is defined by (h, q) ↦ q + b_{M}(h, ·). Then the natural monomorphism G ↪ Ĥ is replaced by the Bockstein-homomorphism G = H^{1}(M, ℤ_{2}) → H^{2}(M, ℤ) = H. In the sequel we consider Q^{c}(M) with this H -action.

Quadratic functions appear in a natural way. In order to see this, let N be as in 2.2. Pick a characteristic element, that is an element k ∈ L′, so that (x, x) + k(x) ∈ 2ℤ for any x ∈ L. Then for any d ∈ L′ with class [d] ∈ H, define

q_{D(L),k}([d]) := \frac{1}{2} (d + k, d)_{ℚ} (mod ℤ).

Then −q_{D(L),k} is a quadratic function associated with b_{M} = −b_{D(L)}. If in addition k ∈ Im(i_{L}), then −q_{D(L),k} is a quadratic refinement of b_{M}.

There are two important examples to consider.

First assume that N is an almost-complex manifold, i.e. its tangent bundle T N carries an almost complex structure. By Wu formula, k = −c_{1}(T N) ∈ L′ is a characteristic element. Hence −q_{D(L),k} is a quadratic function associated with b_{M}. If c_{1}(T N) ∈ Im(i_{L}) then we obtain a quadratic refinement.

Next, assume that N carries a spin structure. Then w_{2}(N) vanishes, hence by Wu formula ( , ) is an even form. Then one can take k = 0, and −q_{D(L),0} is a quadratic refinement of b_{M}.

2.4 The spin structures of M. The 3-manifold M is always spinnable. The set Spin(M) of the possible spin structures of M is a G-torsor. In fact, there is a natural (equivariant) identification of 𝔮 : Spin(M) → Q(M).

In order to see this, fix a spin structure ϵ ∈ Spin(M). Then there exists a simple connected oriented spin 4-manifold N with ∂N = M whose induced spin structure on M is exactly ϵ (see, e.g. [15, 5.7.14]). Set L = (H_{2}(N, ℤ), ( , )). Then the quadratic refinement −q_{D(L),0} (cf. 2.3) of b_{M} depends only on the spin structure ϵ and not on the particular choice of N. The correspondence ϵ ↦ −q_{D(L),0} determines the identification 𝔮 mentioned above.

2.5 The spin^{c} structures on M. We denote by Spin^{c}(M) the space of isomorphism classes of spin^{c} structures on M. Spin^{c}(M) is in a natural way a H = H^{2}(M, ℤ)-torsor. We denote this action H × Spin^{c}(M) → Spin^{c}(M) by (h, σ) ↦ h · σ. For every σ ∈ Spin^{c}(M) we denote by 𝕊_{σ} the associated bundle of complex spinors, and by det σ the associated line bundle, det σ := det 𝕊_{σ}. We set c(σ) := c_{1}(𝕊_{σ}) ∈ H. Note that c(h · σ) = 2h + c(σ).

Spin^{c}(M) is equipped with a natural involution σ ⟷ σ̄ such that

c(σ̄) = −c(σ) and \overline{h · σ} = (−h) · σ̄.

There is a natural injection ϵ ↦ σ(ϵ) of Spin(M) into Spin^{c}(M). The image of Spin(M) in Spin^{c}(M) is

{σ ∈ Spin^{c}(M); c(σ) = 0} = {σ ∈ Spin^{c}(M); σ = σ̄}.

Consider now a 4-manifold N with lattices L and L′ as in 2.2. We prefer to write L′ = H^{2}(N, ℤ), and denote by d ↦ [d] the restriction map L′ = H^{2}(N, ℤ) → H^{2}(M, ℤ) = H.

Then N is automatically a spin^{c} manifold. In fact, the set of spin^{c} structures on N is parametrized by the set of characteristic elements

C_{N} := {k ∈ L′ : k(x) + (x, x) ∈ 2ℤ for all x ∈ L}

via σ̃ ↦ c(σ̃) ∈ C_{N} (see. e.g. [15, 2.4.16]). The set Spin^{c}(N) is an L′ torsor with action (d, σ̃) ↦ d · σ̃. Let r : Spin^{c}(N) → Spin^{c}(M) be the restriction. Then r(d · σ̃) = [d] · r(σ̃) and c(r(σ̃)) = [c(σ̃)].

Moreover, notice that r(σ̃) = r(d · σ̃) if and only if [d] = 0, i.e. d ∈ L. If this is happening then c(σ̃) − c(d · σ̃) ∈ 2L.

2.6 Lemma There is a canonical H -equivariant identification

𝔮^{c} : Spin^{c}(M) → Q^{c}(M).

Moreover, this identification is compatible with the G-equivariant identification 𝔮 : Spin(M) → Q(M) via the inclusions Spin(M) ⊂ Spin^{c}(M) and Q(M) ⊂ Q^{c}(M).

Proof Let N be as above. We first show that r is onto. Indeed, take any σ̃ ∈ Spin^{c}(N) with restriction σ ∈ Spin^{c}(M). Then all the elements in the H -orbit of σ are induced structures. But this orbit is the whole set. Next, define for any σ̃ corresponding to k = c(σ̃) the quadratic function q_{D(L),k}. Then r(σ̃) = r(d · σ̃) if and only if d ∈ L. This means that c(σ̃) − c(d · σ̃) ∈ 2L hence c(σ̃) and c(d · σ̃) induce the same quadratic function. Hence 𝔮^{c}(r(σ̃)) := −q_{D(L),c(σ)} is well-defined. Finally, notice that 𝔮^{c} does not depend on the choice of N, fact which shows its compatibility with 𝔮 as well (by taking convenient spaces N).

2.7 M as a plumbing manifold. Fix a sufficiently small (Stein) representative X of (X, 0) and let π : X̃ → X be a resolution of the singular point 0 ∈ X. In particular, X̃ is smooth, and π is a biholomorphic isomorphism above X \ {0}. We will assume that the exceptional divisor E := π^{−1}(0) is a normal crossing divisor with irreducible components {E_{v}}_{v∈V}. Let Γ(π) be the dual resolution graph associated with π decorated with the self intersection numbers {(E_{v}, E_{v})}_{v}. Since M is a rational homology sphere, all the irreducible components E_{v} of E are rational, and Γ(π) is a tree.

It is clear that H_{1}(X̃, ℤ) = 0 and H_{2}(X̃, ℤ) is freely generated by the fundamental classes {[E_{v}]}_{v}. Let I be the intersection matrix {(E_{v}, E_{w})}_{v,w}. Since π identifies ∂X̃ with M, the results from 2.2 can be applied. In particular, H = coker(I) and b_{M} = −b_{D(I)}. The matrix I is negative definite.

The graph Γ(π) can be identified with a plumbing graph, and M can be considered as an S^{1}-plumbing manifold whose plumbing graph is Γ(π). In particular, any resolution graph Γ(π) determines the oriented 3-manifold M completely.

We say that two plumbing graphs (with negative definite intersection forms) are equivalent if one of them can be obtained from the other by a finite sequence of blowups and/or blowdowns along rational (−1)-curves. Obviously, for a given (X, 0), the resolution π, hence the graph Γ(π) too, is not unique. But different resolutions provide equivalent graphs. By a result of W. Neumann [33], the oriented diffeomorphism type of M determines completely the equivalence class of Γ(π). In particular, any invariant defined from the resolution graph Γ(π) (which is constant in its equivalence class) is, in fact, an invariant of the oriented C^{∞} 3-manifold M. This fact will be crucial in the next discussions.

Now, we fix a resolution π as above and identify M = ∂X̃. Let K be the canonical class (in P ic(X̃)) of X̃. By the adjunction formula,

−K · E_{v} = E_{v} · E_{v} + 2

for any v ∈ V. In fact, K at homological level provides an element kX̃ ∈ L′ which has the obvious property

−kX̃([E_{v}]) = ([E_{v}], [E_{v}]) + 2 for any v ∈ V .

Since the matrix I is non-degenerate, this defines kX̃ uniquely.

−kX̃ is known in the literature as the canonical (rational) cycle of (X, 0) associated with the resolution π. More precisely, let Z_{K} = ∑_{v∈V} r_{v}E_{v}, r_{v} ∈ ℚ, be a rational cycle supported by the exceptional divisor E, defined by

Z_{K} · E_{v} = −K · E_{v} = E_{v} · E_{v} + 2 for any v ∈ V .  (∗)

Then the above linear system has a unique solution, and ∑_{v} r_{v}[E_{v}] ∈ L ⊗ ℚ can be identified with (i_{L} ⊗ ℚ)^{−1}(−kX̃).

It is clear that −kX̃ ∈ Im(i_{L}) if and only if all the coefficients {r_{v}}_{v} of Z_{K} are integers. In this case the singularity (X, 0) is called numerically Gorenstein (and we will also say that “M is numerically Gorenstein”).

In particular, for any normal singularity (X, 0), the resolution π provides a quadratic function −q_{D(I),kX̃} associated with b_{M}, which is a quadratic form if and only if (X, 0) is numerically Gorenstein.

2.8 The universal property of q_{D(I),kX̃}. In [25], Looijenga and Wahl define a quadratic function q_{LW} (denoted by q in [25]) associated with b_{M} from the almost complex structure of the bundle T M ⊕ ℝ_{M} (where T M is the tangent bundle and ℝ_{M} is the trivial bundle of M). By the main universal property of q_{LW} (see [loc. cit.], Theorem 3.7) (and from the fact that any resolution π induces the same almost complex structure on T M ⊕ ℝ_{M}) one gets that for any π as in 2.7, the identity q_{LW} = −q_{D(I),kX̃} is valid. This shows that q_{D(I),kX̃} does not depend on the choice of the resolution π.

This fact can be verified by elementary computation as well: one can prove that q_{D(I),kX̃} is stable with respect to a blow up (of points of E).

2.9 The “canonical” spin^{c} structure of a singularity link. Assume that M is the link of (X, 0). Fix a resolution π : X̃ → X as in 2.7. Then π determines a “canonical” quadratic function q_{can} := −q_{D(I),kX̃} associated with b_{M} which does not depend on the choice of π (cf. 2.8). Then the natural identification 𝔮^{c} : Spin^{c}(M) → Q^{c}(M) (cf. 2.3) provides a well-defined spin^{c} structure (𝔮^{c})^{−1}(q_{can}). Then the “canonical” spin^{c} structure σ_{can} on M is (𝔮^{c})^{−1}(q_{can}) modified by the natural involution of Spin^{c}(M). In particular, c(σ_{can}) = −[kX̃] ∈ H. (Equivalently, σ_{can} is the restriction to M of the spin^{c} structure given by the characteristic element −kX̃ ∈ CX̃.) If (X, 0) is numerically Gorenstein then σ_{can} is a spin structure. In this case we will use the notation ϵ_{can} = σ_{can} as well.

We want to emphasize (again) that σ_{can} depends only on the oriented C^{∞} type of M (cf. also with 2.11). Indeed, one can construct q_{can} as follows. Fix an arbitrary plumbing graph Γ of M with negative definite intersection form (lattice) L. Then determine Z_{K} by 2.7(∗), and take

q_{can}([d]) := −\frac{1}{2}(d − [Z_{K}], d)_{ℚ} (mod ℤ).

Then q_{can} does not depend on the choice of Γ.

It is remarkable that this construction provides an “origin” of the torsor space Spin^{c}(M).

2.10 Compatibility with the (almost) complex structure As we have already mentioned in 2.8, the result of Looijenga and Wahl [25] implies the following: the almost complex structure on X \ {0} determines a spin^{c} structure, whose restriction to M is σ_{can}. Similarly, if π : X̃ → X is a resolution, then the almost complex structure on X̃ gives a spin^{c} structure σX̃ on X̃, whose restriction to M is σ_{can}. Here we would like to add the following discussion. Assume that the intersection form ( , )X̃ is even, hence X̃ has a unique spin structure ϵX̃. The point is that, in general, σX̃ ≠ ϵX̃, and their restrictions can be different as well, even if the restriction of σX̃ is spin.

More precisely: ( , )X̃ is even if and only if kX̃ ∈ 2L′; r(σX̃) ∈ Spin(M) if and only if kX̃ ∈ L; and finally, r(σX̃) = r(ϵX̃) if and only if kX̃ ∈ 2L.

2.11 Remarks. 1) In fact, by the classification theorem of plumbing graphs given by Neumann [33], if M is a rational homology sphere which is not a lens space, then already π_{1}(M) (i.e. the homotopy type of M) determines its orientation class and its canonical spin^{c} structure. Indeed, if one wants to recover the oriented C^{∞} type of M from its fundamental group, then by Neumann’s result the only ambiguity appears for cusp singularities (which are not rational homology spheres) and for cyclic quotient singularities. The links of cyclic quotient singularities are exactly the lens spaces. In fact, if we assume the numerically Gorenstein assumption, even the lens spaces are classified by their fundamental groups (since they are exactly the du Val A_{p}-singularities).

2) If M is a numerically Gorenstein ℤ_{2}–homology sphere, the definition of ϵ_{can} is obviously simpler: it is the unique spin structure of M. If M is an integral homology sphere then it is automatically numerically Gorenstein, hence the above statement applies.

3) Assume that (X, 0) has a smoothing with Milnor fiber F whose homology group H_{1}(F, ℤ) has no torsion. Then the (almost) complex structure of F provides a spin^{c} structure on F whose restriction to M is exactly σ_{can}. This follows (again) by the universal property of q_{LM} ([25, Theorem 3.7], ; cf. also with 2.8).

Moreover, F has a spin structure if and only if the intersection form ( , ) of F is even (see e.g. [15, 5.7.6]); and in this case, the spin structure is unique. If F is spin, then its spin structure ϵ_{F} coincides with the spin^{c} structure induced by the complex structure (since the canonical bundle of F is trivial). In particular, if F is spin, σ_{can} is the restriction of ϵ_{F}, hence it is spin. This also proves that if (X, 0) has a smoothing with even intersection form and without torsion in H_{1}(F, ℤ), then it is necessarily numerically Gorenstein.

Here is worth noticing that the Milnor fiber of a smoothing of a Gorenstein singularity has even intersection form [46].

4) Clearly, q_{can} depends only on Z_{K} (mod 2ℤ).

2.12 The invariant K^{2} + #V. Fix a resolution π : X̃ → X of (X, 0) as in 2.7, and consider Z_{K} or kX̃. The rational number Z_{K} · Z_{K} = (kX̃, kX̃)_{ℚ} will be denoted by K^{2}. Let #V denote the number of irreducible components of E = π^{−1}(0). Then K^{2} + #V does not depend on the choice of the resolution π. In fact, the discussion in 2.7 and 2.9 shows that it is an invariant of M. Obviously, if (X, 0) is numerically Gorenstein, then K^{2} + #V ∈ ℤ.

2.13 Notations. Let X̃ as above. Let {E_{v}}_{v∈V} be the set of irreducible exceptional divisors and D_{v} a small transversal disc to E_{v}. Then {[E_{v}]}_{v} (resp. {[D_{v}]}_{v}) are the free generators of L = H_{2}(X̃, ℤ) (resp. L′ = H_{2}(X̃, M, ℤ)) with [D_{v}] · [E_{w}] = 1 if v = w and = 0 otherwise. Moreover, g_{v} := [∂D_{v}] (v ∈ V) is a generator set of L′/L = H. In fact ∂D_{v} is a generic fiber of the S^{1}-bundle over E_{v} used in the plumbing construction of M. If I is the intersection matrix defined by the resolution (plumbing) graph, then i_{L} written in the bases {[E_{v}]}_{v} and {[D_{v}]}_{v} is exactly I.

Using this notation, kX̃ ∈ L′ can be expressed as ∑_{v}(−e_{v} − 2)[D_{v}], where e_{v} = E_{v} · E_{v}. For the degree of v (i.e. for #{w : E_{w} · E_{v} = 1}) we will use the notation δ_{v}. Obviously

∑_{v} δ_{v} = −2 × Euler characteristic of the plumbing graph + 2#𝒱 = 2#𝒱 − 2.

Most of the examples considered later are star-shaped graphs. In these cases it is convenient to express the corresponding invariants of the Seifert 3-manifold M in terms of their Seifert invariants. In order to eliminate any confusion about the different notations and conventions in the literature, we list briefly the definitions and some of the needed properties.

2.14 The unnormalized Seifert invariants. Consider a Seifert fibration π : M → Σ. In our situation M is a rational homology sphere and the base space Σ is an S^{2} with genus 0 (and we will not emphasize this fact anymore).

Consider a set of points {x_{i}}_{i=1}^{ν} in such a way that the set of fibers {π^{−1}(x_{i})}_{i} contains the set of singular fibers. Set O_{i} := π^{−1}(x_{i}). Let D_{i} be a small disc in X containing x_{i}, Σ′ := Σ \ ∪_{i}D_{i} and M′ := π^{−1}(Σ′). Now, π : M′ → Σ′ admits sections, let s : Σ′ → M′ be one of them. Let Q_{i} := s(∂D_{i}) and let H_{i} be a circle fiber in π^{−1}(∂D_{i}). Then in H_{1}(π^{−1}(D_{i}), ℤ) one has H_{i} ∼ α_{i}O_{i} and Q_{i} ∼ −β_{i}O_{i} for some integers α_{i} > 0 and β_{i} with (α_{i}, β_{i}) = 1. The set ((α_{i}, β_{i})_{i=1}^{ν}) constitute the set of (unnormalized) Seifert invariants. The number

e := − ∑_{i}(β_{i}/α_{i})

is called the (orbifold) Euler number of M. If M is a link of singularity then e < 0.

Replacing the section by another one, a different choice changes each β_{i} within its residue class modulo α_{i} in such a way that the sum e = − ∑_{i}(β_{i}/α_{i}) is constant.

