Algebraic torsion, zeta function and Dirichlet series for graph links in homology 3-spheres

András Némethi

October 11, 1997

Abstract: In this paper, we define the algebraic torsion τ associated with an S-constructible sheaf F^{•} on a topological space X and a ring homomorphism π_{1}(X) → Z (with an acyclicity condition). If (X, S) has a fiber structure over S^{1}, then τ is equal to the zeta function of the monodromy acting on the hypercohomology of the fiber. If X is the complement of a link K_{1} in a homology 3-sphere and S is given by a link K_{2}, such that K_{1} ∪ K_{2} admits a fiberable splice diagram, then this zeta function has a product decomposition (corresponding to the Jaco-Shalen-Johansson decomposition). This can be interpreted as a “Lefschetz type formula” for a dynamical system suitably chosen.

1 Introduction

The aim of this paper is to show that the reduction formula of the zeta function associated with a fiberable graph link, with coefficients in a constructible sheaf ([21, Theorem (2.8)]), can be interpreted as a “Lefschetz type formula: τ_{ρ} = Z_{ρ}(0) ” [6, 7, 8, 16] for a “Lefschetz flow” suitably chosen.

The Whitehead torsion is a direct generalization of the Reidmeister, Franz and de Rham torsion invariant [14]. Moreover, using different acyclic representations ρ , we derive different R-torsion invariants τ_{ρ}(X) [14]. In particular, for the cyclic representation ρ : π_{1}(X) → Z → Q(λ) we recapture the zeta function of the corresponding monodromy (or the Alexander polynomial) as an R-torsion.

In the second section and in (3.11), we extend the definition of the Rtorsion: we define an invariant τ(X, F^{•}, h) associated with a topological space X with a stratification S , with an S-constructible sheaf F^{•} ∈ D_{c}^{b}(X) (defined over a subring R of C ) and a ring homomorphism h : π_{1}(X) → Z. This algebraic torsion has the (test) property: if (X, S) is a locally trivial fibering p over S^{1} with fiber F, and h = π_{1}(p), then τ(X, F^{•}, h) is equal to the zeta function ζ_{(F^{•},h)} of the algebraic monodromy acting on the hypercohomology group H^{•}(F, F^{•}) (section 3).

Let Σ be a homology 3-sphere, K ⊂ Σ a link in Σ, F^{•} ∈ D_{c}^{b}(Σ − K) an object of the category of the bounded complexes, with constructible cohomologies, such that the corresponding stratification S = {S_{i}}_{i∈{0}∪I} is of type:

S_{0} = Σ − (K ∪ ⋃_{i∈I} K_{i}) ; S_{i} = K_{i} i ∈ I

where K_{i} are (virtual) link components in Σ. Suppose that (Σ, K ∪ ⋃_{i} K_{i}) admits a fiberable splice diagram [3] (or a plumbing diagram). Then, the associated zeta function ζ_{(F^{•},h)} has a product decomposition, corresponding to the Jaco–Shalen–Johannson decomposition, with factors of type

det(1 − λ^{m(K)}ρ_{F^{•}}^{q}(K)) = det(1 − ρ_{π}(K) · ρ_{F^{•}}^{q}(K))

(see [21] and section 4)

This reduction formula was extremely useful in the zeta function computations of hypersurface complex singularities [21].

We note that there is a strong similarity between this product decomposition and the Ruelle-type zeta functions R_{ρ}(s) evaluated in s = 0. (R_{ρ}(s) is a product over prime closed geodesics γ of factors det(1 − ρ(γ)e^{−s·l(γ)}).) David Fried [8] proved that for X hyperbolic, the square of the Ray-Singer analytic torsion T_{ρ}(X) is equal to |R_{ρ}(0)^{±1}| for an orthogonal representation ρ. But, by Cheeger [2] and Müller [17], T_{ρ}(X) = τ_{ρ}(X) for acyclic and orthogonal representations. Thus, our decomposition theorem corresponds exactly to the Fried’s result. (See also [4] and [22]).

On the other hand, there is a remarkable connection between the Rtorsion and the Weil-type zeta function of a dynamical system. This idea was investigated by Fried [7], who constructed, for any smooth flow and flat bundle, a formal Dirichlet-type function (see section 5) counting the periodic orbits of the flow. In many important cases this function has an analytic extension at zero and its value in the origin is exactly the corresponding R-torsion [7, 16].

