HYPERSURFACE COMPLEMENTS, ALEXANDER MODULES AND MONODROMY

by Alexandru Dimca and András Némethi

Laboratoire de Mathématiques Pures de Bordeaux Department of Mathematics,
Université Bordeaux I  Ohio State University,
33405 Talence Cedex, FRANCE  Columbus, Ohio 43210, USA
email: dimca@math.u-bordeaux.fr  email: nemethi@math.ohio-state.edu

Abstract. We consider an affine variety U and an affine map f : U → ℂ. Our goal is to provide structure theorems of the Alexander module associated with a complement M_{X} = U \ X (where X is an arbitrary fiber of f), in particular, a complete description of the Alexander invariants of M_{X}.

We show that the Alexander invariants of all the fibers of f are closely related to the monodromy representation of f. This also explains conceptually all the divisibility properties that have recently appeared in the literature connecting the Alexander polynomials and the characteristic polynomials of some special monodromy operators.

The main connection is realized via to new Alexander modules constructed from the monodromy representation of f: the global Alexander module M(f) of f, which can be regarded as a commutative version of the monodromy representation; and the local Alexander module M(f, b) associated with any fiber X = f^{−1}(b), which is a very good approximation of the Alexander module of M_{X}.

The general theory is exemplified explicitly on the case of h-good affine maps (a class of affine maps generalizing polynomials with good properties at infinity).

1. Introduction

Let X ⊂ ℂ^{n+1} (resp. V ⊂ ℙ^{n+1}) be an algebraic hypersurface and set M_{X} = ℂ^{n+1} \ X (resp. M_{V} = ℙ^{n+1} \ V) where we suppose n > 0. The study of the topology of X, V and of their complements M_{X}, M_{V} is a classical subject going back to Zariski. In a sequence of papers Libgober has introduced and studied the Alexander invariants associated to X, V, see for instance [Li0-3]. The case n = 1 was intensively studied by Kulikov, see [K] and the other references there, especially the structure of the fundamental groups of the complements M_{V} and M_{X}.

In the affine case, let f = 0 be a reduced equation for X. One can use the results on the topology of polynomial functions, see for instance [B], [ACD], [NZ], [SiT], to study the topology of the complements M_{X}, as in the recent paper by Libgober and Tibăr [LiT].

In fact, one can consider the Alexander invariants of M_{X} associated with an arbitrary epimorphism e_{M_{X}} : π_{1}(M_{X}) → ℤ. But even then, e_{M_{X}} can be recovered as f_{∗} : π_{1}(M_{X}) → π_{1}(ℂ^{∗}) for some polynomial map f : ℂ^{n+1} → ℂ with X = f^{−1}(0) (with possibly nonreduced structure). Additionally, the presence of f provides more structure and tools. The goal of the present paper is to investigate this point of view.

In fact, we will consider a more general situation f : U → ℂ, where U is an affine variety. In order to simplify the presentation, we will assume that U is smooth, and the generic fiber of f is connected. Section 2 contains a complete description of the Alexander invariants of M_{X} = U \ X for any fiber X of f.

The article has two major messages. First, the most important qualitative properties of the Alexander modules are completely independent of the behaviour of f at infinity, or of the special fibers. Second, the Alexander invariants of all the fibers of f are closely related to the monodromy representation of f. In fact, all the torsion parts of the Alexander modules (associated with all the possible fibers) can be obtained by taking the quotient of a unique universal Alexander module M(f), which is constructed from the monodromy representation, cf. 2.14. This explains nicely and conceptually (cf. 2.15) all the divisibility properties that have recently appeared in the literature connecting the Alexander polynomials of M_{X} and the characteristic polynomials of some special monodromy operators (see e.g. [Li2] and [LiT]). Note that the monodromy considered by Libgober in [Li2], section 2, is associated to a Lefschetz pencil and hence quite different from our monodromy associated to an arbitrary polynomial.

On the other hand, for some classes of affine maps with some additional information about the special fibers or about the behaviour at infinity, one can obtain nice vanishing or connectivity results. In order to exemplify our general theory, and also to generalize some connectivity results already present in the literature (for polynomial maps), we introduce the family of h-good affine maps (and h-good polynomials) which fits perfectly to the study of Alexander invariants. The family of h-good polynomials includes e.g. all the “good” polynomials considered by Neumann and Rudolph [NR], and the polynomials with isolated singularities on the affine space and at infinity in the sense of Siersma-Tibăr [SiT].

The content of the paper is the following. In section 2 we establish some properties of the corresponding fundamental groups which basically will guide all the covering properties considered later. Then we state the main structure theorems about the Alexander modules. Here we also define two new Alexander modules associated with f, namely the global Alexander module M(f) which can be regarded as a commutative version of the monodromy representation, and a local Alexander module M(f, b) associated with any fiber X = f^{−1}(b). This module M(f, b) gives a very good approximation of the Alexander modules of M_{X} (see 2.14 and 2.15).

In particular, the general theory also explains the relationship between any individual monodromy operator and Alexander modules. The two main examples, the monodromy at infinity and the monodromy around the fiber X, are discussed with special care, see 2.16. These two operators have been intensively studied recently using various techniques (mixed Hodge structures, D-modules), see the references given in 2.16. Via our results, all this information on the monodromy operators yields valuable information on the Alexander invariants of M_{X}.

Finally, we give a complete characterization of the homological Alexander modules in terms of monodromy operators and the vanishing cycles, see 2.18.

In section 3 we introduce and discuss the h-good affine maps (and polynomials). In some paragraphs we emphasize the parallelism of h-good polynomials with the case of hypersurfaces with only isolated singularities including at infinity considered by Libgober [Li2] (cf. also with 4.3). In section 4 we relate the Alexander invariants to the cohomology of certain local systems on M_{X}. In fact, the (co)homology of M_{X} with coefficients in such a local system can be computed (or at least estimated) from our structure theorems for the Alexander modules. In the next section we present non-vanishing results: we present four construction which provide in a systematic way examples with non-trivial Alexander invariants. In particular, we emphasize the subtle connections with superabundance properties (5.4) and the structure of the vanishing cycles and variation maps (5.7,5.8). As a convincing example of the power of this new approach, in section 6 we compute the various Alexander modules for a polynomial ℂ^{4} → ℂ. The last section contains examples, remarks and comments about the ℤ-torsion of the corresponding spaces. This latter aspect was somewhat neglected recently in spite of the pioneering work by Libgober [Li0].

We thank D. Arapura, A. Libgober, C. Sabbah and A. Suciu for useful discussions.

2. The case of general affine maps

Before we start to investigate the cyclic coverings associated with the fiber-complements of affine maps, let us remind the basic definitions of Alexander modules and polynomials.

2.1. Definitions. (1) Let Y be a connected CW-complex and e_{Y} : π_{1}(Y) → ℤ be an epimorphism. We denote by Y^{c} the ℤ-cyclic covering associated to the kernel of the morphism e_{Y}. It follows that a generator of ℤ acts on Y^{c} by a certain covering homeomorphism h and all the groups H_{∗}(Y^{c}, A), H^{∗}(Y^{c}, A) and π_{j}(Y^{c}) ⊗ A for j > 1 become in the usual way Λ_{A}-modules, where Λ_{A} = A[t, t^{−1}], for any ring A. These are called the Alexander modules of the pair (Y, e_{Y}) or simply of Y when the choice of e_{Y} is clear.

(2) If Z is a second connected CW-complex, e_{Z} : π_{1}(Z) → ℤ an epimorphism and ϕ : Y → Z is a continuous map such that the induced map at the level of π_{1} is an epimorphism compatible to the two given epimorphisms e_{Y} and e_{Z}, then Y^{c} → Y can be regarded as a pull-back covering obtained via ϕ from the covering Z^{c} → Z. In particular, this gives a lift ϕ^{c} : Y^{c} → Z^{c} which is compatible with the covering transformations, and hence, an induced morphism of Λ_{A}-modules, say ϕ_{∗}^{c} : H_{∗}(Y^{c}, A) → H_{∗}(Z^{c}, A).

(3) If A is a field then the ring Λ_{A} is a PID. Hence any finite type Λ_{A}-module M has a decomposition M = Λ_{A}^{k} ⊕ (⊕_{p}M_{p}), where k is the rank of M and the second sum is over all the prime elements in Λ_{A}, and M_{p} denotes the p-torsion part of M.

In fact, for each prime p with M_{p} ≠ 0, we have a unique decomposition M_{p} = ⊕_{i=1,ℓ_{p}}Λ_{A}/p^{k_{i}} with ℓ_{p} > 0 and k_{1} ≥ k_{2} ≥ .... ≥ k_{ℓ_{p}} ≥ 1.

The sequence (k_{1}, k_{2}, ..., k_{ℓ_{p}}) obtained in this way will be denoted by K(M, p). One can define an order relation on such sequences by saying that

(k_{1}, ...., k_{a}) ≥ (m_{1}, ..., m_{b})

iff a ≥ b and k_{i} ≥ m_{i} for all 1 ≤ i ≤ b.

Δ(M) = ∏_{p} ∏_{i=1}^{ℓ_{p}} p^{k_{i}} is called the the Alexander polynomial of the module M. Sometimes, in order to emphasize the action h (corresponding to t ∈ Λ_{ℤ}) we write Δ(M, h).

With this notation one has the following easy facts (whose proof is left to the reader).

(4) Let u : M → N be an epimorphism of R-modules, where R is a PID and M is torsion of finite type. Then N is torsion of finite type and for any prime p ∈ R one has K(M, p) ≥ K(N, p). In particular Δ(N) divides Δ(M).

(5) For A = ℂ set Λ := Λ_{ℂ}. The prime elements in this case are just the linear forms t − a, for a ∈ ℂ^{∗}. Moreover, a Λ-module (H_{q}(Y^{c}, ℂ), h_{q}) is of finite type and torsion iff the Betti number b_{q}(Y^{c}) is finite. If this is the case, the (t − a)-torsion part of H_{q}(Y^{c}, ℂ) is determined (and determines) the Jordan block structure of h_{q}: a Jordan block of size m corresponding to the eigenvalue a produces a summand Λ/(t − a)^{m}. The corresponding Alexander polynomial Δ(H_{q}(Y^{c}, ℂ), h_{q}) is just the characteristic polynomial Δ(h_{q}) of h_{q}.

When A = ℚ, the prime elements are the irreducible polynomials in ℚ[t] different from t, hence they are a lot more difficult to describe. However, the knowledge of the Λ-module structure of H_{q}(Y^{c}, ℂ) implies easily the Λ_{ℚ}-module structure of H_{q}(Y^{c}, ℚ) just by grouping together the polynomials t − a for those a’s having the same minimal polynomial over ℚ.

2.2. The affine map f. Let U be an irreducible affine complex variety of dim U = n+1 ≥ 2. In order not to distract the reader’s attention with different technicalities from the main message of the paper, we will assume that U is smooth with trivial divisor class group (in fact we just need that the irreducible components of the fibers of f are principal divisors). The interested reader can rewrite the next results for more general cases with some extra work.

We consider an affine map f : U → ℂ. It is well known that there is a (minimal) finite bifurcation set B_{f} in ℂ such that f is a C^{∞}-locally trivial fibration over ℂ \ B_{f}. If b_{0} ∉ B_{f}, then F = f^{−1}(b_{0}) is called the generic fiber of f; otherwise f^{−1}(b) is called a special fiber.

For any b ∈ ℂ we fix a sufficiently small closed disc D_{b} containing b, and a point b′ ∈ ∂D_{b}. We set T_{b} := f^{−1}(D_{b}), T_{b}^{∗} := T_{b} \ f^{−1}(b). Sometimes, it is convenient to identify f^{−1}(b′) with the generic fiber F. Then, we have the obvious inclusions F ⊂ T_{b}^{∗} ⊂ T_{b}. Set v_{∗}(b) := rankH_{∗}(T_{b}, F) (b ∈ B_{f}), the number of vanishing cycles at b. As a general rule, coefficients for (co)homology groups are not specified when they are in ℤ.