The elements q_{i} = [Q_{i}] (1 ≤ i ≤ ν) and the class h of the generic fiber H generate the group H = H_{1}(M, ℤ). By the above construction is clear that:

H = ab⟨q_{1}, … q_{ν}, h | q_{1} … q_{ν} = 1, q_{i}^{α_{i}}h^{β_{i}} = 1, for all i⟩.

Let α := lcm(α_{1}, …, α_{ν}). The order of the group H and of the subgroup ⟨h⟩ can be determined by (cf. [32]):

|H| = α_{1} ⋯ α_{ν}|e|, |⟨h⟩| = α|e|.

2.15 The normalized Seifert invariants and plumbing graph. We write

e = b + ∑ ω_{i}/α_{i}

for some integer b, and 0 ≤ ω_{i} < α_{i} with ω_{i} ≡ −β_{i} (mod α_{i}). Clearly, these properties define {ω_{i}}_{i} uniquely. Notice that b ≤ e < 0. For the uniformity of the notations, in the sequel we assume ν ≥ 3.

For each i, consider the continued fraction α_{i}/ω_{i} = b_{i1} − 1/(b_{i2} − 1/(⋯ − 1/b_{iν_{i}}) ⋯). Then (a possible) plumbing graph of M is a star-shaped graph with ν arms. The central vertex has decoration b and the arm corresponding to the index i has ν_{i} vertices, and they are decorated by b_{i1}, …, b_{iν_{i}} (the vertex decorated by b_{i1} is connected by the central vertex).

We will distinguish those vertices v ∈ V of the graph which have δ_{v} ≠ 2. We will denote by v̄_{0} the central vertex (with δ = ν), and by v̄_{i} the end-vertex of the i^{th} arm (with δ = 1) for all 1 ≤ i ≤ ν. In this notation, gv̄_{0} = h, the class of the generic fiber. Moreover, using the plumbing representation of the group H, we have another presentation for H, namely:

H = ab⟨gv̄_{1}, … gv̄_{ν}, h | h^{−b} = ∏_{i=1}^{ν} gω_{i} ; v̄_{i}, h = gα_{i} ; v̄_{i} for all i⟩.

3 Seiberg-Witten invariants of ℚ-homology spheres

In this section we consider an oriented rational homology 3-sphere M. We set H := H^{2}(M, ℤ). When working with the group algebra ℚ[H] of H it is more convenient to use the multiplicative notation for the group operation of H.

3.1 The Seiberg-Witten invariants of M. To describe the Seiberg-Witten invariants we need to fix some additional geometric data belonging to the space of parameters

𝒫 = {(g, η); g = Riemann metric, η = closed two-form}.

For each spin^{c} structure σ on M (cf. 2.5), we have the space of configurations C_{σ} (associated with σ) consisting of pairs C = (ψ, A), where ψ is a section of 𝕊_{σ} and A is a Hermitian connection on det σ. The gauge group 𝒢 := Map (M, S^{1}) acts on C_{σ}. Moreover, it acts freely on the irreducible part

C_{σ}^{irr} = {(ψ, A) ∈ C_{σ}; ψ ≢ 0},

and the quotient ℬ_{σ}^{irr} := C_{σ}^{irr}/𝒢 can be equipped with a structure of Hilbert manifold. Every parameter u = (g, η) ∈ 𝒫 defines a 𝒢-invariant function ℱ_{σ,u} : C_{σ} → ℝ whose critical points are called the (σ, g, η)-Seiberg-Witten monopoles. In particular, ℱ_{σ,u} descends to a smooth function [ℱ_{σ,u}] : ℬ_{σ}^{irr} → ℝ. We denote by 𝔐_{σ,u}^{irr} its critical set.

The first Chern class c(σ) of 𝕊_{σ} is a torsion element of H^{2}(M, ℤ), and thus the curvature of any connection on det σ is an exact 2-form. In particular we can find an unique 𝒢-equivalence class of connections A on det σ with the property that

F_{A} = iη.  (†)

Using the metric g on M (which is part of our parameter u) and a connection A_{u} satisfying (†), we obtain a spin^{c}-Dirac operator 𝔇_{A_{u}}. To define the Seiberg-Witten invariants we need to work with good parameters, i.e. parameters u such that the following two things happen.

• The Dirac operator 𝔇_{A_{u}} is invertible.

• The function [ℱ_{σ,u}] is Morse, and 𝔐_{σ,u} consists of finitely many points.

The space of good parameters is generic. Fix such a good parameter u. Then each critical point has a well defined ℤ_{2}-valued Morse index

m : 𝔐_{σ,u}^{irr} → {±1}

and we set

sw_{M}(σ, u) = ∑_{x∈𝔐_{σ,u}^{irr}} m(x) ∈ ℤ.

This integer depends on the choice of the parameter u and thus it is not a topological invariant. To obtain an invariant we need to alter this monopole count.

The eta invariant of 𝔇_{A_{u}} depends only on the gauge equivalence class of A_{u}, and we will denote it by η_{dir}(σ, u). The metric g defines an odd signature operator on M whose eta invariant we denote by η_{sign}(u). Now define the Kreck-Stolz invariant associated with the data (σ, u) by

KS_{M}(σ, u) := 4η_{dir}(σ, u) + η_{sign}(u) ∈ ℚ.

We have the following result.

3.2 Theorem. [6, 24, 26] The rational number

\frac{1}{8}KS_{M}(σ, u) + sw_{M}(σ, u)

is independent of u and thus it is a topological invariant of the pair (M, σ). We denote this number by sw_{M}^{0}(σ). Moreover

sw_{M}^{0}(σ) = sw_{M}^{0}(σ̄).  (∗)

It is convenient to rewrite the collection {sw_{M}^{0}(σ)}_{σ} as a function H → ℚ (see e.g. the Fourier calculus below). For every spin^{c} structure σ on M we consider

SW _{M,σ}^{0} := ∑_{h∈H} sw_{M}^{0}(h^{−1} · σ)h ∈ ℚ[H].

Equivalently, SW _{M,σ}^{0} , as a function H → ℚ, is defined by SW _{M,σ}^{0}(h) = sw_{M}^{0}(h^{−1} · σ). The symmetry condition 3.2(∗) implies

SW _{M,σ}^{0}(h) = SW _{M,σ̄}^{0}(h^{−1}) for all h ∈ H.

This description is very difficult to use in concrete computations unless we have very specific information about the geometry of M. This is the case of the Seifert 3-manifolds, see [37, 38] for the complete presentation. In the next subsection we recall some facts needed in our computations. The interested reader is invited to consult [loc. cit.] for more details.

3.3 The Seiberg-Witten invariants of Seifert manifolds. We will use the notations of 2.14 and 2.15; nevertheless, in [11, 27, 38] (and in general, in the gauge theoretic literature) some other notations became generally accepted too. They will be mentioned accordingly.

In [27, 38] a Seifert manifold is regarded as the unit circle sub-bundle of an (orbifold) V -line bundle over a 2-dimensional V -manifold (orbifold) Σ. The 2-dimensional orbifold in our case is ℙ^{1} (with ν conical singularities each with angle \frac{π}{α_{i}}, i = 1, …, ν).

The space of isomorphisms classes of topological V -line bundles over Σ is an Abelian group Pic_{top}^{V}(Σ). Its is a subgroup of ℚ × ∏_{i=1}^{ν} ℤ_{α_{i}}, and correspondingly we denote its elements by (ν + 1)-uples

L(c; \frac{τ_{1}}{α_{1}} , ⋯, \frac{τ_{ν}}{α_{ν}}),

where 0 ≤ τ_{i} < α_{i}, i = 1, ⋯, ν. The number c is called the rational degree, while the fractions \frac{τ_{i}}{α_{i}} are called the singularity data. They are subject to a single compatibility condition

c − ∑_{i=1}^{ν} \frac{τ_{i}}{α_{i}} ∈ ℤ.

To any V -line bundle L(c; \frac{τ_{i}}{α_{i}} , 1 ≤ i ≤ ν) we canonically associate a smooth line bundle |L| → Σ = ℙ^{1} uniquely determined by the condition

deg |L| = c − ∑_{i=1}^{ν} \frac{τ_{i}}{α_{i}} .

The canonical V -line bundle K_{Σ} has singularity data (α_{i} −1)/α_{i} for 1 ≤ i ≤ ν, and deg |K_{Σ}| = −2, hence rational degree

κ := deg^{V} K_{Σ} = −2 + ∑_{i=1}^{ν}(1 − \frac{1}{α_{i}} ).

The Seifert manifold M with non-normalized Seifert invariants ((α_{i}, β_{i})_{i=1}^{ν}) (or, equivalently, with normalized Seifert invariants (b; (α_{i}, ω_{i})_{i=1}^{ν}), cf. 2.15), is the unit circle bundle of the V -line bundle 𝕃_{0} with rational degree ℓ = e, and singularity data

\frac{ω_{i}}{α_{i}} , 1 ≤ i ≤ ν, (ω_{i} ≡ −β_{i} (mod α_{i})).

Denote by ⟨𝕃_{0}⟩ ⊂ Pic_{top}^{V}(Σ) the cyclic group generated by 𝕃_{0}. Then one has the following exact sequence:

0 → ⟨𝕃_{0}⟩ → Pic_{top}^{V}(Σ) →^{π^{∗}} Pic_{top}(M) → 0,

where π^{∗} is the pullback map induced by the natural projection π : M → Σ. Therefore, the above exact sequence identifies for every L ∈ Pic_{top}^{V}(Σ) the pullback π^{∗}(L) with the class [L] ∈ Pic_{top}^{V}(Σ)/⟨𝕃_{0}⟩.

For every L ∈ Pic_{top}^{V}(Σ), c := deg^{V} L we set

ρ(L) := \frac{deg^{V} K_{Σ} − 2c}{2ℓ} = \frac{κ}{2ℓ} − \frac{c}{ℓ} ∈ ℚ.

For every class u ∈ Pic_{top}(M) we can find an unique E_{u} ∈ Pic_{top}^{V}(Σ) such that u = [E_{u}] and ρ(E_{u}) ∈ [0, 1). We say that E_{u} is the canonical representative of u. As explained in [27, 38] there is a natural bijection

Pic_{top}(M) ∋ u ↦ σ(u) ∈ Spin^{c}(M)

with the property that det σ(u) = 2u − [K_{Σ}] ∈ Pic_{top}(M). The canonical spin^{c} structure σ_{can} ∈ Spin^{c}(M) corresponds to u = 0. In fact, Pic_{top}(M) can be identified in a natural way to H via the Chern class. Then σ(u), in terms of the H -action described in 2.5, is given by u · σ_{can}. In this case, if one writes ρ_{0} := ρ(E_{0}) one has

ρ_{0} = \{\frac{κ}{2ℓ}\} and E_{0} := n_{0}𝕃_{0}, with n_{0} := ⌊\frac{κ}{2ℓ}⌋.

We denote the orbifold invariants of E_{0} by \frac{γ_{i}}{α_{i}}. Observe that

\frac{γ_{i}}{α_{i}} = \{\frac{n_{0}ω_{i}}{α_{i}} \}.

The Seifert manifold M admits a natural metric, the so called Thurston metric which we denote by g_{0}. The (σ_{can}, g_{0}, 0)-monopoles were explicitly described in [27, 38].

The space 𝔐_{0}^{∗} of irreducible (σ_{can}, g_{0}, 0) monopoles on M consists of several components parametrized by a subset of

S_{0} = \{E = E_{0} + n𝕃_{0} ∈ π; 0 < |ν(E)| ≤ \frac{1}{2} deg^{V} K_{Σ}\},

where

ν(E) := deg^{V} (E_{0} + n𝕃_{0}) − \frac{1}{2} deg^{V} K_{Σ}.

More precisely, consider the sets

S_{0}^{+} := {E ∈ S_{0}; ν(E) < 0, deg |E| ≥ 0},
S_{0}^{−} = {E ∈ S_{0}; ν(E) > 0, deg |K_{Σ} − E| ≥ 0}.

To every E ∈ S_{0}^{+} there corresponds a component 𝔐_{E}^{+} of 𝔐_{0}^{∗} of dimension 2 deg |E|, and to every E ∈ S_{0}^{−} there corresponds a component 𝔐_{E}^{−} of 𝔐_{0}^{∗} of dimension 2 deg |K_{Σ} − E|.

The Kreck-Stolz invariant KS(σ_{can}, g_{0}, 0) is given by (see [38])

KS_{M}(σ_{can}, g_{0}, 0) = ℓ + 1 − 4ℓρ_{0}(1 − ρ_{0})

+4νρ_{0} − 4 ∑_{i=1}^{ν} s(ω_{i}, α_{i}) − 8 ∑_{i=1}^{ν} s(ω_{i}, α_{i}; \frac{γ_{i} + ρ_{0}ω_{i}}{α_{i}} , −ρ_{0})

+4 \{ \begin{matrix}− ∑_{i=1}^{n} ((\frac{r_{i}γ_{i}}{α_{i}} )) & if & ρ_{0} = 0 \\ \frac{2+κ}{2}(1 − 2ρ_{0}) − ∑_{i=1}^{ν}\{\frac{r_{i}γ_{i}+ρ_{0}}{α_{i}} \} & if & ρ_{0} ≠ 0 \end{matrix} ,

where

r_{i}ω_{i} ≡ 1 (mod α_{i}), i = 1, …, ν.

Above, s(h, k; x, y) is the Dedekind-Rademacher sum defined in Appendix B, where we list some of its basic properties as well.

The following result is a consequence of the analysis carried out in [37, 38].

3.4 Proposition. (a) If ρ_{0} ≠ 0 and 𝔐_{0}^{∗} has only zero dimensional components then (g_{0}, 0) is a good parameter and

sw_{M}^{0}(σ_{can}) = \frac{1}{8}KS_{M}(σ_{can}, g_{0}, 0) + |S_{0}^{+}| + |S_{0}^{−}|.

(b) If g_{0} has positive scalar curvature then (g_{0}, 0) is a good parameter, S_{0}^{+} = S_{0}^{−} = ∅ and

sw_{M}^{0}(σ_{can}) = \frac{1}{8}KS(σ_{can}, g_{0}, 0).

Notice that part (b) can be applied for the links of quotient singularities.

One of the main obstructions is, that in many cases, the above theorem cannot be applied (i.e. the natural parameter provided by the natural Seifert metric is not “good”, cf. 3.1).

Fortunately, the Seiberg-Witten invariant has an alternate combinatorial description as well. To formulate it we need to review a few basic topological facts.

3.5 The Reidemeister-Turaev torsion. According to Turaev [48] a choice of a spin^{c} structure on M is equivalent to a choice of an Euler structure. For every spin^{c} structure σ on M, we denote by

𝒯_{M,σ} = ∑_{h∈H} 𝒯_{M,σ}(h) h ∈ ℚ[H],

the sign refined Reidemeister-Turaev torsion determined by the Euler structure associated to σ. (For its detailed description, see [48].) Again, it is convenient to think of 𝒯_{M,σ} as a function H → ℚ given by h ↦ 𝒯_{M,σ}(h). The Poincaré duality implies that 𝒯_{M,σ} satisfies the symmetry condition

𝒯_{M,σ}(h) = 𝒯_{M,σ̄}(h^{−1}) for all h ∈ H.  (∗)

Recall that the augmentation map 𝔞𝔲𝔤 : ℚ[H] → ℚ is defined by

∑ a_{h} h ↦ ∑ a_{h}.

It is known that 𝔞𝔲𝔤(𝒯_{M,σ}) = 0.

3.6 The Casson-Walker invariant and the modified Reidemeister-Turaev torsion. Denote by λ(M) the Casson-Walker invariant of M normalized as in Lescop’s book (cf. [23, §4.7]), and denote by H ∋ h ↦ 𝒯_{M,σ}^{0}(h) the modified Reidemeister-Turaev torsion

𝒯_{M,σ}^{0}(h) := 𝒯_{M,σ}(h) − λ(M)/|H|.

We have the following result.

3.7 Theorem. [41] SW _{M,σ}^{0}(h) = 𝒯_{M,σ}^{0}(h) for all σ ∈ Spin^{c}(M) and h ∈ H.

3.8 The Fourier transform. Later we will need a dual description of these invariants in terms of Fourier transform. Denote by Ĥ the Pontryagin dual of H, namely Ĥ := Hom(H, U(1)). The Fourier transform of any function f : H → ℂ is the function

f̂: Ĥ → ℂ, f̂(χ) = ∑_{h∈H} f(h)χ̄(h).

The function f can be recovered from its Fourier transform via the Fourier inversion formula

f(h) = \frac{1}{|H|} ∑_{χ∈Ĥ} f̂(χ)χ(h).

Notice that 𝔞𝔲𝔤(f) = f̂(1), in particular 𝒯̂_{M,σ}(1) = 𝔞𝔲𝔤(𝒯_{M,σ}) = 0. By the above identity,

\begin{matrix}sw_{M}^{0}(σ) = SW _{M,σ}^{0}(1) = −\frac{1}{|H|}λ(M) + \frac{1}{|H|} ∑_{χ∈Ĥ} 𝒯̂_{M,σ}(χ) \\ = −\frac{1}{|H|}λ(M) + 𝒯_{M,σ}(1). \end{matrix}  (1)

The symmetry condition 3.5(∗) transforms into

𝒯̂_{M,σ}(χ) = 𝒯̂_{M,σ̄}(χ̄).  (2)

It is convenient to use the notation ∑′_{χ} for a summation where χ runs over all the non-trivial characters of Ĥ.

3.9 The identification Spin^{c}(M) → Q^{c}(M) via the Seiberg-Witten invariant Sometimes it is important to have an efficient way to recover the spin^{c} structure σ (or, equivalently, the quadratic function 𝔮^{c}(σ), cf. 2.6) from the Seiberg-Witten invariant SW _{M,σ}^{0} , or from 𝒯_{M,σ}. In order to do this, we first recall that Turaev in [48, Thm. 4.3.1] proves the following identity for any σ and g, h ∈ H.