In section 6, we construct a vector field on Σ − K, such that the formal Dirichlet function (which in our case is analytic), evaluated in zero, corresponds to the product decomposition of the reduction formula. In particular , the set of the closed orbits can be parametrized by a well-determined splice diagram. Therefore, our reduction formula can be interpreted as a ”Lefschetz-type formula”.(In our case, the coefficient sheaf is more complicated, the space Σ − K is not compact, but we do not need regularization procedure.)

We wish to thank Professors H. Moscovici and R. J. Stanton for various enlightening remarks and references in connection with our work.

2 The algebraic-torsion of constructible sheafs

2.1. R will denote a subring of R (or of C) and A^{∗} will denote the group of invertible elements of a ring A.

Let X be a topological space (embedded in a space R^{N}) with a Whitney stratification S = {S_{i}}_{i∈I}. In particular each S_{i} is a manifold. Let F be an S-constructible sheaf on X, such that each stalk F_{x} (x ∈ X) is a finitely generated free R-module. In particular, the restrictions F|S_{i} are local systems for each i ∈ I.

The geometric model for F is given by the free part of R^{q}f_{∗}R_{Y} for suitable functions f : Y → X.

2.2. Suppose that X admits a triangulation T adapted to the stratification S (i.e. every stratum S_{i} is a union of interiors of simplices of T)[11]. The existence of T is automatically satisfied if we work with algebraic structures.

2.3. We will denote by Δ a closed cell, Δ^{o} denotes its interior. If dimΔ = 0, then, by definition, Δ^{o} = Δ. For each cell Δ, we define the neighbourhood N(Δ) of Δ^{o} by:

N(Δ) = ⋃_{Δ_{1}∩Δ^{o}≠∅} Δ_{1}^{o}

We have the following 2.4 Properties:

a) If Δ_{1} ⊂ ∂Δ_{2} then N(Δ_{2}) ⊂ N(Δ_{1}); b) {N(Δ)}_{Δ} is an open covering of X;

c) The inclusion Δ ⊂ N(Δ) admits a strong deformation retract (in particular N(Δ) is contractible);

d) If Δ^{o} ⊂ S_{i} then F(N(Δ)) = F|S_{i}(Δ^{o}) (use [10, p. 151]);

e) For each Δ, F(N(Δ)) is a finitely generated free R-module. If Δ_{1} ⊂ ∂Δ_{2}, then the restriction map ρ_{Δ_{1}}^{Δ_{2}} : F(N(Δ_{1})) → F(N(Δ_{2})) can be identified with an element of HomR(R^{N_{1}}, R^{N_{2}}).

2.5. We introduce an orientation on each cell Δ. The boundary ∂Δ will inherit its natural induced orientation. If Δ′ ⊂ ∂Δ, then we define

ϵ_{Δ′}^{Δ} = \{ \begin{matrix}1 & if the orientation of Δ′ coincides with the induced orientation \\ −1 & otherwise \end{matrix}

This is summarized in the algebraic relation ∂Δ = ∑_{Δ′⊂∂Δ} ϵ_{Δ′}^{Δ} · Δ′. 2.6. We define the free R-modules

C^{q}(X, F) = ⊕F(N(Δ)) (the sum over the q-cells Δ)

and the operators δ_{F}^{q} : C^{q}(X, F) → C^{q+1}(X, F) by

δ_{F}^{q}(∑_{Δ′} v_{Δ′}) = ∑_{Δ′} ∑_{Δ:Δ′⊂∂Δ} ϵ_{Δ′}^{Δ}ρ_{Δ′}^{Δ}(v_{Δ′}).

Then δ∘δ = 0. In fact, (C^{∗}(X, F), δ) is the Čech-complex (C^{∗}(M, F), δ) corresponding to the covering M = {N(Δ)}_{Δ}; in particular H^{∗}(C^{∗}(X, F), δ) = H^{∗}(X, F) (see [10, p. 208-209]).

2.7. Suppose that we have a ring homomorphism h : π_{1}(X) → Z. This representation defines a locally constant sheaf L with stalk R[Z] = R[λ, λ^{−1}] by the inclusion Z ↪ AutR[Z], n ↦ { multiplication by λ^{n}}. We denote by _{L}ρ_{Δ′}^{Δ} the restriction map L(N(Δ′)) → L(N(Δ)) (Δ′ ⊂ Δ). Then, we can define:

C^{q}(X, F, h) = ⊕F(N(Δ)) ⊗R R[Z] ( the sum over the q-cells)

and δ_{F,h}^{q} : C^{q}(X, F, h) → C^{q+1}(X, F, h) by

δ_{F,h}^{q}(∑_{Δ′} v_{Δ′} ⊗ P_{Δ′}) = ∑_{Δ′} ∑_{Δ′⊂∂Δ} ϵ_{Δ′}^{Δ}ρ_{Δ′}^{Δ}(v_{Δ′}) ⊗ _{L}ρ_{Δ′}^{Δ}(P_{Δ′})

