THOM SEBASTIANI CONSTRUCTION AND MONODROMY OF POLYNOMIALS

by Alexandru Dimca and András Némethi

1. Introduction

Consider a polynomial function f : ℂ^{n} → ℂ with generic fiber 𝔽. Let the finite set B_{f} be its (minimal) bifurcation set, i.e. f is a smooth locally trivial fibration over ℂ \ B_{f} (and B_{f} is minimal with this property), see (2.1).

From the topological properties of f we can extract the following data:

(i) the topology of the special fibers f^{−1}(s), (s ∈ B_{f}), and the inclusion of any special fiber f^{−1}(s) in its “tube” f^{−1}(D_{s}), where D_{s} is a sufficiently small disc centered at s;

(ii) the topology of the generic fiber 𝔽 := f^{−1}(s) for s ∉ B_{f};

(iii) the monodromy representation of the function f (with integer or complex coefficients):

ρ_{f} : π_{1}(ℂ \ B_{f}, pt) → Aut(H̃_{∗}(𝔽)).

The main difficulty in the study of the above data is that for n > 1 the polynomial map f is not proper, so the behaviour at infinity plays an important role. Nevertheless, in some particular cases we have rather complete answer for (i) and (ii). For instance, we say that the polynomial f has a good behaviour at infinity if f is tame in the sense of Broughton [B] (a condition latter generalized to M-tame by Némethi-Zaharia in [NZ]) or if there are no singularities at infinity in the sense of Siersma-Tibăr [ST], p. 773. In both cases, all the fibers 𝔽_{t} = f^{−1}(t) are homotopy equivalent to bouquets of (n − 1)–spheres and the inclusion of a special fiber in its tube is a homotopy equivalence. But for general polynomials (without any restriction about their behaviour at infinity), even (i) and (ii) are extremely difficult.

On the other hand, the third data (iii), even for special good polynomials is a highly non-trivial and very interesting object. It seems that even to provide complete description of non–trivial examples is rather difficult.

Already in the case n = 1 to determine this monodromy representation means to determine the corresponding representation with values in the permutation group of the (finite) fiber 𝔽 which in turn determines and is determined by the topological type of the polynomial f, see [KZ].

In the case n > 1 there are additional difficulties in determining the representation ρ_{f} coming from the higher dimensional topology and the non–properness of f.

In fact, in some cases one may compute the individual monodromy operators corresponding to going around one bifurcation value at a time or even the monodromy operator at infinity M_{∞}(f) (see (7.1) for the definition, and [ACD], [GN] for such examples), but to write these operators in the same base (i.e. to glue them in a global representation) is a more subtle problem.

The goal of our paper is to study the behaviour of the data (i-ii-iii) with respect to the Thom-Sebastiani construction. More precisely, we consider arbitrary polynomials f : ℂ^{n} → ℂ and g : ℂ^{n′} → ℂ (in (iii) we will assume that B_{g} = {0}). The sum h = f ⊕ g given by h(x, y) = f(x) + g(y) is then a polynomial ℂ^{n+n′} → ℂ. The main results of this paper describe the above invariants of h in terms of the corresponding invariants of f and g. Note that in spite of the fact that the relation between 𝔽, 𝔾 and ℍ (the generic fibers of f, g and h respectively) is simple, i.e. ℍ is homotopy equivalent to the join 𝔽 ∗ 𝔾 and that the corresponding monodromy operators at infinity satisfy the equality M_{∞}(h) = M_{∞}(f) ⊗ M_{∞}(g) (see [N1] for a proof), the relation between the special fibers, their tubes, and finally between the full representations, is not easy to describe. In fact, the monodromy representation of h involves the natural decomposition of the homology H_{∗}(𝔽) associated to a system of generators for π_{1}(ℂ \ B_{f}, pt), see (2.3) and our main Theorem (6.3).

This direct sum decomposition is given in section 2 together with the affine Picard-Lefschetz formulas, see (2.4) and (2.5). These formulas which have appeared recently in several preprints, see [NN1], [NN2] and [ST2], though very simple, have a lot of remarkable applications, see for instance [DN2].

In section 3 we review the basic construction of [N1] which describes the topology of the fibers of h, and provides the join ℍ = 𝔽 ∗ 𝔾 as well (this construction is crucial in the sequel). In section 4 we show the compatibility of the constructions of section 2 and 3, and also here we give the main theorems about the behaviour of tubes (see e.g. Theorem 4.3) and special fibers.

Notice that, in general, it is impossible to describe the topology of the special fibers of the polynomial h in terms of joins (see Example (4.6), where a special fiber not homotopically equivalent to any kind of join is exhibited). Nevertheless, in Theorem (4.7) a join decomposition is obtained under some additional conditions. This latter result generalizes a very useful topological property discovered by Kaliman and Zaidenberg in [KaZ]. Here we wish to emphasize that the study of the topology of the special fibers of a polynomial plays a key role in the construction of exotic affine spaces, see [K], [Z],

[KaZ] and [KR]. In fact, in section 5 (which contains some homological consequences of the previous sections), we give new examples of smooth hypersurfaces diffeomorphic to affine spaces (i.e. a potentially exotic affine space), generalizing some examples due to Koras and Russell [KR].

Section 6 contains the most important result of the article, namely Theorem (6.3) (which, in fact, was our main motivation): the presentation of monodromy representation of h. The next section contains some examples. The most important of them is the computation of the monodromy representation of polynomials of type f ⊕ g, where B_{g} = {0}, and f : ℂ → ℂ is a polynomial in one variable with “rigid passport” (see 7.3). In this way, we provide a family of examples with complete description of the monodromy representation.

Our last application is the description of the complex monodromy representation for the Briançon polynomial (the simplest polynomial in a very interesting series of polynomials constructed by Artal-Bartolo, Cassou-Noguès et Luengo in [ACL]).

The present manuscript is the slightly modified version of our preprint [DN]. We notice that in the recent paper [NN2], Neumann and Norbury determine the monodromy representation of the Briançon polynomial using the technique of splice diagrams. Acknowledgements.

The first author would like to thank D. Burghelea and A. Némethi for their kind invitation to visit Ohio State University at Columbus during the fall of 1998 when part of this work was done. He also acknowledges financial support from the INTAS programme 97-1644.

Both authors thank M. Zaidenberg for raising questions on the topology of the special fibers of a polynomial in the form f ⊕ g and on the Example (5.6), and S. Kaliman for his help with the proof of Lemma (8.2) below.

2. Preliminaries on polynomial functions

(2.1) Let f : ℂ^{n} → ℂ be a polynomial function. 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} [P]. If c_{0} ∈ ℂ is not in B_{f}, then 𝔽 = f^{−1}(c_{0}) will be called the generic fiber of f.

In this paper we discuss the monodromy representation

ρ_{f} : π_{1}(ℂ \ B_{f}, c_{0}) → Aut(H̃_{∗}(𝔽))

of the above fibration at the homology level. Coefficients for the homology are in ℤ if not stated otherwise.

Note that the local system (of abelian groups) s ↦ H̃_{∗}(𝔽_{s}), with 𝔽_{s} = f^{−1}(s) determines and is determined by this monodromy representation.

The fundamental group π_{1}(ℂ \ B_{f}, c_{0}) is a free group with t = |B_{f}| generators. Set B_{f} = {b_{1}, ..., b_{t}}. Then a convenient way to construct a set of free generators γ_{i} for this group is the following. First fix some paths p_{i} : [0, 1] → ℂ, i = 1, ..., t such that we have

(i) p_{i}(0) = c_{0}, p_{i}(1) = b_{i};

(ii) (image p_{i})∩ (image p_{j}) = {c_{0}} for any i ≠ j;

(iii) the paths p_{1},...,p_{t} leave c_{0} in the counterclockwise order.

(2.2) Definition. A family of paths (p_{i})_{i=1,...,t} with the properties (i), (ii) and (iii) is called a star with base point c_{0} and end points b_{1},...,b_{t}.

Once we fix a star as above, we have automatically a base (γ_{i})_{i=1,...,t} of π_{1}(ℂ \ B_{f}, c_{0}). Indeed, consider small discs D_{i} centered at b_{i} such that (image p_{i}) ∩ D_{i} is a topological radius in D_{i}. Set (image p_{i}) ∩ ∂D_{i} = c_{i} = p_{i}(t_{i}). Then p̃_{i} : [0, t_{i}] → ℂ, p̃_{i}(t) = p_{i}(t) connects c_{0} to c_{i}. Let ∂D_{i} be the loop with both end points c_{i} and going once around b_{i} counterclockwise.

Then γ_{i} = (p̃_{i})^{−1} ∘ ∂D_{i} ∘ p̃_{i} for i = 1, ..., t are loops based at c_{0} and their homotopy classes (still denoted by γ_{i}) generate the group π_{1}(ℂ \ B_{f}, c_{0}). We emphasize that the ordered set of homotopy classes γ_{1}, ..., γ_{t} is canonically associated with the star p_{1}, ...., p_{t} and it is independent on any other choices.

(2.3) The sum decomposition of H̃_{∗}(𝔽) associated with a star.