𝒯_{M,σ}(1) − 𝒯_{M,σ}(h) − 𝒯_{M,σ}(g) + 𝒯_{M,σ}(gh) = −b_{M}(g, h) (mod ℤ).

Clearly there is a similar identity for 𝒯_{M,σ}^{0} instead of 𝒯_{M,σ}. By Fourier inversion, this reads

\frac{1}{|H|}∑_{χ}′𝒯̂_{M,σ}(χ)(χ(h) − 1)(χ(g) − 1) = −b_{M}(g, h) (mod ℤ).  (1)

This identity has a “refinement” in the following sense (see [40, 3.3]): for any spin^{c} structure σ, the map H ∋ h ↦ q′_{σ}(h), defined by,

\begin{matrix}q′_{σ}(h) := 𝒯_{M,σ}(1) − 𝒯_{M,σ}(h) = 𝒯_{M,σ}^{0}(1) − 𝒯_{M,σ}^{0}(h) \\ = SW _{M,σ}^{0}(1) − SW _{M,σ}^{0}(h) (mod ℤ), \end{matrix}

is a quadratic function associated with b_{M}. Moreover, the correspondence 𝔮_{sw}^{c} : Spin^{c}(M) → Q^{c}(M) given by σ ↦ q′_{σ} is a bijection.

3.10 Proposition. 𝔮_{sw}^{c} = 𝔮^{c}, i.e. the above bijection is exactly the canonical identification 𝔮^{c} considered in 2.6.

Proof Let us denote 𝔮^{c}(σ) by q_{σ}. Since both maps 𝔮^{c} and 𝔮_{sw}^{c} are H - equivariant, it suffices to show that q_{ϵ} = q′_{ϵ} for some spin structure ϵ on M. Fix ϵ ∈ Spin(M), and as in 2.4, pick a simple connected oriented 4-manifold N with ∂N = M together with a ϵ̃ ∈ Spin(N) such that the restriction of ϵ̃ to M is ϵ. In particular, q_{ϵ} = −q_{D(L),0}, cf. 2.4. For every h ∈ H set σ_{h} := h · σ(ϵ). Pick h̃ ∈ H^{2}(N, ℤ) = L′ such that [h̃] = h. For h = 0 we choose h̃ = 0. Set σ̃_{h} := h̃ · σ(ϵ̃) ∈ Spin^{c}(N). Then c_{1}(det σ̃_{h}) = 2h̃. Observe that

SW _{M,ϵ}^{0}(h) = \frac{1}{8}KS(σ_{h}) (mod ℤ).

But the Atiyah-Patodi-Singer index theorem implies (see e.g. [24, p. 197]):

\frac{1}{8}KS(σ_{h}) = \frac{1}{8}(c_{1}(det σ̃_{h}), c_{1}(det σ̃_{h}))_{ℚ} − \frac{1}{8}signature(N) (mod ℤ),

for any h ∈ H. Thus q′_{ϵ}(h) = −\frac{1}{2} (h̃, h̃)_{ℚ} = q_{ϵ}(h) (mod ℤ).

Via the Fourier transform, the above identity is equivalent to the following one, valid for any h ∈ H:

\frac{1}{|H|}∑_{χ}′𝒯̂_{M,σ}(χ)(χ(h) − 1) = −𝔮^{c}(σ)(h) (mod ℤ).  (2)

3.11 Remark. The above discussion can be compared with the following identity. Let us keep the notations of 2.3. Let σ(L) denote the signature of L, and k ∈ L′ a characteristic element. Then the (mod 8)-residue class of σ(L) − (k, k)_{ℚ} ∈ ℚ/8ℤ depends only on the quadratic function q = q_{D(L),k}. In fact one has the following formula of van der Blij [51] for the Gauss sum:

γ(q) := |H|^{−1/2} ∑_{x∈H} e^{2πiq(x)} = e^{\frac{πi}{4} (σ(L)−(k,k)_{ℚ})}.

If k ∈ Im(i_{L}) then σ(L) − (k, k)_{ℚ} = σ(L) − (k, k) ∈ ℤ/8ℤ.

4 Analytic invariants and the main conjecture

4.1 Definitions. Let (X, 0) be a normal surface singularity. Consider the holomorphic line bundle Ω_{X\{0}}^{2} of holomorphic 2-forms on X \ {0}. If this line bundle is holomorphically trivial then we say that (X, 0) is Gorenstein. If some power of this line bundle is holomorphically trivial then we say that (X, 0) is ℚ-Gorenstein. If Ω_{X\{0}}^{2} is topologically trivial we say that (X, 0) is numerically Gorenstein. The first two conditions are analytic, the third depends only on the link M (cf. 2.7).

4.2 The geometric genus. Fix a resolution π : X̃ → X over a sufficiently small Stein representative X of the germ (X, 0). Then p_{g} := dim H^{1}(X̃, OX̃) is finite and independent of the choice of π. It is called the geometric genus of (X, 0). If p_{g}(X, 0) = 0 then the singularity (X, 0) is called rational.

4.3 Smoothing invariants. Let (X, 0) be as above. By a smoothing of (X, 0) we mean a proper flat analytic germ f : (X , 0) → (ℂ, 0) with an isomorphism (f^{−1}(0), 0) → (X, 0). Moreover, we assume that 0 is an isolated singular point of the germ (X , 0).

If X is a sufficiently small contractible Stein representative of (X , 0), then for sufficiently small η (0 < |η| ≪ 1) the fiber F := f^{−1}(η) ∩ X is smooth, and its diffeomorphism type is independent of the choices. It is a connected oriented real 4-manifold with boundary ∂F which can be identified with the link M of (X, 0).

We will use the following notations: μ(F) = rank H_{2}(F, ℤ) (called the Milnor number); ( , )_{F} = the intersection form of F on H_{2}(F, ℤ); (μ_{0}, μ_{+}, μ_{−}) the Sylvester invariant of ( , )_{F}; σ(F) := μ_{+} − μ_{−} the signature of F. Notice that the Milnor fiber F, hence its invariants too, in general depend on the choice of the (irreducible component) of the smoothing.

If M = ∂F is a rational homology sphere then μ_{0} = 0, hence μ(F) = μ_{+} + μ_{−}. It is known that for a smoothing of a Gorenstein singularity rank H_{1}(F, ℤ) = 0 [13]. Therefore, in this case μ(F) + 1 is the topological Euler characteristic χ_{top}(F) of F.

The following relations connect the invariants p_{g}, μ(F) and σ(F). The next statement is formulated for rational homology sphere links, for the general statements the reader can consult the original sources [9, 22, 47] (cf. also with [25]).

4.4 Theorem. Assume that the link M is a rational homology sphere. Then the following identities hold.

(1) [Wahl, Durfee, Steenbrink] 4p_{g} = μ(F) + σ(F).

In addition, if (X, 0) is Gorenstein, then

(2) [Laufer, Steenbrink] μ(F) = 12p_{g} + K^{2} + #V, where K^{2} + #V is the topological invariant of M introduced in 2.12.

In particular, for Gorenstein singularities, (1) and (2) give σ(F) + 8p_{g} + K^{2} + #V = 0.

This shows that modulo the link-invariant K^{2}+#V there are two (independent) relations connecting p_{g}, μ(F) and σ(F), provided that (X, 0) is Gorenstein. So, if by some other argument one can recover one of them from the topology of M, then all of them can be determined from M.

In general, these invariants cannot be computed from M. Here one has to emphasize two facts. First, if M is not a rational homology sphere, then one can construct easily (even hypersurface) singularities with the same link but different (μ, σ, p_{g}). On the other hand, even if we restrict ourselves to rational homology links, if we consider all the possible analytic structures of (X, 0), then again p_{g} can vary. For example, in the case of “weakly” elliptic singularities, there is a topological upper bound of p_{g} (namely, the length of the elliptic sequence, found by Laufer and S. S.-T. Yau) which equals p_{g} for Gorenstein singularities; but p_{g} drops to 1 for a generic analytic structure (fact proved by Laufer). For more details and examples, see the series of articles of S. S.-T. Yau (e.g. [53]), or [28]. On the other hand, the first author in [28] conjectured that for Gorenstein singularities with rational homology sphere links the invariants (μ, σ, p_{g}) can be determined from the topology of (X, 0) (i.e. from the link M); (cf. also with the list of conjectures in [36]). The conjecture is true for rational singularities [3, 4], minimally elliptic singularities [21], “weakly” elliptic singularities [28], and some special hypersurface singularities [10, 35], and special complete intersections [35]; in all cases with explicit formulae for p_{g}. But in general, even a conjectural topological candidate (computed from M) for p_{g} was completely open. The next conjecture provides exactly this topological candidate (which is also a “good” topological upper bound, cf. introduction).

4.5 The Main Conjecture. Assume that (X, 0) is a normal surface singularity whose link M is a rational homology sphere. Let σ_{can} be the canonical spin^{c} structure on M. Then, conjecturally, the following facts hold:

(1) For any (X, 0), there is a topological upper bound for p_{g} given by:

sw_{M}^{0}(σ_{can}) − \frac{K^{2} + #V}{8} ≥ p_{g}.

(2) If (X, 0) is ℚ-Gorenstein, then in (1) one has equality.

(3) In particular, if (X , 0) is a smoothing of a Gorenstein singularity (X, 0) with Milnor fiber F, then

−sw_{M}^{0}(ϵ_{can}) = \frac{σ(F)}{8}.

If (X, 0) is numerically Gorenstein and M is a ℤ_{2}–homology sphere then ϵ_{can} = σ_{can} is the unique spin structure of M; if M is an integral homology sphere then in the above formulae −sw_{M}^{0}(ϵ_{can}) = λ(M), the Casson invariant of M.

4.6 Remarks. (1) Assume that (X, 0) is a hypersurface Brieskorn singularity whose link is an integral homology sphere. Then λ(M) = σ(F)/8 by a result of Fintushel and Stern [10]. This fact was generalized for Brieskorn-Hamm complete intersections and for suspension hypersurface singularities ((X, 0) = {g(x, y) + z^{n} = 0}) with H_{1}(M, ℤ) = 0 by Neumann and Wahl [35]. In fact, for a normal complete intersection surface singularity with H = H_{1}(M, ℤ) = 0, Neumann and Wahl conjectured λ(M) = σ(F)/8. This conjecture was one of the starting points of our investigation.

The result of Neumann-Wahl [35] was re-proved and reinterpreted by Collin and Saveliev (see [7] and [8]) using equivariant Casson invariant and cyclic covering techniques.

(2) The family of ℚ-Gorenstein singularities is rather large: it contains e.g. the rational singularities [3, 4], the singularities with good ℂ^{∗}-actions and with rational homology sphere links [32], the minimally elliptic singularities [21], and all the isolated complete intersection singularities. Neumann-Wahl have conjectured in [36] that all the singularities in 4.5 (2) are finite abelian quotients of complete intersection singularities.

(3) If one wants to test the Conjecture for rational or elliptic singularities (or in any example where p_{g} is known), one should compute the corresponding Seiberg-Witten invariant. But, in some cases, even if all the terms in the main conjecture can (in principle) be computed, the identification of these contributions in the main formula can create difficulties (e.g. involving complicated identities of Dedekind sums and lattice point counts).

4.7 Remark. Notice, that in the above Conjecture, we have automatically built in the following statements as well.

(1) For any normal singularity (X, 0) the topological invariant

sw_{M}^{0}(σ_{can}) − \frac{K^{2} + #V}{8}

is non-negative. Moreover, this topological invariant is zero if and only if (X, 0) is rational. This provides a new topological characterization of the rational singularities.

(2) Assume that (X, 0) (equivalently, the link) is numerically Gorenstein. Then the above topological invariant is 1 if and only if (X, 0) is minimally elliptic (in the sense of Laufer). Again, this is a new topological characterization of minimally elliptic singularities.

4.8 Remark The invariant K^{2} + #V appears not only in the type of results listed in 4.4, but also in other topological contexts. For example, it can be identified with the Gompf invariant θ(ξ) defined in [14, 4.2] (see also [15, 11.3.3]). This appears as an “index defect” (similarly to the signature defect of Hirzebruch) (cf. also with [9] and [25]).

More precisely, the almost complex structure on T M ⊕ℝ_{M} (cf. 2.8) determines a contact structure ξ_{can} on M (see e.g. [15, p. 420]), with c_{1}(ξ_{can}) torsion element. Then the Gompf invariant θ(ξ_{can}), computed via X̃, is K^{2} − 2χ_{top}(X̃) − 3σ(X̃) = K^{2} + #V − 2.

In fact, in our situation, by 2.12, θ(ξ_{can}) can be recovered from the oriented C^{∞} type of M completely. In the Gorenstein case, in the presence of a smoothing, θ(ξ_{can}) computed from the Milnor fiber F, equals −2 − 2μ(F) − 3σ(F). The identity K^{2} + #V + 2μ(F) + 3σ(F) = 0 can be deduced from 4.4 as well.

The goal of the remaining part of the present paper is to describe the needed topological invariants in terms of the plumbing graphs of the link, and finally, to provide a list of examples supporting the Main Conjecture.

5 Invariants computed from the plumbing graph

5.1 Notations. The goal of this section is to list some formulae for the invariants K^{2} + #V, λ(M) and 𝒯_{M,σ} from the resolution graph of M (or, equivalently, from any negative definite plumbing). The formulae are made explicit for star-shaped graphs in terms of their Seifert invariants. For notations, see 2.13, 2.14 and 2.15.

Let I^{−1} be the inverse of the intersection matrix I. For any v, w ∈ V, I_{vw}^{−1} denotes the (v, w)-entry of I^{−1}. Since I is negative definite, and the graph is connected, I_{vw}^{−1} < 0 for each entry v, w. Since I is described by a tree, these entries have the following interpretation as well. For any two vertex v, w ∈ V, let p_{vw} be the unique minimal path in the graph connecting v and w, and let I_{(vw)} be the matrix obtained from I by deleting all the lines and columns corresponding to the vertices on the path p_{vw} (i.e. I_{(vw)} is the intersection matrix of the complement graph of the path). Then I_{vw}^{−1} = −| det(I_{(vw)})/ det(I)|.

For simplicity we will write E_{v}, D_{v}, … instead of [E_{v}], [D_{v}] . . ..

5.2 The invariant K^{2} + #V from the plumbing graph. Let Z_{K} = ∑_{v} r_{v}E_{v}. Since Z_{K} = (i_{L} ⊗ ℚ)^{−1}(∑(e_{v} + 2)D_{v}) (cf. 2.13), one has clearly r_{v} = ∑_{w}(e_{w} + 2)I_{vw}^{−1}.

Then a “naive formula” of K^{2} = Z_{K}^{2} is

Z_{K}^{2} = ∑_{v} r_{v}Z_{K} · E_{v} = ∑_{v} r_{v}(e_{v} + 2) = ∑_{v,w}(e_{v} + 2)(e_{w} + 2)I_{vw}^{−1}.

However, we prefer a different form for r_{v} and Z_{K}^{2} which involves only a small part of the entries of I^{−1}. Indeed, let us consider the class

D = ∑_{v} i_{L}(E_{v}) + ∑_{v}(2 − δ_{v})D_{v} ∈ L′.

Then clearly D · E_{v} = e_{v} + δ_{v} + 2 − δ_{v} = e_{v} + 2, hence D = (i_{L} ⊗ ℚ)Z_{K}. Therefore,

Z_{K} = ∑_{v} E_{v} + ∑_{v}(2 − δ_{v})(i_{L} ⊗ ℚ)^{−1}(D_{v}),

hence

r_{v} = 1 + ∑_{w}(2 − δ_{w})I_{vw}^{−1}.

Moreover,

Z_{K}^{2} = Z_{K} · D = ∑_{v} Z_{K} · E_{v} + ∑_{v}(2 − δ_{v})Z_{K} · D_{v} = ∑_{v}(e_{v} + 2) + ∑_{v}(2 − δ_{v})r_{v},

hence by the second formula for r_{v} we deduce

K^{2} + #V = ∑_{v} e_{v} + 3#V + 2 + ∑_{v,w}(2 − δ_{v})(2 − δ_{w})I_{vw}^{−1}.

In particular, this number depends only of those entries of I^{−1} whose index set runs over the rupture points (δ_{v} ≥ 3) and the end-vertices (δ_{v} = 1) of the graph.

For cyclic quotient singularities, the above formula for K^{2} goes back to the work of Hirzebruch. In fact, the right hand side can also be expressed in terms of Dedekind sums, see e.g. [25, 5.7] and [18] (or 7.1 here).

5.3 The Casson-Walker invariant from plumbing. We recall a formula for the Casson-Walker invariant for plumbing 3-manifolds proved by A. Ratiu in his dissertation [45]. In fact, the formula can also be recovered from the surgery formulae of Lescop [23] (since any plumbing graph can be transformed into a precise surgery data, see e.g. A1). The first author thanks Christine Lescop for providing him all the details and information about it. We have

−\frac{24}{|H|} λ(M) = ∑_{v} e_{v} + 3#V + ∑_{v}(2 − δ_{v}) I_{vv}^{−1}.

If M = L(p, q), then this can be transformed into λ(L(p, q)) = p · s(q, p)/2. (Here we emphasize that by our notations, L(p, q) is obtained by −p/q-surgery on the unknot in S^{3}, as in [15, page 158], and not by p/q-surgery as in [52, page 108].)

5.4 The Casson-Walker invariant for Seifert manifolds. Assume that M is a Seifert manifold as in 2.14 and 2.15. Using [23], Proposition 6.1.1, one has the following expression:

−\frac{24}{|H|}λ(M) = \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}^{2}} ) + e + 3 + 12 ∑_{i=1}^{ν} s(β_{i}, α_{i}).

(Warning: our notations for the Seifert invariants differ slightly from those used in [23]; and also, our e and b have opposite signs.)