Obviously δ ∘ δ = 0. Let R_{Q} be the quotient field of R. Then the inclusion R[Z] = R[λ, λ^{−1}] ↪ R_{Q}(λ) gives a R_{Q}(λ)-complex (C^{∗}(X, F, h) ⊗_{R[Z]} R_{Q}(λ), δ). We note that this complex, in general, is not acyclic. 2.8. Example: Take X = S^{1} ∨ S^{1} and F = C_{X}. The fundamental group < g_{1}, g_{2} :> of X is free and is generated by the two circles. Define h : π_{1}(X) ⟶ Z by h(g_{1}) = 1 and h(g_{2}) = 0. Then H^{1}(C^{∗} ⊗ C(λ)) = C(λ).

2.9. The definition of the torsion of a complex, in the Milnor paper [14] , is given for complexes C_{n} → … → C_{0}. This can be modified for complexes with δ-operator with degree +1. For the convenience of the reader, we give the definition (but only for vector spaces over a field F).

Let (C^{∗}, δ^{∗}) be an acyclic complex (δ of degree +1) with preferred bases c^{q} in C^{q}. Take a base b^{q} in B^{q} = imδ^{q+1}. Then, the torsion is defined by

τ(C^{∗}) = ∏_{q} det[b^{q}b^{q+1}/c^{q}]^{(−1)^{q+1}} ∈ K̄_{1}F = F^{∗}/{±1}.

Moreover, if (C^{∗}, δ^{∗}) is not acyclic, but with preferred bases {h^{q}}^{q} in the cohomology groups, then

τ(C^{∗}) = ∏_{q} det[b^{q}h^{q}b^{q+1}/c^{q}]^{(−1)^{q+1}} ∈ F^{∗}/{±1}.

(for the notations see [14]). The analog of the theorem 3.1 and 3.2 [14] is the following: 2.10. Lemma:

Let 0 ⟶ C^{′∗} ⟶ C^{∗} ⟶ C^{′′∗} ⟶ 0 be a short exact sequence (over a field ). Then, τ(C) = τ(C′) · τ(C′′) · τ(H), where H is the associated long exact cohomology sequence. 2.11. Remark: If C_{1} = C_{0} = F^{n} (F^{n} with its natural base) then the torsion of 0 ⟶ C_{1} ⟶^{A} C_{0} ⟶ 0 is [detA] ∈ F^{∗}/{±1}. Similarly, if C^{1} = C^{0} = F^{n}, then the torsion of 0 ⟶ C^{0} ⟶^{A} C^{1} ⟶ 0 is [detA]. In the definition we chose the exponent of the q-factor (−1)^{q+1} in order to have this coincidence. 2.12. Definition: Suppose that (C^{∗}(X, F, h) ⊗ R_{Q}(λ), δ) is acyclic. If b^{q} is a base over R in C^{q}(X, F), then the collection of bases b^{q} ⊗ 1 defines a torsion in R_{Q}(λ)^{∗}/{±1}. If we take another collection of bases, then the torsion will be modified by some determinants of invertible matrices over R. In conclusion, the (algebraic) torsion in R_{Q}(λ)^{∗}/R^{∗} is independent of the choice of the base.

Moreover, the acyclicity condition H^{∗}(C^{∗} ⊗R_{Q}(λ)) = 0 and the algebraic torsion τ(X, F, h) ∈ R_{Q}(λ)^{∗}/R^{∗} do not depend on the choice of the triangulation. 2.13. Remarks:

a) If R = Z, then τ ∈ Q(λ)^{∗}/{±1}. If F = Z_{X}, then τ corresponds to the R-torsion τ_{h} given by the cyclic representation π_{1}(X) → Z → Q(λ) (or Alexander polynomial) defined by Milnor [14, 15] in C(λ)^{∗}/_{{±λ^{i}}}.

b) If F is a local system over R ⊂ R, then τ ∈ R(λ)^{∗}/R^{∗}. Milnor, in his definition of R-torsion corresponding to flat bundles, used “rigid” systems given by orthogonal representations. In his case the ambiguity of the base choice resulted in terms: det(orthogonal matrix)= ±1. In our case, however, we have an “arithmetic rigidity”: if F is defined over R, then the multiplicative correction is in R^{∗}.

c) The divisor in Z[C] associated with τ ∈ C(λ)^{∗}/R^{∗} (i.e. the collection of the zeros and poles with their multiplicities) is independent of the choice of R ⊂ C.