Consider as above a polynomial f : ℂ^{n} → ℂ with a bifurcation set B_{f}, a point c_{0} not in B_{f} and a star with base point c_{0} and end points in B_{f}. This provides a direct sum decomposition of H̃_{∗}(𝔽), (see [B], and [DN], [NN1], [ST], [ST2] for various degrees of generality). Indeed, first we have the obvious isomorphism ∂^{−1} : H̃_{∗}(𝔽) → H_{∗+1}(ℂ^{n}, 𝔽), next we can replace the second group by H_{∗+1}(f^{−1}(∪_{i}p_{i}([0, 1]) ∪ ∪_{i}D_{i}, 𝔽) using a deformation retract and finally by excision we may replace the last group by ⊕_{i}H_{∗+1}(f^{−1}(D_{i}), 𝔽_{c_{i}}). Next we can define a subgroup H_{∗,i} in H̃_{∗}(𝔽) for i = 1, ..., t by pushing the subgroup ∂H_{∗+1}(f^{−1}(D_{i}), 𝔽_{c_{i}}) in H̃_{∗}(𝔽_{c_{i}}) to H̃_{∗}(𝔽) along the path p_{i}.

Notice that while rank H_{∗,i} (or equivalently, rank H_{∗+1}(f^{−1}(D_{i}), 𝔽_{c_{i}}), or even the abstract group H_{∗,i}) is independent of the choice of the star, the direct sum decomposition

H̃_{∗}(𝔽) = ⊕_{i=1}^{t} H_{∗,i}

depends essentially on the choice of the star. The following result provides some special elements in H_{∗,i}.

(2.4) Lemma. Let m_{j} = ρ_{f}(γ_{j}) ∈ Aut(H̃_{∗}(𝔽)) be the monodromy operators associated with a star as above. Then, for any homology class [c] ∈ H̃_{∗}(𝔽) one has m_{j}([c])−[c] ∈ H_{∗,j}. Proof. The geometric monodromy along the path γ_{j} is defined as follows. Take a qcycle c and move it along the path γ_{j} to get a cycle m_{geom}(c) in 𝔽. This means that there is a (q + 1)-cycle d such that ∂d = m_{geom}(c) − c and f(d) ⊂ γ_{j}([0, 1]). But then m_{j}([c]) − [c] = ∂[d] ∈ H_{q,j}. □

(2.5) The above Lemma shows that in the direct sum decomposition ⊕_{i=1}^{t} H_{∗,i}, the monodromy m_{j} = ρ_{f}(γ_{j}) (1 ≤ j ≤ t) has a block-decomposition of the following form:

m_{j} = ( \begin{matrix}1 \\ . & . & . \\ & & 1 \\ v_{j1} & ⋯ & v_{j,j−1} & m_{jj} & v_{j,j+1} & ⋯ & v_{jt} \\ & & & & 1 \\ . & . & . \\ & & & & & & 1 \end{matrix} ) ,

where all the entries, excepting the diagonal and the j-th line, are trivial. Above m_{jj} : H_{∗,j} → H_{∗,j} is m_{jj}(a_{j}) = m_{j}(a_{j}); and for i ≠ j (the variation mappings) v_{ji} : H_{∗,i} → H_{∗,j} are given by v_{ji}(a_{i}) = m_{j}(a_{i}) − a_{i}.

(2.6) Remark. Lemma (2.4) is a weak global affine Picard-Lefschetz formula (corresponding to the result of Picard) and it says that m_{j}([c])−[c] can be localized in f^{−1}(D_{j}). In the case of projective Lefschetz pencils the precise form of m_{j}([c]) − [c] was given by Lefschetz in terms of vanishing cycles and intersection forms. In our affine case the theory of vanishing cycles is not yet completely developed, and in general m_{j}([c]) − [c] cannot be computed in terms of the intersection form (see [BM]).

In general, it is very difficult to determine the decomposition (2.3). Nevertheless, in some cases the decomposition (2.3) can be recovered from the monodromy representation ρ_{f}. This is explained in the next example.

(2.7) Example. For simplicity (in this example) we will work with (co)homology groups with ℚ− or ℂ−coefficients. Let ρ_{f,k} : π_{1}(ℂ \ B_{f}, c_{0}) → Aut(H_{k}(𝔽)) (resp. ρ_{f,k}^{∗}) be the corresponding homology (resp. cohomology) representation for a fixed k = 1, 2, ..., n − 1.

Fix a star for B_{f} as above and set m_{i} = ρ_{f,k}(γ_{i}) and m_{i}^{∗} = ρ_{f,k}^{∗}(γ_{i}). Let S_{i} be the span of all the vectors m_{i}(c) − c, where c ∈ H_{k}(𝔽). By (2.4) S_{i} ⊂ H_{k,i}. Then, obviously H^{k}(𝔽) = ⊕_{i=1}^{t}H_{i}^{k}, where H_{i}^{k} = {ω ∈ H^{k}(𝔽); ω|H_{k,j} = 0 for all j ≠ i} and ⊕_{j≠i}H_{j}^{k} ⊂ Ker(m_{i}^{∗} − 1).

Then the following are equivalent (the proof is easy and it is left to the reader):

(i) S_{i} = H_{k,i} for all i = 1, ..., t;

(ii) ⊕_{j≠i}H_{j}^{k} = Ker(m_{i}^{∗} − 1);

(iii) H_{i}^{k} = ∩_{j≠i}Ker(m_{j}^{∗} − 1);

(iv) there is no nonzero vector in H^{k}(𝔽) invariant under ρ_{f,k}^{∗}.

This shows that if one of the above conditions is satisfied, then the direct sum decomposition (2.4) is determined by the representation ρ_{f,k}^{∗}, or equivalently by ρ_{f,k}. On the other hand the condition (iv) can be checked in practice in many cases using the results in [ACD], Theorem 2, and [NN1].

3. The generic fiber of h = f ⊕ g (review)

(3.1) Let f : ℂ^{n} → ℂ and g : ℂ^{n′} → ℂ be two polynomial functions. Define h = f ⊕ g : ℂ^{n} × ℂ^{n′} → ℂ by (x, y) ↦ f(x) + g(y). In [N1] it is proved that the generic fiber ℍ of h is homotopy equivalent to the join 𝔽 ∗ 𝔾 of the generic fibers of f and g. This joindecomposition is not canonical, it depends on some choices in a geometrical construction. This fact is crucial when we have to make this construction compatible with the choice of some stars providing the decomposition (2.3) and with the monodromy representation. That is why in this section we review the basic construction of [N1].

First we recall the following.

(3.2) Theorem. [N1] The bifurcation set B_{h} of the polynomial h is contained in the sum B_{f} + B_{g} := {λ + μ | λ ∈ B_{f}, μ ∈ B_{g}} of the bifurcation sets for f and g.

Now consider the map ϕ = (f, g) : ℂ^{n} × ℂ^{n′} → ℂ × ℂ defined by (x, y) ↦ (f(x), g(y)). For any λ ∈ ℂ we consider the complex line L_{λ} = {(c, d) ∈ ℂ^{2} : c + d = λ}. Then we obviously have ℍ_{λ} = ϕ^{−1}(L_{λ}). It is convenient to introduce the sets B_{λ,f} = L_{λ} ∩(B_{f} ×ℂ), B_{λ,g} = L_{λ} ∩ (ℂ × B_{g}) and B_{λ,ϕ} = B_{λ,f} ∪ B_{λ,g}.

By Theorem (3.2) λ ∉ B_{h} if B_{λ,f} ∩ B_{λ,g} = ∅. Therefore, for a fixed generic λ, not in B_{h}, we can construct (using the transitivity of the diffeomorphism group of the plane on finite subsets) a diffeomorphism α : ℝ × ℝ → L_{λ} such that:

(i) L_{λ}^{+} := α(ℝ × (0, +∞)) contains B_{λ,f}; (ii) L_{λ}^{−} := α(ℝ × (−∞, 0)) contains B_{λ,g}.

Let L̄_{λ}^{+/−} be the closure of L_{λ}^{+/−} in L_{λ} and l_{λ} = L̄_{λ}^{+} ∩ L̄_{λ}^{−} = α(ℝ × {0}). Then the following facts hold.

(3.3) Theorem. [N1] (Below ≈ means “diffeomorphic”.)

(i) ϕ is a smooth locally trivial fibration over the spaces ℂ^{2} \ (B_{f} × ℂ ∪ ℂ × B_{g}) and L_{λ} \ B_{λ,ϕ};

(ii) ϕ^{−1}(L̄_{λ}^{+}) ≈ ℂ^{n} × 𝔾;

(iii) ϕ^{−1}(L̄_{λ}^{−}) ≈ 𝔽 × ℂ^{n′};

(iv) ϕ^{−1}(l_{λ}) ≈ 𝔽 × 𝔾 × l_{λ}.

Now we are ready to explain the join decomposition of the generic fiber ℍ. Let c_{0} ∈ l_{λ} be a fixed point. The strong deformation retract l_{λ} → {c_{0}} can be lifted using (3.3.i) and (3.3.iv) and this gives

ℍ = ℍ_{λ} = ϕ^{−1}(L_{λ}) = ϕ^{−1}(L̄_{λ}^{+}) ∪_{ϕ^{−1}(l_{λ})} ϕ^{−1}(L̄_{λ}^{−}) ≈ 𝔽 × ℂ^{n′} ∪_{𝔽×𝔾} ℂ^{n} × 𝔾 ∼ 𝔽 ∗ 𝔾,

where ∼ denotes the homotopy equivalence.