5.5 K^{2} + #V for Seifert manifolds. Using 5.3 we deduce

−\frac{24}{|H|}λ(M) = ∑_{v} e_{v} + 3#V + ∑_{i=1}^{ν} I_{v̄_{i}v̄_{i}}^{−1} + (2 − ν)I_{v̄_{0}v̄_{0}}^{−1}.

For Seifert manifolds, 5.2 can be rewritten as

K^{2}+#V = 2+∑_{v} e_{v}+3#V+∑_{i=1}^{ν} I_{v̄_{i}v̄_{i}}^{−1} +(2−ν)^{2}I_{v̄_{0}v̄_{0}}^{−1} +2 ∑_{i=1}^{ν}(2−ν)I_{v̄_{0}v̄_{i}}^{−1} +∑_{i≠j} I_{v̄_{i}v̄_{j}}^{−1}.

Using the interpretation of the entries of I^{−1} given in 5.1, one gets easily that

I_{v̄_{0}v̄_{0}}^{−1} = \frac{1}{e}; I_{v̄_{i}v̄_{0}}^{−1} = \frac{1}{eα_{i}}; I_{v̄_{i}v̄_{j}}^{−1} = \frac{1}{eα_{i}α_{j}} (i ≠ j; 1 ≤ i, j ≤ ν).  (1)

Therefore, these identities and 5.4 give:

K^{2} + #V = \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}} )^{2} + e + 5 + 12 ∑_{i=1}^{ν} s(β_{i}, α_{i}).

It is instructive to compare this expression with 5.4 and also with the coefficient r_{0} of Z_{K}, namely with r_{0} = 1 + (2 − ν + ∑_{i} 1/α_{i})/e.

5.6 The Reidemeister-Turaev torsion. In the remaining part of this section we provide a formula for the torsion 𝒯_{M} of M using the plumbing representation of M. We decided not to distract the reader’s attention from the main message of the paper and we deferred its proof to Appendix A.

5.7 Theorem. Let M be an oriented rational homology 3-manifold represented by a negative definite plumbing graph Γ. (E.g., let M be the link of a normal surface singularity (X, 0), and Γ = Γ(π) be one of its resolution graphs.) In the sequel, we keep the notations used above (cf. 2.13 and 5.1). For any spin^{c} structure σ ∈ Spin^{c}(M), consider the unique element h_{σ} ∈ H such that h_{σ} · σ_{can} = σ. Then for any χ ∈ Ĥ, χ ≠ 1, the following identity holds:

𝒯̂_{M,σ}(χ̄) = χ̄(h_{σ}) · ∏_{v∈V} (χ(g_{v}) − 1)^{δ_{v}−2}.

The right hand side should be understood as follows. If χ(g_{v}) ≠ 1 for all v (with δ_{v} ≠ 2), then the expression is well-defined. Otherwise, the right hand side is computed via a limit (regularization procedure). More precisely, fix a vertex v_{∗} ∈ V so that χ(g_{v_{∗}}) ≠ 1, and let b⃗^{∗} be the column vector with entries =1 on the place v_{∗} and zero otherwise. Then find w^{∗} ∈ ℤ^{n} with entries {w_{v}^{∗}}_{v} in such a way that Iw^{∗} = −m^{∗} · b⃗^{∗} for some integer m^{∗} > 0. Then

𝒯̂_{M,σ}(χ̄) = χ̄(h_{σ}) · lim_{t→1} ∏_{v∈V} (t^{w_{v}^{∗}} χ(g_{v}) − 1)^{δ_{v}−2}.

The above limit always exists. Moreover, once v_{∗} is fixed, the vector w^{∗} is unique modulo a positive multiplicative factor (which does nor alter the limit). In fact, the above limit is independent even of the choice of v_{∗} (as long as χ(g_{v_{∗}}) ≠ 1). This follows also from the general theory (cf. also A.3 and A.4), but it also has an elementary combinatorial proof given in Lemma A.7. (This can be read independently from the other parts of the proof.) In fact, by Lemma A.7, the set of vertices v_{∗}, providing a suitable w^{∗} in the limit expression, is even larger than the set identified in the theorem: one can take any v_{∗} which satisfy either χ(g_{v_{∗}}) ≠ 1 or it has an adjacent vertex u with χ(g_{u}) ≠ 1.

5.8 The torsion of Seifert manifolds. In this paragraph we use the notations of 2.14 and 2.15. Recall that we introduced ν + 1 distinguished vertices v̄_{i}, 0 ≤ i ≤ ν, whose degree is ≠ 2 (and gv̄_{0} is the central vertex). Fix χ and first assume that χ(gv̄_{i}) ≠ 1 for all 0 ≤ i ≤ ν. Then, the above theorem reads as

𝒯̂_{M,σ_{can}}(χ̄) = \frac{(χ(gv̄_{0}) − 1)^{ν−2}}{∏_{i=1}^{ν} (χ(gv̄_{i}) − 1)}.

If there is one index 1 ≤ i ≤ ν with χ(gv̄_{i}) = 1, then necessarily χ(gv̄_{0}) = 1 as well. If χ(gv̄_{0}) = 1, then either 𝒯_{M,σ_{can}}(χ̄) = 0, or for exactly ν − 2 indices i (1 ≤ i ≤ ν) one has χ(gv̄_{i}) = 1. In this later case the limit is non-zero. Let us analyze this case more closely. Assume that χ(gv̄_{i}) ≠ 1 for i = 1, 2. Using the last statement of the previous subsection, it is not difficult to verify that w^{∗} computed from any vertex on these two arms provide the same limit. In fact, by the same argument (cf. A.7), one gets that even the central vertex v̄_{0} provides a suitable set of weight w^{∗} (for any χ). The relevant weights can be computed via 5.5(1), and with the notation α := lcm(α_{1}, …, α_{ν}) one has:

𝒯̂_{M,σ_{can}}(χ̄) = lim_{t→1} \frac{(t^{α}χ(gv̄_{0}) − 1)^{ν−2}}{∏_{i=1}^{ν} (t^{α/α_{i}} χ(gv̄_{i}) − 1)} for any χ ∈ Ĥ \ \{1\}.

Notice the mysterious similarity of this expression with the Poincaré series of the graded affine ring associated with the universal abelian cover of (X, 0), provided that (X, 0) admits a good ℂ^{∗}-action, cf. [32].

6 Brieskorn-Hamm rational homology spheres

6.1 Notations. Fix n ≥ 3 positive integers a_{i} ≥ 2 (i = 1, …, n). For any set of complex numbers C = {c_{j,i}}, i = 1, …, n; j = 1, …, n − 2, one can consider the affine variety

X_{C}(a_{1}, …, a_{n}) := {z ∈ ℂ^{n} : c_{j,1}z_{1}^{a_{1}} + ⋯ + c_{j,n}z_{n}^{a_{n}} = 0 for all 1 ≤ j ≤ n − 2} .

It is well-known (see [16]) that for generic C, the intersection of X_{C}(a_{1}, …, a_{n}) with the unit sphere S^{2n−1} ⊂ ℂ^{n} is an oriented smooth 3-manifold whose diffeomorphism type is independent of the choice of the coefficients C. It is denoted by M = Σ(a_{1}, …, a_{n}).

6.2 In fact (cf. [19, 34]), M is an oriented Seifert 3-manifold with Seifert invariants

(g; \underbrace{(α_{1}, β_{1}), ⋯, (α_{1}, β_{1})}_{s_{1}}; ⋯; \underbrace{(α_{n}, β_{n}), ⋯, (α_{n}, β_{n})}_{s_{n}})

where g denotes the genus of the base of the Seifert fibration, and the pairs of coprime positive integers (α_{i}, β_{i}) (each considered s_{i} times) are the orbit invariants (cf. 2.14 and 2.15 for notations). Recall that the rational degree of this Seifert fibration is

e = − ∑_{i=1}^{n} s_{i} · \frac{β_{i}}{α_{i}} < 0.  (e)

Set

a := lcm (a_{i}; 1 ≤ i ≤ n), q_{i} := \frac{a}{a_{i}} , 1 ≤ i ≤ n; A := ∏_{i=1}^{n} a_{i}.

The Seifert invariants are as follows (see [19, Sec. 7] or [34]):

α_{i} := \frac{a}{lcm (a_{j}; j ≠ i)}, s_{i} := \frac{∏_{j≠i} a_{j}}{lcm (a_{j}; j ≠ i)} = \frac{Aα_{i}}{aa_{i}} , (1 ≤ i ≤ n);

g := \frac{1}{2}(2 + (n − 2)\frac{A}{a} − ∑_{i} s_{i}).  (g)

By these notations −e = A/(a^{2}). Notice that the integers {α_{j}}_{j=1}^{n} are pairwise coprime. Therefore the integers β_{j} are determined from (e). In fact ∑_{j} q_{j}β_{j} = 1, hence β_{i}q_{j} ≡ 1 (mod α_{j}) for any j. Similarly as above, we set α := lcm(α_{1}, …, α_{n}).

It is clear that M is a ℚ-homology sphere if and only if g = 0. In order to be able to compute the Reidemeister-Turaev torsion, we need a good characterization of g = 0 in terms of the integers {a_{i}}_{i}. For hypersurface singularities this is given in [5]. This characterization was partially extended for complete intersections in [16]. The next proposition provides a complete characterization (for the case when the link is 3-dimensional).

6.3 Proposition Assume that X_{C}(a_{1}, …, a_{n}) is a Brieskorn-Hamm isolated complete intersection singularity as in 6.1 such that its link M is a 3dimensional rational homology sphere. Then (a_{1}, …, a_{n}) (after a possible permutation) has (exactly) one of the following forms:

(i) (a_{1}, …, a_{n}) = (db_{1}, db_{2}, b_{3}, …, b_{n}), where the integers {b_{j}}_{j=1}^{n} are pairwise coprime, and gcd(d, b_{j}) = 1 for any j ≥ 3;

(ii) (a_{1}, …, a_{n}) = (2^{c}b_{1}, 2b_{2}, 2b_{3}, b_{4} …, b_{n}), where the integers {b_{j}}_{j=1}^{n} are odd and pairwise coprime, and c ≥ 1.

Proof The proof will be carried out in several steps.

1. Fix any four distinct indices i, j, k, l. Then d_{ijkl} := gcd(a_{i}, a_{j}, a_{k}, a_{l}) = 1. Indeed, if a prime p divides d_{ijkl}, then p^{2}|(A/a) and p^{2}|s_{i} for all i. Hence by 6.2(g) one has p^{2}|2.

2. Fix any three distinct indices i, j, k and set d_{ijk} := gcd(a_{i}, a_{j}, a_{k}). If a prime p divides d_{ijk} then p = 2. Indeed, if p|d_{ijk} then p|(A/a) and p|s_{j} for all j, hence p|2 by 6.2(g).

3. There is at most one triple i < j < k with d_{ijk} ≠ 1. This follows from steps 1 and 2.

4. Assume that d_{ijk} = 1 for all triples i, j, k. For any i ≠ k set d_{ik} := gcd(a_{i}, a_{k}). Then A/a = ∏_{i<k} d_{ik}, and there are similar identities for each s_{j}. Then 6.2(g) reads as

2 + (n − 2) ∏_{i<k} d_{ik} − ∑_{j=1}^{n} ∏_{i<k;i,k≠j} d_{ik} = 0.  (eq1)

5. Assume that d_{123} ≠ 1, hence (a_{1}, …, a_{n}) = (2^{c}b_{1}, 2^{u}b_{2}, 2^{v}b_{3}, b_{4} …, b_{n}), with c ≥ u ≥ v, and all b_{i} odd numbers (after a permutation of the indices). Then u = v = 1. For this use similar argument as above with 4|(A/a), 4|s_{i} for i ≥ 4, s_{1} and s_{2} (resp. s_{3}) is divisible exactly by the v^{th} (resp. u^{th}) power of 2.

Using this fact, write for each par i ≠ k, d_{ik} := gcd(b_{i}, b_{k}). We deduce as above that A/a = 4 ∏_{i<k} d_{ik}, and there are similar identities for each s_{j}. Then 6.2(b) transforms into

\frac{1}{2} + (n − 2) ∏_{i<k} d_{ik} − ∑_{j=1}^{n} ε_{j} ∏_{i<k;i,k≠j} d_{ik} = 0,  (eq2)

where ε_{j} = 1/2 for j ≤ 3 and = 1 for j ≥ 4.

6. The equation (eq2) has only one solution with all d_{ik} strict positive integer, namely d_{ik} = 1 for all i, k. Similarly, any set of solutions of (eq1) has at most one d_{ik} strict greater than 1, all the others being equal to 1.

This can be proved e.g. by induction. For example, in the case of (eq2), if one replaces the set of integers d_{ik} in the left hand side of the equation with the same set but in which one of them is increased by one unit, then the new expression is strictly greater than the old one. A similar argument works for (eq1) as well. The details are left to the reader.

6.4 Verification of the conjecture in the case 6.3(i). We start to list the properties of Brieskorn-Hamm complete intersections of the form (i).

• α_{j} = b_{j} for j = 1, …, n;

• s_{1} = 1, s_{2} = 1, and s_{j} = d for j ≥ 3; in particular, the number of “arms” is ν = 2 + (n − 2)d;

• α = B := ∏_{j=1}^{n} b_{j}, and −e · B = 1, hence by 2.14 the generic orbit h is homologically trivial.

• Using the group representation 2.15, and the fact that h is trivial, one has

H = ab⟨ g_{ij}, 1 ≤ j ≤ n, 1 ≤ i ≤ s_{j} | g_{ij}^{α_{j}} = 1 for all i, j; ∏_{i,j} g_{ij}^{ω_{j}} = 1 ⟩.

Since the integers α_{j} are pairwise coprime, taking the α/α_{j} power of the last relation, and using that gcd(α_{j}, ω_{j}) = 1, one obtains that

H = ⨁_{j≥3} ab⟨ g_{ij}, 1 ≤ i ≤ d | g_{ij}^{b_{j}} = 1 for all i; ∏_{i} g_{ij} = 1 ⟩ ≅ ⨁_{j≥3}(ℤ_{b_{j}})^{d−1}.

In particular, |H| = ∏_{j≥3} b_{j}^{d−1}.

• The Reidemeister-Turaev torsion of M. By 5.8

𝒯̂_{M,σ_{can}}(χ) = lim_{t→1} \frac{(t^{α} − 1)^{d(n−2)}}{(t^{α/α_{1}} − 1)(t^{α/α_{2}} − 1) ∏_{j≥3} ∏_{i=1}^{d}(t^{α/α_{j}} χ̄(g_{ij}) − 1)}.

As explained in 5.8, for a fixed χ, the expression 𝒯̂_{M,σ_{can}}(χ) is nonzero if and only if for exactly two pairs (i, j) (where j ≥ 3 and 1 ≤ i ≤ b) χ(g_{ij}) ≠ 1. Analyzing the group structure, one gets easily that these two pairs must have the same j. For a fixed j, there are d(d − 1)/2 choices for the set of indices {i_{1}, i_{2}}. For fixed j, the set of nontrivial characters of the group

ab⟨ g_{ij}, 1 ≤ i ≤ d | g_{ij}^{b_{j}} = 1 for all i; ∏_{i} g_{ij} = 1 ⟩,

satisfying χ(g_{ij}) = 1 for all i ≠ i_{1}, i_{2} is clearly ℤ̂_{b_{j}} \ {1}, and in this case χ(g_{i_{1}j})χ(g_{i_{2}j}) = 1 as well. Therefore

𝒯_{M,σ_{can}}(1) = \frac{1}{|H|} · ∑_{j≥3} \frac{α^{d(n−2)}}{\frac{α}{α_{1}} \frac{α}{α_{2}} (\frac{α}{α_{3}} )^{d} ⋯ (\frac{α}{α_{j}})^{d−2} ⋯ (\frac{α}{α_{n}})^{d}}

×\frac{d(d − 1)}{2} · ∑_{ℤ_{b_{j}}}′ \frac{1}{(ζ − 1)(ζ̄ − 1)}.

Recall that |H| = ∏_{j≥3} b_{j}^{d−1}. Hence, by (B.9) of Appendix B and an easy computation:

𝒯_{M,σ_{can}}(1) = \frac{B · d(d − 1)}{24} ∑_{j≥3}(1 − \frac{1}{b_{j}^{2}} ).

• The Casson-Walker invariant. From 5.4 we get

−\frac{λ(M)}{|H|} = −\frac{B}{24}(−d(n − 2) + ∑_{j=1}^{n} \frac{s_{j}}{b_{j}^{2}} ) − \frac{1}{24B} + \frac{1}{8} + \frac{1}{2} ∑_{j=1}^{n} s_{j} s(β_{j}, b_{j}).

• The signature of the Milnor fiber. Since X_{C} is an isolated complete intersection singularity, its singular point is Gorenstein. Hence, by 4.4, it is enough to verify only part (3) of the main conjecture, part (2) will follow automatically.

The signature σ(F) = σ(a_{1}, …, a_{n}) of the Milnor fiber F of a Brieskorn-Hamm singularity is computed by Hirzebruch [17] in terms of cotangent sums. Nevertheless, we will use the version proved in [35, 1.12]. This, in the case (i), (via B.10) reads as

σ(F) = −1 + \frac{1}{3B} (1 − (n − 2)d^{2}B^{2} + B^{2} ∑_{j=1}^{n} \frac{s_{j}^{2}}{b_{j}^{2}} ) − 4 ∑_{j=1}^{n} s_{j} s(q_{j}, b_{j}),

where q_{j} = s_{j}B/b_{j} for all 1 ≤ j ≤ n. Since β_{j}q_{j} ≡ 1 (mod b_{j}) (cf. 6.2), one has s(q_{j}, b_{j}) = s(β_{j}, b_{j}) for all j. Now, by a simple computation one can verify the conjecture.