By a well-known deformation retract argument (see e.g. (2.3) in [DN1]), the pair (U, F) has the homotopy type of the space (Y, F) obtained by gluing all the pairs (T_{b}, F) (b ∈ B_{f}) along F. We denote this fact by

(U, F) ∼ ⋁_{F}(T_{b}, F) (b ∈ B_{f}).  (1)

Next, we fix a distinguished fiber X := f^{−1}(0) and we write M_{X} := U \ X. It is convenient to introduce the notation B_{f}^{∗} := B_{f} \ {0}. If 0 ∈ B_{f} then X is a special fiber, otherwise it is a generic one (and B_{f}^{∗} = B_{f}).

2.3. Proposition. The following facts are equivalent:

(i) the generic fiber F is connected.

(ii) π_{1}(T_{b}, F) is trivial for any b ∈ B_{f}.

(iii) π_{1}(T_{0}^{∗}) → π_{1}(M_{X}) (induced by the inclusion) is onto, and π_{1}(F) ⟶^{i_{X}} π_{1}(M_{X}) ⟶^{f_{∗}} ℤ → (1) is an exact sequence (i.e. im(i_{X}) = ker(f_{∗})), where i_{X} is induced by the inclusion F ⊂ M_{X}, and f_{∗} by f : M_{X} → ℂ^{∗}.

The claim (ii) above for n = 1 is exactly Theorem 2. in [K].

Proof. (i) ⇔ (ii) If π_{1}(T_{b}, F) = 1 for all b, then H_{1}(U, F) = ⊕_{b∈B_{f}} H_{1}(T_{b}, F) = 0 by (1), hence H̃_{0}(F) = 0 as well. Conversely, assume that F is connected and fix a b ∈ B_{f}. Then we have to show that j : π_{1}(F) → π_{1}(T_{b}) is onto. Since T_{b} is smooth and f^{−1}(b) ⊂ T_{b} has real codimension two, one obtains that i : π_{1}(T_{b}^{∗}) → π_{1}(T_{b}) is onto. Since f restricted to T_{b}^{∗} is a fiber bundle, the kernel of f_{∗} : π_{1}(T_{b}^{∗}) → ℤ is π_{1}(F). Assume that f^{−1}(b) has r irreducible components, and f − b = ∏_{i=1}^{r} g_{i}^{m_{i}}. Then one can construct easily elementary loops in T_{b}^{∗} around the component {g_{i} = 0} representing x_{i} ∈ π_{1}(T_{b}^{∗}) with properties f_{∗}(x_{i}) = m_{i} and i(x_{i}) = 1. Set m := gcd_{i}{m_{i}}. Then a combination of the x_{i}’s provides an x ∈ π_{1}(T_{b}^{∗}) with f_{∗}(x) = m and i(x) = 1. But m = 1 (otherwise f − b would be an m-power of a function with non-connected generic fiber). The existence of such an x and the surjectivity of i implies the surjectivity of j.

(i) ⇒ (iii) Similarly as in (1), M_{X} has the homotopy type of a space obtained by gluing T_{0}^{∗} and all the “tubes” Tb̄ (b̄ ∈ B_{f}^{∗}) along F. Then the first claim follows from van Kampen theorem and from the surjectivity of π_{1}(F) → π_{1}(Tb̄) for each b̄ (cf. part (ii)). This, and the the exact sequence π_{1}(F) → π_{1}(T_{0}^{∗}) → ℤ → (1) implies the second claim.

(iii) ⇒ (i) In the homotopy exact sequence of T_{0}^{∗} the map f_{∗} : π_{1}(T_{0}^{∗}) → ℤ is onto. □

2.4. Assumption. In the sequel we will assume that F is connected.

Let p : 𝔽 := M_{X}^{c} → M_{X} be the ℤ-cyclic covering associated to the kernel of the epimorphism e_{M_{X}} = f_{∗} : π_{1}(M_{X}) → π_{1}(ℂ^{∗}) = ℤ. The notation 𝔽 is chosen because

(i) 𝔽 is the homotopy fiber of f : M_{X} → ℂ^{∗}, regarded as a homotopy fibration; and (ii) the spaces 𝔽 and F are nicely comparable (see below). Let h be the covering transformation of 𝔽 corresponding to the generator 1_{ℤ}. Sometimes

we prefer to denote the homological Alexander Λ_{ℤ}-module H_{∗}(M_{X}^{c}) by (H_{∗}(𝔽), h_{∗}).

2.5. Fix a base-point ∗ ∈ 𝔽 with p(∗) = b_{0} ∈ F. Since f(F) is a point, there is a natural section s : F → 𝔽 of p above F with s(b_{0}) = ∗, hence we can regard F as a subspace of 𝔽.

2.6. Corollary. s_{∗} : π_{1}(F) → π_{1}(𝔽) is onto, or equivalently, π_{1}(𝔽, F) is trivial.

Proof. Compare the exact sequences (1) → π_{1}(F) → π_{1}(T_{0}^{∗}) → ℤ → (1) and (1) → π_{1}(𝔽) → π_{1}(M_{X}) → ℤ → (1) via 2.3(iii). □

2.7. Loops and their monodromies. In order to relate the Alexander modules of M_{X} with the monodromy representation of f, we need some more notation. Fix an orientation of S^{1}, and consider a smooth loop γ : S^{1} → ℂ^{∗} \ B_{f}. Denote by q : γ^{−1}(f) → S^{1} the pull-back of f by γ, i.e. γ^{−1}(f) = {(t, x) ∈ S^{1} × U : γ(t) = f(x)}, and q(t, x) = t. Obviously, q is a fiber bundle over S^{1} with fiber F. The map which covers γ is denoted by γ̄ : γ^{−1}(f) → M_{X}, hence f ∘ γ̄ = γ ∘ q. We can consider the epimorphism e_{γ} : π_{1}(γ^{−1}(f)) → π_{1}(S^{1}) = ℤ. The associated ℤ-covering has a total space isomorphic to F ×ℝ and the covering transformation (corresponding to 1_{ℤ}) can be identified with the geometric monodromy of f associated with the oriented loop γ. The (homological) Alexander Λ_{ℤ}-module associated with e_{γ} will be denoted by (H_{∗}(F), T_{γ,∗}).

2.8. Corollary. Assume that ℓ is the winding number of γ with respect to 0. Then one has the following commutative diagram with all the lines and columns exact:

(1) → π_{1}(F) → π_{1}(γ^{−1}(f)) → ℤ → (1)
↓ ↓ ↓
(1) → π_{1}(𝔽) → π_{1}(M_{X}) → ℤ → (1)
↓  ↓  ↓
(1) →  ℤ/ℓℤ  → ℤ/ℓℤ → (1)
↓  ↓
(1)  (1)

Proof. The first two lines are the homotopy exact sequences of the corresponding fibrations. Then use 2.6 and note that γ_{∗} : π_{1}(S^{1}) → π_{1}(ℂ^{∗}) (i.e. γ_{∗} : ℤ → ℤ) is multiplication by the integer ℓ. □

The above corollary allows us to connect the two Alexander modules. First, assume that γ_{∗} : ℤ → ℤ (cf. 2.8) satisfies γ_{∗}(1) = 1. Then by 2.8, π_{1}(γ̄) : π_{1}(γ^{−1}(f)) → π_{1}(M_{X}) is onto, and e_{M_{X}} ∘ π_{1}(γ̄) = e_{γ}. Therefore, by 2.1(2) there exists a morphism of Λ_{ℤ}-modules

γ_{Λ,∗}^{c} : (H_{∗}(F, ℤ), T_{γ,∗}) → (H_{∗}(𝔽, ℤ), h_{∗})

(in the sense that h_{∗} ∘ γ_{Λ,∗}^{c} = γ_{Λ,∗}^{c} ∘ T_{γ,∗}).

More generally, assume that γ_{∗} : ℤ → ℤ is multiplication with an integer ℓ. Then we can replace M_{X} by the ℤ/ℓℤ-cyclic covering M_{X,|ℓ|}, with projection pr : M_{X,|ℓ|} → M_{X}, determined by the subgroup e_{M_{X}}^{−1} (ℓℤ). By 2.8 and the general theory of covering spaces (see e.g. [S]), γ̄ : γ^{−1}(f) → M_{X} can be lifted to γ̄′ : γ^{−1}(f) → M_{X,|ℓ|}, with pr ∘ γ̄′ = γ̄. Obviously, 𝔽 is the cyclic covering of M_{X,|ℓ|} with transformation group ℓℤ, and π_{1}(γ̄′) is onto. Hence again we obtain a morphism of Λ_{ℤ}-modules

γ_{Λ,∗}^{c} : (H_{∗}(F, ℤ), T_{γ,∗}) → (H_{∗}(𝔽, ℤ), h_{∗}^{ℓ})

(i.e. h_{∗}^{ℓ} ∘ γ_{Λ,∗}^{c} = γ_{Λ,∗}^{c} ∘ T_{γ,∗}. Above, if ℓ = 0 then M_{X,|ℓ|} is obviously 𝔽 itself).

The above constructions can be made compatible with some base points. (Since all the spaces are connected, this is not really relevant, the details are left to the interested reader).

Notice that the target module (H_{∗}(𝔽, ℤ), h_{∗}) is completely independent of γ. The main point is that even the map H_{∗}(F, ℤ) → H_{∗}(𝔽, ℤ), as a ℤ-module, is independent of γ. Indeed, γ^{−1}(f)^{c} has the homotopy type of F and γ̄^{c} is up to homotopy the same as our fixed embedding s : F → 𝔽. The above discussion is summarized in the following theorem.

2.9. Theorem. Assume that the generic fiber F of f is connected. Fix an embedding s : F → 𝔽 as above. Let γ : S^{1} → ℂ^{∗}\B_{f} be a smooth loop such that γ_{∗} : π_{1}(S^{1}) → π_{1}(ℂ^{∗}) is multiplication by ℓ = ℓ(γ). Then s_{∗} : H_{∗}(F, ℤ) → H_{∗}(𝔽, ℤ) is compatible with the Alexander Λ_{ℤ}-module structures in the sense that s_{∗} ∘ T_{γ,∗} = h_{∗}^{ℓ} ∘ s_{∗}.

Recall that F is an n-dimensional affine variety, hence it has the homotopy type of a finite CW-complex of dimension ≤ n. The above theorem is optimal exactly when s_{q} is onto in the non-trivial cases q ≤ n. As we will see, this is the case e.g. for h-good affine maps (cf. 3.3 below). On the other hand, we cannot hope that for an arbitrary f (even if U = ℂ^{n+1}) s_{q} : H_{q}(F) → H_{q}(𝔽) (q ≤ n) is onto, since H_{q}(F, ℚ) is a Λ_{ℚ}-torsion module, but H_{q}(𝔽, ℚ) may have a non zero Λ_{ℚ}-free part. Nevertheless, the next theorem shows that im(s_{∗}) is “as large as possible”. Below Δ_{γ,∗}(t) denotes the characteristic polynomial of the monodromy operator T_{γ,∗}.

2.10. Theorem. Assume that F is connected, and s : F → 𝔽 is fixed as in 2.9. Let (T_{∗}, h_{∗}) be the torsion part of the Λ_{ℚ}-module (H_{∗}(𝔽, ℚ), h_{∗}). Then:

(i) im(s_{∗} ⊗ 1_{ℚ}) = T_{∗}.

(ii) For any loop γ : S^{1} → ℂ \ B_{f} with ℓ := ℓ(γ) one has the following epimorphism of Λ_{ℚ}-modules:

s_{∗} : (H_{∗}(F, ℚ), T_{γ,∗}) → (T_{∗}, h_{∗}^{ℓ}).

(iii) In particular, Δ(H_{q}(𝔽, ℚ), h_{q}^{ℓ})(t) divides Δ_{γ,q}(t) for any q ≥ 0 (cf. with 2.11).

