Hypersurface singularities with 2–dimensional critical locus

András Némethi^{1}

Consider an analytic germ f : (C^{m}, 0) → (C, 0) (m ≥ 3) whose critical locus is a 2–dimensional complete intersection with an isolated singularity (icis). We prove that the homotopy type of the Milnor fiber of f is a bouquet of spheres, provided that the extended codimension of the germ f is finite. This result generalizes the cases when the dimension of the critical locus is zero [8], respectively one [12]. Notice that if the critical locus is not an icis, then the Milnor fiber, in general, is not homotopically equivalent to a wedge of spheres. For example, the Milnor fiber of the germ f : (C^{4}, 0) → (C, 0), defined by f(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}x_{2}x_{3}x_{4} has the homotopy type of S^{1} × S^{1} × S^{1}. On the other hand, the finiteness of the extended codimension seems to be the right generalization of the isolated singularity condition, see e.g. [9, 10, 11, 12, 17, 18].

We mention that in the last years different types of “bouquet theorems” appeared. A part of them deal with germs f : (X, x) → (C, 0) where f defines an isolated singularity. Then, in some cases, similarly as in the Milnor case [8], F has the homotopy type of a bouquet of (dim X − 1)-spheres, e.g. when X is an icis [2], or X is a complete intersection [5]. Moreover, in [13] Siersma proved that F has a bouquet decomposition F ∼ F_{0} ∨ S^{n} ∨ ⋯ ∨ S^{n} (where F_{0} is the complex link of (X, x)), provided that both (X, x) and f have an isolated singularity. Actually, Siersma conjectured, and M. Tibăr proved [16] a more general bouquet theorem for the case when (X, x) is a stratified space and f defines an isolated singularity (in the sense of the stratified spaces). In this case F ∼ ∨_{i}F_{i}, where the F_{i}’s are repeated suspensions of complex links of strata of X. (If (X, x) has “Milnor property”, then the result has been proved by Lê, for details see [6].)

In our situation, the space–germ (X, x) is smooth, but f has big singular locus. Surprisingly, for dim Singf^{−1}(0) ≤ 2, the Milnor fiber is again a bouquet (actually, a bouquet of spheres, maybe of different dimensions). This result is in the spirit of Siersma’s paper [12], where dim Singf^{−1}(0) = 1. In that case, there is only a rather small topological obstruction for the Milnor fiber to be homotopically equivalent to a bouquet of spheres (as it is explained in (2.4)). In the present paper, we attack the dim Singf^{−1}(0) = 2 case. In our investigation some results of A. Zaharia are crucial [17, 18].

1Partially supported by NSF Grant No. DMS-9622724 and OSU Seed Grant 1991 Mathematics Subject Classification: 32S05, 14B05, 32S25, 32S50, 32S55

1. The main result.

Let O be the ring of holomorphic germs f : (C^{m}, 0) → (C, 0), (m ≥ 3). Let I ⊂ O be a reduced ideal defining an icis Σ of dimension 2. Let D_{I} be the group of local analytic isomorphisms ϕ : (C^{m}, 0) → (C^{m}, 0) such that ϕ^{∗}(I) = I [10, 11].

Let f ∈ O be a germ whose critical set contains Σ. Then, by [loc.cit.], f ∈ I^{2}. The group D_{I} acts on I^{2}, and the extended codimension of the orbit of f with respect to this action is

c_{e}(f) = dimC \frac{I^{2}}{I^{2} ∩ J(f)},

where J(f) is the Jacobian ideal of f.

In this note, we determine the homotopy type of the Milnor fiber of f, provided that c_{e}(f) < ∞.

By [17] or [18], c_{e}(f) < ∞ if and only if the critical locus of f is exactly Σ and the germ of f at every point of Σ − {0} is a D(2, p)–singularity, where p ≤ 1.

The D(2, p)–singularities were introduced in [10] and [11]. Their local equations for p ≤ 1, in a suitable coordinate system x = (x_{1}, …, x_{m}), are:

D(2, 0) : x ↦ ∑_{i=3}^{m} x_{i}^{2},
D(2, 1) : x ↦ x_{2}x_{3}^{2} + ∑_{i=4}^{m} x_{i}^{2}.

If m = 4, then D(2, 0) is the union of two transverse hyperplanes (and stably equivalent to this when m > 4), and D(2, 1) is a pinch–point×C (and stably equivalent to this when m > 4).