6.5 Verification of the conjecture in the case 6.3(ii). The discussion is rather similar to the previous case, the only difference (which is not absolutely negligible) is that now h is not trivial. This creates some extra work in the torsion computation. In the sequel we write B := ∏_{j} b_{j}.

• a = 2^{c}B and A = 2^{c+2}B. Moreover, α_{1} = 2^{c−1}b_{1} and α_{j} = b_{j} for j ≥ 2.

• s_{j} = 2 for j ≤ 3 and s_{j} = 4 for j ≥ 4. The number of “arms” is ν = 4n − 6.

• α = 2^{c−1}B, hence −e^{−1} = 2^{c−2}B. Therefore |⟨h⟩| = 2 and |H| = 2^{c}B^{3}/(b_{1}b_{2}b_{3})^{2}.

• The self intersection number b of the central exceptional divisor is even. Indeed, equation (e) implies that 1 ≡ β_{1}b_{2} ⋯ b_{n} (mod α_{1}). Since ω_{1} ≡ −β_{1} (mod α_{1}), one has 2^{c−1}|1 + ω_{1}b_{2} ⋯ b_{n}. This, and the first formula of 2.15 implies that b is even.

• Using 2.15, and the fact that h^{−b} = 1 is automatically satisfied, we obtain the following presentation for H:

ab⟨g_{ij}, 1 ≤ j ≤ n, 1 ≤ i ≤ s_{j}; h| g_{ij}^{α_{j}} = h for all i, j; ∏_{i,j} g_{ij}^{ω_{j}} = 1, h^{2} = 1⟩.

Clearly ⟨h⟩ ≈ ℤ_{2}, and there is an exact sequence 0 → ⟨h⟩ → H → Q → 0 with

Q = ⨁_{j≥1} ab⟨ g_{ij}, 1 ≤ i ≤ s_{j} | g_{ij}^{α_{j}} = 1 for all i; ∏_{i} g_{ij} = 1 ⟩ ≅ ⨁_{j≥1}(ℤ_{α_{j}})^{s_{j}−1}.

• The Reidemeister-Turaev torsion of M. We have to distinguish two types of characters χ ∈ Ĥ \ {1} since χ(h) is either +1 or −1. The sum over characters χ with χ(h) = 1 (i.e. over Q̂ \ {1}) can be computed similarly as in case (i), namely it is

\frac{1}{|H|} · ∑_{j≥1} \frac{α^{ν−2}}{(\frac{α}{α_{1}} )^{s_{1}} ⋯ (\frac{α}{α_{j−1}} )^{s_{j−1}}(\frac{α}{α_{j}})^{s_{j}−2}(\frac{α}{α_{j+1}} )^{s_{j+1}} ⋯ (\frac{α}{α_{n}})^{s_{n}}} · ∑_{ℤ_{α_{j}}}′ \frac{s_{j}(s_{j} − 1)}{2(ζ − 1)(ζ̄ − 1)}

= \frac{2^{c−1}B}{24} ∑_{j≥1} \frac{s_{j}(s_{j} − 1)}{2}(1 − \frac{1}{α_{j}^{2}} ).

The sum over characters χ with χ(h) = −1 requires no “limit regularization”, hence it is (−2)^{ν−2}/(∏_{j} P_{j}), where for any fixed j the expression P_{j} has the form ∏_{∗}(ζ_{i} − 1), where the product runs over 1 ≤ i ≤ s_{j}, ζ_{i}^{α_{j}} = −1 with restriction ∏_{i} ζ_{i} = 1.

Using the identity −2 = ζ_{i}^{α_{j}} − 1 one gets

\frac{1}{P_{j}} = \frac{1}{(−2)^{s_{j}}} ∏_{∗}(1 + ζ_{i} + ⋯ + ζ_{i}^{α_{j}−1}).

By an elementary argument, this is exactly (α_{j}/2)^{s_{j}}. Therefore, this second contribution is 2^{c−1}B/8, hence

𝒯_{M,σ_{can}}(1) = \frac{2^{c−1}B}{8} + \frac{2^{c−1}B}{24} ∑_{j≥1} \frac{s_{j}(s_{j} − 1)}{2}(1 − \frac{1}{α_{j}^{2}} ).

• The Casson-Walker invariant. From 5.4 and ν = 4n − 6 one gets

−\frac{λ(M)}{|H|} = −\frac{2^{c−2}B}{24} (−4(n − 2) + ∑_{j=1}^{n} \frac{s_{j}}{α_{j}^{2}} ) − \frac{1}{3 · 2^{c+1}B} + \frac{1}{8} + \frac{1}{2} ∑_{j=1}^{n} s_{j} s(β_{j}, α_{j}).

• The signature of the Milnor fiber. Using the above identities about the Seifert invariants (and a_{j} = 4α_{j}/s_{j} too), [35, 1.12] reads as

σ(F) = −1 + \frac{1}{3 · 2^{c−2}B} (1 − (n − 2)2^{2c}B^{2} + 2^{2c−4}B^{2} ∑_{j=1}^{n} \frac{s_{j}^{2}}{α_{j}^{2}} ) − 4 ∑_{j=1}^{n} s_{j} s(β_{j}, α_{j}).

Now, the verification of the statement of the conjecture is elementary.

7 Some rational singularities

7.1 Cyclic quotient singularities. The link of a cyclic quotient singularity (X_{p,q}, 0) (0 < q < p, (p, q) = 1) is the lens space L(p, q). X_{p,q} is numerically Gorenstein if and only if q = p − 1, case which will be considered in 7.2. In all other cases σ_{can} is not spin. In all the cases p_{g} = 0. Moreover (see e.g. [25, 5.9], or [18], or 5.2):

K^{2} + #V = \frac{2(p − 1)}{p} − 12 · s(q, p).

On the other hand, the Seiberg-Witten invariants of L(p, q) are computed in [39] (where a careful reading will identify sw_{M}^{0}(σ_{can}) as well). In fact, cf. ([39, 3.16]):

𝒯̂_{M,σ_{can}}(χ) = \frac{1}{(χ̄ − 1)(χ̄^{q} − 1)},

fact which follows also from 5.7. Therefore, using [44, 18a] (or B.8), one gets that

𝒯_{M,σ_{can}}(1) = \frac{p − 1}{4p} − s(q, p).

The Casson-Walker contribution is λ(L(p, q))/p = s(q, p)/2 (cf. 5.3). Hence one has equality in part (1) of the Conjecture.

7.2 Particular case: the A_{p−1}-singularities. Assume that (X, 0) = ({x^{2} + y^{2} + z^{p} = 0}, 0). Then by 7.1 (or [39, §2.2.A]) one has sw_{M}^{0}(ϵ_{can}) = (p − 1)/8 (in [39] this spin structure is denoted by σ_{spin}). On the other hand, ( , )_{F} is negative definite of rank p − 1, hence σ(F) = −(p − 1). (Obviously, the A_{p−1}-case can also be deduced from section 6.)

7.3 The D_{n}-singularities For each n ≥ 4, one denotes by D_{n} the singularity at the origin of the weighted homogeneous complex hypersurface x^{2}y + y^{n−1} + z^{2} = 0. It is convenient to write p := n − 2. We invite the reader to recall the notations of 3.3 about orbifold invariants.

The normalized Seifert invariants are

(b, (ω_{1}, α_{1}), (ω_{2}, α_{2}), (ω_{3}, α_{3})) = (−2, (p, p − 1), (2, 1), (2, 1)).

Its rational degree is ℓ = −1/p. Observe that κ = ℓ.

The link M is the unit circle bundle of the V -line bundle 𝕃_{0} with rational degree ℓ and singularity data ((p−1)/p, 1/2, 1/2). Therefore, 𝕃_{0} = K_{Σ}. Hence, ρ(σ_{can}) = {κ/2ℓ} = 0. The canonical representative of σ_{can} is then the trivial line bundle E_{0}. It has rational degree 0 and singularity data γ⃗ = (0, 0, 0). The Kreck-Stolz invariant is then

KS_{M}(σ_{can}, g_{0}, 0) = 7−4 ∑_{i=1}^{3} s(ω_{i}, α_{i})−8 ∑_{i=1}^{3} s(ω_{i}, α_{i}; \frac{ω_{i}}{2α_{i}} , −1/2)−4 ∑_{i=1}^{3}\{\frac{1}{2α_{i}} \} .

Using the fact that s(1, 2; 1/4, −1/2) = 0, this expression equals:

6 + \frac{p}{3} + \frac{2}{3p} + 8s(1, p, \frac{1}{2p}, 1/2).

Now using the reciprocity formula for the generalized Dedekind sum, one has

8s(1, p, \frac{1}{2p}, 1/2) = \frac{4/3 − 2p + 2p^{2}/3}{p} = \frac{4}{3p} − 2 + \frac{2p}{3}.

Thus

8sw_{M}^{0}(σ_{can}) = KS_{M}(σ_{can}, g_{0}) = 4 + \frac{p}{3} − \frac{4}{3p} + \frac{4}{3p} − 2 + \frac{2p}{3} = 2 + p.

On the other hand, the signature of the Milnor fiber is −n = −(p + 2), confirming again the Main Conjecture.

7.4 The E_{6} and E_{8} singularities. Both E_{6} (i.e. x^{4} + y^{3} + z^{2} = 0) and E_{8} (i.e. x^{5} + y^{3} + z^{2} = 0) are Brieskorn (hypersurface) singularities, hence the result of section 6 can be applied. The link of E_{8} is an integral homology sphere, hence the validity of the conjecture in this case was proved in [10]. The interested reader can verify the conjecture using the machinery of 3.3 and 3.4 as well.

7.5 The E_{7} singularity It is given by the complex hypersurface x^{3} + xy^{3} + z^{2} = 0. The group H is ℤ_{2}. The normalized Seifert invariants are (−2, (2, 1), (3, 2), (4, 3)), the rational degree is −1/12. We deduce as above that𝕃_{0} = K_{Σ}, with ρ_{0} = ρ(σ_{can}) = 1/2. The canonical representative is again the trivial line bundle E_{0}. Its singularity data are trivial. The Seiberg-Witten invariant of σ_{can} is determined by the Kreck-Stolz invariant alone. A direct computation shows that KS_{M}(σ_{can}, g_{0}) = 7. But the signature of the Milnor fiber is σ(E_{7}) = −7 as well, hence the statement of the conjecture is true.

7.6 Another family of rational singularities Consider a singularity (X, 0) whose link M is described by the negative definite plumbing given in Figure 1. (It is clear that in this case M it is not numerically Gorenstein.)

The number of −2 spheres on any branch is m − 1, where m ≥ 2. It is easy to verify that the (X, 0) is a rational (with Artin cycle ∑_{v} E_{v}) (see, e.g. [29]). M is Seifert manifold with normalized Seifert invariants (−3, (m, m − 1), (m, m − 1), (m, m − 1)) and rational degree l = −3/m.

Figure 1: The resolution graph of the rational singularity (X, 0)

To compute the Seiberg-Witten invariant of M associated with σ_{can} we use again 3.4.

The canonical V -line bundle of Σ has singularity data (m−1)/m (three times) and rational degree κ = 1 + ℓ. The link M is the unit circle bundle of the V - line bundle 𝕃_{0} with rational degree ℓ and singularity data (m − 1)/m (three times). Therefore

ρ_{0} = \{−\frac{m − 3}{6}\}.

To apply 3.4 we need ρ_{0} ≠ 0, i.e.

m ≢ 3 (mod 6).

The canonical representative of σ_{can} is the V -line bundle E_{0} with

E_{0} = n_{0}𝕃_{0}, n_{0} = ⌊\frac{3 − m}{6}⌋.

The reader is invited to recall the definition of S_{0}^{±}. We start with the computation of S_{0}^{+}. Notice that

−\frac{1}{2} deg^{V} K_{Σ} ≤ ν(E) < 0 ⟺ 0 ≤ nℓ < \frac{1}{2} deg^{V} K_{Σ}.

Hence

\frac{1}{2ℓ} deg^{V} K_{Σ} < n ≤ 0, i.e. − \frac{m − 3}{6} < n ≤ 0.  (1)

The singularity data of n𝕃_{0} are all equal to {−n/m} (three times). We deduce

deg |n𝕃_{0}| = deg^{V} (n𝕃_{0}) − 3\{\frac{−n}{m}\} = 3⌊\frac{−n}{m}⌋.

Now observe that (1) implies

0 ≤ \frac{−n}{m} < \frac{m − 3}{6m}, hence ⌊\frac{−n}{m}⌋ = 0

for every n subject to the condition (1). Since m ≢ 3 (mod 6), we deduce

|S_{0}^{+}| = ⌊\frac{m − 3}{6}⌋ + 1 = −n_{0}.  (2)

Moreover, all the connected components corresponding to the elements in S_{0}^{+} are points. Similarly, the condition 0 < ν(E) ≤ \frac{1}{2} deg^{V} K_{Σ} implies

\frac{1}{2} deg^{V} K_{Σ} < nℓ ≤ deg^{V} K_{Σ} ⟹ \frac{1}{ℓ} deg^{V} K_{Σ} ≤ n < \frac{1}{2ℓ} deg^{V} K_{Σ}.

Hence

−\frac{m − 3}{3} ≤ n < −\frac{m − 3}{6}.  (3)

The singularity data of the V -line bundle K_{Σ}−n𝕃_{0} are all equal to {(n−1)/m}. We deduce

deg |K_{Σ} − n𝕃_{0}| = 1 + 3\frac{n − 1}{m} − 3\{(n − 1)/m\} = 1 + 3⌊\frac{n − 1}{m}⌋.

But this number is negative (because of (3)), hence S_{0}^{−} = ∅. These considerations show that Proposition 3.4 is applicable. Set

−\frac{m − 3}{6} = −k + ρ_{0}, 0 < ρ_{0} < 1, k non-negative integer.

Then n_{0} = −k. The canonical representative is E_{0} = −k𝕃_{0}. It has degree −kℓ. Its singularity data are all equal to γ_{i}/α_{i} = k/m. Then in the formula of KS_{M} one has ω_{i} = m − 1, r_{i} = −1, γ_{i} = k for all i. Hence

KS_{M}(σ_{can}) = ℓ + 1 − 4ℓρ_{0}(1 − ρ_{0}) + 12ρ_{0} + 2(3 + ℓ)(1 − 2ρ_{0}) − 12\{\frac{ρ_{0} − k}{m} \}

−12s(m − 1, m) − 24s(m − 1, m; \frac{k + ρ_{0}(m − 1)}{m}, −ρ_{0}).

Observe now that

−s(m − 1, m) = s(1, m) = \frac{m}{12} + \frac{1}{6m} − \frac{1}{4}.

−s(m − 1, m; \frac{k + ρ_{0}(m − 1)}{m}, −ρ_{0})) = −s(−1, m, \frac{k + ρ_{0}(m − 1)}{m} − ρ_{0}, −ρ_{0})

= s(1, m, \frac{ρ_{0} − k}{m} , −ρ_{0}).

Moreover, from the definition of Dedekind sum we obtain

s(1, m, \frac{ρ_{0} − k}{m} , −ρ_{0}) = s(1, m, \frac{k − ρ_{0}}{m} , 0) = s(1, m) + \frac{k(k − 1)}{2m} − \frac{k − 1}{2}.

Finally, by an elementary but tedious computation we get

KS_{M}(σ_{can}) = 3m − \frac{m}{3} − 2 − 8k.

The Seiberg-Witten invariant of the canonical spin^{c} structure is then

8sw_{M}^{0}(σ_{can}) = KS_{M}(σ_{can}) + 8|S_{0}^{+}| = KS_{M}(σ_{can}) + 8k = 3m − \frac{m}{3} − 2.

The coefficients of Z_{K} are labelled on the graph, where the unknown x is determined from the adjunction formula applied to the central −3-sphere; namely −3mx + 3(m − 1)x = −1, hence x = 1/3. Then Z_{K}^{2} = ∑ r_{v}(e_{v} + 2) = −r_{0} = −m/3. The number of vertices of this graph is 3m − 2 so

8p_{g} + K^{2} + #V = 3m − 2 − \frac{m}{3}.  (4)

This confirms once again the Main Conjecture.

7.7 The case m = 3 In the previous example we verified the conjecture for all m ≢ 3 (mod 6). For the other values the method given by 3.4 is not working. But this fact does not contradict the conjecture. In order to show this, we indicate briefly how one can verify the conjecture in the case m = 3 by the torsion computation.

In this case |H| = 27 and h has order 3. First consider the set of characters χ with χ(gv̄_{0}) = 1 (there are 9 altogether). They satisfy ∏_{i} χ(gv̄_{i}) = 1. If χ(gv̄_{i}) ≠ 1 for all i (2 cases), or if χ = 1 (1 case) then 𝒯̂(χ) = 0. If χ(gv̄_{i}) = 1 for exactly one index i, then the contribution in ∑_{χ} 𝒯̂(χ) is 2 for each choice of the index, hence altogether 6.

Then, we consider those characters χ for which χ(gv̄_{0}) ≠ 1 (18 cases). Then one has to compute the sum

∑ \frac{1 − ζ_{1}^{3}}{(1 − ζ_{1})(1 − ζ_{2})(1 − ζ_{1}^{−1}ζ_{2}^{−1})},

where the sum runs over ζ_{1}, ζ_{2} ∈ ℤ_{9}, ζ_{1}^{3} = ζ_{2}^{3} ≠ 1. A computation shows that this is 9. Therefore, 𝒯_{M}(1)/|H| = (6 + 9)/27 = 5/9.

The Casson-Walker invariant can be computed easily from the Seifert invariants, the result is λ(M)/|H| = −7/36. Therefore, the Seiberg-Witten invariant is 5/9+7/36=3/4. But this number equals (K^{2} + #V)/8 (cf. 7.6(4) for m = 3 and p_{g} = 0).

8 Some minimally elliptic singularities

8.1 “Polygonal” singularities. Let (X, 0) be a normal surface singularity with resolution graph given by Figure 2.