(iv) The (t − a)-torsion of H_{∗}(𝔽, ℂ) is zero if a is not a root of unity.

(v) The Λ_{ℚ}-rank of H_{q}(𝔽, ℚ) is ∑_{b̄∈B_{f}^{∗}} v_{q}(b̄) and

∑_{q}(−1)^{q} rank_{Λ_{ℚ}}H_{q}(𝔽, ℚ) = χ(M_{X}).

(vi) H_{q}(𝔽, ℤ) is trivial for q > n + 1 and Λ_{ℤ}-free for q = n + 1.

Proof. Notice that part (i) and 2.9 imply (ii), hence (iii) as well via 2.1(4). Moreover, the statement of (i) is independent of the choice of any loop. In order to prove it, we take a special loop, namely the oriented boundary of D_{0}. Notice (cf. 2.2(1) and the proof of 2.3) that M_{X} has the homotopy type of a space obtained by gluing to T_{0}^{∗} along F all the tubes Tb̄ for b̄ ∈ B_{f}^{∗}. Moreover, p^{−1}(T_{0}^{∗}) has the homotopy type of F, and its embedding into 𝔽 is homotopically equivalent to the embedding s : F → 𝔽. Therefore, by excision:

H_{q}(𝔽, F) = H_{q}(p^{−1}(T_{0}^{∗} ∨_{F} (Tb̄)), p^{−1}(T_{0}^{∗})) = ⊕b̄H_{q}(p^{−1}(Tb̄), p^{−1}(F)),  (2)

where b̄ ∈ B_{f}^{∗}. But (p^{−1}(Tb̄), p^{−1}(F)) = ℤ × (Tb̄, F); in particular its homology is Λ_{ℤ}-free. This implies (v), and (i) follows from the homological exact sequence of the pair (𝔽, F) (considered as a sequence of Λ-modules). For (iv) apply (iii) for the monodromy operator T_{0,∗} above ∂D_{0}, whose eigenvalues are roots of unity by general (Hodge) theory. For (vi) notice that M_{X} has the homotopy type of a CW-complex of dimension ≤ (n + 1). □

2.11. Remark. If M = (H, h) is a torsion Λ-module with Alexander polynomial Δ(H, h)(t) (i.e., if h acts on H with characteristic polynomial Δ(H, h)(t)), then for any ℓ, the polynomial Δ(H, h)(t) clearly determines Δ(H, h^{ℓ})(t). Indeed, if for any polynomial P (t) = ∏_{a}(t − a)^{n_{a}} (n_{a} ∈ ℕ) we write P (t)^{(ℓ)} := ∏_{a}(t − a^{ℓ})^{n_{a}}, then Δ(H, h^{ℓ})(t) = Δ(H, h)(t)^{(ℓ)}.

2.12. Example. Assume that 0 ∉ B_{f}, i.e. X is a generic fiber of f. Then the above theorem says (applied e.g. for an embedding γ with image ∂D_{0}) that the restriction of h_{∗} to im(s_{∗}) is trivial. Additionally, s_{∗} ∘ T_{γ,∗} = s_{∗} for any γ.

In fact, H_{∗}(𝔽, ℤ) can be determined very precisely. Indeed, first notice that T_{0}^{∗} ∼ S^{1} × F, hence M_{X} ∼ S^{1}×F ∐_{F} U (cf. 2.2(1) or the proof of 2.10). Therefore, 𝔽 ∼ ℝ×F ∐_{ℤ×F} ℤ×U, where the identification F → U is the natural inclusion j. Notice also that the map s : F → 𝔽 can be identified with the inclusion x ↦ (0, x) ∈ ℝ × F ⊂ 𝔽. Consider also the inclusion i : U → 𝔽, i(x) = (0, x) ∈ ℤ × U ⊂ 𝔽. Since there is (a natural) projection pr : 𝔽 → U with pr ∘ i = 1_{U}, one gets that i_{∗} : H_{∗}(U, ℤ) → H_{∗}(𝔽, ℤ) is injective. Then, by the identity i ∘ j = s one obtains that

im(s_{∗}) ≈ im(j_{∗} : H_{∗}(F, ℤ) → H_{∗}(U, ℤ)).

Moreover, using e.g. 2.2(1), the Λ-rank of H_{∗}(𝔽, ℚ) is b_{∗}(U, F).

In particular, if the inclusion of F into U is homologically trivial (e.g. when U = ℂ^{n+1}), then for any q > 0, s_{q} = 0 and H_{q}(𝔽, ℤ) is free of rank b_{q}(U) + b̃_{q−1}(F). (In fact, if the inclusion F → U is homotopically trivial, then M_{X} ∼ S^{1} ∨S(F)∨U, and 𝔽 ∼ ∨_{ℤ}(S(F)∨U), where S(F) denotes the suspension of F; cf. with [LiT] for the case U = ℂ^{n+1}).

In conclusion, if 0 ∉ B_{f} then (T_{∗}, h_{∗}) is not really interesting.

In the next paragraphs we will assume that 0 ∈ B_{f}, i.e. X is a special fiber of f. First, we reinterpret 2.9 in terms of the monodromy representation of f. Let G := π_{1}(ℂ \ B_{f}, b_{0}) and ρ_{∗} : G → Aut(H_{∗}) be the monodromy representation, where H_{∗} := H_{∗}(F, ℤ) with F = f^{−1}(b_{0}). Consider the natural morphism p_{0} : G = π_{1}(ℂ \ B_{f}) → π_{1}(ℂ^{∗}) defined by [γ] ↦ ℓ(γ), the winding number of γ with respect to 0. Then 2.9 reads as follows:

2.13. Corollary. Consider both H_{∗} and H_{∗}(𝔽, ℤ) as G-modules (the last one via p_{0}). Then s_{∗} is a morphism of G-modules.

In other words, the (complicated) monodromy representation, when it is mapped via s_{∗} into H_{∗}(𝔽, ℤ), is collapsed into the modest cyclic (Alexander-) action.

Recall, that for any G-module M and a normal subgroup H ⊂ G, one denotes by M_{H} the group of coinvariants, namely the quotient of M by the subgroup spanned by all elements h(m) − m for h ∈ H and m ∈ M, see Brown [Br]. Then M_{H} has a natural G-module structure, which, in fact, factorizes through G/H.

In our case, we will distinguish two subgroups of G = π_{1}(ℂ \ B_{f}, b_{0}). The first is the commutator H = [G, G] of G, the second is K_{0} = ker(p_{0}).

Let us analyze again the G-morphism s_{∗} (cf. 2.9). Since the action in the target is abelian, this already shows that s_{∗} has a factorization through (H_{∗})_{H}. Clearly, (H_{∗})_{H} is a G/H = ℤ^{g}-module, where g := |B_{f}|. For any ring A, denote by Λ_{A,g} = A[ℤ^{g}] the Laurent polynomial ring in g indeterminates t_{1}, …, t_{g}. We call the Λ_{ℤ,g}-module M_{∗}(f) := (H_{∗})_{H} the global Alexander Λ_{ℤ,g}-module associated with the affine map f. Notice that it can be determined completely from the monodromy representation ρ_{∗}; in fact, it can be regarded as the commutative version of the monodromy representation.

Next, notice that K_{0} = ker(p_{0}) consists of loops (with base points) γ with ℓ(γ) = 0. Theorem 2.9, applied for such a loop γ and for the trivial loop, guarantees that ρ_{∗}[γ]m − m ∈ ker(s_{∗}) for any m. In particular, s_{∗} has a factorization through (H_{∗})_{K_{0}}. Obviously, (H_{∗})_{K_{0}} = ((H_{∗})_{H})_{K_{0}/H}, hence (H_{∗})_{K_{0}} can be determined completely from the global Λ_{ℤ,g}- module M_{∗}(f) of f. Its action has a factorization through G/K_{0} = ℤ. The Λ_{ℤ}-module M_{∗}(f, 0) := (H_{∗})_{K_{0}} will be called the local Alexander Λ_{ℤ}-module associated with the affine map f and bifurcation point 0 ∈ B_{f}. Clearly, for any b ∈ B_{f} one can consider a similar Λ_{ℤ}-module M_{∗}(f, b) constructed from ρ_{∗} or from M_{∗}(f).

The above results are summarized in the following theorem.

2.14. Theorem. Let f : U → ℂ be an affine map with generic fiber F connected. Consider the monodromy representation G → Aut(H_{∗}), which determines the global Alexander module M_{∗}(f) = (H_{∗})_{H}, and the local Alexander module M_{∗}(f, 0) = (H_{∗})_{K_{0}} associated with the bifurcation point 0 ∈ B_{f}. If we consider H_{∗}(𝔽, ℤ) as a G-module via p_{0}, then s_{∗} : H_{∗} → H_{∗}(𝔽, ℤ) has the following factorization into morphisms of G-modules:

s_{∗} : H_{∗} → (H_{∗})_{H} → (H_{∗})_{K_{0}} → H_{∗}(𝔽, ℤ).

If we tensor this tower by ℚ, the last term H_{∗}(𝔽, ℚ) can be replaced by T_{∗} = T ors(H_{∗}(𝔽, ℚ), h_{∗}), which is the image of s_{∗} ⊗ 1_{ℚ}.

2.15. Remark. Notice that any γ with ℓ(γ) = +1 induces the same operator \overline{ρ_{q}}([γ]) acting on (H_{q})_{K_{0}}; and this operator is the positive generator of the cyclic action on (H_{q})_{K_{0}}. E.g., one can take \overline{T}_{0,q}, or \overline{T}_{∞,q}, where T_{0,q} (resp. T_{∞,q}) is the monodromy operator around the special fiber f^{−1}(0) (respectively at infinity) (cf. 2.16). Then, for an arbitrary [γ] ∈ G with ℓ := ℓ(γ) = p_{0}([γ]), one has the Λ_{ℤ}-module epimorphisms:

(H_{∗}(F, ℤ), ρ_{∗}(γ)) → ((H_{∗})_{K_{0}}, \overline{T} _{0,∗}^{ℓ}) → (H_{∗}(𝔽), h_{∗}^{ℓ}).

Evidently, this provides the following divisibilities of the corresponding Alexander (or characteristic) polynomials:

Δ(H_{q}(𝔽), h_{q})(t)^{(ℓ)} | Δ((H_{q})_{K_{0}}, \overline{T}_{0,q})(t)^{(ℓ)} | Δ_{γ,q}(t).

2.16. Examples. The monodromy around the origin and at infinity. (A) To any f (and to a distinguished atypical value 0 ∈ B_{f}) one can associate two distinguished monodromy operators:

(i) the local monodromy T_{0,∗} : H_{∗}(F, ℤ) → H_{∗}(F, ℤ) of f at 0, provided by a loop γ going around the bifurcation point 0 on the oriented boundary of a small disc D_{0};

(ii) the monodromy at infinity T_{∞,∗} : H_{∗}(F, ℤ) → H_{∗}(F, ℤ) of f provided by a loop γ going around the oriented boundary of a large disc D_{∞} containing all the bifurcation points B_{f} inside.

They provide two (Alexander) Λ_{ℤ}-module structures on H_{∗}(F, ℤ) denoted by A_{∗}(f, 0), respectively A_{∗}(f, ∞). The corresponding Alexander polynomials, or equivalently, the characteristic polynomials of the monodromy operators T_{0,∗} and T_{∞,∗}, are denoted by Δ(T_{0,∗}) resp. Δ(T_{∞,∗}). Note that the complexified Alexander module A_{∗}(f, 0) (resp. A_{∗}(f, ∞)) encodes exactly the Jordan block structure of T_{0,∗} (resp. T_{∞.∗}) and that this Jordan structure was studied in several papers, e.g. [ACD], [D2], [DN1-2], [DS], [Do], [GN1-3], [Sa].