We emphasize that in this join decomposition ℍ ∼ 𝔽 ∗ 𝔾 the choice of l_{λ} is essential: different choices provide different decompositions.

At the homology level, the Mayer-Vietoris exact sequence associated to the covering (ϕ^{−1}(L̄_{λ}^{+/−}))_{+/−} of ℍ = ℍ_{λ} gives the following (functorially split) exact sequence

(3.4)  0 → H̃_{∗+1}(ℍ) ⟶^{δ} H̃_{∗}(𝔽 × 𝔾) → H̃_{∗}(ℂ^{n} × 𝔾) ⊕ H̃_{∗}(𝔽 × ℂ^{n′}) → 0

or equivalently an isomorphism δ

(3.5)  H̃_{r}(ℍ) → (⊕_{p+q=r−1}H̃_{p}(𝔽) ⊗ H̃_{q}(𝔾)) ⊕ (⊕_{p+q=r−2}T or(H̃_{p}(𝔽), H̃_{q}(𝔾)))

which comes from a (non-functorial) splitting in the Künneth formula, see Milnor [M]. If in (3.5) the T or-part is trivial, then the direct sum decomposition is functorial.

This boundary morphism δ can be described geometrically as follows. Take a cycle [c] ∈ H_{∗+1}(ℍ), consider a transversal intersection c ∩ ϕ^{−1}(l_{λ}) (i.e. take c to be representative which is transversal to the smooth hypersurface ϕ^{−1}(l_{λ})) and retract this intersection cycle into ϕ^{−1}(c_{0}) = 𝔽 × 𝔾 via the lift of the retraction l_{λ} → {c_{0}}. The result is the cycle δ([c]). Notice that δ([c]) depends on the choice of the separating real line l_{λ} ⊂ L_{λ}.

4. Special fibers and their “tubes”

(4.1) Although the main goal of the present article is the study of the monodromy of the local system provided by the generic fibers of h = f ⊕ g, in this section we study the properties of the special fibers and their tubes associated with h. This also has an independent interest, but establishes some compatibility properties between the constructions of sections 2 and 3 as well.

In the first part we discuss the compatibility of the decomposition (2.3) with respect to the Thom Sebastiani construction. [In particular, it also follows the compatibility of the “vanishing cycles” with respect to the Thom Sebastiani construction; for the definition of these cycles, see e.g. [NN1] and [NN2].] To do this we need to describe the relations among the tubes about the special fibers of f, g and h respectively.

Fix a point λ_{0} ∈ B_{f} + B_{g} and set

I(λ_{0}) = {(i, j) : b_{i} + b′_{j} = λ_{0}, b_{i} ∈ B_{f}, b′_{j} ∈ B_{g}}.

Let D_{i} (resp. D′_{j}) be a small disc centered at b_{i} (resp. b′_{j}) and fix c_{i} ∈ ∂D_{i} (resp. c′_{j} ∈ ∂D′_{j}). Similarly we fix a small disc D_{0} with center λ_{0} and ξ ∈ ∂D_{0}.

(4.2) Theorem.

H_{∗}(h^{−1}(D_{0}), ℍ_{ξ}) = ⨁_{(i,j)∈I(λ_{0})} H_{∗}((f^{−1}(D_{i}), 𝔽_{c_{i}}) × (g^{−1}(D′_{j}), 𝔾_{c′_{j}})).

Proof. For any (i, j) ∈ I(λ_{0}) fix a small closed ball B(i, j) in ℂ^{2} with center (b_{i}, b′_{j}) such that any two of them are disjoint and B(i, j)∩(B_{f} ×B_{g}) = {(b_{i}, b′_{j})}. Fix also a sufficiently small ε such that T = {(c, d) : |c + d − λ_{0}| ≤ ε} intersects the boundaries ∂B(i, j) transversally and T ∩(ℂ×{b′_{j}}∪{b_{i}}×ℂ) ⊂ intB(i, j) for any (i, j) ∈ I(λ_{0}). Let D_{0} be the disc of radius ε and center λ_{0}. Then we obviously have (h^{−1}(D_{0}), ℍ_{ξ}) = (ϕ^{−1}(T), ϕ^{−1}(L_{ξ})), hence

H_{∗}(h^{−1}(D_{0}), ℍ_{ξ}) = H_{∗}(ϕ^{−1}(T), ϕ^{−1}(L_{ξ})).

But L_{ξ} \ ∪_{(i,j)}B(i, j) is a strong deformation retract of T \ ∪_{(i,j)}B(i, j). Since ϕ provides a locally trivial fibration over T \ ∪_{(i,j)}B(i, j) (cf. 3.3.i), by excision one has:

H_{∗}(ϕ^{−1}(T), ϕ^{−1}(L_{ξ})) = ⊕_{(i,j)∈I(λ_{0})}H_{∗}(ϕ^{−1}(T ∩ B(i, j)), ϕ^{−1}(L_{ξ} ∩ B(i, j))).

Again by (3.3.i) this last group is equal to

⊕_{(i,j)∈I(λ_{0})}H_{∗}(ϕ^{−1}(D_{i} × D′_{j}), ϕ^{−1}(D_{i} × {c′_{j}} ∪ {c_{i}} × D′_{j}))

and the result follows. □

In fact, we will prove even a stronger homotopical result about the tubes in the case #I(λ_{0}) = 1, e.g. when B_{g} = {0}. Below, X ∐_{U} Y ∐_{V} Z, where U ⊂ X∩Y and V ⊂ Y ∩Z, denotes the space obtained from the disjoint union of X, Y and Z after identification of the subspace U in X and Y, respectively of V in Y and Z.

(4.3) Theorem. Fix λ_{0} ∈ B_{f} + B_{g} such that #I(λ_{0}) = 1. This means that λ_{0} can be written uniquely as λ_{0} = b_{i} + b′_{j} with b_{i} ∈ B_{f} and b′_{j} ∈ B_{g}. Let D_{0}, D_{i} and D′_{j} be small discs as above. Then

(i) the tube h^{−1}(D_{0}) has the homotopy type of

ℂ^{n} × 𝔾 ∐_{f^{−1}(D_{i})×𝔾}f^{−1}(D_{i}) × g^{−1}(D′_{j}) ∐_{𝔽×g^{−1}(D′_{j})}𝔽 × ℂ^{n′},

where the inclusion 𝔾 ↪ g^{−1}(D′_{j}) is given by the inclusion of a generic fiber 𝔾 = g^{−1}(c′_{j}) ↪ g^{−1}(D′_{j}); similarly for 𝔽.

(ii) the above decomposition is compatible with the inclusion ℍ_{ξ} ↪ h^{−1}(D_{0}): realize the space ℍ_{ξ} as

ℂ^{n} × 𝔾 ∐_{𝔽×𝔾}𝔽 × 𝔾 ∐_{𝔽×𝔾}𝔽 × ℂ^{n′},

(where the maps are induced by the natural inclusions) and embed ℍ_{ξ} in the tube h^{−1}(D_{0}) by the map induced by the five natural inclusions ℂ^{n}×𝔾 → ℂ^{n}×𝔾, 𝔽×𝔾 → f^{−1}(D_{i})×𝔾, etc. Proof. We can assume that b_{i} = 0, b′_{j} = 0, hence λ_{0} = 0 as well. Then consider T = {(c, d) : |c + d| ≤ ε} and L_{ε} = {(c, d) : c + d = ε} as above (with ξ = ε). Then D_{0} = {λ : |λ| ≤ ε}, and h^{−1}(D_{0}) = ϕ^{−1}(T).

L_{ε} intersects {b_{i}} × ℂ, respectively ℂ × {b′_{j}}, in (0, ε), respectively in (ε, 0). We construct l_{ε} ⊂ L_{ε} in such a way that for any (c, d) ∈ l_{ε} one has: |c| ≥ ε/2, |d| ≥ ε/2, and |(c, d) − (u, v)| >> ε for any (u, v) ∈ B_{ε,ϕ} \ {(0, ε), (ε, 0)}. (For example, in the neighborhood of the origin, l_{ε} can be defined by Re(c) = Re(d) = ε/2 with Re the real part of a complex number, and extended to a real curve which is “far” from the other bifurcation points.)

Moreover, we will assume that if a closed disc with radius ε intersects l_{ε} (resp. L̄_{ε}^{±}), then this intersection is contractible. We define the following five subsets of T:

T_{f} = {(c, d) ∈ T : d ∈ pr_{2}L̄_{ε}^{+}}, T_{f,l} = {(c, d) ∈ T : d ∈ pr_{2}l_{ε}},

T_{g} = {(c, d) ∈ T : c ∈ pr_{1}L̄_{ε}^{−}}, T_{g,l} = {(c, d) ∈ T : c ∈ pr_{1}l_{ε}},

and T_{f,g} = {(c, d) ∈ T : d ∈ pr_{2}L̄_{ε}^{−}, c ∈ pr_{1}L̄_{ε}^{+}}. Obviously

T = T_{f} ∐_{T_{f,l}} T_{f,g} ∐_{T_{g,l}} T_{g},

which gives:

h^{−1}(D_{0}) = ϕ^{−1}(T) = ϕ^{−1}(T_{f}) ∐_{ϕ^{−1}(T_{f,l})} ϕ^{−1}(T_{f,g}) ∐_{ϕ^{−1}(T_{g,l})} ϕ^{−1}(T_{g}).