Figure 2: The resolution graph of a “polygonal” singularity

Here we assume that ν > 2. The negative definiteness of the intersection form implies that the integers (a_{1}, ⋯, a_{ν}) ∈ ℤ_{>1}^{ν} satisfy

ℓ := 2 − ν + ∑_{i=1}^{ν} \frac{1}{a_{i}} < 0.

An elementary computation shows that Z_{K} = 2E_{0} + ∑_{i=1}^{ν} E_{i}. If ν > 3 then this cycle is exactly the minimal cycle Z_{min} of Artin. If ν = 3 then the graph is not minimal, but after blowing down the central irreducible exceptional divisor one gets the identity Z_{K} = Z_{min}. In particular, (X, 0) is minimally elliptic (by Laufer’s criterion, see [21]). Hence p_{g} = 1. Moreover, by a calculation K^{2} = 8 − ∑_{i=1}^{ν} a_{1}, and thus

8p_{g} + (K^{2} + ν + 1) = 17 + ν − ∑_{i=1}^{ν} a_{i}.

Now we will compute the Seiberg-Witten invariant via 3.4. The Seifert manifold M is the unit circle bundle of the V -line bundle 𝕃_{0} with rational degree ℓ, and singularity data 1/a_{i}, 1 ≤ i ≤ ν.

The canonical V -line bundle K_{Σ} has singularity data (a_{i} − 1)/a_{i}, 1 ≤ i ≤ ν, and rational degree κ := −ℓ > 0. Note that K_{0} = −𝕃_{0} ∈ Pic_{top}^{V}(Σ). We have ρ_{0} = 1/2, n_{0} = ⌊−1/2⌋ = −1. The canonical representative of σ_{can} is the V -line bundle E_{0} = −𝕃_{0} = K_{Σ}.

Resolving the inequalities for S_{0}^{±}, one gets

S_{0}^{+} = {n𝕃_{0} ∈ ℤ; ℓ/2 ≤ (n + 1/2)ℓ < 0} = {0 · 𝕃_{0}},
S_{0}^{−} = {n𝕃_{0} ∈ ℤ; 0 < (n + 1/2)ℓ ≤ −ℓ/2} = {−1 · 𝕃_{0}}.

Hence, we have only two components 𝔐_{0}^{+}, 𝔐_{K_{Σ}}^{−}, both of dimension 0. Thus 𝔐_{0}^{∗} consists of only two monopoles. Thus (g_{0}, 0) is a good parameter. The Kreck-Stolz invariant of σ_{can} is

KS_{M}(σ_{can}) = 1 − 2ν − 8 ∑_{i=1}^{ν} s(1, a_{i}, \frac{a_{i} − 1/2}{a_{i}}, −1/2) − 4 ∑_{i=1}^{ν} s(1, a_{i}) + 2 ∑_{i=1}^{3} \frac{1}{a_{i}} .

The last identity can be further simplified using the identities from the Appendix B, namely

s(1, a_{i}, \frac{a_{i} − 1/2}{a_{i}}, −1/2) = s(1, a_{i}) = \frac{a_{i}}{12} + \frac{1}{6a_{i}} − \frac{1}{4}.

Therefore

8sw_{M}^{0}(σ_{can}) = 17 − 2ν − 12(∑_{i=1}^{ν}(\frac{a_{i}}{12} + \frac{1}{6a_{i}} − \frac{1}{4})) + 2 ∑_{i=1}^{ν} \frac{1}{a_{i}} = 17 + ν − ∑_{i=1}^{ν} a_{i}.

We have thus verified the conjecture in this case too.

8.2 A singularity whose graph is not star-shaped. All the examples we have analyzed so far had star shaped resolution graphs. In this section we consider a different situations which will indicate that the validity of the Main Conjecture extends beyond singularities whose link is a Seifert manifold. [In this subsection we will use some standard result about hypersurface singularities. For these result and the terminology, the interested reader can consult [2].]

Consider the isolated plane curve singularity given by the local equation g(x, y) := (x^{2} + y^{3})(x^{3} + y^{2}) = 0. We define the surface singularity (X, 0) as the 3-fold cyclic cover of f, namely (X, 0) is a hypersurface singularity in (ℂ^{3}, 0) given by f(x, y, z) := g(x, y) + z^{3} = 0.

The singularity (smoothing) invariants of f can be computed in many different ways. First notice that it is not difficult to draw the embedded resolution graph of g, which gives all the numerical smoothing invariants of g. For example, by A’Campo’s formula [1] one gets that the Milnor number of g is 11. Then by Thom-Sebastiani theorem (see, e.g. [2], page 60) μ(f) = 11 · 2 = 22. The signature σ(F) of the Milnor fiber of F can be computed by the method described in [30] or [31]; and it is −18. Now, by the relations 4.4 one gets p_{g}(X, 0) = 1 and K^{2} + #V = 10.

In fact, by the algorithm given in [30], one can compute easily the resolution graph of (X, 0) as well (see Figure 3).

Figure 3: The resolution graph of (x^{2} + y^{3})(x^{3} + y^{2}) = 0

Then it is not difficult to verify that the graph satisfies Laufer’s criterion for a minimally elliptic singularity, in particular this also gives that p_{g} = 1.

Using either way, finally one obtains p_{g} + (K^{2} + #V)/8 = 9/4. Using the correspondence between the characteristic polynomial Δ(t) of the monodromy action (which can be again easily computed from the Thom-Sebastiani theorem) and the torsion of H (namely that |Δ(1)| = |H|), one obtains H = ℤ_{3}.

Using the formula for the Casson-Walker invariant from the plumbing graph one gets λ(M)/|H| = −49/36.

Finally we have to compute the torsion. There are only two non-trivial characters. One of them appears one the resolution graph (i.e. χ(g_{v}) = ζ^{n_{v}} with ζ^{3} = 1). The other is its conjugate. Using the general formula for plumbing graphs, one gets 𝒯_{M,σ_{can}}(1) = 8/9.

Since 8/9 + 49/36 = 9/4, the conjecture is true.

References

[1] N A’Campo, La fonction zeta d’une monodromy, Com. Math. Helvetici, 50 (1975), 233-248.

[2] V I Arnold, S M Gusein-Zade, A N Varchenko, Singularities of Differentiable Mappings, Vol. 2 , Birkhauser, Boston, 1988.

[3] M Artin, Some numerical criteria for contractibility of curves on algebraic surfaces, Amer. J. of Math., 84(1962), 485-496.

[4] , On isolated rational singularities of surfaces, Amer. J. of Math., 88 (1966), 129-136.

[5] E Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Inventiones math. 2(1969), 1-14.

[6] W Chen, Casson invariant and Seiberg-Witten gauge theory, Turkish J. Math., 21(1997), 61-81.

[7] O Collin, N Saveliev, A geometric proof of the Fintushel-Stern formula, Adv. in Math., 147(1999), 304-314 .

[8] O Collin, Equivariant Casson invariant for knots and the Neumann-Wahl formula, Osaka J. Math., 37 (2000), 57-71.

[9] A Durfee, The Signature of Smoothings of Complex Surface Singularities, Math. Ann., 232(1978), 85-98.

[10] R Fintushel, R Stern Instanton homology of Seifert fibered homology three spheres, Proc. London Math. Soc., 61(1990), 109-137.

[11] M Furuta,B Steer, Seifert fibered homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. in Math., 96(1992), 38-102.

[12] H Grauert, Über Modifikationen und exceptionelle analytische Mengen, Math. Ann., 146 (1962), 331-368.

[13] G -M Greuel, J H M Steenbrink, On the topology of smoothable singularities, Proc. Symp. of Pure Math., 40, Part 1 (1983), 535-545.

[14] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math., 148(1998), 619-693.

[15] R E Gompf, A I Stipsicz, An Introduction to 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20, Amer. Math. Soc., 1999.

[16] H A Hamm, Exotische Sphären als Umgebungsrn̈der in speziellen komplexen Räumen, Math. Ann, 197 (1972), 44-56.

[17] F Hirzebruch, Pontryagin classes of rational homology manifolds and the signature of some affine hypersurfaces, Proceedings of the Liverpool Singularities Symposium II (ed. C.T.C. Wall), Lecture Notes in Math. 209, Springer Verlag 1971, 207-212.

[18] S Ishii, The invariant −K^{2} and continued fractions for 2-dimensional cyclic quotient singularities, preprint.

[19] M Jankins, W D Neumann, Lectures on Seifert Manifolds, Brandeis Lecture Notes, 1983.

[20] F Hirzebruch, D Zagier, The Atiyah-Singer Index Theorem and Elementary Number Theory, Math. Lect. Series 3, Publish or Perish Inc., Boston, 1974.

[21] H Laufer, On minimally elliptic singularities, Amer. J. of Math., 99 (1977), 1257-1295.

[22] H B Laufer, On μ for surface singularities, Proc. of Symp. in Pure Math., 30(1977), 45-49.

[23] C Lescop, Global Surgery Formula for the Casson-Walker Invariant, Annals of Math. Studies, vol. 140, Princeton University Press, 1996.

[24] Y Lim Seiberg-Witten invariants for 3-manifolds in the case b_{1} = 0 or 1, Pacific J. of Math., 195(2000), 179-204.

[25] E Looijenga, J Wahl, Quadratic functions and smoothing surface singularities, Topology, 25(1986), 261-291.

[26] M Marcolli, B L Wang, Seiberg-Witten invariant and the Casson-Walker invariant for rational homology 3-spheres, math.DG/0101127, Geometriæ Dedicata, to appear.

[27] T Mrowka, P Ozsvath, B Yu, Seiberg-Witten monopoles on Seifert fibered spaces, Comm. Anal. and Geom., 5(1997), 685-791.

[28] A Némethi, “Weakly” elliptic Gorenstein singularities of surfaces , Invent. math. 137 (1999), 145-167.

[29] , Five lectures on normal surface singularities, Proceedings of the summer school, Bolyai Society Mathematical Studies 8, Low Dimensional Topology, 1999.

[30] , The signature of f(x, y) + z^{n}, Proceedings of Real and Complex Singularities , (C.T.C Wall’s 60th birthday meeting), Liverpool (England), August 1996; London Math. Soc. Lecture Note Series, 263(1999), 131-149.

[31] , Dedekind sums and the signature of z^{N} + f(x, y), Selecta Math., 4(1998), 361-376.

[32] W Neumann Abelian covers of quasihomogeneous surface singularities, Proc. of Symposia in Pure Mathematics, vol. 40, Part 2, 233-244.

[33] , A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Transactions of the AMS, 268(1981), 299-344.

[34] W D Neumann, F Raymond, Seifert manifolds, plumbing, μ-invariant and orientation reserving maps, Algebraic and Geometric Topology (Proceedings, Santa Barbara 1977), Lecture Notes in Math. 664, 161-196.

[35] W Neumann, J Wahl, Casson invariant of links of singularities, Comment. Math. Helvetici, 65(1990), 58-78.

[36] , Universal abelian covers of surface singularities, math.AG/0110167.

[37] L I Nicolaescu, Adiabatic limits of the Seiberg-Witten equations on Seifert manifolds, Communication in Analysis and Geometry, 6(1998), 331-392.

[38] , Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations, Comm. Anal. Geom. 8(2000), 1027-1096.

[39] , Seiberg-Witten invariants of lens spaces, Canad J. of Math., 53(2001), 780-808.

[40] , On the Reidemeister torsion of rational homology spheres, Int. J. of Math. and Math. Sci., 25(2001), 11-17.

[41] , Seiberg-Witten invariants of rational homology spheres, math.DG/0103020.

[42] , Notes on the Reidemeister Torsion, Walter de Gruyter, to appear.

[43] H Rademacher, Some remarks on certain generalized Dedekind sums, Acta Arithmetica, 9(1964), 97-105.

[44] H Rademacher,E Grosswald, Dedekind Sums, The Carus Math. Monographs, MAA, 1972.

[45] A Ratiu, Ph. D. Thesis, Paris VII.

[46] J A Seade, A cobordism invariant for surface singularities, Proc. of Symp. in Pre Math., Vol. 40, Part 2, 479-484, 1983.

[47] J H M Steenbrink, Mixed Hodge structures associated with isolated singularities, Proc. Sumpos. Pure Math., 40, Part 2, 513-536, 1983.

[48] V G Turaev, Torsion invariants of Spin^{c} -structures on 3-manifolds, Math. Res. Letters, 4(1997), 679-695.

[49] , Surgery formula for torsions and Seiberg-Witten invariants of 3manifolds, math.GT/0101108.

[50] , Introduction to Combinatorial Torsions, Lectures in Mathematics, ETH Zurich, Birkhäuser, 2001.

[51] F Van der Blij, An invariant of quadratic forms mod 8, Indag. Math., 21(1959), 291-293.

[52] K Walker, An extension of Casson’s invariant, Annals of Math. Studies, vol. 126, Princeton University Press, 1996.

[53] S S-T Yau, On maximally elliptic singularities, Transact. AMS, 257(1980), 269-329.

A The Reidemeister-Turaev torsion for plumbings

In this section we prove Theorem 5.7, which describes the torsion 𝒯_{M} of M in terms of plumbing data. The proof has two parts.

In the first part we use Turaev’s surgery results [49] formulated in Fourier theoretic terms which will allow us to replace formal objects (elements in group algebras) by analytic ones (meromorphic functions of several variables). We obtain a first rough description of 𝒯_{M} in terms of surgery data which has a spin^{c} structure ambiguity.

In the second part, we eliminate the ambiguity about the spin^{c} structure using Turaev’s structure theorem [48, Thm. 4.2.1], and the identities 3.9 (1) and (2) from our section 3, which completely determine the spin^{c} structure from the Fourier transform of a sign-refined torsion.

A.1 The surgery data. We consider an integral surgery data: M is a rational homology 3-sphere described by the Dehn surgery on the oriented link L = L_{1} ∪ ⋯ ∪ L_{n} ⊂ S^{3} with integral surgery coefficients. We will assume that n > 1. We denote by E the complement of this link. The manifold M is obtained from E by attaching n solid tori Z_{1}, ⋯, Z_{n}. We denote by μ_{i} ∈ H_{1}(E, ℤ) the meridian of L_{i}. Similarly as in the case of plumbing, we can construct lattices G := H_{2}(M, E, ℤ) and G′ := Hom_{ℤ}(G, ℤ) and a presentation P : G → G′ for H := H_{1}(M, ℤ).

Indeed, the exact sequence 0 → H_{2}(M, E, ℤ) → H_{1}(E, ℤ) → H_{1}(M, ℤ) → 0 produces a short exact sequence 0 → G → G′ → H → 0. Moreover, we can identify G = ℤ^{n} via the canonical basis consisting of classes [D × ∗], one for each solid torus Z_{i} = D × S^{1} ; and G′ = ℤ^{n} via the canonical basis determined by the oriented meridians {μ_{i}}_{i}. Sometimes we regard P as a matrix written in these bases.

Recall that a plumbing graph provides a canonical surgery presentation in such a way that the 3-manifolds obtained by plumbing, respectively by the surgery, are the same. This presentation is the following: the components of the link L ∈ S^{3} are in one-to-one correspondence with the vertices of the graph (in particular, the index set I_{n} = {1, …, n} is identified with V); all these components are trivial knots in S^{3} ; their framings are the decorations of the corresponding vertices; two knots corresponding to two vertices connected by an edge form a Hopf link, otherwise the link is “the simplest possible”. In this way we obtain an integral surgery data in such a way that the matrix P becomes exactly the intersection matrix I.

If π : G′ → H is the natural projection, then the cores K_{i} of the attached solid tori Z_{i} determine the homology classes π(K_{i}) in H, and π(K_{i}) = π(−μ_{i}).

For every fixed i ∈ I_{n} we denote by E_{i} the manifold obtained by performing the surgery only along the knots K_{j}, j ∈ I_{n} \ {i}. Equivalently, E_{i} is the exterior of K_{i} in M. We set G_{i} := H_{2}(M, E_{i}, ℤ) and G′_{i} := H_{1}(E_{i}, ℤ). G′_{i} is generated by the set {μ_{j}}_{j=1}^{n} subject to the relations provided by the j -th columns of P for each j ≠ i. There is a natural projection G′ → G′_{i} denoted by π_{i}. Sometimes, for simplicity, we write K_{j} (j ≠ i) for its projections as well. We write also G̃ := Hom(G′, ℂ^{∗}) and G̃_{i} := Hom(G′_{i}, ℂ^{∗}). It is natural and convenient to introduce the following definition.

A.2 Definition. A surgery presentation of a rational homology sphere is called non-degenerate if the homology class π_{i}(K_{j}) has infinite order in G′_{i} for any j ≠ i.

The non-degenerate surgeries can be recognized as follows: the surgery is non-degenerate if and only if every off diagonal element of P^{−1} is nontrivial.

Indeed, the fact that for some j ≠ i the class π_{i}(μ_{j}) has finite order in G′_{i} is equivalent with the existence of n ∈ ℤ^{∗} and v⃗ ∈ G with i-th component v_{i} = 0 such that nμ_{j} = P · v⃗. But this says v_{i} = nP_{ij}^{−1} . Notice that in our case, when the matrix P is exactly the negative definite intersection matrix I associated with a connected (resolution) graph, by a well-known result, the surgery presentation is non-degenerate.

We can now begin the presentation of the surgery formula for the Reidemeister-Turaev torsion.

A.3 Proposition. Suppose that the rational homology 3-sphere M is described by a non-degenerate Dehn surgery. Fix a relative spin^{c} structure σ̃ on E. For any j, it induces a relative spin^{c} structures σ_{j} on E_{j} and a spin^{c} structure σ on M. Let 𝒯_{E_{j},σ} be the (sign-refined) Reidemeister-Turaev torsion of E_{j} determined by σ_{j}. Fix χ ∈ Ĥ \ {1} and i ∈ I_{n} such that χ(K_{i}) ≠ 1. Then the following hold.