Obviously, in both cases ℓ = 1. Therefore, 2.10 guarantees that for any f with F connected, and for any q ≥ 0, the Alexander polynomial

Δ(H_{q}(𝔽, ℚ), h_{q}) divides the characteristic polynomials Δ(T_{0,q}) and Δ(T_{∞,q}). (3)

(B) (3) above is a generalization of Theorem 4 in Kulikov [K], where U = ℂ^{2} but also of Libgober result [Li2] (4.3), where U = ℂ^{n+1}. Let us explain more precisely the relation between our divisibility result (3) and the divisibility results in [Li2].

First consider the local monodromy T_{0,∗}. In [Li2], X has only isolated singularities including at infinity. In that case only the case q = n is relevant (cf. 4.3). Let V be the closure of X in ℙ^{n+1}. It is known that by the localization of the monodromy (see e.g. [NN]) one has

Δ(T_{0,n})(t) = (t − 1)^{k} · ∏ Δ_{i}(t),  (4)

where Δ_{i} are the characteristic polynomials of the local monodromies associated to the isolated singularities on V, and k = b_{n}(F) − ∑ deg(Δ_{i}) = rank_{Λ}H_{n+1}(𝔽, ℂ), cf. also with 7.5(2). In fact, the singularities at infinity (i.e. on V ∩ H) should be treated slightly different, as explained in [Li2], [LiT]. Moreover, in general k ≥ 0, and in the case of isolated singularities at infinity we have k = 0 iff B_{f} = {0} (fact which follows from 7.5(2) and 2.19(5)). Note that (3) (for q = n) and (4) give a similar result to Theorem (4.3) in [Li2], yielding in addition a precise value for k.

The discussion for the monodromy at infinity T_{∞,∗} is more involved. It was shown by Neumann and Norbury [NN] that the total space of the fibration (∗) f : f^{−1}(S_{r}^{1}) → S_{r}^{1} for r ≫ 0 (which provides T_{∞,∗}) can be embedded in a natural way as an open subset of S^{2n+1} \ f^{−1}(0), where S^{2n+1} is a large sphere in ℂ^{n+1}. Moreover, it was shown in [NZ] that for an M-tame polynomial this fibration (∗) is equivalent to the Milnor fibration at infinity ϕ : S^{2n+1} \ f^{−1}(0) → S^{1}, ϕ(x) = f(x)/|f(x)|. Note that in general S^{2n+1} \ f^{−1}(0) is not the total space of a fibration over the circle, or, even when it is, it may happen that the corresponding fiber is not F, see the case of semi-tame polynomials in [PZ].

In [Li2], Libgober considers an infinite cyclic covering U_{∞} of the knot complement S^{2n+1}\ f^{−1}(0) and takes the associated Alexander module H_{n}(U_{∞}, ℂ) as the Alexander module at infinity for f. From our discussion above it seems that, in general, one cannot hope to identify easily the structure of this module H_{n}(U_{∞}, ℂ). However, in the case of M-tame polynomials, the module H_{n}(U_{∞}, ℂ) is exactly our A_{n}(f, ∞). (C) By the above general theory, one gets the epimorphisms of Λ_{ℚ}-modules A_{∗}(f, 0) ⊗ ℚ → T_{∗} (and similarly for A_{∗}(f, ∞)). But, in general, the kernel of this map is rather big, hence the modules A_{∗} provide rather weak “approximations” of the torsion parts T_{∗}, whose description is our ultimate goal. Moreover, studying only the operators T_{0,∗} (or T_{∞,∗}) there is no hope to find any quotient of the modules A_{∗}, which would provide a better approximation of T_{∗}. On the other hand, using the whole monodromy representation, A_{∗} firstly can be divided by the commutator of the monodromy group (which in general is rather substantial, since the monodromy representation, in general, is rather non-commutative). Secondly, this can be divided further (using loops winding around the other bifurcation point), which again can be substantial if the number of bifurcation points is large. Hence, the module (H_{∗})_{K_{0}} is a substantially better approximation of T_{∗} than any module of type A_{∗}. Moreover, it can be computed from the monodromy representation without any knowledge about the singular fibers.

Nevertheless, in general, the computation of the monodromy representation of f can be a serious task (cf. [DN2], [NN] or §6 below).

2.17. The connection with the vanishing cycles. In general, the epimorphism (H_{q})_{K_{0}}⊗ ℚ → T_{q} is not an isomorphism (see e.g. §6). In fact, we believe that, in general, T_{q} cannot be recovered from the monodromy representation alone (e.g. one cannot decide how to distribute the G-invariant subspace of H in the different T_{q}’s associated with different bifurcation points, provided that |B_{f}| > 1; see e.g. §6); cf. also with 5.8(1). On the other hand, using some extra information about the structure of the vanishing cycles in H_{q}(F, ℚ), the representation ρ_{q} determines T_{q}. This is explained in the next paragraphs.

We recall briefly the definition of the vanishing cycles (see e.g. [NN] or [DN1-2]). Fix b_{0} ∈ ℂ \ B_{f} and F := f^{−1}(b_{0}). For each b ∈ B_{f} let γ_{b} be a simple path (in ℂ \ int(∪_{b}D_{b})) connecting b_{0} and b′ ∈ ∂D_{b}. Then γ_{b} defines an identification γ_{b,∗} : H_{∗}(f^{−1}(b′), ℤ) → H_{∗}(F, ℤ), hence a map γ_{b,∗} ∘ ∂ : H_{∗+1}(f^{−1}(D_{b}), f^{−1}(b′), ℤ) → H_{∗}(F, ℤ). The image of this map is denoted by V_{∗}(b, γ_{b}), and it is called the subspace of vanishing cycles associated with b and the path γ_{b}. Clearly, V_{∗}(b, γ_{b}), as an abstract group, it is independent of the choice of γ_{b}, but its embedding into H_{∗}(F, ℤ) depends on the path in a rather subtle way.

2.18. Proposition. Consider the morphism s_{∗} : H_{∗}(F, ℤ) → H_{∗}(𝔽, ℤ) as above. Then

ker(s_{∗}) = ∑ V_{∗}(b̄, γb̄),

where the sum is over all the bifurcation points b̄ ∈ B_{f}^{∗} = B_{f} \ {0} and all the possible paths γb̄. This determines completely T_{∗} = H_{∗}(F, ℚ)/ker(s_{∗}) ⊗ ℚ.

Proof. Using the homology exact sequence of the pair (𝔽, F), notice that ker(s_{∗}) is exactly the image of H_{∗+1}(𝔽, F) → H_{∗}(F). Then use the construction from the proof of 2.10 and the (homotopy) lifting property along the paths with respect to p. □

2.19. Example. Let us make the above statement more precise. For simplicity we will assume that U = ℂ^{n+1}. We fix a “star” {γ_{b}}_{b∈B_{f}} of simple paths with base point b_{0}. (For details see [DN1-2].) Then, using the notation V_{q}(b) := V_{q}(b, γ_{b}) for any fixed q and b ∈ B_{f}, one has

H_{q}(F, ℤ) = V_{q}(0) ⊕ V_{q}(b_{1}) ⊕ ⋯ ⊕ V_{q}(b_{g−1}).  (5)

Let T_{0,q} be the monodromy around 0 ∈ B_{f}, associated with the path γ_{0}^{−1} ∘ ∂D_{0} ∘ γ_{0}. Then the image of V_{q}(0) := T_{0,q} − Id is in V_{q}(0). Using the sum-decomposition (5), V_{q}(0) : H_{q}(F, ℤ) → V_{q}(0) can be written in the form V_{q}(0) = (m_{0} − Id, m_{1}, …, m_{g−1}) for some operators m_{0} : V_{q}(0) → V_{q}(0) and m_{i} : V_{q}(b_{i}) → V_{q}(0) (1 ≤ i ≤ g − 1) (see e.g. [loc. cit.]).

Now, notice that by 2.18, V_{q}(b_{i}) ⊂ ker(s_{q}) for any 1 ≤ i ≤ g − 1. Moreover, ker(s_{q}) is G-invariant (cf. 2.13), hence also T_{0,q}-invariant. Therefore, for any x ∈ V_{q}(b_{i}), m_{i}(x) = (T_{0,q} −Id)(0, …, x, …, 0) ∈ ker(s_{q}) and m_{0}^{k}m_{i}(x) ∈ ker(s_{q}) for any k ≥ 0 and 1 ≤ i ≤ g −1. Proposition 2.18 reads as

T_{q} = \frac{V_{q}(0)}{V_{q}(0) ∩ ker(s_{q})} = \frac{V_{q}(0)}{∑_{k≥0} ∑_{1≤i≤g−1} im(m_{0}^{k}m_{i})}.

3. The case of h-good affine maps

Sometimes it is convenient to work with special maps with nice behaviour around the special fibers or at infinity. First, we recall the definition of Neumann and Rudolph [NR] of good polynomials f : ℂ^{n+1} → ℂ. A fiber f^{−1}(b) is called “regular at infinity” if there exist a small disc D containing b and a compact set K such that the restriction of f to f^{−1}(D) \ K is a trivial C^{∞}-fibration. The polynomial f is called good (or “topologically good”) if all its fibers are regular at infinity.

For example, the tame polynomials introduced by Broughton [B], the larger class of Mtame polynomials introduced in [NZ] are good. We recall that any fiber of a good polynomial is a bouquet of spheres S^{n}, B_{f} is the set of critical values of f, for any b ∈ B_{f} the “tube” T_{b} has the homotopy type of f^{−1}(b), and f^{−1}(b) (homotopically) is obtained from F by attaching some cells of dimension n + 1.

For our purpose here it is enough and more natural a much weaker assumption.

3.1. Definition. An affine map f : U → ℂ is called “homotopically good” (h-good) if for any b ∈ B_{f}, the pair (T_{b}, F) is n-connected. If U = ℂ^{n+1}, then we say that f is a h-good polynomial.

From the above discussion it follows easily that all the good polynomials are h-good. Another example is provided by the polynomials with isolated singularities on the affine space and at infinity in the sense of Siersma-Tibăr (see p.776 in [SiT]).

In general, for any affine map, it is much easier to handle the properties of the generic fiber and the “tubes” T_{b} than the properties of the special fibers (see e.g. [DN2] for polynomial maps). One of the advantages of the above definition 3.1 is that it requires information only about F and T_{b}’s. (Conversely, this fact also explains that for h-good maps one can say very little about the special fibers. E.g. the special fibers of h-good polynomials, in general, are not even reduced, as it happens e.g. for f(x, y) = x^{2}y. For a non-trivial example of a h-good polynomial which has non-isolated singularities, see the polynomial f_{d,a} constructed by tom Dieck and Petrie, cf. [D1], p.175.)

The second advantage of definition 3.1 is that, in fact, it is almost homological. Indeed, for n = 1, f is h-good iff F is connected (by 2.3). For n > 1 the polynomial f is h-good if F is simply-connected and H_{q}(T_{b}, F, ℤ) = 0 for all b and q ≤ n. This follows from 2.3 and the relative Hurewicz isomorphism theorem (see e.g. [S], p.397). (If π_{1}(U) = (1), then even the converse is true: by Hurewicz theorem H_{q}(T_{b}, F, ℤ) = 0, and by the proof of 3.2 F is simply-connected.)

We list now the properties of the map s_{∗} in the case of h-good affine maps using the notation of 2.9.

3.2. Proposition. If f is h-good then the pairs (U, F), (𝔽, F) and (M_{X}, T_{0}^{∗}) are homologically n-connected. Moreover, if π_{1}(U) is trivial, then all these pairs are n-connected.

Proof. For the homological n-connectedness of (U, F) use 2.2(1); for (𝔽, F) use (2) from the proof of 2.10; the same proof works for (M_{X}, T_{0}^{∗}) (but neglecting p). Moreover, π_{1}(M_{X}, T_{0}^{∗}) = (1) by 2.3(iii), π_{1}(𝔽, F) = (1) by 2.6. If π_{1}(U) = (1) then π_{1}(U, F) = (1) as well. This ends the case n = 1.