Now, we have to analyze these constituent spaces.

I. The space ϕ^{−1}(T_{f,l}). Fix an isomorphism ψ : g^{−1}(pr_{2}l_{ε}) → pr_{2}l_{ε} × 𝔾 such that pr_{1} ∘ ψ = g. Then the space

ϕ^{−1}(T_{f,l}) = {(x, y) ∈ f^{−1}(pr_{1}T_{f,l}) × g^{−1}(pr_{2}l_{ε}) : (f(x), g(y)) ∈ T_{f,l}},

via the isomorphism id × ψ (here id is the identity of f^{−1}(pr_{1}T_{f,l})), can be identified with the space:

{(x, d) ∈ f^{−1}(pr_{1}T_{f,l}) × pr_{2}l_{ε} : (f(x), d) ∈ T_{f,l}} × 𝔾 := P × 𝔾.

But P is exactly the fibered product of the maps pr_{1} : T_{f,l} → pr_{1}T_{f,l} and f : f^{−1}(pr_{1}T_{f,l}) → pr_{1}T_{f,l}. Consider the first map pr_{1}. It can be verified that

pr_{1}T_{f,l} = {c| ∃c′ ∈ pr_{1}l_{ε} : |c − c′ + ε| ≤ ε}.

Moreover, for any c from this set, pr_{1}^{−1}(c) is exactly the set {(c, ε−c′) : |c−c′ +ε| ≤ ε, c′ ∈ pr_{1}l_{ε}}, which is contractible for any c (by one of our assumptions at the beginning of the construction). Therefore, pr_{1} : T_{f,l} → pr_{1}T_{f,l} is a homotopy equivalence. This implies that the fibered product P has the same homotopy type as f^{−1}(pr_{1}T_{f,l}). Now, it is easy to verify that pr_{1}T_{f,l} is a “band”, containing pr_{1}l_{ε} and of thickness 2ε, containing b_{i} = 0, but not the other bifurcation points of f. Therefore, f^{−1}(pr_{1}T_{f,l}) ≈ f^{−1}(D_{i}). This shows that ϕ^{−1}(T_{f,l}) ∼ f^{−1}(D_{i}) × 𝔾. II. The space ϕ^{−1}(T_{f}). This part is very similar to part I. Fix a diffeomorphism ψ : g^{−1}pr_{2}T_{f} → pr_{2}T_{f} × 𝔾 with pr_{1} ∘ ψ = g. Similarly as above ψ^{−1}(T_{f}) ≈ P × 𝔾, where P is the fibered product of pr_{1} : T_{f} → pr_{1}T_{f} and f : f^{−1}(pr_{1}T_{f}) → pr_{1}T_{f}. But

pr_{1}T_{f} = {c| ∃c′ ∈ pr_{1}L̄_{ε}^{+} : |c − c′ + ε| ≤ ε},

and pr_{1}^{−1}(c) is contractible. Hence P ∼ f^{−1}(pr_{1}T_{f}). But pr_{1}T_{f} is a “topological semiplane” containing B_{f}, hence f^{−1}(pr_{1}T_{f}) ≈ f^{−1}(ℂ) = ℂ^{n}. Therefore, ϕ^{−1}(T_{f}) ∼ ℂ^{n} × 𝔾.

Obviously, by a symmetric argument, ϕ^{−1}(T_{g}) ∼ 𝔽 × ℂ^{n′} and ϕ^{−1}(T_{g,l}) ∼ 𝔽 × g^{−1}(D′_{j}). III. The space ϕ^{−1}(T_{f,g}). It is easy to verify that the multidisc D_{r} × D′_{r} = {(c, d) : |c| ≤ r, |d| ≤ r} is in T_{f,g} for r sufficiently small (e.g. if l_{ε} in a neighbourhood of the origin is given by Re(c) = Re(d) = ε/2 then one can take r = ε/2). Moreover, the inclusion D_{r} × D′_{r} ↪ T_{f,g} admits a strong deformation retract compatible with the bifurcation set cd = 0. Therefore, ϕ^{−1}(T_{f,g}) = f^{−1}(D_{r}) × g^{−1}(D′_{r}). This ends the proof of the theorem.□

(4.4) Corollary. Assume that B_{g} = {0}. Then g^{−1}(D′_{j}) ≈ ℂ^{n′}, therefore, for any b_{i} ∈ B_{f}:

h^{−1}(D_{0}) ∼ ℂ^{n} × 𝔾 ∐_{f^{−1}(D_{i})×𝔾}f^{−1}(D_{i}) × ℂ^{n′} ∼ f^{−1}(D_{i}) ∗ 𝔾

for sufficiently small discs D_{0} and D_{i} centered at b_{i}. This homotopy equivalence h^{−1}(D_{0}) ∼ f^{−1}(D_{i})∗𝔾 is compatible with the inclusion of ℍ_{ξ} in h^{−1}(D_{0}), respectively of the inclusion of 𝔽 in f^{−1}(D_{i}), and the join decomposition ℍ_{ξ} ∼ 𝔽 ∗ 𝔾.

(4.5) Actually, in many questions, e.g. in the construction of exotic affine spaces, see [Z], the ultimate aim is not the topology of the tubes, but the topology of the special fibers. Recall that for the generic fibers ℍ_{ξ} ∼ 𝔽 ∗ 𝔾, and in the case B_{g} = {0} for the tubes of the special fibers h^{−1}(D_{i}) ∼ f^{−1}(D_{i}) ∗ 𝔾 as well. Therefore, it is natural to ask if for the special fibers

(∗) ℍ_{b_{i}} ∼ 𝔽_{b_{i}} ∗ 𝔾

holds or not (provided that B_{g} = {0}). Unfortunately, in general, the answer is negative.

(4.6) Example. If f = x(xy +1) and g = z^{2}, then 𝔽_{0} ≈ ℂ ∐ ℂ^{∗}, 𝔾 consists of two points, 𝔽_{0} ∗ 𝔾 ∼ S^{1} ∨ S^{2}, but ℍ_{0} ∼ ℙ_{ℝ}^{2} (the real projective plane). (Cf. also Example 5.6.)

Actually, since π_{1}(ℍ_{0}) = ℤ_{2}, ℍ_{0} cannot be any kind of join !

On the other hand, we can list a few situations when (∗) does hold. The next result generalizes Proposition 4.1 of Kaliman and Zaidenberg [KaZ] (which corresponds to the case g(u, v) = uv).

(4.7) Theorem. Assume that B_{g} = {0} and one of the following conditions hold: (i) f and h both have good behaviour at infinity above a small disc D_{i} centered at b_{i}; (ii) the geometric monodromy m_{geom}(g) of g is homotopically trivial, and the special

fiber 𝔽_{b_{i}} of f is reduced and smooth.

Then ℍ_{b_{i}} ∼ 𝔽_{b_{i}} ∗ 𝔾.

Proof. The proof depends on which condition holds. If the condition (i) holds, then the claim is an easy consequence of (4.4) as the inclusion of a special fiber in its tube is a homotopy equivalence (see our Introduction).

Assume that condition (ii) holds. Notice that if f has a bad behaviour at infinity above D_{i} then the tube f^{−1}(D_{i}) is not a “good tubular neighbourhood” of 𝔽_{b_{i}}, e.g. their homotopy types might be different, the tube in general has no retract to the central fiber, etc.

Since the fiber 𝔽_{b_{i}} is smooth, it has a closed tubular neighborhood p : T_{i} → 𝔽_{b_{i}}, where T_{i} is a smooth fiber bundle with base space 𝔽_{b_{i}} and fibers real two-dimensional closed discs. In particular, the inclusion 𝔽_{b_{i}} ↪ T_{i} is a homotopy equivalence. Let CT_{i} be the closure of the complement of T_{i} in ℂ^{n}.

The projection pr_{1} : ℍ_{b_{i}} → ℂ^{n}, induced by (x, y) ↦ x, and the covering {T_{i}, CT_{i}} of ℂ^{n} realizes ℍ_{b_{i}} as a fibered product

pr_{1}^{−1}(CT_{i}) ∐_{pr_{1}^{−1}(∂T_{i})} pr_{1}^{−1}(T_{i}).

Notice that ℍ_{b_{i}} \ pr_{1}^{−1}(𝔽_{b_{i}}) → ℂ^{n} \ 𝔽_{b_{i}}, given by (x, y) ↦ x, is a fiber bundle with fiber 𝔾, but with trivial geometric monodromy (since m_{geom}(g) ∼ 1). Therefore

ℍ_{b_{i}} ∼ CT_{i} × 𝔾 ∐_{∂T_{i}×𝔾} pr_{1}^{−1}(T_{i}).