Fix a set of generators (g_{i})_{1,...,n}, (n = m − 2) of the ideal I. Then, since f ∈ I^{2}, there is a decomposition f = ∑_{i,j}^{n} h_{ij}g_{i}g_{j} with h_{ij} = h_{ji}. Define D(x) = det(h_{ij}(x)) ∈ O. Then D(0) ≠ 0 if and only if f at any point of Σ − {0} is of type D(2, 0). In this case, with a good choice of the generators (g_{i})_{i}, f can be written as g_{1}^{2} + ⋯ + g_{n}^{2} (see, for example, [17], or [18] (5.5.7)).

If D(0) = 0, then Δ = Σ ∩ D^{−1}(0) is a 1–dimensional icis [17]. Δ − {0} corresponds exactly to those points of Σ − {0} where f is of type D(2, 1).

Consider a deformation f_{s} of f with the following properties (for the proof of its existence, see [17](3.3)):

a) the critical locus of f_{s} is the disjoint union Σ_{s} ∐ S_{s}, where Σ_{s} is the Milnor fiber of Σ, S_{s} consists of some isolated points, and the singularities of f_{s} at the points of S_{s} are of type A_{1},

b) f_{s} at any point of Σ_{s} is of type D(2, p) with p ≤ 1,

c) the set Δ_{s} of points of Σ_{s} where f_{s} has D(2, 1)–type corresponds to a Milnor fiber of Δ.

In particular, a suitable restriction of f_{s} defines a family of D(2, 1)– singularities over the base space Δ_{s}. Therefore, {f_{s}(x) = ϵ} induces a fibration over Δ_{s} such that its fiber is the Milnor fiber of a D(2, 1)–singularity. More precisely, there is a tubular neighbourhood T_{s} of Δ_{s} such that the natural projection pr : T_{s} → Δ_{s} induces a fibration T_{s} ∩ {x : f_{s}(x) = ϵ} → Δ_{s} (0 < ϵ << 1), whose fiber is (diffeomorphic to) the Milnor fiber of (x_{2}, …, x_{m}) ↦ x_{2}x_{3}^{2} + ∑_{i≥4} x_{i}^{2} (∼ S^{n}). This defines a monodromy map:

Ξ : π_{1}(Δ_{s}, ∗) → Aut(H_{n}(S^{n}, Z)) = Z_{2}.

Now, we are able to formulate our main result: Main Theorem: Let f ∈ I^{2} be a germ so that c_{e}(f) < ∞. Then, the homotopy type of the Milnor fiber F of f is given in the following list: If D(0) ≠ 0 and n = 1, then

F ∼ S^{0} × ⋁_{μ_{Σ}} S^{m−1}.

If D(0) ≠ 0 and n > 1, then

F ∼ S^{n−1} ⋁( ⋁_{μ_{Σ}+σ} S^{m−1}).

If D(0) = 0 and Ξ is trivial, then

F ∼ S^{n} ⋁( ⋁_{2μ_{Δ}+μ_{Σ}+σ} S^{m−1}).

If D(0) = 0 and Ξ is not trivial, then

F ∼ ⋁_{2μ_{Δ}+μ_{Σ}+σ−1} S^{m−1}.

The numerical invariants are: μ_{Σ} (resp. μ_{Δ}) is the Milnor number of Σ (resp. of Δ), and σ is the number of A_{1}–points which occur in a ”good” deformation of f (i.e. the cardinality of S_{s}).

Notice that in the case D(0) ≠ 0 and n = 1 the Milnor fiber is not connected. Indeed, in this case, f can be written in the form f = g^{2}, where Σ = {g = 0}, in particular, F consists of two disjoint copies of the Milnor fiber of the isolated singularity g.

In all other cases the homotopy type of f is a bouquet of spheres.

Examples.

a) If f(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}^{2} + (x_{2}^{2} + x_{3}^{2} + x_{4}^{2})^{2}, then F ∼ S^{1} ⋁ S^{3} ⋁ S^{3}.

b) If f(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}^{2}(x_{3}x_{4} + x_{1}^{2}) + x_{2}^{2}, then F ∼ S^{2} ⋁(⋁_{4} S^{3}).

c) If f(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}^{2}x_{3} + x_{2}^{2}x_{4}, then F ∼ S^{3}.