3 The torsion of a cyclic fibering and the zeta function of the monodromy

In this section, we prove that if (X, S) has a fiber structure over S^{1} and h is induced by the fibration map, then the R_{Q}(λ)-complex constructed in the section 2 is acyclic and its torsion can be identified with the zeta function of the monodromy. 3.1. Let (X, S, F) be as in (2.1). Suppose that X is the total space of a fibration p : X → S^{1} with fiber F = p^{−1}(1) and geometric monodromy m_{g} : F ⟶ F (i.e. X = F × [0, 1]/_{(m_{g}(x),0)∼(x,1)} ). Suppose that the fiber structure is compatible with the stratification of X : for any i ∈ I we have S_{i} = (S_{i} ∩ F) × [0, 1]/_{∼}. 3.2. If T_{F} is a triangulation of F adapted to the stratification S ∩ F, such that m_{g} is compatible with it, then T_{F} defines a natural triangulation of X. Each q-cell Δ of F defines two q-cells in X: Δ_{0} = Δ × {0}/_{∼} and Δ_{1/2} = Δ × {1/2}/_{∼}; each (q–1)-cell Δ of F defines two q-prisms in X: Δ_{[0,1/2]} = Δ × [0, 1/2]/_{∼} and Δ_{[1/2,1]} = Δ × [1/2, 1]/_{∼} . Each prism has a natural decomposition in cells of X. This is the triangulation T_{X} of X. 3.3. If Δ is a q-cell of F, then we can consider the sequence of neighbourhoods

N(Δ_{0}) ⊃ N(Δ_{[0,1/2]}) ⊂ N(Δ_{1/2}) ⊃ N(Δ_{[1/2,1]}) ⊂ N(m_{g}(Δ)_{0})

Since F(N(Δ_{0})) ≈ … ≈ F(N(m_{g}(Δ)_{0})), this sequence of isomorphisms defines a monodromy action m_{Δ}^{q} : F(N(Δ_{0})) ⟶ F(N(m_{g}(Δ)_{0})), or globaly m^{q} = ⊕m_{Δ}^{q} acting on ⊕F(N(Δ_{0})). Recall that N(Δ) is a neighbourhood of Δ^{o} in X; if we denote the corresponding neighbourhood N_{F} (Δ) in F, defined similarly, then F(N_{F} (Δ)) = F(N(Δ_{0})). Therefore, m^{∗} can be considered as the monodromy at C^{∗}(F, F)-level.

Since R^{∗}p_{∗}F is a local system on S^{1}, it defines a monodromy m_{a}^{∗} on (R^{∗}p_{∗}F)_{1} = H^{∗}(F, F). Obviously, m^{∗} induces at the cohomology level exactly the algebraic monodromy m_{a}^{∗}. 3.4. It can be verified that the torsion computed by the triangulation T_{X}, and the torsion computed by the “prism-triangulation” :

{Δ_{0}, Δ_{1/2}}_{Δ} ∪ {Δ_{[0,1/2]}, Δ_{[1/2,1]}}_{Δ}

is the same. Hence, we will use the later one.

In the comparison of C^{∗}(X, F) with the complex of F appears an ambiguity of choice ( ambiguity of lifting ). In our subsequent computations, we will use the identification of the following modules

F(N(Δ_{0})), F(N(Δ_{1/2})), F(N(Δ_{[0,1/2]})) and F(N(Δ_{[1/2,1]}))

by identity. Different identifications modify the torsion by elements of type det( (m^{q})^{n_{1}}λ^{n_{2}} ) ∈ R[Z]^{∗} = {rλ^{±n}|r ∈ R^{∗}}.

By the above identification:

C^{q}(X, F) = C^{q}(F, F) ⊕ C^{q}(F, F) ⊕ C^{q−1}(F, F) ⊕ C^{q−1}(F, F) and

δ_{X,F}^{q} = | \begin{matrix}δ_{F}^{q} & 0 & 0 & 0 \\ 0 & δ_{F}^{q} & 0 & 0 \\ −1 & +1 & −δ_{F}^{q−1} & 0 \\ (m^{q})^{−1} & −1 & 0 & −δ_{F}^{q−1} \end{matrix} |

Consider C^{∗}(X, F)′ = C^{∗}(X, F)′′ = C^{∗}(F, F) ⊕ C^{∗−1}(F, F) with

(δ^{q})′ = | \begin{matrix}δ_{F}^{q} & 0 \\ (m^{q})^{−1} − 1 & −δ_{F}^{q−1} \end{matrix} | resp. (δ^{q})′′ = | \begin{matrix}δ_{F}^{q} & 0 \\ −1 & −δ_{F}^{q−1} \end{matrix} |