Since 𝔽_{b_{i}} is smooth and reduced, for any point P ∈ 𝔽_{b_{i}}, in some small neighborhood U_{P} of P with local coordinates (x_{1}, …, x_{n}), the function f has local equation f(x) = x_{1} and p(x) = (x_{2}, …, x_{n}). Therefore ℍ_{b_{i}} above U_{P} has equation x_{1} + g(y) = b_{i}. This shows that if we consider the composite map p ∘ pr_{1} : pr_{1}^{−1}(T_{i}) → 𝔽_{b_{i}}, then all its fibers are homeomorphic to g^{−1}(D′_{j}) ≈ ℂ^{n′}, hence they are contractible. In particular, p ∘ pr_{1} is a homotopy equivalence. Therefore the above fibered product can be replaced by

CT_{i} × 𝔾 ∐_{∂T_{i}×𝔾} T_{i} × Cone(𝔾) ∼ ℂ^{n} × 𝔾 ∐_{T_{i}×𝔾} T_{i} × Cone(𝔾),

which is 𝔾 ∗ T_{i} ∼ 𝔾 ∗ 𝔽_{b_{i}}. □

Notice again that above it is necessary to have 𝔽_{b_{i}} reduced. Indeed, take again the example g = x(xy + 1) and f = z^{2} (where 𝔽_{b_{i}} is not reduced, but its reduced space is smooth).

The last result has the following interesting consequence. Consider the family G = {g : ℂ^{n′} → ℂ for some non-fixed n′, B_{g} = {0}, 𝔾_{0} is reduced and smooth, and m_{geom}(g) ∼ 1}.

(4.8) Corollary. Assume that g ∈ G. Then (pt ∐ S^{1})∗𝔾 ∼ S^{1}∗𝔾_{0}, where pt ∐ S^{1} denotes the disjoint union of a point with S^{1}. In other words (S^{0} ∗ 𝔾) ∨ (S^{1} ∗ 𝔾) ∼ S^{1} ∗ 𝔾_{0}. This at the homology level reads as H̃^{q}(𝔾_{0}) = H̃^{q}(𝔾) ⊕ H̃^{q+1}(𝔾).

In particular, if g_{i} ∈ G for i = 1, 2, then (pt ∐ S^{1}) ∗ 𝔾(g_{1}) ∼ (pt ∐ S^{1}) ∗ 𝔾(g_{2}) if and only if S^{1} ∗ 𝔾_{0}(g_{1}) ∼ S^{1} ∗ 𝔾_{0}(g_{2}).

Proof. Consider f = x(xy + 1) with 𝔽 ∼ S^{1} and 𝔽_{0} ∼ (pt ∐ S^{1}). Then for any g ∈ G, (4.7) can be applied in two different ways, i.e. ℍ_{0} ∼ 𝔽_{0} ∗ 𝔾 ∼ 𝔽 ∗ 𝔾_{0}.

5. Applications

(5.1) This section has a double role. First, it contains some immediate homological applications of the previous sections. On the other hand, after we review briefly a connection between the monodromy operator around a special smooth fiber and the topology of the special fiber itself, we construct polynomials with special fibers as a potentially exotic affine spaces (cf. 5.6).

In the sequel, for simplicity of the discussion, we consider the class C = C_{1} ∩ C_{2} of polynomials f : ℂ^{n} → ℂ defined by the following conditions:

(C_{1}) The reduced Betti numbers b̃_{j}(𝔽) of the generic fiber of f vanish for j ≠ n − 1, and

(C_{2}) f (at finite distances, i.e. in ℂ^{n}) has only isolated singularities.

The class C is suited for our study, since if two polynomials f and g satisfy C_{1} and/or C_{2}, the same is true for their sum h = f ⊕ g. To state some basic properties for the polynomials in the class C we introduce some notations.

For a polynomial f in C_{2} we can define its total Milnor number μ_{f} to be the sum of the local Milnor numbers corresponding to all the singularities of f. We let μ_{f}(b) denote the similar sum, but this time over the singularities on the fiber 𝔽_{b} of the polynomial f. With the notation from (2.3) we also define for any b_{i} ∈ B_{f}

ν_{f}(b_{i}) = rank H_{n}(f^{−1}(D_{i}), 𝔽_{c_{i}}) − μ_{f}(b_{i}),

and we set ν_{f} = ∑_{i=1}^{t} ν_{f}(b_{i}).

Then we have the following known result about the homology of tubes and special fibers (see for instance [ACD], in particular Lemma 1 and the discussion on page 322).

(5.2) Theorem. Let f : ℂ^{n} → ℂ (where n > 1) be a polynomial in the class C. Then (i) b_{n−1}(𝔽) = μ_{f} + ν_{f}; (ii) χ(𝔽_{b_{i}}) = χ(𝔽) + (−1)^{n}(μ_{f}(b_{i}) + ν_{f}(b_{i})); (iii) b̃_{j}(𝔽_{t}) = 0 for any t ∈ ℂ and any j < n − 2; (iv) H_{j}(f^{−1}(D_{i})) = 0 for any j ≥ n and b̃_{j}(f^{−1}(D_{i})) = 0 for j < n − 1.

Our first application says that the μ- and ν-invariants behave in a natural way with respect the Thom-Sebastiani construction. Indeed, (4.2) implies:

(5.3) Corollary. Let f : ℂ^{n} → ℂ and g : ℂ^{n′} → ℂ be two polynomials in the class C and let h = f ⊕ g be their sum. Then

(i) μ_{h}(λ_{0}) = ∑_{(i,j)∈I(λ_{0})} μ_{f}(b_{i})μ_{g}(b′_{j});

(ii) ν_{h}(λ_{0}) = ∑_{(i,j)∈I(λ_{0})}(μ_{f}(b_{i})ν_{g}(b′_{j}) + ν_{f}(b_{i})μ_{g}(b′_{j}) + ν_{f}(b_{i})ν_{g}(b′_{j})).

Before we formulate a consequence of (5.3), we recall that in general there is no (topological) characterization of the (minimal) bifurcation sets. Moreover, in the Thom-Sebastiani construction, even if we know that B_{h} ⊂ B_{f} + B_{g} (cf. 3.2), and we really expect an equality here, because of the lack of a good characterization of B_{h}, there is no proof of the equality B_{h} = B_{f} + B_{g}.

(5.4) Corollary. Assume that B_{f} = {t ∈ ℂ : χ(𝔽_{t}) ≠ χ(𝔽)} and B_{g} = {t ∈ ℂ : χ(𝔾_{t}) ≠ χ(𝔾)} (this holds for instance if f, resp. g, have isolated singularities at infinity, see [ST]). Then

B_{h} = {t ∈ ℂ : χ(ℍ_{t}) ≠ χ(ℍ)} = B_{f} + B_{g}.

(5.5) Some technical remarks. Assume that f : ℂ^{n} → ℂ is a polynomial in the class C such that the fiber 𝔽_{0} is special but smooth. Let D_{0} be a small disc in ℂ centered at the origin, T_{0} = f^{−1}(D_{0}) the corresponding tube, and m_{1} (resp. m_{1}^{∗}) the monodromy operator of f corresponding to the oriented loop ∂D_{0} acting on H_{n−1}(𝔽) (resp. on H^{n−1}(𝔽)).

Then the Gysin sequence of the hypersurface 𝔽_{0} in T_{0} (see [D1] p. 46) and (5.2.iv) above give an isomorphism H_{n−1}(𝔽_{0}) →^{≃} H_{n}(T_{0} \ 𝔽_{0}). On the other hand, the Wang sequence of the corresponding locally trivial fibration over ∂D_{0} contains the sequence

0 → H_{n}(T_{0} \ 𝔽_{0}) → H_{n−1}(𝔽) ⟶^{m_{1}−1} H_{n−1}(𝔽).

Therefore, H_{n−1}(𝔽_{0}) ≃ ker(m_{1} − 1), and it is free over ℤ.

To determine the torsion part in the (co)homology of the fibers of f, it is more convenient to use the cohomological analog of the above isomorphisms, namely:

H^{n−1}(𝔽_{0}) ≃ H^{n}(T_{0} \ 𝔽_{0}) ≃ coker(m_{1}^{∗} − 1).

When the special fiber 𝔽_{0} is singular, a similar argument can give the Betti numbers for the regular part 𝔽_{0}^{∗} of 𝔽_{0}, since b̃_{j}(𝔽_{0}^{∗}) = 0 for j < n − 2 by [ACD], p. 322.

In the next example we construct hypersurfaces with interesting topology: they have either torsion in the co(homology) or they are diffeomorphic to the affine spaces.

(5.6) Example. Let f : ℂ^{2} → ℂ be given by f(x_{1}, x_{2}) = x_{1}(x_{1}x_{2} + u_{1}) ⋯ (x_{1}x_{2} + u_{k}) where u_{i} ∈ ℂ^{∗} are k distinct complex numbers. The equality f(tx_{1}, t^{−1}x_{2}) = tf(x_{1}, x_{2}) implies that B_{f} = {0} and m_{1} = 1. (Obviously, m_{1} is the only monodromy operator of f.) Moreover, μ_{f} = 0 (since 𝔽_{0} is smooth), and H_{1}(𝔽) = ℤ^{k}, for instance by [ACD].