If n > 1 we have to verify that π_{1}(F) = (1), then the result follows from the relative Hurewicz isomorphism theorem. The connectivity assumption assures that for each b ∈ B_{f}, π_{1}(F) → π_{1}(T_{b}) is an isomorphism. We denote all these fundamental groups by π. If the cardinality |B_{f}| of B_{f} is one, then this implies that π_{1}(F) = π is trivial (since in this case, T_{b} ∼ U). If |B_{f}| > 1, then by van Kampen theorem, applied for Y = ∨_{F} (T_{b}) (cf. 2.2(1)), and induction over |B_{f}|, we get π_{1}(Y) = π. But by 2.2(1), π_{1}(Y) = π_{1}(U) = (1). □

3.3. Corollary. Assume that f is h-good. Then one has the following facts:

(a) If q < n, then H_{q}(𝔽, ℤ) = H_{q}(F, ℤ) = H_{q}(U, ℤ), ρ_{q} and h_{q} are trivial, and s_{q} is an isomorphism. Hence H_{q}(𝔽, ℤ) is a torsion module with trivial action.

(b) If q = n, then s_{n} is onto, H_{n}(𝔽, ℤ) is a torsion module, and one has the following tower of epimorphisms of G-modules:

H_{n} → (H_{n})_{H} → (H_{n})_{K_{0}} → H_{n}(𝔽, ℤ).

(c) If q = n + 1, then H_{n+1}(𝔽, ℤ) is Λ_{ℤ}-free of rank |χ(M_{X})|. If U is (n − 1)-connected (e.g. if U = ℂ^{n+1}), then one has the following additional data:

(i) F is a bouquet of spheres S^{n}.

(ii) H_{q}(M_{X}, ℤ) = π_{q}(M_{X}) = 0 for 1 < q < n.

(iii) 𝔽 is (n − 1)-connected, and for n > 1 one has the isomorphism of the Alexander Λ_{ℤ}-modules H_{n}(𝔽, ℤ) = π_{n}(𝔽) = π_{n}(M_{X}).

Proof. (a)-(b)-(c) follows from the results of section 2 and 3.2. (i) is clear for n = 1. If n > 1 then by 3.2 F is homologically (n − 1)-connected, and by the proof of 3.2 it is simplyconnected. Being a finite CW-complex of dimension n, the result follows from Whitehead theorem. This implies (iii) by 3.2. For (ii) notice that 𝔽 is the cyclic covering of M_{X} (or use the connectivity of (M_{X}, T_{0}^{∗}) and the Wang exact sequence). □

For special cases of the above connectivity results, see also [Li2] and [LiT].

3.4. Application. Since the case of h-good polynomials with all the involved numerical invariants represents a special interest, we present the geometric interpretation of the homology groups of the G-modules H and H_{H}. We set S := ℂ \ B_{f}, E := f^{−1}(S), and b_{k}^{c}(Y) = dim H_{c}^{k}(Y, ℂ) (where H_{c}^{k} denotes the cohomology with compact supports).

We recall the following result, which shows how the G-module H determines the homology of E. Its proof is based on the homological Leray spectral sequence of the fibration F → E → S and on basic facts on group homology (see e.g. [Br]). The claim about the rank of H_{G} follows from the long exact sequence ... → H_{c}^{k}(E) → H_{c}^{k}(ℂ^{n+1}) → H_{c}^{k}(∪F_{b}) → ....

3.5. Proposition. The reduced homology groups H̃_{k}(E, ℤ) are trivial except at most for k = 1, k = n and k = n + 1. For these values of k one has the following.

(i) For n = 1 one has H_{2}(E, ℤ) = H_{1}(G, H) and an exact sequence of groups

0 → H_{G} → H_{1}(E, ℤ) → ℤ^{g} → 0.

In particular, H_{G} is a free ℤ-module with rank(H_{G}) = ∑_{b∈B_{f}} (n(F_{b}) − 1), where n(Y) denotes the number of irreducible components of a curve Y.

(ii) For n > 1 one has H_{1}(E, ℤ) = ℤ^{g}, H_{n}(E, ℤ) = H_{0}(G, H) = H_{G} and H_{n+1}(E, ℤ) = H_{1}(G, H). In particular, rank(H_{G}) = ∑_{b∈B_{f}} b_{n+1}^{c}(F_{b}) = ∑_{b∈B_{f}} b_{n+1}(T_{b}, ∂T_{b}).

Set H = [G, G] = G′ as above, and let S′ → S be the corresponding covering space. Let f′ : E′ → S′ be the fibration (with fiber F) obtained from the fibration f : E → S by pull-back. Then the monodromy of the fibration f′ corresponds exactly to the H-module H obtained by restriction of ρ to H. On the other hand, we can regard E′ → E as being the covering space corresponding to the kernel of the composition π_{1}(E) → π_{1}(S) = G → G/H = ℤ^{g} → 0. It follows that the deck transformation group of E′ → E is ℤ^{g} and hence we can regard H_{n}(E′, ℤ) as a Λ_{ℤ,g}-module.

3.6. Proposition. The reduced homology groups H̃_{k}(E′, ℤ) are trivial except at most for k = 1, k = n and k = n + 1. For these values of k one has the following.

(i) For n = 1 one has H_{2}(E′, ℤ) = H_{1}(H, H) and an exact sequence of groups

0 → H_{H} → H_{1}(E′, ℤ) → H_{1}(H, ℤ) → 0.

(ii) For n > 1 one has H_{1}(E′, ℤ) = 0 for g = 1 and H_{1}(E′, ℤ) = H_{1}(H, ℤ) for g > 1, H_{n}(E′, ℤ) = H_{0}(H, H) = H_{H} and H_{n+1}(E′, ℤ) = H_{1}(H, H).

Using the description of the K(H, 1) and of the associated chain complex given in [Li4], (1.2.2.1), it follows that H_{1}(H, ℤ) = G′/G′′ is a submodule of Λ_{ℤ,g}^{g} and hence H_{1}(H, ℤ) is Λ_{ℤ,g}-torsion free. This implies that in both cases (i) and (ii) in 3.6, we have H_{H} = Tors(H_{n}(E′, ℤ)) (as a Λ_{ℤ,g}-module).

4. Local systems on M_{X}

4.1. Definitions. Coming back to our general situation, we recall that f : U → ℂ is an arbitrary affine map and X = f^{−1}(0). We assume that F is connected, and we consider the epimorphism e_{M_{X}} = f_{∗} : π_{1}(M_{X}) → ℤ induced by f : M_{X} → ℂ^{∗} (cf. 2.3).

For a complex number a ∈ ℂ^{∗}, we consider the rank one local system L_{a} on M_{X} defined by the composed map π_{1}(M_{X}) ⟶^{e_{M_{X}}} ℤ → ℂ^{∗}, where the last map is defined by 1_{ℤ} ↦ a. Obviously, if a = 1, then L_{a} = ℂ.

Then, exactly as in [Li3], we have the following long exact sequence

... → H_{q}(𝔽, ℂ) → H_{q}(𝔽, ℂ) → H_{q}(M_{X}, L_{a}) → H_{q−1}(𝔽, ℂ) → ...  (1)

where the first morphism is multiplication by t − a. In fact, for a = 1, (1) is also valid over ℤ. Indeed, the ℤ-covering 𝔽 → M_{X} homotopically can be identified with the inclusion of the fiber into the total space of a fibration M_{X} → K(ℤ, 1) = S^{1} (with fiber 𝔽). Hence (1) (with a = 1 and over ℤ) is exactly the (homotopy) Wang exact sequence of the covering 𝔽 → M_{X}. More generally, let M_{X,e} (e > 0) be the cyclic covering of M_{X} of degree e (i.e. the covering determined by the subgroup e_{M_{X}}^{−1} (eℤ)). If ℤ denotes the transformation group of 𝔽 above M_{X}, then 𝔽 is a ℤ-covering of M_{X,e} with transformation group eℤ. Hence, we have the (Wang) exact sequence

... → H_{q}(𝔽, ℤ) → H_{q}(𝔽, ℤ) → H_{q}(M_{X,e}, ℤ) → H_{q−1}(𝔽, ℤ) → ...  (2)

where the first morphism is multiplication by t^{e} − 1.

In particular, the Λ_{ℤ}-module H_{∗}(𝔽, ℤ) contains all the homological information about the local systems L_{a} and the homology of M_{X,e} (cf. also with 5.4 and 7.5(3) below).

Then 2.10 and 4.1(1) imply:

4.2. Theorem. Denote by N(a, q) the number of direct summands in the (t − a)-torsion part of H_{q}(𝔽, ℂ).

(i) H_{q}(M_{X}, L_{a}) = 0 for any a ∈ ℂ^{∗} and q > n + 1.

(ii) dim H_{q}(M_{X}, L_{a}) = rank_{Λ}H_{q}(𝔽, ℂ) + N(a, q) + N(a, q − 1) for any q.

(iii) In particular, the Alexander modules H_{q}(𝔽, ℂ) are torsion for q ≤ n iff H_{q}(M_{X}, L_{a}) = 0 for any q ≠ n + 1 and generic a (e.g. for non-roots of unity). If we are in this case, then dim H_{n+1}(M_{X}, L_{a}) = N(a, n) + |χ(M_{X})| for any a.

Note that 2.10 gives a topological way to compute the ranks rank_{Λ}H_{q}(𝔽, ℂ), while 2.14, 2.15 give a lot of information on the numbers N(a, q).

4.3. Example. Here we wish to emphasize the parallelism of h-good polynomials with the case of hypersurfaces with only isolated singularities including at infinity considered by Libgober [Li2]. (In this later case, e_{M_{X}} is defined by f_{∗} where f is the reduced equation of X.) Assume that U = ℂ^{n+1}. Then, if either

(I) X is the fiber of a h-good polynomial map, or

(II) X has only isolated singularities including at infinity, or

(III) X is an essential hyperplane arrangement (i.e. the minimal dimension of the nonempty intersections of hyperplanes in the arrangement is zero), then the equivalent conditions in (iii) do hold. Additionally, in the cases (I) and (II), H̃_{q}(𝔽, ℤ) = 0 for k < n, H_{n}(𝔽, ℤ) is a Λ_{ℤ}-torsion module, and, for n > 1, π_{n}(M_{X}) = π_{n}(𝔽) = H_{n}(𝔽, ℤ) are isomorphic Alexander Λ_{ℤ}-modules. See 3.3 for (I), [Li2] for (II), and [CS] for (III).

The proof of the above results is topological. It follows that the same proof can be applied to coefficients in a finite field K, as soon as |K| is large enough. Therefore, Theorem 4.2 also holds in such a case (of course without any reference to roots of unity).

4.4. Example. Assume that f is a weighted homogeneous polynomial with respect to an integer set of weights (non necessarily strictly positive) such that d = deg(f) ≠ 0. Denote by F the associated affine Milnor fiber. Then it is easy to see that 𝔽 and F are homotopically equivalent, and the covering transformation h corresponds to the Milnor monodromy. In particular h_{∗}^{d} = Id. It follows that N(a, q) = dim ker(h_{∗}−aId|H_{q}(𝔽, ℂ)) is trivial for a^{d} ≠ 1. Moreover, in this case, due to the presence of S^{1}-actions, we have χ(M_{X}) = 0.

In the case of a central hyperplane arrangement D. Cohen [C] has shown that for any rank one local system L on M_{X} one has dim H_{m}(M_{X}, L) ≤ dim H_{m}(M_{X}, ℂ) for any m.

The example of the A_{1}-singularity f : ℂ^{3} → ℂ, f = x^{2} + y^{2} + z^{2} , m = 2 and a = −1, shows that this inequality is not true for a general homogeneous polynomial.

4.5. Next, we present another case (as a generalization of Theorem 3.1 in [Li2]) satisfying some equivalent conditions similar to 4.2(iii).