Then, we have an exact sequence of complexes:

0 ⟶ C^{′∗} ⟶^{i} C^{∗}(X, F) ⟶^{pr} C^{′′∗} ⟶ 0

where i(x, t) = (x, x, 0, t) and pr(x, y, z, t) = (x − y, z). Moreover, (C^{′′∗}, δ′′) is acyclic. 3.5. Now, p : X → S^{1} defines a representation π_{1}(p). The corresponding twisted complexes are:

C^{∗}(X, F) ⊗ R_{Q}(λ) with

δ(λ) = | \begin{matrix}δ & 0 & 0 & 0 \\ 0 & δ & 0 & 0 \\ −1 & +1 & −δ & 0 \\ (λm)^{−1} & −1 & 0 & −δ \end{matrix} |

C^{′∗} ⊗ R_{Q}(λ) resp. C^{′′∗} ⊗ R_{Q}(λ) with

δ′(λ) = | \begin{matrix}δ & 0 \\ (λm)^{−1} − 1 & −δ \end{matrix} | resp. δ′′(λ) = | \begin{matrix}δ & 0 \\ −1 & −δ \end{matrix} |

3.6. Lemma: The twisted complexes are acyclic. Proof: (λm)^{−1} − 1 is invertible over R_{Q}(λ).

In consequence, the torsions τ, τ′ and τ′′ are well-defined. But τ′′ is in R^{∗}, therefore, by (2.10), we get: 3.7. Lemma

τ(C^{∗}(X, F) ⊗ R_{Q}(λ)) = τ(C^{′∗} ⊗ R_{Q}(λ)) in R_{Q}(λ)^{∗}/R^{∗}.

3.8. Consider the exact sequence of complexes:

0 → (C^{∗−1}(F) ⊗ R_{Q}(λ), −δ^{∗−1}) →^{j} C^{′∗} ⊗ R_{Q}(λ) →^{q} (C^{∗}(F) ⊗ R_{Q}(λ), δ^{∗}) → 0

where j(y) = (0, y), q(x, y) = x.

In general, (C^{∗}(F, F)⊗R_{Q}(λ)) is not acyclic: its cohomology is H^{∗}(F, F)⊗ R_{Q}(λ). Take a base over R in H^{∗}(F, F) . Then the torsion τ_{F} of C^{∗}(F, F) ⊗ R_{Q}(λ) is well-defined. Moreover , the torsion of C^{∗−1} ⊗R_{Q}(λ) is τ_{F}^{−1}. Therefore, by (2.10), τ′ = { the torsion of the long exact sequence }. This sequence has the form:

H^{0}(F, F)⊗R_{Q}(λ) ⟶^{h^{0}} H^{0}(F, F)⊗R_{Q}(λ) ⟶ 0 ⟶ H^{1}(F, F)⊗R_{Q}(λ) ⟶^{h^{1}} …

where h^{q} = (λm_{a}^{q})^{−1} − 1.

In consequence, τ′ = ∏_{q} det[ (λm_{a}^{q})^{−1} − 1 ]^{(−1)^{q}} and this is independent of the base chosen in H^{∗}(F, F). Taking into account the ambiguity of the identification from (3.4), we get: 3.9. Theorem: Let F ↪ X ⟶^{p} S^{1} be a fibration compatible with the stratification S of X. Let F be an S-constructible sheaf. Then:

a) The torsion τ(X, F, h), corresponding to the representation h = π_{1}(p), is well-defined in R_{Q}(λ)^{∗}/R^{∗}.

b) If we define the zeta function of the algebraic monodromy m_{a}^{∗} : H^{∗}(F, F) → H^{∗}(F, F) by

ζ_{(p,F)}(λ) = ∏_{q} det(1 − λm_{a}^{q})^{(−1)^{q}}

then

τ(X, F, h) = ζ_{(p,F)}(λ) in R_{Q}(λ)^{∗}/_{R[Z]^{∗}}.

3.10. If F = Z_{X}, then τ_{X} = ζ_{p} in Q(λ)^{∗}/_{{±λ^{i}}}. This is Assertion 7 of the Milnor paper [15]. 3.11. Definition: Let F^{•} ∈ D_{c}^{b}(X) be an S-constructible sheaf in the derived category (defined over R), where (X, S) is as above. Then we define the algebraic torsion in R_{Q}(λ)^{∗}/R^{∗} by:

τ(X, F^{•}, h) = ∏_{q} τ(X, H^{q}(F^{•}), h)^{(−1)^{q}}.