The proof of the main theorem has several steps. The starting point is the construction of the “good deformation” f_{s} of f, which gives a nice geometric description of the Milnor fiber F. Then homology exact sequences give some immediate information about the homology of F. These facts basically are given in [17] and [18]. We recall them in section 3.

The real difficult part of the proof is the extraction from the geometry of F the necessary homological and homotopical ingredients which give the main result. In section 2, we prove that it is enough to verify that H_{∗}(F, Z) is free and the Hurewicz maps are onto. The last sections contain the verification of these facts in different cases which appear. Actually, in the case D(0) ≠ 0, the result proved in the body of the paper is more general than the result stated in the introduction (cf. Theorem 4.1).

As far as the author knows, the map Ξ (as an invariant which separates the different cases) is new. It gives a geometric support of the classification. It is similar to the “vertical monodromy” introduced by Siersma and others in the context of singularities with 1–dimensional critical locus. Notation: in this paper the homology groups are with integer coefficients.

2. When is a CW–complex a bouquet of spheres?

In this section F is a finite 1–connected CW–complex. By a theorem of Whitehead ([14] §7.6), if one can construct a continuous map h : ⋁_{i≥2}(⋁_{b_{i}} S^{i}) → F such that h induces either at the level of the homotopy groups , or at the level of the homology groups an isomorphism, then h is a homotopy equivalence.

In general, it is easier to compute the homology groups of F. An obvious necessary condition for the existence of a homotopy equivalence as above is that H_{∗}(F, Z) is a free Z–module. In general, this condition is not enough. 2.1. Example. Fix a non–zero element π of π_{n+1}(S^{n}), where n ≥ 2. (If n ≥ 3 then π is the generator of π_{n+1}(S^{n}) ≈ Z_{2}.) Let f : ∂D^{n+2} → S^{n} be a representative of π, where D^{n+2} is the (n + 2)–dimensional ball. Construct F = S^{n} ∪_{f} D^{n+2} (i.e. in the disjoint union of S^{n} and D^{n+2} one identifies x ∈ ∂D^{n+2} with f(x) ∈ S^{n}). Then F is 1–connected, and H̃_{q}(F) = Z if q = n, n + 2 and = 0 otherwise. On the other hand, F and S^{n} ⋁ S^{n+2} have different homotopy types.

In this paper we will use the following criterion:

2.2. Criterion. Let F be a 1–connected CW–complex such that H_{∗}(F, Z) is free and the Hurewicz map Hur_{∗} : π_{∗}(F) → H_{∗}(F, Z) is onto. Then there exists a homotopy equivalence F ∼ ⋁_{i≥2}(⋁_{b_{i}} S^{i}), where b_{i} is the i-th Betti number of F.

Proof. Construct a map h : ⋁_{i≥2}(⋁_{b_{i}} S^{i}) → F such that the restrictions {h|S^{i}}_{i=1}^{b_{i}} represent a Hur_{i}–lifted base of H_{i}(F). Then, by the theorem of Whitehead mentioned above, h is a homotopy equivalence. □

Notice that if F ∼ ⋁_{i≥2}(⋁_{b_{i}} S^{i}) then Hur_{∗} is onto, in particular both hypotheses of the criterion are necessary. 2.3. Remark. Let f be a finite 1–connected CW–complex such that b_{i} = 0 for i ∉ {l, l − 1} (l ≥ 3). Then, by a theorem of Hurewicz (see, for example, [15], p. 185), Hur_{l−1} is an isomorphism and Hur_{l} is onto. Therefore, F has the homotopy type of a bouquet of spheres if and only if H_{∗}(F, Z) is free.

2.4. Corollary. Let f : (C^{n+1}, 0) → (C, 0) (n ≥ 3) be the germ of an analytic function with one–dimensional critical locus. Then its Milnor fiber F has the homotopy type of a bouquet of spheres if and only if H_{n−1}(F, Z) is free.

Proof. Use (2.2) and (2.3) and the fact that H_{n}(F, Z) is always free (because F has the homotopy type of a CW–complex of dimension ≤ n). The (n−2)– connectivity of F follows from [3]. □

3. The deformation f_{s} of f.

In this section we recall the construction of a “good” deformation f_{s} of f and the geometry of the Milnor fiber F. We follow section 3 of [17] (or section 5.3 of [18]).

Let I = (g_{1}, …, g_{n}) be a 2–dimensional icis and f = ∑_{i,j=1}^{n} h_{ij}g_{i}g_{j} (h_{ij} = h_{ji}) with c_{e}(f) < ∞.