Let g : ℂ^{n} → ℂ be given by g(y_{1}, ..., y_{n}) = y_{1}^{a_{1}} + ⋯ + y_{n}^{a_{n}} for some positive integers a_{j}, i.e. g is a Brieskorn-Pham polynomial. Hence again B_{g} = {0}. Let m denote the monodromy of g.

If the link K_{g} associated to the singularity of g is a ℤ-homology sphere (and this is determined by the arithmetic properties of the a_{j}’s, and requires n > 1, see [Bri] or [D1], p. 94), then H̃_{∗}(ℍ_{0}) = 0. Indeed, the monodromy M_{1} of h is m_{1} ⊗ m by [N1] (or by (7.1) below), but m − 1 is an isomorphism by the assumption about a_{j}’s, hence the statement follows from (5.5).

Using the connectivity properties in [DP], it is also clear that ℍ_{0} is simply-connected, and hence contractible. But then ℍ_{0} is actually diffeomorphic to the affine space ℂ^{n+1}, see [D1], p.28 and p. 175.

Special cases of this construction, namely for k = 1 and n = 2, have been considered by Koras and Russell in [KR]. In fact the topological properties in their Example (7.8) can be obtained in exactly the same way as here since the corresponding polynomial is f_{1}(x, y) = x + x^{r}y and it enjoys exactly the same properties as our f, e.g. one has f_{1}(tx, t^{−r+1}y) = tf_{1}(x, y).

Next, take in our example n = 1 i.e. g(y) = y^{a}. Then coker(m^{∗} − 1) = ℤ/aℤ. Therefore, using (5.5), H^{2}(ℍ_{0}) = coker (M_{1} − 1) = (ℤ/aℤ)^{k}.

6. The monodromy representation of h = f ⊕ g

(6.1) Consider two polynomial functions f and g as in sections 3 and 4. From now on in this paper we assume that the bifurcation set for g consists of just one point, say B_{g} = {0}. This means that the monodromy representation of g is completely determined by one monodromy operator m = m_{g} corresponding to going once around the origin counterclockwise.

On the other hand the bifurcation set B_{f} = {b_{1}, …, b_{t}} can be a rich set and the corresponding monodromy representation ρ_{f} can be very complicated. By (3.2) we know that B_{h} ⊂ B_{f}. Our goal is to describe the monodromy representation ρ_{h} associated to h in terms of ρ_{f} and m = m_{g}.

Recall that a set of free generators of the fundamental group depends essentially on the choice of a star with end points in the bifurcation set. In order to compare ρ_{h} and ρ_{f} we have to construct compatible stars for f and h. These stars provide automatically sum decompositions in homology, hence our monodromy operators will have blocks corresponding to these decompositions. On the other hand the geometric construction in section 3 decomposes ℍ as a join and we have to be sure that this decomposition is compatible with the direct sum decompositions with respect to the stars. This compatibility is established in the next discussion.

(6.2) First, fix a complex number c_{0} ∉ {b_{1}, …, b_{t}}. Then consider an arbitrary star p_{1}, …, p_{t} with base point c_{0} and end points b_{1}, ⋯, b_{t}. This is the “star of f”. Moreover, we can fix a real line l_{f} ⊂ ℂ (where l_{f} = α(ℝ×{0}) for some diffeomorphism α : ℝ^{2} → ℂ) such that l_{f} separates b_{1}, ⋯, b_{t} from c_{0} as follows.

In order to visualize this (and any similar picture in the next pages) we will use a “model”. This “model” is the plane ℂ = ℝ^{2}, with usual coordinates (x, y), where c_{0} = (0, −1), b_{i} = (−i, 1) for i = 1, 2, …, t; the star consists of segments which connect b_{i} and c_{0}, and finally l_{f} is the x-axis. Then, going back to the original situation, we assume that our star and l_{f} in ℂ is topologically equivalent to this “model”. (See Fig. 1)

[figure: c_{0}; b_{t}; b_{t−1}; ⋯; ⋯; b_{i}; b_{2}; b_{1}; l_{f}; (Fig. 1)]

Next, we wish to construct an associated star for h. We define the basepoint by λ_{0} := c_{0}. This means that B_{λ_{0},g} = (c_{0}, 0) and B_{λ_{0},f} = {(b_{i}, λ_{0} − b_{i})_{i=1,...,t}}. Moreover, we define l_{λ_{0}} ⊂ L_{λ_{0}} by pr_{1}^{−1}(l_{f}) ∩ L_{λ_{0}}. Similarly as above, let L_{λ_{0}}^{+} be the “topological semiplane” which contains {(b_{i}, λ_{0} − b_{i})_{i=1,...,t}}; and L_{λ_{0}}^{−} the other, containing (c_{0}, 0). The star of h is defined by the paths p′_{i}(t) := p_{i}(t), connecting λ_{0} and b_{i}. Therefore, the stars of f and h are tautologically related (in some sense, they are the same).

In the sequel we will fix such a tautological pair of stars of f and h, and the partition of L_{λ_{0}} by l_{λ_{0}} as above. The stars provide sum decompositions in the homology groups of 𝔽 and ℍ, l_{λ_{0}} provide a join decomposition of ℍ (inducing a tensor product decomposition in the homology).

(6.3) Theorem. Assume as above that B_{g} = {0} and we fix a ring of coefficients R ⊃ ℤ in such a way that H̃_{∗}(𝔾, R) is R−free. Identify H̃_{∗}(ℍ) with H̃_{∗}(𝔽)⊗H̃_{∗}(𝔾) (cf. 3.5), and consider in H̃_{∗}(ℍ) the two natural direct sum decompositions: one of them provided by the star of h, the other given by ⊕_{i=1}^{t}H_{∗,i} ⊗H̃_{∗}(𝔾), where the decomposition H̃_{∗}(𝔽) = ⊕_{i=1}^{t}H_{∗,i} is provided by the star of f. Then the following facts hold:

(i) The two decompositions are the same.

(ii) For any 1 ≤ j ≤ t, set m_{j} = ρ_{f}(γ_{j}) : ⊕_{i=1}^{t}H_{∗,i} → ⊕_{i=1}^{t} H_{∗,i} and consider its block decomposition as it is described in (2.5). Then the monodromy operator M_{j} = ρ_{h}(γ_{j}), as an automorphism of ⊕_{i=1}^{t} H_{∗,i} ⊗ H̃_{∗}(𝔾), has the following block-decomposition:

M_{j} = ( \begin{matrix}1 \\ . & . & . \\ & & 1 \\ v_{j1} ⊗ 1 & ⋯ & v_{j,j−1} ⊗ 1 & m_{jj} ⊗ m & v_{j,j+1} ⊗ m & ⋯ & v_{jt} ⊗ m \\ & & & & 1 \\ . & . & . \\ & & & & & & 1 \end{matrix} ) .

Here m = m_{g}, 1 stands for various identity morphisms, and if A = (A_{p}) ∈ Aut(H̃_{∗}(𝔽)) and B = (B_{q}) ∈ Aut(H̃_{∗}(𝔾)), then A ⊗ B ∈ Aut(H̃_{∗}(𝔽 ∗ 𝔾)) has graded components given by (A ⊗ B)_{r} = ⊕_{p+q=r−1}A_{p} ⊗ B_{q}. Proof. The next proof of part (ii) implies also (i) (cf. also with (4.2) and (4.4)). Let γ′_{j} be the loops associated with the paths p′_{j}. If we move λ along p′_{j}, via the first projection pr_{1} : L_{λ} → ℂ, the points pr_{1}(B_{λ,f}) are constant, and pr_{1}(B_{λ,g}) = λ is moving from c_{0} to b_{j} along the path p_{j}. If we move λ along γ′_{j} then pr_{1}(B_{λ,g}) is moving along γ_{j}.

Now, we need a local trivialization of the geometric picture of (6.2) (i.e. for the embedding of the star and the separating real line in ℂ) for the family {L_{λ}}_{λ}. It is convenient to describe all the data in L_{λ} via its first projection pr_{1}.

First of all, let’s fix a point c′_{0} on l_{f} and a deformation retract r of l_{f} to c′_{0}. This c′_{0} will be our base point, and in our local trivialization procedure we have to be careful not to move this point. Take a star q_{1}, …, q_{t} with base point c′_{0} and end points b_{1}, …, b_{t} such that this new star is topologically equivalent with the old star p_{1}, …, p_{t}, and also the image of the paths {q_{j}}_{j} are in pr_{1}(L̄_{λ_{0}}^{+}). Moreover, fix a simple path q′ connecting c′_{0} and c_{0} in pr_{1}(L̄_{λ_{0}}^{−}). (In the above “model”, put c′_{0} = (−t, 0) and construct the star q_{1}, …, q_{t} and q′ using segments.) It is clear that H_{∗}(𝔽_{c_{0}}) and H_{∗}(𝔽_{c′_{0}}) can be identified via q′ and the direct sum decompositions given by the two stars {p_{j}}_{j} and {q_{j}}_{j} are the same. So, we will write H̃_{∗}(𝔽_{c′_{0}}) = ⊕_{i} H_{∗,i}. It is clear that all these data can be lifted from ℂ to L_{λ_{0}} via pr_{1}.