Let X ⊂ U be a reduced affine hypersurface and let (Y, V) be a smooth compactification of M_{X} = U \ X, i.e. Y is projective and smooth, U is a Zariski open subset in Y such that W = Y \ U is a divisor, V is the closure of X in Y. Additionally, let (Z, D) → (Y, V ∪ W) be an embedded resolution of singularities, which is an isomorphism over Y \ Sing(V ∪ W). Let D = ∪_{i=1}^{r}D_{i} be a decomposition of D as a union of irreducible smooth hypersurfaces. We denote by {D_{i}}_{i∈I} the strict transforms of the irreducible components of V (which are in one-to-one correspondence with the irreducible components of X). Let D′ = ∪_{i∈I}D_{i}.

4.6. Theorem. Assume that Z \D′ is affine. Then the following equivalent conditions hold: (i) The Alexander modules H_{q}(𝔽, ℂ) are (t − 1)-torsion for q ≤ n; (ii) H_{q}(M_{X}, L_{a}) = 0 for all a ≠ 1 and q ≤ n.

Proof. The equivalence (i) ⇔ (ii) is clear. We will prove below the claim (ii). By duality we can pass to the cohomology groups, hence we have to show that

(ii′) H^{q}M_{X}, L_{a}) = 0 for all a ≠ 1 and q ≤ n.

We look at L_{a} as a local system on Z \D. As such, it has a monodromy along any irreducible component D_{i} of D, which for i ∈ I is clearly a. Consider now the diagram of inclusions

M_{X} ⟶^{j_{1}} U′ ⟶^{j_{2}} Z,

where U′ = Z \ D′. Let j = j_{2} ∘ j_{1}. Then H^{q}(M_{X}, L_{a}) = ℍ^{q}(Z, Rj_{∗}L_{a}). On the other hand Rj_{∗} = Rj_{2∗} ∘ Rj_{1∗}. Set F = Rj_{1∗}(L_{a}). Then F(n + 1) is a perverse sheaf on U′. Indeed, L_{a}(n + 1) is perverse on M_{X}, and Rj_{1∗} preserves the perverse sheaves, since j_{1} is an affine morphism with finite fibers (cf. [BBD] (2.2.5) and (4.1.1)). The key fact is that Rj_{2∗}F = Rj_{2!}F. Indeed, due to the existence of a natural morphism Rj_{2!} = j_{2!} → Rj_{2∗}, it is enough to check the equality at a point x ∈ D′ and after taking the homology sheaves. I.e., we have to verify that (R^{q}j_{2∗}F)_{x} = 0 for all q. Note that R^{q}j_{2∗}F = R^{q}j_{∗}F, hence the above vanishing follows from Künneth Theorem, exactly as in [EV] (1.6). It follows that

H^{q}(M_{X}, L_{a}) = ℍ^{q}(Z, j_{!}F) = ℍ_{c}^{q}(U′, F) = ℍ_{c}^{q−n−1}(U′, F(n + 1)).

Since U′ is affine, ℍ_{c}^{k}(U′, P) = 0 for any perverse sheaf P and all k < 0. Indeed, by Verdier duality one has ℍ_{c}^{k}(U′, P) = ℍ^{−k}(U′, DP)^{∨}, and the last group is zero by Artin’s Theorem, see [KS] (10.3.8). □

4.7. Example. Assume that V ∪ W is a normal crossing divisor (e.g. X is a generic affine hyperplane arrangement in ℂ^{n+1}). Then 4.6 clearly holds.

4.8. Application: Twisted de Rham cohomology and Alexander invariants. First notice that by duality, one has dim H_{q}(M_{X}, L_{a}) = dim H^{q}(M_{X}, L_{b}), where b = a^{−1}. Due to the fact that our Alexander modules have torsion only associated with roots of unity, one also has dim H_{q}(M_{X}, L_{a}) = dim H_{q}(M_{X}, L_{b}) (and the same for cohomology).

Fix now a ∈ ℂ^{∗} and write a = exp(2πiλ) for some λ ∈ ℂ with λ ∉ ℕ^{∗}. Consider the algebraic twisted de Rham complex (Ω^{•}(M_{X}), ∇_{λ}), where Ω^{k}(M_{X}) is the space of global k-forms on M_{X} and ∇_{λ}ω = dω − (λ · df/f) ∧ ω. The following Comparision Theorem by Grothendieck-Deligne relates the topology to the algebraic differential forms, see [De]:

4.9. Theorem. H^{∗}(Ω^{•}(M_{X}), ∇_{λ}) = H^{∗}(M_{X}, L_{b}).

This result combined with our 4.2 yields the following fact (which generalizes the corresponding result in [Li3]).

4.10. Corollary. dim H^{q}(Ω^{•}(M_{X}), ∇_{λ}) = rank_{Λ}H_{q}(𝔽, ℂ) + N(b, q) + N(b, q − 1).

Same remarks as after 4.2 apply here.

5. Non-vanishing results

5.1. In the previous sections we concentrated our atention on structure theorems and vanishing results. In this section our goal is to convince the reader that non-trivial Alexander modules really do appear. In fact, already the case U = ℂ^{n+1} provides a large variety of possibilities, so in the sequel we assume this. Moreover, we will focus on the non-vanishing H_{n}(M_{X}, ℤ) ≠ 0. This is also motivated by the following facts:

(1) Assume that 𝔽 is (n − 1)-connected and n > 1 (cf. e.g. 4.3). Then by 4.1(1) (with a = 1 and over ℤ) one has an epimorphism H_{n}(𝔽, ℤ) → H_{n}(M_{X}, ℤ). In particular, the nonvanishing of H_{n}(M_{X}, ℤ) implies the non-vanishing of the Alexander modules H_{n}(𝔽, ℤ) = π_{n}(𝔽) = π_{n}(M_{X}).

(2) By 2.12, if F is the generic fiber of f, then (∗) H̃_{q}(M_{F} , ℤ) = H̃_{q}(S^{1}) ⊕ H̃_{q−1}(F, ℤ) for any q. More generally, (∗) is also valid for any smooth fiber X by the Gysin sequence, see [D1], p.46. Hence, for n > 1, H_{n}(M_{X}, ℤ) = H_{n−1}(X, ℤ), where the second group has an independent interest (e.g. it is the first obstruction for X to be ∨S^{n}).

(3) If X has isolated singularities including at infinity [Li2], or if it is the special fiber of a h-good polynomial, then H_{q}(M_{X}, ℤ) = 0 for 1 < q < n (cf. 3.3, 4.3, [Li2], see also [LiT]). Hence the first interesting homology group occurs in degree n.

In this section we present four constructions which provide in a systematic way hypersurfaces X with H_{n}(M_{X}, ℤ) ≠ 0.

5.2. The first construction (using duality). Let V denotes the projective closure of X in ℙ^{n+1} and H the hyperplane at infinity. Following [Li2], section 1, we consider the isomorphism

H_{q}(M_{X}) = H^{2n−q+1}(V, V ∩ H) for all q.

Assume that V and V ∩ H have only isolated singularities (this is exactly the condition on X to have isolated singularities including at infinity in [Li2]). The exact sequence

H^{n}(V) → H^{n}(V ∩ H) → H^{n+1}(V, V ∩ H) → H^{n+1}(V) → H^{n+1}(V ∩ H)  (1)

and the isomorphism H^{n+1}(V ∩ H) = H^{n+1}(ℙ^{n−1}) (cf. [D1], p.161) imply the following. If H^{n+1}(V, ℤ) ≠ H^{n+1}(ℙ^{n−1}, ℤ) then H_{n}(M_{X}, ℤ) ≠ 0.

5.3. Example. Let V be a cubic surface in ℙ^{3} having two singularities, one of type A_{1} and the other of type A_{5}. First we take H a generic plane. In this case, using [D1], p.165, we see that the exact sequence 5.2(1) becomes

ℤ → ℤ → H_{2}(M_{X}, ℤ) → ℤ/2ℤ → 0,

where the first morphism is multiplication by deg(V) = 3. It follows that H_{2}(M_{X}, ℤ) = ℤ/6ℤ, X is singular and the associated polynomial f is tame.

Secondly, take H to be any plane passing through the 2 singularities on V. By the above discussion H^{3}(V, ℤ) = ℤ/2ℤ, hence 5.2 can be applied. Notice that X is smooth, but it cannot be the generic fiber of a h-good polynomial. In fact, by 5.1(2) H_{2}(X, ℤ) ≠ 0.

5.4. The second construction (using finite cyclic coverings and defect). The second approach uses M_{X,e}, the cyclic covering of M_{X} of degree e (cf. 4.1). It is clear that

M_{X,e} = {(x, u) ∈ ℂ^{n+1} × ℂ^{∗}|f(x) − u^{e} = 0}.

In some cases we can get a useful approximation of M_{X,e} as follows. Fix a system of positive integer weights w = (w_{0}, ..., w_{n}), and let e be the top degree term in f with respect to w. Introduce a new variable t of weight 1 and let f̃(x, t) be the homogenization of f with respect to the weights (w, 1). Consider the affine Milnor fiber F′ := {f̃(x, t) = 1}, which is a smooth hypersurface in ℂ^{n+2}. One has an embedding j : M_{X,e} → F′ given by j(x, u) = (u^{−1} ∗ x, u^{−1}), where ∗ denotes the multiplication associated to the system of weights w. The complement F′ \ j(M_{X,e}) is characterized by {t = 0}, hence it can be identified with the affine Milnor fiber F_{e} := {f_{e}(x) = 1} (considered din ℂ^{n+1}) defined by the top homogeneous component f_{e} of f.

If f_{e} defines an isolated singularity at the origin, then f is M-tame (see e.g. [DP]), in particular good. Hence 4.3 can be applied, in particular 𝔽 is (n − 1)-connected.

Moreover, in this case, f_{e} is a good polynomial, hence F_{e} is (n − 1)-connected as well, and the Gysin sequence of the divisor F_{e} implies the isomorphisms H_{q}(M_{X,e}) → H_{q}(F′) for 1 < q ≤ n. Note that under this isomorphism the action of the natural generator of the covering transformation group on M_{X,e} corresponds to multiplication by exp(−2πi/e) (of all the coordinates) on F′.

Finally notice that the monodromy of the singularity f_{e} = 0 is semisimple and all the eigenvalues are e-roots of unity. Therefore, in the exact sequence 4.1(2), written for q = n, the first morphism is trivial. Hence, by all these discussions, for n > 1 one has:

π_{n}(M_{X}) = H_{n}(𝔽, ℤ) = H_{n}(M_{X,e}, ℤ) = H_{n}(F′, ℤ)

are isomorphic Λ_{ℤ}-modules. This results should be compared to Corollary (4.9) in [Li2].

Notice also that H_{q}(M_{X}, ℚ) is isomorphic to the group of invariants of H_{q}(M_{X,e}, ℚ) with respect to the action given by the covering transformation.

The dimension of H_{n}(F′) is closely related to the defect associated with the singular points of the projective hypersurface f̃ = 0 considered in ℙ^{n+1} (for details, see e.g. [D1]). Hence, this construction emphasizes the connection between H_{n}(M_{X}), Alexander modules and superabundance properties.

5.5. Example. Let f : ℂ^{4} → ℂ be the tame polynomial given by f(x, y, u, v) = x^{3} + y^{3} + xy − u^{3} − v^{3} − uv. Set X = {f = 0}. Then f is a tame polynomial and X has 10 nodes. It follows from [D1], p.208-209, (with all the weights w_{i} = 1) that dim H_{3}(F′, ℚ) = 5 and the multiplication by exp(−2πi/3) on F′ induces the trivial action on H_{3}(F′, ℚ). It follows that dim H_{3}(M_{X}, ℚ) = dim H_{3}(M_{X,3}, ℚ) = dim H_{3}(F′, ℚ) = 5. In fact,

π_{3}(M_{X}) ⊗ ℚ = H_{3}(𝔽, ℚ) = H_{3}(M_{X,3}, ℚ) = H_{3}(F′, ℚ) = (Λ_{ℚ}/(t − 1))^{5}.