Let G : (C^{m}×C^{r}, 0) → (C^{n}×C^{r}, 0) be the miniversal deformation G(x, v) = (G_{1}(x, v), …, G_{n}(x, v), v) of g = (g_{1}, …, g_{n}) (see [7] (6.7)). Consider the deformation

f : (C^{m} × S, 0) = (C^{m} × C^{r} × C^{n} × C^{n(n+1)/2} × C^{m·n}, 0) → C

given by f_{s}(x) = f(x, v, u, a, b) = ∑_{i,j=1}^{n}(h_{ij}(x)+a_{ij}+∑_{t=1}^{m} b_{tj}x_{t}δ_{ij})(G_{i}(x, v)− u_{i})(G_{j}(x, v) − u_{j}), where a_{ij} = a_{ji}. 3.1. Lemma. [17] There exists an ϵ–ball B_{ϵ} with center 0 ∈C^{m}, a proper analytic set (A, 0) ⊂ (S, 0), and a neighbourhood U of 0 ∈ S, such that for any s ∈ U − A one has:

a) Σ_{s} = {x ∈ B_{ϵ} : G_{i}(x, v) = u_{i}; i = 1, …, n} is the Milnor fiber of Σ;

b) the zero set of D(s) = det(h_{ij}(x) + a_{ij} + ∑_{t} b_{tj}x_{t}δ_{ij}) intersects Σ_{s} transversally, hence Δ_{s} = D_{s}^{−1}(0) ∩ Σ_{s} is smooth. In particular, Δ_{s} is (diffeomorphic to) the Milnor fiber of Δ;

c) the singularities of f_{s} in B_{ϵ} − Σ_{s} are of type A_{1};

d) the germ of f_{s} at any point of Σ_{s} − Δ_{s} is of type D(2, 0), and at any point of Δ_{s} is of type D(2, 1).

e) Fix ϵ sufficiently small and δ sufficiently small with respect to ϵ. Then, if U is sufficiently small with respect to ϵ and δ, then f_{s}^{−1}(t) (as a stratified set) intersects ∂B_{ϵ} transversally for any s ∈ U and t ∈ Λ = {z : |z| ≤ δ}. In particular, the topological type of the smooth fibers of the maps

f_{s} : X_{s} = f_{s}^{−1}(Λ) ∩ B_{ϵ} → Λ (s ∈ U)

is independent of the parameter s ∈ U. (In fact, even the fibrations f_{s} : f_{s}^{−1}(∂Λ) ∩ B_{ϵ} → ∂Λ are equivalent to the fibration f : f^{−1}(∂Λ) ∩ B_{ϵ} → ∂Λ. In particular, the corresponding fibers are homotopically equivalent.)

f) The spaces X_{s} (s ∈ U) are contractible.

3.2. Remarks.

1) In [17] a)–d) is proved for points s in a dense set U′. But, it is not difficult to see that U′ can be chosen as U′ = U − A as above. (Actually, this specification is not used in this paper.)

2) The Milnor fiber of a D(2, p)–singularity has the homotopy type of S^{n+p−1}.

Fix an element s ∈ U − A, and denote a smooth, generic fiber f_{s}^{−1}(t) ∩ B_{ϵ} by X_{s,t}. Then, by (3.1.f) and the usual decomposition of the homology group H_{∗}(X_{s}, X_{s,t}) (see also [12], (2.8)), one has:

(3.3)  H̃_{∗−1}(X_{s,t}) = H_{∗}(X_{s}, X_{s,t}) = \{ \begin{matrix}H_{∗}(B_{s}^{0}, F_{s}^{0}) & if ∗ ≠ m \\ H_{m}(B_{s}^{0}, F_{s}^{0}) ⊕ Z^{σ} & if ∗ = m, \end{matrix}

where:

a) B_{s}^{0} is a small tubular neighbourhood {x ∈ B_{ϵ} : ∑_{i=1}^{n} |G_{i}(x, v) − u_{i}|^{2} ≤ ρ} of Σ_{s} ∩B_{ϵ}, which is diffeomorphic to the product (Σ_{s} ∩B_{ϵ})×B_{ρ}(n). (Here B_{ρ}(n) is the closed ρ–ball in C^{n} with center at the origin.)

b) F_{s}^{0} = B_{s}^{0} ∩ X_{s,t}, where t is sufficiently small with respect to ρ;

c) σ is the number of A_{1}–type singular points of f_{s} described in (3.1.c). In particular, Z^{σ} ⊂ H_{m−1}(X_{s,t}) is generated by σ vanishing cycles [∂e_{m}], where e_{m} are the corresponding Lefschetz “thimbles” in H_{m}(X_{s}, X_{s,t}) associated with the A_{1}–type singular points. (For the definition and nice presentation of the Lefschetz thimbles, see [4].)