(a) The Fourier transform 𝒯̂_{E_{i},σ} of the torsion of E_{i} extends to a holomorphic function on G̃_{i} \ {1} uniquely determined by the equality

𝒯̂_{E_{i},σ}(χ) · ∏_{j≠i}(χ^{−1}(K_{j}) − 1) = 𝒯̂_{E,σ̃}(χ), for all χ ∈ G̃_{i}.

Here 𝒯̂_{E,σ̃} is the holomorphic extension of Fourier transform of the Alexander-Conway polynomial 𝒯_{E,σ̃} of the link complement E, associated with the spin^{c} structure σ (normalized as in [49, section 8]).

(b)

𝒯̂_{M,σ}(χ) = \frac{𝒯̂_{E_{i},σ}(χ)}{χ̄(K_{i}) − 1}.

Proof G̃ is complex n-dimensional torus, and the Fourier transform of the torsion of E extends to a holomorphic function χ ↦ 𝒯̂_{E}(χ) on G̃. The elements K_{j} also define holomorphic functions on G̃ by χ ↦ χ(K_{j})^{−1} − 1. Moreover, G̃_{i} is an union of 1-dimensional complex tori and the Fourier transform of 𝒯_{E_{i}} extends a holomorphic function χ ↦ 𝒯̂_{E_{i}}(χ) on G̃_{i}. Since the elements K_{j} (j ≠ i) have infinite orders in G′_{i}, we deduce from [50, Lemma 17.1], [42, §2.5], that 𝒯̂_{E_{i}} is the unique holomorphic extension of the meromorphic function

G̃_{i} \ \{1\} ∋ χ ↦ \frac{𝒯̂_{E,σ̃}(χ)}{∏_{j≠i}(χ^{−1}(K_{j}) − 1)}.

Part (b) follows from the surgery formula [49, Lemma 5.1].

A.4 The “limit” expression. Let us now explain how we will use the above theoretical results. For each χ ∈ Ĥ \ {1} pick an arbitrary i with χ(K_{i}) ≠ 1. Then χ belongs to G̃_{i} too. The group G̃_{i} is an union of complex tori, we denote by 𝕋_{χ,i} the irreducible component containing χ. In fact, there exists w^{i} ∈ G such that 𝕋_{χ,i} = {t ∗_{w^{i}} χ; t ∈ ℂ^{∗}}, where

t ∗_{w^{i}} χ(v) := t^{v(w^{i})}χ(v) for all t ∈ ℂ^{∗} and v ∈ G′.

A possible set of “weights” w^{i} can be determined easily. G_{i} is a free Abelian group of rank 1 which injects into G. We can choose w^{i} to be an arbitrary non-trivial element of G_{i}. Obviously, w^{i} depends on the index i. In general, there is no universal choice of the index i which is suitable for any character χ.

Using the matrix notation, w^{i} can be regarded as a vector w^{i} so that Iw^{i} is an (integer) multiple of b⃗^{i}, where b⃗^{i} is the vector whose i-th entry is 1, all the other entries are zero. Then the above proposition reads as follows:

𝒯̂_{M,σ}(χ) = \frac{1}{χ̄(K_{i}) − 1} · lim_{t→1} \frac{𝒯̂_{E,σ̃}(t ∗_{w^{i}} χ)}{∏_{j≠i}(t ∗_{w^{i}} χ(K_{j})^{−1} − 1)}.

In particular, by switching the index set I_{n} to V, if χ(μ_{v}) ≠ 1 for all v, one has:

𝒯̂_{M,σ}(χ) = \frac{𝒯̂_{E,σ̃}(χ)}{∏_{v}(χ(μ_{v}) − 1)}.

A.5 The first part of the proof. According to Turaev [49], for any σ̃ ∈ Spin^{c}(E, ∂E) the Alexander-Conway polynomial 𝒯_{E,σ̃} has the form

𝒯_{E,σ̃} = g ∏_{v} (μ_{v} − 1)^{δ_{v}−1},

where g ∈ G′ depends on σ̃ by a normalization rule established by Turaev [49, section 8] (described in terms of “charges”). The generator set {g_{v}}_{v∈V} of H (defined via the plumbing) and {[μ_{v}]}_{v∈V} (defined via the surgery) can be identified as follows. (Here we will identify their Poincaré duals.) Consider a resolution X̃ → X of (X, 0) as above. The lattice inclusion I : L → L′ (i.e. H^{2}(X̃, ∂X̃, ℤ) → H^{2}(X̃, ℤ)) can be identified with the lattice inclusion P = G → G′ (i.e. H^{2}(E, ℤ) → H^{2}(E, ∂E, ℤ)). Indeed, let D^{4} be the 4-dimensional ball with boundary S^{3} with L ∈ S^{3} . Then X̃ can be obtained from D^{4} by attaching n = #V copies of 2-handles D^{2} × D^{2} . Let the union of these handles be denoted by H. Clearly, S^{3} \int(E) is a union T of solid tori. Then the isomorphism L′ → G′ is given by the following sequence of isomorphism:

H^{2}(E, ∂E) ⟵^{(1)} H^{2}(S^{3}, T) ⟵^{(2)} H^{2}(D^{4}, T) ⟵^{(3)} H^{2}(X̃, H) ⟶^{(4)} H^{2}(X̃).

Above, (1) is an excision, (2) is given by the triple (D^{4}, S^{3}, T), (3) is excision, and (4) is a restriction isomorphism. Moreover, under this isomorphism, the basis {μ_{v}}_{v} correspond exactly to the basis {D_{v}}_{v}. This also shows that g_{v} = [μ_{v}] for all v. Now, fix a character χ ∈ Ĥ, χ ≠ 1. Set v_{∗} and w^{∗} ∈ ℤ^{n} as in Theorem 5.7, i.e. with Iw^{∗} = −m^{∗}b⃗^{∗}. Then, using the notations of A.4, clearly w^{∗} ∈ ℤ^{n} = G, μ_{v} ∈ G′, and μ_{v}(w^{∗}) is exactly the v-component w_{v}^{∗} of w^{∗}. Hence t ∗_{}_{w^{∗}} χ(μ_{v}) = t^{w_{v}^{∗}}χ(g_{v}). Then by the above notations and A.4 we conclude that for any σ ∈ Spin^{c}(M)

𝒯̂_{M,σ}(χ) = χ(h) · lim_{t→1} ∏_{v∈V} (t^{w_{v}^{∗}} χ(g_{v}) − 1)^{δ_{v}−2},

for some h = h(σ) ∈ H which depends (bijectively) on σ. (Clearly, the limit is not effected by the choice of m^{∗}.) Now, notice that if we use the identity 𝒯̂_{M,σ}(χ̄) = 𝒯̂_{M,σ̄}(χ) (cf. 3.8(3)), Theorem 5.7 is equivalent with the following identity

𝒯̂_{M,σ̄}(χ) = χ̄(h_{σ}) · lim_{t→1} ∏_{v∈V}(t^{w_{v}^{∗}} χ(g_{v}) − 1)^{δ_{v}−2}.  (t)

The above discussion shows clearly that this is true, modulo the ambiguity about h_{σ}. This ambiguity (i.e. the fact that in the above expression exactly h_{σ} should be inserted) is verified via 3.9(2) (since there is exactly one h which satisfies 3.9(2) with a fixed spin^{c} structure σ).

A.6 Additional discussion about the “weights”. Before we start the

second part, we clarify an important fact about the behavior of the weights considered above. Recall that above, for a fixed χ ≠ 1, we chose v_{∗} with χ(g_{v_{∗}}) ≠ 1. This can be rather unpleasant in any Fourier formula, since for different characters we have to take different vertices v_{∗}. Therefore, we also wish to analyze the case of an arbitrary v_{0} (disregarding the fact that χ(g_{v_{0}}) is 1 or not) instead of v_{∗}.

A.7 Lemma. Fix a character χ ∈ Ĥ \ {1}.

(a) For an arbitrary vertex v_{0} , consider a vector w^{0} , with components {w_{v}^{0}}_{v}, satisfying Iw^{0} = −m^{0}b⃗^{0} for some positive m^{0} . Then the limit

lim_{t→1} ∏_{v∈V} (t^{w_{v}^{0}} χ(g_{v}) − 1)^{δ_{v}−2}

exists and it is finite.

(b) Let I_{χ} := {v : either χ(g_{v}) ≠ 1 or v has an adjacent vertex u with χ(g_{u}) ≠ 1}. Then the above limit is the same for any v_{0} ∈ I_{χ}.

Proof First we fix some notations. We say that

• a subgraph Γ′ of the plumbing graph Γ satisfies the property (P) if ∑_{Γ′}(δ_{v} −2) ≥ 0, where the sum ∑_{Γ′} runs over the vertices of Γ′ (and δ_{v} is the degree of v in Γ).

• Γ′ is a “full” subgraph of Γ if any two vertices of Γ′ adjacent in Γ are adjacent in Γ′ as well. For any subgraph Γ′, we denote by V(Γ′) its set of vertices.

• a full proper subgraph Γ′ of Γ has property (C) if it has a unique vertex (say v_{end}) which is connected by an edge of Γ with a vertex in V(Γ) \ V(Γ′). For any χ ∈ Ĥ, let Γ^{1} be the full subgraph of Γ with set of vertices {v ∈ V | χ(g_{v}) = 1}. Next, fix a character χ ∈ Ĥ \ {1} and a vertex v of Γ. Then

e_{v}g_{v} + ∑ g_{u} = 0 in H, hence χ(g_{v})^{e_{v}} · ∏ χ(g_{u}) = 1,  (1)

where the sum (resp. product) runs over the adjacent vertices u of v in Γ. Therefore, if v is in Γ^{1} then

#{u : u adjacent to v and u ∉ V(Γ^{1})} ≠ 1.  (2)

The proof of (a). We have to show that Γ^{1} satisfies (P). Let Γ^{1,c} be one of its connected components, and denote by δ_{v}^{1,c} the degree of v in Γ^{1,c}. Since Γ^{1,c} is a tree, one has ∑_{Γ^{1,c}}(δ_{v}^{1,c} − 2) = −2. Since Γ^{1,c} is a proper subgraph of the connected graph Γ, there exists at least one edge of Γ which is not an edge of Γ^{1,c}, but it has one of its endvertices in Γ^{1,c}. In fact, (2) shows that there are at least two such edges. Therefore, Γ^{1,c} satisfies (P).

The proof of (b). First we claim the following fact.

(F) Let Γ′ be a full proper subgraph of Γ which satisfies (C). Then for any

v_{0} ∈ (V(Γ) \ V(Γ′)) ∪ {v_{end}}

the solution {w_{v}^{0}}_{v} of Iw^{0} = −m^{0}b⃗^{0} has the following special property: the subset {v_{v}^{0}}_{v∈V(Γ′)} , modulo a multiplicative constant, is independent of the choice of v_{0} .

Indeed, the subset {v_{v}^{0}}_{v∈V(Γ′)} , modulo a multiplicative constant, is completely determined by the set of relations of type (1) considered for vertices v ∈ V(Γ′) \ {v_{end}}. Since the intersection form associated with Γ′ is non-degenerate, this system has a maximal rank. Now, we make a partition of V(Γ^{1,c}) (cf. part (a) for the notation). Each set S of the partition defines a full subgraph Γ^{1,c,j} of Γ^{1,c} with S = V(Γ^{1,c,j}). The partition is defined in such a way that each Γ^{1,c,j} is a maximal subgraph satisfying both properties (P) and (C). One way to construct such a partition is the following.

Let us start with Γ^{1,c}. By (a), it satisfies (P). If it does not satisfies (C), then take two of its vertices, both having adjacent vertices outside of Γ^{1,c}. Eliminate next all the edges of Γ^{1,c} situated on the path connecting these two vertices, and then, if necessary, repeat the above procedure for the connected components of the remaining graph. After a finite number of steps all the connected components will satisfy both properties (P) and (C).

Now, fact (F) can be applied for all these subgraphs Γ^{1,c,j}. In the limit we regroup the product corresponding to the subsets V(Γ^{1,c,j}), and the result follows.

A.8 The second part of the proof. Preliminaries. Our next goal is to

show that the right hand side of (t) satisfies the formulae 3.9(1) and (2) for the spin^{c} structure σ̄. This clearly ends our proof.

For this, let us fix a vertex v_{0} ∈ V and we plan to verify 3.9 (1) and (2) for h = g_{v_{0}} . In the sequel we prefer to fix m^{0} in the equation of v_{0} , namely we let m > 0 be the smallest positive integer so that

Iw^{0} = −mb⃗^{0} has a solution w^{0} = {w_{v}^{0}}_{v∈V} ∈ ℤ^{#V}.  (3)

Clearly gcd({w_{v}^{0}}_{v}) = 1 (and each w_{v}^{0} > 0, fact not really important here). For a non-trivial character χ ∈ Ĥ with χ(g_{v_{0}}) ≠ 1, the vertex v_{0} is a good candidate for v_{∗} (or, at least, the weights w^{∗} in (t) can be replaced by the weights w^{0} since they provide the same limit, cf. A.7). But for characters χ with χ(g_{v_{0}}) = 1 the limit in A.7 (consider for v_{0} ) can be different from the limit needed in (t) (where one has v_{∗}). Nevertheless, the products of these (probably different) limits with χ(g_{v_{0}}) − 1 are the same (namely zero) (and in 3.9(1) and (2) we need only these type of products !). More precisely, for any χ ∈ Ĥ \ {1}:

(χ(g_{v_{0}}) − 1) · lim_{t→1} ∏_{v∈V} (t^{w_{v}^{∗}} χ(g_{v}) − 1)^{δ_{v}−2}= (χ(g_{v_{0}}) − 1) · lim_{t→1} ∏_{v∈V} (t^{w_{v}^{0}} χ(g_{v}) − 1)^{δ_{v}−2}.

Therefore, in all our verifications, we can use only one set of weights, namely w^{0} = {w_{v}^{0}}_{v}, given exactly by the vertex v_{0} , and this is good for all χ ∈ Ĥ \ {1}. In the sequel we drop the upper index 0, and we simply write w_{v} instead w_{v}^{0} . Let us introduce the notation

R̂_{χ}(t) := ∏_{v∈V} (t^{w_{v}} χ(g_{v}) − 1)^{δ_{v}−2}.

We have to show that lim_{t→1} χ̄(h_{σ})R̂_{χ}(t) satisfies the formulae 3.9(1) and (2) for the spin^{c} structure σ̄. Since in these formulae we need the product of this limit with χ(g_{v_{0}}) − 1, we set δ̄_{v} := δ_{v} for any v ≠ v_{0} , but δ̄_{v_{0}} := δ_{v_{0}} + 1, and define

P̂_{χ}(t) := R̂_{χ}(t) · (t^{w_{v_{0}}} χ(g_{v_{0}}) − 1)= ∏_{v∈V} (t^{w_{v}} χ(g_{v}) − 1)^{δ̄_{v}−2}.

In the case of the trivial character χ = 1, we define Δ(t) via the identity:

\frac{Δ(t)}{t − 1} := P̂_{1}(t) = ∏_{v∈V} (t^{w_{v}} − 1)^{δ̄_{v}−2}.  (4)

Since ∑_{v}(δ̄_{v} − 2) = −1, one gets that Δ(t) has no pole or zero at t = 1, in fact:

Δ(t) = ∏_{v∈V} (t^{w_{v}−1} + ⋯ + t + 1)^{δ̄_{v}−2}.  (5)

Let L_{0} be a fixed generic fiber of the S^{1} -bundle over E_{v_{0}} used in the plumbing construction of M (cf. 2.13). Set G_{0} := H_{1}(M \ L_{0}, ℤ).

The reader familiar with the theorem of A’Campo about the zeta function associated with the monodromy action of a Milnor fibration, certainly realizes that P̂_{1}(t) is such a zeta function, and Δ(t) is a characteristic polynomial of a monodromy operator. The next proof will not use this possible interpretation. Nevertheless, in A.10 we will show that Δ(t) ∈ ℤ[t], and Δ(1) is the order of the torsion subgroup of G_{0} .

Since H_{2}(M, M \ L_{0}, ℤ) = ℤ, one has the exact sequence:

0 → ℤ →^{i} G_{0} →^{p} H → 0,

where i(1_{ℤ}) = g̃_{∞}:=the homology class in M \ L_{0} of the meridian of L_{0} viewed as a knot in M. Let g̃_{v} be the homology class in G_{0} of ∂D_{v}, defined similarly as g_{v} ∈ H, cf. 2.13. Obviously, {g̃_{v}}_{v∈V} is a generator set for G_{0} . Define φ : G_{0} → ℤ by g̃_{v} ↦ w_{v}. The equations (3) guarantee that this is well-defined. Moreover, since gcd({w_{v}}_{v}) = 1, φ is onto. Then clearly, its kernel T is exactly the subgroup of torsion elements of G_{0} . Let j : T → G_{0} be the natural inclusion. Again by (3), φ(g̃_{∞}) = m, hence the composition φ ∘ i in multiplication by m. These facts can be summarized in the following diagram (where r is induced by φ):

T ⟷^{1} T
↓j  ↓j′
ℤ ↪^{i} G_{0} ⟶^{p} H
↕1  ↓φ  ↓r
ℤ ↪^{·m} ℤ ⟶^{q} ℤ_{m}

It is convenient to identify ℤ_{m} with a subgroup of ℚ/ℤ via ℤ_{m} ∋ â ↦ \frac{a}{m} ∈ ℚ/ℤ.

A.9 Lemma. For any h ∈ H, r(h) = b_{M}(g_{v_{0}}, h) via the above identification.

Proof It is enough to verify the identity for each g_{v}, v ∈ V. In that case, r(g_{v}) = q(φ(g̃_{v})) = ŵ_{v} = w_{v}/m ∈ ℚ/ℤ. But by 2.2 and A.8(3), b_{M}(g_{v_{0}}, g_{v}) = −(μ_{v_{0}}, μ_{v})_{ℚ} = −μ_{v}(I^{−1}μ_{v_{0}}) = w_{v}/m as well.