Start with a cycle [a_{i} ⊗ s] ∈ H_{∗,i} ⊗ H̃_{∗}(𝔾). This means that a_{i}, representing [a_{i}] ∈ H_{∗,i}, is the boundary of a cycle Cone(a_{i}) which is above image(q̃_{i}) ∪ D_{i} (where D_{i} is a small disc centered at b_{i}), and Cone(a_{i}) has a product structure a_{i} × [0, t_{i}] above image(q̃_{i})=image(q_{i}) − D_{i} (cf. section 2).

Similarly, s is the boundary of Cone(s) which stays above the union of pr_{2}pr_{1}^{−1}(image(r)) and a small disc centered at 0. [Here we used the notation Cone(a_{i}) in order to suggest that in nice cases (when H_{∗,i} represents the vanishing cycles associated with isolated singularities situated at finite distance) this cycle is exactly the topological cone over a_{i}, but in general this fact is not true anymore.]

Geometrically one has the cycle C_{i} = Cone(a_{i}) × s ∪_{a_{i}×s} a_{i} × Cone(s). Notice that C_{i} intersected with ϕ^{−1}(l_{λ_{0}}) is exactly a_{i} × s, i.e. C_{i} represents a cycle in H_{∗+1}(ℍ) such that δ([C_{i}]) = [a_{i} ⊗ s], cf (3.4)-(3.5).

Next, we want to see the effect of the geometric monodromy.

Assume that i > j and move λ along γ′_{j}. Then we construct the paths {q_{i}(λ)}_{i} and q′(λ) in pr_{1}(L_{λ}) in a locally trivial way, such that we keep c′_{0} constant. The effect of the geometric monodromy can be described in the “model” as follows. As above, c_{0} = (0, −1), b_{i} = (−i, 1), l_{f} = {y = 0}, c′_{0} = (−t, 0). Moreover, in this case i > j, in the “model” picture q_{i}(1) is again the segment connecting c′_{0} and b_{i}. On the other hand, q′(1) is a curve formed by five segments connecting consecutively the points (−t, 0), (0, −1 − ϵ), (ϵ, 0), (−j, 1 + ϵ), (−j − ϵ, 0) and (0, −1), where 0 < ϵ << 1. (see Fig. 2)

[figure: c_{0}; c′_{0}; b_{t}; q_{i}(1); q′(1); b_{i}; b_{j+1}; b_{j−1}; b_{j}; b_{2}; b_{1}; l_{f}; (Fig. 2; case i > j); a_{i} × s; m_{j}(a_{i}) × m(s); −a_{i} × m(s)]

[Intuitively, this can be explained as follows. In the “model” consider the segment S connecting c_{0} and b_{j}. Now, topologically, in a small neighbourhood N(S) of S, rotate S with 2π keeping all the complement of N(S) unchanged. Then see what is the effect of this transformation on the paths q_{i} and q′, and q_{i}(1) and q′(1) are this transformed paths. Obviously, l_{f} will also be modified by this rotation, but since we want to describe the domain and the target of the monodromy operator in the same base, we consider even in the new picture the old l_{f} = {y = 0}.]

Consider the cycle m_{geom}(C_{i}), intersect with l_{λ_{0}} and retract (via the lifted retraction r) to pr_{1}^{−1}(c′_{0}). This corresponds to the monodromy of h. The intersection cycle is a_{i} × s + m_{j}(a_{i}) × m(s) − a_{i} × m(s), i.e. M_{j}([a_{i} ⊗ s]) = [a_{i} ⊗ s] + v_{ji} ⊗ m([a_{i} ⊗ s]).

Similar pictures can be drawn in the two other cases.

If i = j, then the intersection cycle is a_{j}×s−a_{j}×s+m_{j}(a_{j})×m(s)+a_{j}×m(s)−a_{j}×m(s), i.e. M_{j}([a_{j} ⊗ s]) = m_{jj} ⊗ m([a_{j} ⊗ s]). (See Fig. 3)

[figure: c_{0}; c′_{0}; b_{t}; q_{j}(1); q′(1); b_{j+1}; b_{j−1}; b_{j}; b_{1}; l_{f}; (Fig. 3; case i = j)]

If i < j, then the intersection cycle is: a_{i} × s − a_{i} × s + m_{j}(a_{i}) × s + m_{j}(a_{i}) × m(s) − m_{j}(a_{i}) × m(s) + a_{i} × m(s) − a_{i} × m(s) = m_{j}(a_{i}) × s = a_{i} × s + (m_{j}(a_{i}) − a_{i}) × s. In particular, M_{j}([a_{i} ⊗ s]) = [a_{i} ⊗ s] + v_{ji} ⊗ 1([a_{i} ⊗ s]). (See Fig. 4) □

[figure: c_{0}; c′_{0}; b_{t}; q_{i}(1); q′(1); b_{j+1}; b_{j−1}; b_{j}; b_{i}; b_{1}; l_{f}; (Fig. 4; case i < j)]

7. Examples

(7.1) Example. The monodromy at infinity. When B_{f} = {0} = B_{g}, then B_{h} ⊂ {0} and the only monodromy operator for h is given by the formula M_{1} = m_{1} ⊗ m.

More generally, for any polynomial f (with arbitrary B_{f}), define its monodromy at infinity M_{∞}(f) to be the monodromy along any simple closed path containing all the bifurcation points of f inside. This curve is oriented counterclockwise and in particular we have M_{∞}(f) = m_{t} ⋯ m_{1}. Then M_{∞}(h) = M_{∞}(f) ⊗ M_{∞}(g) is valid for any two polynomials, see [N1]. This equality can also be verified (for the case B_{g} = {0}) using (6.3).

(7.2) Example. The case m = 1. Assume that m = m_{g} = 1. Then in (6.3) we have M_{j} = m_{j} ⊗ 1, i.e. ρ_{h} = ρ_{f} ⊗ 1, and this relation is independent of the direct sum decompositions provided by the stars.

(7.3) Example. The case f : ℂ → ℂ. In this example we discuss the case when f : ℂ → ℂ is a polynomial of degree k. First we review some basic facts about the topological classification of such polynomials, then we indicate how one can obtain by Thom-Sebastiani construction large families of polynomials with interesting monodromy representations.

In [KZ] is shown that quite often the monodromy representation ρ_{f} associated to a polynomial f : ℂ → ℂ can be determined from a local simple data which we describe now.

Note that in this case 𝔽 is just a set of k points and to know the monodromy representation of f is equivalent to knowing the permutations σ_{i} associated to going around the loops γ_{i}, i.e. a representation ρ′_{f} of the free group π_{1}(ℂ \ B_{f}) in the symmetric group S_{k}.

Assume that f has t critical values b_{1}, ..., b_{t}. For each b_{i} the multiplicities of the roots for the equation f(x) = b_{i} give a partition d_{i} = (d_{i1}, ..., d_{ik_{i}}) of k. The list Σ = (d_{1}, ..., d_{t}) of all these partitions is called the passport of f. The integers d_{ij} satisfy the condition ∑_{ij}(d_{ij} − 1) = k − 1 and conversely, for any passport Σ (i.e. any list of partitions of k) satisfying this condition there is a polynomial f_{Σ} having this list as its passport. Note that the proof given in [KZ] of this claim (see Thm. 2) produces in fact an explicit description of the monodromy representation ρ′_{f} for this polynomial f_{Σ}.

The main result in [KZ] is that for many passports Σ (let us call them rigid), any two polynomials having the same (rigid) passport are topologically equivalent. In particular, they have the same monodromy representation as the polynomial f_{Σ}, which is known. Here are two examples of such rigid passports.

(i) A partition d is called nonsimple if it is different from the partition (2, 1, 1, ..., 1). The defect def(Σ) of a passport Σ is by definition the sum of all d_{ij} > 1 from all the nonsimple partitions in Σ. Then if def(Σ) ≤ k + 1, then Σ is rigid.

(ii) If in any partition in Σ there is exactly one d_{ij} > 1, then the passport Σ is rigid. In fact, for t = 2 the list of all the rigid passports is known, see [EHZZ] (where

the result is attributed to N. M. Adrianov) and the first non rigid passport occur for k = 6, see [KZ]. Moreover, the complete list of topological types (and of their monodromy representations ρ′_{f}) is now known for k ≤ 9 see [EHZZ].

After this introduction, we will exemplify how one can compute the monodromy representation of the polynomials of type f ⊕ g, where B_{g} = {0} and f : ℂ → ℂ has a rigid passport. Even if this family looks very particular, we emphasize that in the literature there are only a very restricted number of examples with complete description of their monodromy representations, and maybe the present algorithm for f ⊕ g is the first which provides a large family of examples.