The corresponding zeta function ζ_{(p,F^{•})}(λ) is defined by the same formula as above, but now the algebraic monodromy m_{a}^{∗} acts on H^{∗}(Rp_{∗}F^{•})_{1} = H^{∗}(F, F^{•}). Then, by 3.9, we obtain: 3.12. Theorem:

τ(X, F^{•}, h) = ζ_{(p,F^{•})}(λ) in R_{Q}(λ)^{∗}/_{R[Z]^{∗}}.

4 The reduction formula for the zeta function in the case of graph links

4.1. Let Σ be an oriented homology 3–sphere and let

(Σ, m_{1} S_{1} ∪ … ∪ m_{r} S_{r} ) = L(m)

be a fibered graph multilink associated with the splice diagram Γ(m) (not necessary minimal)[3].

Let Γ^{•}(m) be the graph diagram obtained from Γ(m) by replacing to boundary vertices the arrowhead vertices with zero multiplicities. We denote by V_{n} the set of nodes of Γ^{•}(m) and by V_{b} the set of boundary vertices of Γ^{•}(m) .

For each i ∈ V_{n} , we define K_{i} as a nonsingular oriented fiber of the corresponding Seifert component ; for each i ∈ V_{b} , we define K_{i} as the (possibly singular ) oriented fiber in the corresponding Seifert component.

For each i ∈ V_{nb} = V_{n} ∪ V_{b}, we define the number

m_{Γ}(K_{i}) = ∑_{j=1}^{r} m_{j} · l(S_{j}, K_{i})

where l(,) denotes the linking number. For the geometric interpretation of these numbers, see [3, 21].

4.2. Suppose that m_{Γ}(K_{i}) ≠ 0 for each i ∈ V_{nb}. (This condition implies the fiberability of the multilink.) By a suitable orientation of the virtual link components K_{i} , i ∈ V_{nb}, we will have m_{Γ}(K_{i}) > 0.

4.3. The set {K_{i}}_{i∈V_{b}} defines a stratification S = {S_{i} }_{i∈{0}∪V_{b}} on Σ − K ( K = ⋃_{m_{j}≠0} S_{j} ), with strata:

S_{0} = Σ − (K ∪ ⋃_{i∈V_{b}} K_{i}) and S_{i} = K_{i} if i ∈ V_{b}.

Let F^{•} ∈ D_{c}^{b}(Σ−K) be an S-constructible bounded sheaf over R. Then the projection map p : Σ − K → S^{1} of the fibered multilink L^{•}(m), associated with Γ^{•}(m) defines the algebraic monodromy m_{a}^{∗} acting on H^{∗}(p^{−1}(1), F^{•}) with zeta function ζ_{(p,F^{•})}(λ). 4.4. Since H^{∗}(F^{•} | K_{i}) is a locally constant sheaf on K_{i}, for each i ∈ {0}∪V_{nb}, it defines a monodromy ρ_{F^{•}}^{∗}(K_{i}) ∈ Aut(H^{∗}(F^{•} | K_{i})) and a zeta function

ζ_{(K_{i},F^{•})}(λ) = ∏_{q} det(1 − λρ_{F^{•}}^{q}(K_{i}) )^{(−1)^{q}} (i ∈ V_{nb})

The reduction formula can be formulated as follows: 4.5. Theorem [21]

ζ_{(p,F^{•})}(λ) = ∏_{i∈V_{nb}}[ζ_{(K_{i},F^{•})}(λ^{m_{Γ}(K_{i})})]^{2−δ(K_{i})}

where δ(K_{i}) is the degree of the corresponding vertices of Γ^{•}(m) . If the composition of h : π_{1}(Σ−K) ⟶ Z with the inclusion Z ↪ Q(λ), (n ↦ λ^{n}) is denoted by ρ_{π}, then

ζ_{(p,F^{•})}(λ) = ∏_{i∈V_{nb}}[ ∏_{q} det( 1 − ρ_{π}([K_{i}])ρ_{F^{•}}^{q}(K_{i}) )^{(−1)^{q}}]^{2−δ(K_{i})}

4.6. Remarks

a) If F^{•} is a local system given by a representation ρ_{F^{•}}^{q} : π_{1}(Σ − K) → GL_{N}(R), then ρ_{F^{•}}^{∗}(K_{i}) = ρ_{F^{•}}^{∗}([K_{i}]).

b) Similar formulas appear in the computations of the torsion using (onedimensional) closed orbits of some dynamical systems [4, 5, 6, 7, 8, 16]. They are in fact generalised Lefschetz-type formulas. Our aim is to construct a dynamical system , in which this reduction formula can be interpreted as a Lefschetz-type formula.

c) For several applications of the reduction formula in the singularity theory see [21].