5.6. The third construction (“counting” the Milnor numbers). For simplicity we will assume that f is a (topologically) good polynomial. Assume that X has n_{X} singular points with Milnor fibers F_{i} and Milnor numbers μ_{i} for 1 ≤ i ≤ n_{X}. For each i, let μ_{0,i} denote the rank of H_{n}(∂F_{i}). Set μ_{X} := ∑_{i} μ_{i} and μ_{0,X} := ∑_{i} μ_{0,i}. Below, ⊕_{X} means ⊕_{i=1}^{n_{X}}. Finally, let μ be the sum of all the Milnor numbers of the singularities of f (situated on all the singular fibers). With these notations, one has:

The groups H_{q}(M_{X}, ℤ) (q = n, n + 1) are inserted in the following commutative diagram (with ℤ-coefficients), where the first two vertical arrows are monomorphisms, the third is the identity, and the last one is an epimorphism:

0 → ⊕_{X}H_{n}(∂F_{i}) → ⊕_{X}H_{n}(F_{i}) → ⊕_{X}H_{n}(F_{i}, ∂F_{i}) → ⊕_{X} H̃_{n−1}(∂F_{i}) → 0
↓  ↓  ↓  ↓
0 → H_{n+1}(M_{X}) → H_{n}(F) → ⊕_{X}H_{n}(F_{i}, ∂F_{i}) →  H_{n}(M_{X})  → 0

In particular:

μ_{0,X} ≥ dim H_{n}(M_{X}) ≥ μ_{X} + μ_{0,X} − μ.

Above, the first inequality is not new, and it is not very sharp (cf. e.g. with [GN2] (2.30), or with [Si2] (5.4) and §9; it also follows from 5.7 and 7.5(3) below).

The second inequality is more interesting, and it can be used in two different ways. First, using the integers μ_{X}, μ_{0,X} and μ, if the sum of the first two is strictly larger than the third, one gets a non-vanishing criteria for H_{n}(M_{X}). For example, in the case 5.5, μ = 16, μ_{X} = μ_{0,X} = 10, hence the second inequality reads as dim H_{n}(M_{X}) ≥ 4.

Similarly, if one knows μ_{X}, μ_{0,X} and dim H_{3}(M_{X}) about the hypersurface X, then these numbers may impose serious conditions about singularities of some other singular fibers of f (e.g. about their existence).

Proof. Assume that X = f^{−1}(b), and let T_{b}^{o} be the interior of T_{b}. Clearly, M_{X} has the homotopy type of ℂ^{n+1} \ T_{b}^{o}. Let F be a fixed generic fiber of f inside of T_{b}^{o}. Then one can write the homological long exact sequence of the pair (ℂ^{n+1} \ F, ℂ^{n+1} \ T_{b}^{o}). Notice that H_{q} := H_{q}(ℂ^{n+1} \ F, ℂ^{n+1} \ T_{b}^{o}) equals H_{q}(T_{b} \ F, ∂T_{b}) by excision. Let B_{i} be a small Milnor ball of the i-th singular point of X. Then using the “good”-property of f and excision one gets H_{q} = ⊕_{X}H_{q}(B_{i} \ F_{i}, ∂B_{i} \ ∂F_{i}). But this is isomorphic to ⊕_{X}H_{q−1}(F_{i}, ∂F_{i}) by Gysin isomorphism. Use these facts and the Gysin isomorphism H_{n}(F) → H_{n+1}(M_{F}) to obtain the second line of the above diagram.

Next, assume that the disc D_{b} is sufficiently small with respect to the balls B_{i}, and consider for any ball B_{i} the local analog of the above picture; namely the homological long exact sequence of the pair (B_{i} \ F, B_{i} \ T_{b}^{o}). This homological sequence admits a natural map to the previous sequence induced by the inclusion. Finally, this “local sequence” is modified by dualities and Gysin isomorphisms. □

5.7. The forth construction (using monodromy operators). If one wants to determine the homology groups of M_{X}, either one needs some information about the “tubes” Tb̄ for b̄ ∈ B_{f} \{0}, or (using 5.2) one needs to know the behaviour at infinity of X; both rather subtle problems. Therefore, it is rather surprising that, in some cases, all these information is carried by only one monodromy operator T_{0,n} (and the position of the subspace V_{n}(0) in H_{n}(F, ℤ)). Here we present the case of h-good polynomials.

Consider the terminology of 2.19 for q = n; namely for a fixed “star” take the “variation map” V_{n}(0) : H_{n}(F, ℤ) → V_{n}(0). Let D′ be a closed topological disc in ℂ such that B_{f}^{∗} ⊂ int(D′) but 0 ∉ D′. Set U′ := f^{−1}(D′). Then the homology exact sequence of the pair (M_{X}, U′) provides

0 → H_{n+1}(M_{X}, ℤ) → H_{n}(F, ℤ) ⟶^{V_{n}(0)} V_{n}(0) → H_{n}(M_{X}, ℤ) ⟶^{f_{∗}} H_{n}(ℂ^{∗}, ℤ) → 0.

In particular, the homology groups of M_{X} are trivial (cf. also with 3.3) except possibly for: Case n = 1: H_{0}(M_{X}, ℤ) = ℤ; H_{2}(M_{X}, ℤ) is ℤ-torsion free of rank b_{n}(F) − rank(V_{n}(0)); and H_{1}(M_{X}, ℤ) = coker(V_{n}(0)) ⊕ ℤ, a ℤ-torsion free group of rank equal to the number of irreducible components of X. Case n > 1: H_{0}(M_{X}, ℤ) = H_{1}(M_{X}, ℤ) = ℤ; H_{n+1}(M_{X}, ℤ) is ℤ-torsion free of rank b_{n}(F) − rank(V_{n}(0)); and H_{n}(M_{X}, ℤ) = coker(V_{n}(0)).

For any b ∈ B_{f} one can take similar “variation maps” V_{n}(b) : H_{n}(F, ℤ) → V_{n}(b). Set M_{F_{b}} = ℂ^{n+1} \ f^{−1}(b). In [DN2] it is proved the following equivalence: V_{n}(b) ⊗ ℚ is onto for all b ∈ B_{f} if and only if the cohomology monodromy representation ρ^{n} : G → Aut(H^{n}(F, ℚ)) has no non-trivial invariant vector. In particular:

H^{n}(F, ℚ)^{G} ≠ 0 ⇔ ⊕_{b∈B_{f}} H_{n}(M_{F_{b}}, ℚ) ≠ 0.

In fact, in this criterion we do not really need to know all the representation ρ^{n}. Indeed, assume n > 1 (the other case is similar). Denote by T_{b,n} the (individual) monodromy operator around b ∈ B_{f}. Then dim H_{n}(M_{F_{b}}) = dim V_{n}(b) − rank(T_{b,n} − Id), which via 2.19(5) gives:

∑_{b∈B_{f}} dim H_{n}(M_{F_{b}}) = dim H_{n}(F) − ∑_{b∈B_{f}} rank(T_{b,n} − Id).

5.8. Remarks. (1) In [DN2] we explained that the monodromy representation determines the structure of the vanishing cycles (over ℚ) if and only if H^{n}(F, ℚ)^{G} = 0. Notice that this is exacly the non-interesting case case from the point of view of (all) the complements M_{F_{b}} (b ∈ B_{f}). In particular, in all the non-trivial cases, in the precise characterization of T_{n} and H_{n}(M_{X}, ℤ) we need to know some extra information about the vanishing cycles (e.g. the sum-decomposition 2.19(5); see e.g. (2) below).

(2) 2.19 and the above discussion gives the following commutative diagram:

[figure: H_{n}(F, ℤ) ⟶^{pr} V_{n}(0); H_{n}(𝔽, ℤ); H_{n}(M_{X}, ℤ)/H_{n}(ℂ^{∗}, ℤ); s_{n}; s̄_{n}; r̄_{n}; r_{n}; p_{n}]

Here pr is the natural projection with kernel ⊕_{b̄∈B_{f}^{∗}}V_{n}(b̄). Notice that r_{n}, r̄_{n} and p_{n} are

onto. As we already know (cf. 2.19) ker(s̄_{n}) = ∑_{k≥0} ∑_{i>0} im(m_{0}^{k}m_{i}) ⊂ ker(r̄_{n}) = im(m_{0}− Id) + ∑_{i>0} im(m_{i}). If we work over ℚ, then H_{n}(𝔽, ℤ) can be replaced by T_{n} = im(s_{n} ⊗ ℚ). In particular, the kernel of the natural epimorphism T_{n} → H_{n}(M_{X}, ℚ)/H_{n}(ℂ^{∗}, ℚ) is completely determined in terms of {m_{i}}_{i≥0}.

6. An Example

In this section we determine all the modules considered above for a polynomial map h : ℂ^{4} → ℂ. Since some information about the monodromy representation of f is over ℂ, we choose in all our discussions this coefficient ring.

We start with the polynomial f : ℂ^{2} → ℂ given by f = x + x^{2}y^{2} + x^{2}y^{3}. Then B_{f} = {b_{1}, b_{2}} with b_{1} = −27/16 and b_{2} = 0. The fiber F_{b_{1}} is irreducible, has a node as a singularity and is regular at infinity. On the other hand, the fiber F_{0} = F_{b_{2}} is smooth, has two irreducible components, one a copy of ℂ the other ℂ \ {0, −1}, and has a singularity at infinity with a Milnor number equal to 3. It follows that b_{1}(F) = 4 and the Jordan normal form for the monodromy operators m_{1}, m_{2} and T_{∞} was obtained in [BM]:

m_{1} ≈ (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) m_{2} ≈ (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & j & 0 \\ 0 & 0 & 0 & j^{2} \end{matrix}) T_{∞} ≈ (\begin{matrix}−1 & 1 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})

with j = exp(2πi/3).

Let g : ℂ^{2} → ℂ be given by g = u + u^{2}v. Then B_{g} = {0}, the generic fiber is homotopy equivalent to S^{1} and the corresponding monodromy m is the identity. Consider the polynomial h : ℂ^{4} → ℂ given by h(x, y, u, v) = f(x, y) + g(u, v).

It follows from [DN2] that B_{h} ⊂ B_{f}. Notice that f and g are not good, but both are h-good. The fact that h is also a h-good polynomial follows from [DN2], Corollary (4.4), which basically says that the “Thom-Sebastiani sum” of two h-good polynomials is h-good.

Our goal is to determine the various Alexander modules associated with h and to its special fibers. The generic fiber of h is the join (∨_{4}S^{1}) ∗ S^{1}, hence it is ∨_{4}S^{3}. Since m = Id,

[DN2] guarantees that the monodromy representation of h can be identified to that of f.