Assume, for example, that f : ℂ → ℂ is a polynomial of degree k = 5 with passport ((3, 1, 1), (3, 1, 1)) (in particular, t = 2). Such a polynomial is for instance f(x) = (x + 3)^{3}(x^{2} − 9x + 24) = 1296 + (x − 3)^{3}(x^{2} + 9x + 24), see [Bi]. This passport is rigid (cf. above) and it is easy to determine the corresponding monodromy operators m_{i} starting from the associated permutations σ_{i}. More precisely we have σ_{1} = (123), σ_{2} = (345) (both are cycles of length 3 acting on 𝔽 = {1, 2, 3, 4, 5}) and with respect to the basis (1 − 2, 2 − 3, 3 − 4, 4 − 5) for H̃_{0}(𝔽) we have

m_{1} = ( \begin{matrix}0 & −1 & 1 & 0 \\ 1 & −1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , m_{2} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & −1 \\ 0 & 0 & 1 & −1 \end{matrix} ) ,

(where the block decomposition consists of the standard 2 × 2 blocks).

Then the formulas for M_{1} and M_{2} given by (6.3) yield the following matrices (where each entry is a square block of size rank H̃_{∗}(𝔾)):

M_{1} = ( \begin{matrix}0 & −m & m & 0 \\ m & −m & m & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , M_{2} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & −m \\ 0 & 0 & m & −m \end{matrix} ) .

8. The monodromy representation of the Briançon polynomial

As a final application we determine the complex monodromy representation of the Briançon’s polynomial f (of degree 10); see [ACD] or [D2] for an explicit formula of it. We emphasize that in this section the philosophy is rather different: we do not plan to determine the representation of a Thom-Sebastiani type polynomial as above (f is not of that type), but we apply our results for the suspension h = f + z^{2}, and using some extra information about h we recover some important facts about f.

(8.1) Theorem. The set B_{f} associated with the Briançon polynomial f has two elements. The monodromy representation is given by the two matrices (ρ_{f}) (see below). The entries of these matrices can be determined as follows: t_{1} and t_{2} are the roots of the equation t^{2} − t + 1 = 0, a = 2(1 − t_{1})(t_{1} − t_{2})^{−1}, b = 2(t_{2} − 1)(t_{1} − t_{2})^{−1} and γ = 1. Proof. The following facts about f follow from [ACD] or [D2]: the bifurcation set B_{f} consists of two points: b_{1} = −16/9 and b_{2} = 0, and all the fibers of f are smooth and irreducible. Moreover, f satisfies the condition (iv) in (2.7), and it is in the class C with ν_{f}(b_{1}) = 3 and ν_{f}(b_{2}) = 1 (cf. section 5).

The Jordan normal form for the monodromy operators m_{i}^{∗} is given in [ACD]:

m_{1}^{∗} ≈ diag(1, −1, t_{1}, t_{2}) and m_{2}^{∗} ≈ ( \begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) ,

where t_{i} (i = 1, 2) are the roots of the equation t^{2} − t + 1 = 0.

Let H_{2}^{1} = Ker(m_{1}^{∗} − 1) and H_{1}^{1} = Ker(m_{2}^{∗} − 1) (cf. 2.7 for the notations). Then H^{1}(𝔽) = H_{1}^{1} ⊕ H_{2}^{1} and we may choose a basis e_{1}, e_{2}, e_{3} for H_{1}^{1} and a basis e_{4} for H_{2}^{1} such that m_{1}^{∗} and m_{2}^{∗} have the following associated matrices with respect to the basis e given by e_{1}, e_{2}, e_{3}, e_{4} for H^{1}(𝔽):

m_{1}^{∗} = ( \begin{matrix}−1 & 0 & 0 & 0 \\ 0 & t_{2} & 0 & 0 \\ 0 & 0 & t_{1} & 0 \\ γ & β & α & 1 \end{matrix} ) m_{2}^{∗} = ( \begin{matrix}1 & 0 & 0 & c \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & a \\ 0 & 0 & 0 & 1 \end{matrix} )

where α, β and γ are 0 or 1. The reader may refer to [BM] for the construction of such basis, which are called there “special basis”.

Using the fact that the cohomological monodromy operator at infinity M_{∞}(f)^{∗} = m_{1}^{∗}m_{2}^{∗} is known to be diag(1, −1, s_{1}, s_{2}) with s_{i} the roots of s^{2} + s + 1 = 0, see [D2], we get by taking traces aα + bβ + cγ = −2.

Now, one can verify that 𝔽 has a projective model 𝔽̃ which is an elliptic curve and it is the generic fiber of an elliptic fibration whose special fiber at b_{1} is of type II in the Kodaira’s classification, see [D2] for more details. Using this, it follows that there is a 2-dimensional subspace E ⊂ H^{1}(𝔽) which is ρ_{f,1}^{∗}–stable and such that m_{1}^{∗}|E = diag(t_{1}, t_{2}) and

m_{2}^{∗}|E = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

The existence of such a subspace E implies by some direct computation that

c = 0, α = β = 1, a = 2(1 − t_{1})(t_{1} − t_{2})^{−1}, b = 2(t_{2} − 1)(t_{1} − t_{2})^{−1}.

At this point there is just one more entry to determine, namely γ ∈ {0, 1}. Moreover, it is easy to check that the two different values for γ give rise to two non-equivalent representations.

We will determine γ by our Thom–Sebastiani construction. More precisely, using the dual basis to e for the homology H_{1}(𝔽), we get the following associated matrices for the monodromy operators m_{1} and m_{2}:

(ρ_{f})  m_{1} = ( \begin{matrix}−1 & 0 & 0 & γ \\ 0 & t_{2} & 0 & 1 \\ 0 & 0 & t_{1} & 1 \\ 0 & 0 & 0 & 1 \end{matrix} )  m_{2} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & b & a & 1 \end{matrix} ) .

By (2.7) the block decompositions in these two matrices coincide precisely with the block decompositions discussed in (6.3). We apply now (6.3) to the function h(x, y, z) = f(x, y) + z^{2}. Here the monodromy operator m is just multiplication by (−1) on ℂ so we get

M_{1} = ( \begin{matrix}1 & 0 & 0 & −γ \\ 0 & −t_{2} & 0 & −1 \\ 0 & 0 & −t_{1} & −1 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

It is clear that rang(M_{1} −1) = 2+γ. On the other hand by (5.5) we have rang(M_{1} −1) = 4 − b_{2}(ℍ_{b_{1}}), i.e. b_{2}(ℍ_{b_{1}}) = 2 − γ ∈ {1, 2}. To end we need the following result the proof of which was kindly supplied to us by Shulim Kaliman.

(8.2) Lemma. b_{2}(ℍ_{b_{1}}) = 1. Proof. To simplify the notation we set X = ℍ_{b_{1}}, F = 𝔽_{b_{1}} and ℤ_{2} = ℤ/2ℤ. Since the natural projection X → ℂ^{2} is a 2-covering with the ramification locus F we can consider the Smith homology sequence, see [Bre], Ch. III:

→ H_{3}(ℂ^{2}, F, ℤ_{2}) → H_{2}(F, ℤ_{2}) ⊕ H_{2}(ℂ^{2}, F, ℤ_{2}) → H_{2}(X, ℤ_{2}) → H_{2}(ℂ^{2}, F, ℤ_{2}) →
→ H_{1}(F, ℤ_{2}) ⊕ H_{1}(ℂ^{2}, F, ℤ_{2}) → H_{1}(X, ℤ_{2}) → H_{1}(ℂ^{2}, F, ℤ_{2}) → .

Since H_{i}(ℂ^{2}, F, ℤ_{2}) = H̃_{i−1}(F, ℤ_{2}) we have 0 → ℤ_{2} → H_{2}(X, ℤ_{2}) → ℤ_{2} → ℤ_{2} → H_{1}(X, ℤ_{2}) → 0. This implies that H_{2}(X, ℤ_{2}) is either ℤ_{2} ⊕ ℤ_{2} or ℤ_{2}.

In order to exclude the first possibility consider the hypersurface Y given by t^{2} + z^{2} = f(x, y) in ℂ^{4}. Obviously it is isomorphic to the hypersurface Z given by vu = f(x, y) in another sample of ℂ^{4}. Recall that for reduced homology we have H_{i}(Z) = H_{i−2}(F) (see Prop. 4.1 in [KaZ] or our Thm. (4.7)). In particular, H_{3}(Y, ℤ_{2}) = ℤ_{2}. Since the projection Y → ℂ^{3} is a ramified 2-covering with the ramification locus X we have again the Smith sequence

→ H_{4}(ℂ^{3}, X, ℤ_{2}) → H_{3}(X, ℤ_{2}) ⊕ H_{3}(ℂ^{3}, X, ℤ_{2}) → H_{3}(Y, ℤ_{2}) → .

This implies 0 → H_{2}(X) → ℤ_{2}, hence H_{2}(X) cannot be ℤ_{2} ⊕ ℤ_{2}. Thus H_{2}(X, ℤ_{2}) = ℤ_{2} hence γ = 1. This completes the determination of the complex monodromy representation for the Briançon’s polynomial.

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Laboratoire de Mathématiques Pures de Bordeaux Université Bordeaux I 33405 Talence Cedex, FRANCE e-mail: dimca@math.u-bordeaux.fr

Department of Mathematics, Ohio State University, 231 W 18th Ave., Columbus, OH 43210, USA e-mail: nemethi@math.ohio-state.edu