5 Flows with Lefschetz property and their Dirichlet series

In this section we remind the definition of the “Lefschetz property” [6]. 5.1. Let X be a compact smooth manifold with a flow ϕ. In the most important cases, the set Pe of the periods of closed orbits of ϕ is discrete (e.g.: if ϕ admits an invariant complementary distribution (as the geodesic flows), or if ϕ is a suspension flow ,when Pe = N).

The period–set P ⊂ M × (0, ∞) consists of all pairs (x, t) so that ϕ_{t}(x) = x. If Pe is discrete, then the period–set breaks up into connected components {C}, which are compact and relative open in P. Hence they have a Fuller index [9] ind_{F} (C) ∈ Q. All orbits in C have the same period l(C), and are freely homotopic, thus determining a conjugacy class [o_{C}] in π_{1}(X).

Given a finite dimensional representation ρ : π_{1}(X) → A, we assemble this closed orbit data into the following formal Dirichlet series:

Z_{ρ}(s) = exp(− ∑_{C} ind_{F} (C)Trρ(o_{C})e^{−s·l(C)})

5.2. We say that the triplet (X, ϕ, ρ) is of Lefschetz type if: a.) the sum over C converges absolutely for Res ≫ 0 ; b.) Z_{ρ}(s) extends to a meromorphic function near [0, ∞) with s = 0 a

regular point ;

c.) Z_{ρ}(0) = τ_{ρ}(X). (Here τ_{ρ}(X) = τ(X, F), where F is the local system associated with the representation ρ). 5.3. Remark The homotopy invariance of the Fuller index suggests that Z_{ρ}(0) depends only on the smooth manifold X ( and ρ) and not on ϕ. Part c.) clarifies exactly this fact. 5.4. Examples

a.) Set F → X ⟶^{p} S^{1} a fibration with geometric monodromy m_{g} and cyclic representation ρ_{π} : π_{1}(X) → Q(λ). Then

Z_{ρ}(s) = exp(− ∑_{r≥1} \frac{L(m_{g}^{r})}{r}λ^{r}e^{−sr})

(Here L(m) is the Lefschetz number of m.) Z_{ρ}(s) is analytic for all s ∈ C, and

Z_{ρ}(0) = exp(− ∑_{r≥1} \frac{L(m_{g}^{r})}{r}λ^{r}) = ζ_{p}(λ) [Weil’s formula]

i.e. by (3.12) (X, suspension flow of m_{g}, ρ) is of Lefschetz type.

Other Lefschetz-type triplets:

b.) Constant time suspensions X with acyclic representations ρ : π_{1}(X) → O(n) [6].

c.) (Unit tangent bundle of an oriented hyperbolic manifold, geodesic flow, acyclic representation ρ : π_{1}(X) → O(n) )[7].

d.) (Unit tangent bundle of a locally symmetric manifold of nonpositive sectional curvature, geodesic flow, acyclic unitary representation)[16].

6 The Lefschetz type formula for graph links

6.1. Suppose that we can realize the geometric monodromy m_{g} of p by a flow, such that:

a) ϕ preserves the stratification S of F^{•} (i.e. the link components {K_{i}}_{i∈V_{b}}).

b) each periodic set is contained in a strata. Then, we can define the monodromies on o_{C}: ρ_{F^{•}}^{∗}(o_{C}) ∈ Aut(H^{∗}(F^{•}|o_{C})) (o_{C} an orbit in C); and by the cyclic representation ρ_{π} : π_{1}(X) → Q(λ)^{∗} the elements ρ_{π}([o_{C}]) in Q(λ)^{∗}, for any periodic set C. Our formal Dirichlet series is:

Z_{(F^{•},h)}(s) = exp(− ∑_{C} ∑_{q}(−1)^{q}ind_{F} (C)ρ_{π}([o_{C}])Trρ_{F^{•}}^{q}(o_{C})e^{−s·l(C)})

6.2. Theorem There exists a (suspension) flow (with Pe = N), such that:

a) 6.1.a) and b) are satisfied, thus Z_{(F^{•},h)} can be defined. Moreover, (since {C} is finite) this Dirichlet function is analytic for any s ∈ C. In particular:

Z_{(F^{•},h)}(0) = exp(− ∑_{C} ∑_{q}(−1)^{q}ind_{F} (C)ρ_{π}([o_{C}])Trρ_{F^{•}}^{q}(o_{C}) )

b.) Z_{(F^{•},h)}(0) can be identified with the right hand-side of the reduction formula. In particular, the nontrivial elements in the product decomposition over the periodic sets {C} are exactly the factors in the product decomposition over the non–arrowhead vertices of Γ^{•}(m) (i.e. the periodic sets can be parametrized by an Eisenbud–Neumann diagram and they correspond to the Jaco–Shalen–Johannson decomposition of the link component ). Proof: Consider the splice decomposition of Σ − K given by the diagram Γ(m) . Each edge of the diagram defines a torus in Σ − K, such that the restriction of p on these tori is also a locally trivial fibration. Hence, the geometric monodromy m_{g} can be chosen compatible with this decomposition.

Each vector field V on F with (m_{g})_{∗}V = V defines a flow ϕ on Σ − K by the identifications:

a) Σ − K = F × [0, 1]/_{(m_{g}(x),0)∼(x,1)}

b) ϕ is given by (V, 1).

Since the geometric monodromy preserves the decomposition, it is enough to construct invariant vector fields V on the pieces (in F) with V zero on the collar of their boundary. Then, these local fields define a global one on F ( which is zero on a tubular neighbourhood of the separating annuli).

The geometric monodromy of a piece in F corresponding to a node can be identified with a generator of the transformation group of the cyclic covering S_{i} → S^{2} − {δ_{i} points}, (δ_{i} = δ(K_{i})) [3]. A monodromy invariant vector field on S_{i} (which is zero near the boundary) is equivalent to a vector field on S^{2} − {δ_{i} points} (which is zero on some small punctured discs D_{j}^{·} centered at these points). We construct a vector field V on S^{2} − {δ_{i} points}, such that:

a) V (x) = 0 ⟺ x ∈ ⋃_{j=1}^{δ_{i}} D_{j}^{·} ∪ {s_{i}} s_{i} ∉ ⋃_{j} D_{j}^{·};

b) V (x) has index (−δ_{i} + 2) in s_{i}; c) the periodic points are exactly the fix points. A piece corresponding to a boundary vertex i ∈ V_{b} is a collection of discs

with centers corresponding to the strata K_{i}; m_{g} is a cyclic permutation of these discs. An invariant vector field V which is zero near the boundary and with property that ϕ preserves K_{i}, is given in the polar coordinates (r, α) of the discs by (φ(r), 0), where φ : [0, 1] → [0, ∞) is a C^{∞} map with: φ(x) = 0 ⇔ x = {0} ∪ [1 − ϵ, 1]; (0 < ϵ < 1).

In this case, the periodic orbits of the global vector field (V, 1) on Σ − K corresponds to the fixed points of V. The periodic components, which corresponds to the separating annuli, have Fuller index ind_{F} (C) = 0.

Corresponding to the set of nodes, for each n ∈ N, we have an isolated closed orbit C_{i,n} ( with geometric support K_{i} ), with period l(C_{i,n}) = n and

ind_{F} (C_{i,n}) = \frac{index(Poincaré map, s_{i})}{n} = \frac{−δ_{i} + 2}{n}.

Similarly, for i ∈ V_{b} and n ∈ N, we have an isolated closed orbit C_{i,n} with period n and Fuller index 1/n = (−δ_{i} + 2)/n. In both cases

ρ_{F^{•}}^{q}(C_{i,n}) = ρ_{F^{•}}^{q}(K_{i})^{n} , ρ_{π}(C_{i,n}) = λ^{n·m_{Γ}(K_{i})} = ρ_{π}(K_{i})^{n}

Thus:

Z_{(F^{•},h)}(0) = exp − ∑_{i∈V_{nb}} ∑_{q} ∑_{n≥1}(−1)^{q} \frac{−δ_{i} + 2}{n}ρ_{π}(K_{i})^{n}Trρ_{F^{•}}^{q}(K_{i})^{n} =

= ∏_{i∈V_{nb}}[∏_{q} det( 1 − ρ_{π}(K_{i})ρ_{F^{•}}^{q}(K_{i}) )^{(−1)^{q}}]^{2−δ_{i}}

6.3. Using 3.12. and 6.2., we get that the reduction formula 4.5. is equivalent to the following Lefschetz-type formula for the flow ϕ constructed above: 6.4. Theorem:

τ(Σ − K, F^{•}, h) = Z_{(F^{•},h)}(0) in C(λ)^{∗}/_{C[Z]^{∗}}

.

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Institute of Mathematics of the Romanian Academy, and Department of Mathematics, The Ohio State University, Columbus, Ohio.

e-mail: nemethi@function.mps.ohio-state.edu