Using the above Jordan forms, one obtains:

A_{3}(h, b_{1}) = Λ/(t − 1) ⊕ Λ/(t − 1) ⊕ Λ/(t − 1)^{2}.  (1)
A_{3}(h, 0) = Λ/(t − 1) ⊕ Λ/(t − 1) ⊕ Λ/(t − j) ⊕ Λ/(t − j^{2});  (2)
A_{3}(h, ∞) = Λ/(t − 1) ⊕ Λ/(t − 1) ⊕ Λ/(t + 1)^{2};  (3)

Now, we consider the space M_{X} corresponding to the two special fibers. First, let X = h^{−1}(0). Then X is smooth and applying Theorem (4.7), (ii) in [DN2], we get that X has the homotopy type of S^{2} ∨ S^{3} ∨ S^{3}. It follows that b_{3}(M_{X}) = b_{2}(X) = 1. Using 2.16(3), 3.3 and 4.1 one has:

π_{3}(M_{X}) ⊗ ℂ = π_{3}(𝔽) ⊗ ℂ = H_{3}(𝔽, ℂ) = Λ/(t − 1).  (4)

Next, let X = h^{−1}(b_{1}). Note that X is again smooth but we can no longer apply Theorem (4.7) in [DN2] since F_{b_{1}} is not smooth. Using the equality χ(X) = χ_{c}(X) we get b_{3}^{c}(X) − b_{4}^{c}(X) = 3. Moreover, it is known that b_{3}^{c}(X) = dim coker(m_{1} − 1) = 3, see [ACD] or [DN1]. It follows that b_{4}^{c}(X) = 0. The exact sequence

... → H_{c}^{k}(M_{X}) → H_{c}^{k}(ℂ^{n+1}) → H_{c}^{k}(X) → H_{c}^{k+1}(M_{X}) → ...

and duality for M_{X} imply that b_{3}(M_{X}) = b_{4}^{c}(X) = 0. Moreover, from (1) follows that only (t − 1)-torsion is possible. Hence, via 3.3 and 4.1, b_{3}(M_{X}) = 0 implies that

π_{3}(M_{X}) ⊗ ℂ = π_{3}(𝔽) ⊗ ℂ = H_{3}(𝔽, ℂ) = 0.  (5)

Our next aim is to compute the global Alexander module M(h). First, we determine the monodromy representation ρ_{3} : G → Aut(H), where H = H_{3}(F, ℂ). In terms of a special basis e_{1}, e_{2}, e_{3}, e_{4} for H, as in [DN1] (2.5), we can write the monodromy operators m_{1} and m_{2} of h in the form

m_{1} = (\begin{matrix}1 & a & b & c \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) m_{2} = (\begin{matrix}1 & 0 & 0 & 0 \\ α & 1 & 0 & 0 \\ β & 0 & j & 0 \\ γ & 0 & 0 & j^{2} \end{matrix}) .  (6)

Checking that m_{1}m_{2} is conjugate to the Jordan normal form for m_{∞} given above, we get several information. First, a = α = 0 and bc ≠ 0. By an obvious change of base we may take b = c = 1. Then β and γ are determined by the equations β + γ = −1, βj + γj^{2} = 2, hence β = j^{2} and γ = j. This ends the complete description of the monodromy representation.

An elementary computation shows that the vector subspace in H spanned by all the vectors z(v) − v for z ∈ H and v ∈ H is generated by e_{1}, e_{3}, e_{4}. Hence H_{H} = ℂ with a trivial ℤ^{2}-action. This implies the following.

6.1. Proposition. The polynomial h : ℂ^{4} → ℂ has the following Alexander modules. (i) For the fiber X = h^{−1}(0) one has π_{3}(M_{X}) ⊗ ℂ = H_{3}(M_{X}^{c}, ℂ) = M(f, 0) = Λ/(t − 1). (ii) For the fiber Y = h^{−1}(b_{1}) one has π_{3}(M_{Y}) ⊗ ℂ = H_{3}(M_{Y}^{c} , ℂ) = 0. Moreover in this

case M(f, b_{1}) = Λ/(t − 1) ≠ H_{3}(M_{Y}^{c} , ℂ) = 0.

(iii) The global Alexander module M(h) is isomorphic to Λ_{2}/(t_{1} − 1, t_{2} − 1).

Note that in the case of Y, we have M(h, b_{1}) ≠ H_{3}(M_{Y}^{c} , ℂ), in particular we cannot expect isomorphism for the last arrow in 3.3(b). Nevertheless, the approximation of H_{3}(M_{Y}^{c} , ℂ) = 0 given by M(h, b_{1}) is better than that given by A_{3}(h, b_{1}) since dim_{ℂ} M(h, b_{1}) = 1 while dim_{ℂ} A_{3}(h, b_{1}) = 4.

7. On the ℤ-torsion of X and M_{X}

7.1. One of the most difficult problems about the homology of the spaces X and M_{X} is to provide structure theorems, or existence/non-existence results about their ℤ-torsion. This section gives an insight into this subject. We start with the following remarks:

(1) (Cf. 3.3.) If X is the generic fiber of a h-good polynomial then X ∼ ∨S^{n} and both H_{∗}(X, ℤ) and H_{∗}(M_{X}, ℤ) are torsion free. This can be compared with the following result of Lê’s [Lê]:

(2) For any projective hypersurface V and a generic hyperplane H, the affine hypersurface X = V \ H is homotopy equivalent to a bouquet of spheres S^{n}. [For the computation of the number of spheres in this bouquet, in terms of the degree of gradient mappings, see [DPp].] In particular, generically, an affine hypersurface X has no torsion in homology.

(3) If a fiber X is smooth in U = ℂ^{n+1}, then by Gysin isomorphism the ℤ-torsion parts of H_{q}(M_{X}, ℤ) and H_{q−1}(X, ℤ) are the same. (This, in particular, can be applied for the generic fiber F.)

On the other hand, the examples from §5, and also the next paragraphs, show that the relationship between the torsion parts of the homologies of X, M_{X}, F (resp. M_{F}) are not simple at all, even for a homogeneous polynomial f.

7.2. Example. Consider the homogeneous polynomial f = x^{2}y^{2} + y^{2}z^{2} + z^{2}x^{2} − 2xyz(x + y + z) defined on ℂ^{3}. It is known that its Milnor fiber F has torsion in homology, more precisely H_{1}(F, ℤ) = H_{2}(M_{F} , ℤ) = ℤ/3ℤ and H_{2}(F, ℤ) = ℤ^{3} see [Li0], [DN1] and [Si]. Let C be the 3-cuspidal quartic in ℙ^{2} defined by {f = 0}. It satisfies H_{1}(ℙ^{2} \ C, ℤ) = ℤ_{4} and H_{2}(ℙ^{2} \ C, ℤ) = 0 (cf. [loc. cit.]). Set X = f^{−1}(0). The Gysin sequence of the fibration ℂ^{∗} → M_{X} → ℙ^{2} \ C yields H_{1}(M_{X}, ℤ) = ℤ, H_{2}(M_{X}, ℤ) = ℤ/4ℤ and H_{3}(M_{X}) = 0. In conclusion, both M_{F} and M_{X} have torsions in H_{2}, but these torsions are different.

In fact, if we put together the examples from §5, 7.1 and 7.2, we see that for a homogeneous polynomial f and for X = f^{−1}(0) the only case not covered is the following.

7.3. Question. Find an example of a homogeneous polynomial f such that M_{X} has no torsion but F has torsion.

Even in the case when f is a product of linear forms, the existence of such an example is an open question. (It is known in this latter case that the hyperplane arrangement complement M_{X} is torsion free, see [OT], and the corresponding Milnor fiber can be identified to a cyclic covering of ℙ^{n} \ {f = 0}, see [CS], [CO]). Notice also, that if we allow f to be a product of powers of linear forms, then A. Suciu has examples with torsion in the homology of the associated Milnor fiber F.

The following result gives some conditions on the possible torsions that may arise in such a case.

7.4. Proposition. Assume that for a homogeneous polynomial f of degree d the complement M_{X} has no p-torsion for some prime p and that the p-torsion in a homology group H_{q}(F, ℤ) has the structure

ℤ/p^{k_{1}}ℤ ⊕ ℤ/p^{k_{2}}ℤ ⊕ ... ⊕ ℤ/p^{k_{m}}ℤ

with m ≥ 1 and k_{1} ≥ k_{2} ≥ ... ≥ k_{m} ≥ 1. Then (i) (p − 1, d) = 1 implies m ≥ 2 and k_{1} = k_{2}; (ii) ((p − 1)p(p + 1), d) = 1 implies m ≥ 3 and k_{1} = k_{2} = k_{3}.

Proof. Consider the Wang sequence in homology associated to the fibration F → M_{X} → ℂ^{∗} and the monodromy T := T_{0,q}. The fact that M_{X} has no p-torsion implies that T − Id is an isomorphism when restricted to the p-torsion part.

To prove (i) note that since such an isomorphism preserves the orders of the elements, unless the claim (i) above holds, we get an induced automorphism of ℤ/pℤ, where this latter group is regarded as the quotient of the p-torsion part by the subgroup of elements killed by multiplication by p^{k_{1}−1}. The same is true for the monodromy transformation T.

Denote by T_{p} the induced automorphism of ℤ/pℤ. It follows that T_{p} is not the identity, T_{p}^{d} = Id (since f is homogeneous of degree d) and T_{p}^{p−1} = 1 (since |Aut(ℤ/pℤ)| = p − 1), a contradiction.

For (ii) use the same argument, plus the equality |Aut((ℤ/pℤ)^{2})| = (p − 1)^{2}p(p + 1). □

7.5. Final remarks. Most of the results considered and proved in the previous sections hide non-trivial facts about the ℤ-torsion of the corresponding groups. Here we bring to the light some of them.

(1) Consider the case when 𝔽 is (n − 1)-connected and H_{n}(𝔽, ℤ) is a Λ_{ℤ}-torsion module (see e.g. 4.3). Assume that H_{n}(𝔽, ℤ) is ℤ-torsion free. Then the Alexander polynomial Δ(H_{n}(𝔽, ℤ), h_{n}) is the characteristic polynomial Δ(h_{n}) (cf. 2.1(5)). It follows that H_{n}(M_{X}, ℤ) is finite iff Δ(h_{n})(1) ≠ 0. Moreover, if H_{n}(M_{X}, ℤ) is finite then its order is exactly |Δ(h_{n})(1)|. [However, we do not know whether the assumptions from 4.3 imply that H_{n}(𝔽, ℤ) is ℤ-torsion free.]

Some authors (see e.g. [Li2]) call the Alexander polynomial Δ(M) of some module M the order of M. The above discussion motivates this terminology.

(2) Consider again the notations of 2.19 and 5.7 for a h-good polynomial f. In fact, by 5.7, the ℤ-torsion parts of H_{n}(M_{X}, ℤ) and coker(V_{n}(0)) agree. Even more, one has the following divisibility.

First notice that the characteristic polynomial det(m_{0} − tId) of m_{0} is exactly ∏ Δ_{i}(t) considered in 2.16(4). By 5.7, if n > 1, then m_{0} − Id : V_{n}(0) → V_{n}(0) is injective, if and only if ∏ Δ_{i}(1) ≠ 0. In this case obviously H_{n}(M_{X}, ℤ) is a finite group. Moreover, the epimorphism coker(m_{0} − Id) → coker(V_{n}(0)) implies that the order |H_{n}(M_{X}, ℤ)| divides | ∏ Δ_{i}(1)| = |coker(m_{0} − Id)|.

(3) Let g : ℂ^{n+1} × ℂ → ℂ be the d-suspension of the polynomial f, namely g(x, y) = f(x) − y^{d}. Let Y = g^{−1}(0). Then writing the Gysin sequences in homology associated to a smooth divisor D in a complex manifold Z for the pairs (Z, D) = (M_{Y} , M_{X}), (ℂ^{n+1} × ℂ^{∗}, M_{X,d}) and resp. (M_{X} × ℂ^{∗}, graph(f)), and comparing the associated morphisms, we get for all q > 0 the exact sequence

0 → H_{q}(M_{X}, ℤ) → H_{q}(M_{X,d}, ℤ) → H_{q+1}(M_{Y} , ℤ) → 0.  (3.q)

The exact sequence (3.n + 1) is split since the last group in it is free according to 5.7. The exact sequence (3.n) is not split, as can be seen in the case n = 1, f = x_{1}x_{2}, d = 3 when we get 0 → ℤ^{2} → ℤ^{2} → ℤ/3ℤ → 0. This example shows the difficulty of the question 7.3.

Finally, assume that n > 1 and ∏_{i} Δ_{i}(α^{k}) ≠ 0 for α = exp(2πi/d) and for any k ∈ ℤ (cf. 2.16(4)). Then one has:

(a) all the groups H_{n}(M_{X}, ℤ), H_{n}(M_{X,d}, ℤ) and H_{n+1}(M_{Y} , ℤ) are finite; and

(b) the order |H_{n}(M_{X,d}, ℤ)| divides the product | ∏_{1≤k≤d} ∏_{i} Δ_{i}(α^{k})|.

The proof of these claims follows from the exact sequence (3.q) and the description 5.7 once we know how to compute the variation maps associated to the special fiber Y. This, in turn, is explained in [DN2]. Note that the claim (b) is similar to Theorem 3 in [Li0].

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