Fix g ∈ G_{0} so that φ(g) = 1_{ℤ}. This provides automatically a splitting of the exact sequence 0 → T → G_{0} → ℤ → 0, i.e. a morphism s : G_{0} → T with s ∘ j = id_{T} and s(g) = 1_{T}. In the sequel, we extend any morphism and character to the corresponding group-algebras over ℤ (and we denote them by the same symbol). For any character ξ of G_{0} we define the representation ξ_{t} : G_{0} → ℂ[[t, t^{−1}]]^{∗} given by ξ_{t}(x) = ξ(x)t^{φ(x)} . (This can be identified with a family of characters. Indeed, for any fixed t ∈ ℂ^{∗} and character ξ, one can define the character ξ_{t} given by x ↦ ξ(x)t^{φ(x)} . E.g., for ξ = p∘χ, ξ_{t} is just a more convenient notation for the action t ∗_{}_{w^{0}} χ, cf. A.4 and A.5.)

Now, the point is that the identity (4) has a generalization in the following sense.

A.10 Theorem. For any character ξ ∈ Ĝ_{0} define

P̂_{ξ}(t) := ∏_{v∈V}(ξ_{t}(g̃_{v}) − 1)^{δ̄_{v}−2}= ∏_{v∈V}(t^{w_{v}} ξ(g̃_{v}) − 1)^{δ̄_{v}−2}.

Then there exist an element δ ∈ ℤ[G_{0}] such that the following hold.

(a) for any ξ ∈ Ĝ_{0}

P̂_{ξ}(t) = \frac{ξ_{t}(δ)}{ξ_{t}(g) − 1}.

(b) 1_{t}(δ) = Δ(t); 1(δ) = 𝔞𝔲𝔤(δ) = Δ(1) = |T |.

(c) s(δ) = Σ_{T}, where Σ_{T} := ∑_{x∈T} x ∈ ℤ[T].

Proof From the first part of the proof (cf. A.3.(b) and A.5(t)) follows that lim_{t→1} P̂_{ξ}(t), modulo a multiplicative factor of type ∓ξ(x), for some x ∈ G_{0} , is the Fourier transform of the Reidemeister-Turaev torsion 𝒯 on M \ L_{0} associated with some spin^{c} structure (whose identification is not needed here). By [48, 4.2.1], 𝒯−Σ_{T} /(1−g) ∈ ℤ[G_{0}], identity valid in Q(G_{0}), the ring of quotients of ℚ[G_{0}]. By the first statement P̂_{ξ}(t) = ∓ξ_{t}(x𝒯) for any ξ ∈ Ĝ_{0} . Hence, for some A ∈ ℤ[G_{0}] one has:

P̂_{ξ}(t) = ξ_{t}(A) ± ξ_{t}(\frac{x · Σ_{T}}{g − 1} ).  (∗)

This identity multiplied by ξ_{t}(g − 1), for ξ = 1 and t → 1, and via (4), provides Δ(1) = ±|T |. By (5), Δ(1) is positive, hence in (∗) ±1 = +1. Moreover, Δ(1) = |T |. Now, if one defines δ := A(g−1)+xΣ_{T}, then (a) and (c) follow easily, and 1_{t}(δ) = Δ(t) is exactly (4).

In order to verify 3.9(1) and (2), we will apply the above theorem to special characters of the type ξ = χ ∘ p, where χ ∈ Ĥ. It is clear that for any y ∈ ℤ[G_{0}] and h ∈ H, the sum ∑ χ(h) · χ ∘ p(y) = ∑ χ(h · p(y)) over χ ∈ Ĥ is an integer multiple of |H|. Hence:

\frac{1}{|H|} ∑_{χ∈Ĥ}′χ(h) · χ ∘ p(y) = −\frac{1}{|H|}1(h) · 1(y) = −\frac{1}{|H|}𝔞𝔲𝔤(y) (mod ℤ).  (6)

Using the splitting of G_{0} into T × ℤ given by g, one can easily verify that in Q(G_{0})

\frac{y − s(y)}{g − 1} ∈ ℤ[G_{0}] for any y ∈ ℤ[G_{0}].  (7)

In the sequel, we write simply χ_{t} for (χ ∘ p)_{t}.

A.11 Verification of 3.9(1). Now we will verify that χ̄(h)R̂_{χ}(t) (χ ∈ Ĥ \{1}) satisfies 3.9(1) for h = h_{σ} (in fact, for any h ∈ H). For this, fix a vertex v_{0} , and g ∈ G_{0} with φ(g) = 1 as above. Take an arbitrary x ∈ G_{0} . Then we have to show that

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h)R̂_{χ}(t)(t^{w_{v_{0}}} χ(g_{v_{0}}) − 1)(χ_{t}(x) − 1) = −b_{M}(g_{v_{0}}, p(x)) (mod ℤ). (8)

Via A.10, the left hand side of (8) is

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h) · χ_{t}(δ) · \frac{χ_{t}(x) − 1}{χ_{t}(g) − 1}.  (9)

Set a := φ(x). Since s(δx) = s(δ), (9) transforms as follows (use (6), (7) and A.10):

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h)·χ_{t}(\frac{δx − s(δx)}{g − 1}−\frac{δ − s(δ)}{g − 1}) = −\frac{1}{|H|} lim_{t→1} 1(h)·1_{t}(\frac{δx − s(δx)}{g − 1}−\frac{δ − s(δ)}{g − 1})

= −\frac{1}{|H|} lim_{t→1} \frac{Δ(t)t^{a} − |T | − Δ(t) + |T |}{t − 1} = −\frac{1}{|H|}Δ(1)a = −\frac{a}{m} (mod ℤ).

But the right hand side of (8), via A.9, is the same −a/m ∈ ℚ/ℤ.

A.12 Verification of 3.9(2). Now we will verify

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h_{σ})R̂_{χ}(t)(t^{w_{v_{0}}} χ(g_{v_{0}}) − 1) = −𝔮^{c}(σ̄)(g_{v_{0}}) (mod ℤ).  (10)

The left hand side is

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h_{σ})\frac{χ_{t}(δ)}{χ_{t}(g) − 1}.

The fraction in this expression can be written as (cf. (7))

χ_{t}(\frac{δ − s(δ)}{g − 1}) + \frac{χ(s(δ))}{χ_{t}(g) − 1}.  (11)

This sum-decomposition provides two contributions. The first via (6), (7) and A.10 gives:

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h_{σ}) · χ_{t}(\frac{δ − s(δ)}{g − 1}) = −\frac{1}{|H|} lim_{t→1} \frac{Δ(t) − Δ(1)}{t − 1} = −\frac{1}{|H|}Δ′(1) (mod ℤ),

where Δ′(t) denoted the derivative of Δ with respect to t. On the other hand, cf. (5),

\frac{Δ′(t)}{Δ(t)} = ∑_{v}(δ̄_{v} − 2)\frac{(t^{w_{v}−1} + ⋯ + t + 1)′}{t^{w_{v}−1} + ⋯ + t + 1} ,

hence

\frac{Δ′(1)}{Δ(1)} = \frac{1}{2} ∑_{v}(δ̄_{v} − 2)(w_{v} − 1).

Since Δ(1) = |T | = |H|/m, the first contribution is

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h_{σ}) · χ_{t}(\frac{δ − s(δ)}{g − 1}) = −\frac{1}{2m} ∑_{v}(δ̄_{v} − 2)(w_{v} − 1).

For the second contribution, notice that χ(s(δ)) = χ(Σ_{T}) is zero unless χ is in the image of r̂ : ℤ̂_{m} → Ĥ; if χ is in this image then χ(Σ_{T}) = |T |. For any χ ∈ ℤ̂_{m} we write χ(ˆ1) = ζ. Assume that r(h_{σ}) = −â (or equivalently, r(h_{σ}) = −\frac{a}{m} ∈ ℚ/ℤ). Then

\frac{1}{|H|} lim_{t→1} ∑_{χ∈Ĥ}′χ̄(h_{σ}) · \frac{χ(Σ_{T})}{χ_{t}(g) − 1} = \frac{1}{|H|} lim_{t→1} ∑_{χ∈ℤ̂_{m}}′χ̄(h_{σ}) · \frac{|T |}{χ_{t}(g) − 1} = \frac{1}{m} ∑_{ζ∈ℤ_{m}}′ζ^{a} · \frac{1}{ζ − 1}.

Since

\frac{1}{m} ∑_{ζ∈ℤ_{m}} \frac{ζ^{a} − 1}{ζ − 1} = 0 (mod ℤ),

one gets that the second contribution is (cf. B.6):

\frac{1}{m} ∑_{ζ∈ℤ_{m}}′ζ^{a} · \frac{1}{ζ − 1} = −\frac{a}{m} + \frac{1}{m} ∑_{ζ∈ℤ_{m}}′ \frac{1}{ζ − 1} = −\frac{a}{m} − \frac{1}{2m}(m − 1) (mod ℤ).

Therefore, the left hand side of (10), modulo ℤ, is

−\frac{1}{2m} ∑_{v}(δ̄_{v} − 2)(w_{v} − 1) − \frac{a}{m} − \frac{1}{2m}(m − 1).

Notice that ∑_{v}(δ̄_{v} − 2) = −1. Moreover w_{v} = −mI_{vv_{0}}^{−1} , and the coefficient r_{v_{0}} of Z_{K} equals 1 − ∑_{v}(δ_{v} − 2)I_{vv_{0}}^{−1} (cf. 5.2), hence the above expression can be transformed into

−\frac{1}{2} − \frac{1}{2m} ∑_{v}(δ̄_{v} − 2)w_{v} − \frac{a}{m} = −\frac{1}{2}r_{v_{0}} + \frac{1}{2}I_{v_{0}v_{0}}^{−1} − \frac{a}{m}.

Now, let us compute the right hand side of (10). Since h_{σ}·σ_{can} = σ one has 2h_{σ}+Z_{K} = c(σ). Then the characteristic element which provides σ̄ is −c(σ) = −2h_{σ} − Z_{K}. Therefore

−𝔮^{c}(σ̄)(g_{v_{0}}) = \frac{1}{2}(D_{v_{0}} −Z_{K} −2h_{σ}, D_{v_{0}})_{ℚ} = \frac{1}{2}(∑_{v} I_{vv_{0}}^{−1}E_{v} −∑_{v} r_{v}E_{v}, D_{v_{0}})_{ℚ}−(h_{σ}, D_{v_{0}})_{ℚ}

= \frac{1}{2}I_{v_{0}v_{0}}^{−1} − \frac{1}{2}r_{v_{0}} − (h_{σ}, D_{v_{0}})_{ℚ}.

But, using 2.2 and A.9, (h_{σ}, D_{v_{0}})_{ℚ} = −b_{M}(h_{σ}, g_{v_{0}}) = −r(h) = \frac{a}{m}. This proves A.12(10). At this point we invoke the following elementary fact.

Suppose q_{1} , q_{2} are two quadratic functions on the finite abelian group H associated with the bilinear forms b_{1}, b_{2} ; and S ⊂ H is a generating set such that q_{1}(s) = q_{2}(s) and b_{1}(s, h) = b_{2}(s, h) for all s ∈ S and h ∈ H. Then q_{1}(h) = q_{2}(h) for all h ∈ H.

Using A.11(8), A.12(10) and the above fact we obtain 3.9(2), for any h. The identity 3.9(2) implies that χ̄(h_{σ})R̂_{χ}(t) = 𝒯̂_{M,σ̄}(χ). This concludes the proof of Theorem 5.7.

[Notice that, in fact, we verified even more. First recall, cf. [49], that the sign of the (sign-refined) torsion is decided by universal rules. In some cases its identification is rather involved. The point is that the above verification also reassures us that in (t) we have the right sign.]

B Basic facts concerning the Dedekind-Rademacher sums

In this Appendix we collected some facts about (generalized) Dedekind sums which constitute a necessary minimum in the concrete computation of the Seiberg-Witten invariants (and in the understanding of the relationship between Dedekind sums and Fourier analysis). Let ⌊x⌋ be the integer part of x, and {x} := x − ⌊x⌋ its fractional part. In the paper [43], Rademacher introduces for every pair of coprime integers h, k and any real numbers x, y the following generalization of the classical Dedekind sum

s(h, k; x, y) = ∑_{μ=0}^{k−1} ((\frac{μ + y}{k} )) ((\frac{h(μ + y)}{k} + x)) = −s(−h, k; −x, y),

where ((x)) denotes the Dedekind symbol

((x)) = \{ \begin{matrix}\{x\} − 1/2 & if & x ∈ ℝ \ ℤ \\ 0 & if & x ∈ ℤ. \end{matrix}

A simple computations shows that s(h, k; x, y) depends only on x, y (mod 1). Additionally

s(h, k; x, y) = s(h − mk, k; x + my, y).

Moreover, have the following result

s(1, k; 0, y) = \{ \begin{matrix}\frac{k}{12} + \frac{1}{6k} − \frac{1}{4} = \frac{(k−1)(k−2)}{12k} & y ∈ ℤ \\ \frac{k}{12} + \frac{1}{k} B_{2}(\{y\}) & y ∈ ℝ \ ℤ , \end{matrix}  (B.1)

where B_{2}(t) = t^{2} − t + 1/6 is the second Bernoulli polynomial. If x = y = 0 then we simply write s(h, k). Perhaps the most important property of these Dedekind-Rademacher sums is their reciprocity law which makes them computationally very friendly: their computational complexity is comparable with the complexity of the classical Euclid’s algorithm. To formulate it we must distinguish two cases.

• Both x and y are integers. Then

s(h, k) + s(k, h) = −\frac{1}{4} + \frac{h^{2} + k^{2} + 1}{12hk}.  (B.2)

• x and/or y is not an integer. Then

s(h, k; x, y) + s(k, h; y, x)
= ((x)) · ((y)) + \frac{h^{2}ψ_{2}(y) + ψ_{2}(hy + kx) + k^{2}ψ_{2}(x)}{2hk}  (B.3)

where ψ_{2}(x) := B_{2}({x}). In particular, if x, y ∈ ℝ are not both integers we deduce

s(1, m; x, y) = −((x))·((mx+y))+((x))((y))+ \frac{ψ_{2}(y) + ψ_{2}(y + mx) + m^{2}ψ_{2}(x)}{2m}. (B.4)

An important ingredient behind the reciprocity law is the following identity ([43, Lemma 1])

∑_{μ=0}^{k−1} ((\frac{μ + w}{k} )) = ((w)) for any w ∈ ℝ.  (B.5)

The various Fourier-Dedekind sums we use in this paper can be expressed in terms of Dedekind-Rademacher sums. This follows from the identity ([20, p.170])

\frac{1}{p} ∑_{ζ^{p}=1}′ \frac{ζ^{t}}{1 − ζ} = ((\frac{2t − 1}{2p})) , for all p, q ∈ ℤ, p > 1.  (B.6)

In other words, the function

\{ζ^{p} = 1\} ⊂ ℂ^{∗} → ℂ, ζ ↦ \{ \begin{matrix}0 & if & ζ = 1 \\ \frac{1}{1−ζ} & if & ζ ≠ 1 \end{matrix}

is the Fourier transform of the function

ℤ_{p} → ℂ, t̂↦ ((\frac{2t − 1}{2p})) .

The identity (B.6) implies that

\frac{1}{p} ∑_{ζ^{p}=1}′ \frac{ζ^{tq′}}{1 − ζ^{q}} = ((\frac{2tq′ − 1}{2p})) , for all p, q ∈ ℤ, p > 1, (p, q) = 1,

where q′ = q^{−1} (mod p). Using the fact that Fourier transform of the convolution product of two functions ℤ_{p} → ℂ is the pointwise product of the Fourier transforms of these functions we deduce after some simple manipulations the following identity.

\frac{1}{p} ∑_{ζ^{p}=1}′ \frac{ζ^{t}}{(ζ − 1)(ζ^{q} − 1)} = −s(q, p; \frac{q + 1 − 2t}{2p}, −\frac{1}{2}).  (B.7)

If t = 0 then by (B.5) (and a computation), or by [44, 18a], one has

\frac{1}{p} ∑_{ζ^{p}=1}′ \frac{1}{(ζ − 1)(ζ^{q} − 1)} = −s(q, p) + \frac{p − 1}{4p}.  (B.8)

By setting q = −1 and t = 0 in the above equality we deduce

\frac{1}{p} ∑_{ζ^{p}=1}′ \frac{1}{|ζ − 1|^{2}} = −s(−1, p, 0, −1/2) = s(1, p, 0, 1/2) =^{(B.1)} \frac{p}{12} − \frac{1}{12p}.  (B.9)

The Fourier transform of the function 𝔡_{p,q} : ℤ_{p} → ℂ, t̂↦ ((qt/p)) is the function (see [44, Chap. 2, Sec. C])

\{ζ^{p} = 1\} → ℂ, ζ ↦ \{ \begin{matrix}\frac{1}{2} − \frac{ζ^{q}}{ζ^{q}−1} & if & ζ ≠ 1 \\ 0 & if & ζ = 1. \end{matrix}

Then

s(−q, p) = ∑_{t+s=0 (mod p)} 𝔡_{p,1}(t)𝔡_{p,q}(s) = (𝔡_{p,1} ∗ 𝔡_{p,q})(0)

= \frac{1}{p} ∑_{ζ^{p}=1}′(\frac{1}{2} − \frac{ζ}{ζ − 1})(\frac{1}{2} − \frac{ζ^{q}}{ζ^{q} − 1}).

This implies (cf. also with [44])

\frac{1}{p} ∑_{ζ^{p}=1}′ \frac{ζ + 1}{ζ − 1} · \frac{ζ^{q} + 1}{ζ^{q} − 1} = −4s(q, p).  (B.10)