3.4. The topology of the pair (B_{s}^{0}, F_{s}^{0}) is the following ([18], 5.4.3).

Fix a product structure (Σ_{s} ∩ B_{ϵ}) × B_{ρ}(n) of B_{s}^{0} and let π : B_{s}^{0} → Σ_{s} be the first projection.

If D(0) ≠ 0, then the induced map π : (B_{s}^{0}, F_{s}^{0}) → Σ_{s} is a fibration, where the fiber has the homotopy type of (B_{ρ}(n), S^{n−1}). Since π_{1}(Σ_{s}) = 0, the fibration F_{s}^{0} → Σ_{s} is orientable (in the sense of [14]).

Assume that D(0) = 0. Let T_{s} ⊂ Σ_{s} be a small tubular neighbourhood of Δ_{s} with a projection pr : T_{s} → Δ_{s}. Then:

a) π : (B_{s}^{0}, F_{s}^{0}) ∩ π^{−1}(Σ_{s} − Δ_{s}) → Σ_{s} − Δ_{s} is a fibration, where the fiber has the homotopy type of (B_{ρ}(n), S^{n−1});

b) the composed map pr ∘ π : π^{−1}(T_{s}) ∩ (B_{s}^{0}, F_{s}^{0}) → Δ_{s} is a fibration, where the fiber has the homotopy type of (B(n + 1), S^{n}).

3.5. Proposition. (cf. [18]) Let f be a germ as above, and n ≥ 3. (a) If D(0) ≠ 0, then

H_{∗}(F) = \{ \begin{matrix}Z & if ∗ = 0 or ∗ = n − 1 \\ Z^{μ_{Σ}+σ} & if ∗ = m − 1 \\ 0 & otherwise. \end{matrix}

(b) If D(0) = 0, then F is (n − 1)–connected and has the n–th Betti number equal to 0 or 1. Its Euler–characteristic is χ(F) = 1+(−1)^{n}·(1−2μ_{Δ}−μ_{Σ}−σ). Proof. The statement of (3.5) is identical with the statement of [18] (5.5.4-5.5.5) (or 5.5.2-5.5.4 in [17]), but the proof in [loc.cit.] has a gap: the Gysin exact sequence is applied for all fibrations, even for not–orientable ones!

In the case D(0) ≠ 0, π is an orintable fibration, therefore the proof of (a) in [loc.cit.] is correct. In the case D(0) = 0, the fibration (3.4.b) is not an orientable fibration (cf. 5.4). Therefore, the proof in [18] gives only the Euler–characteristic part of the result. The (n − 1)–connectedness and the n^{th}–Betti number of F will be given in Corollary 5.10. □ 3.6. Remark. Let g = (g_{1}, …, g_{n}) : (C^{m}, 0) → (C^{n}, 0) be an icis (where k = m − n ≥ 2), and consider the ideal I = (g_{1}, …, g_{n}) ⊂ O.

If f ∈ I^{2} satisfies the conditions c_{e}(f) < ∞ and D(0) ≠ 0 then, with a good choice of the local coordinates of (C^{n}, 0), f can be written as f = P ∘g, where P : (C^{n}, 0) → (C, 0) is the map P (w_{1}, …, w_{n}) = ∑_{i=1}^{n} w_{i}^{2} (see, for example, [18] (5.5.7)).

Conversely, any composed map f = P ∘ g = ∑_{i=1}^{n} g_{i}^{2} has finite extended codimension c_{e}(f) < ∞ if and only if the critical locus of f is exactly g^{−1}(0).

If f ∈ I^{2} satisfies these conditions, then all the properties listed above for the case D(0) ≠ 0 and k = 2 are valid for arbitrary k ≥ 2 (cf. [18], section 5). In particular, (3.3) is true for any k ≥ 2, and there exists an (oriented) fibration π : (B_{s}^{0}, F_{s}^{0}) → Σ_{s} with fiber (B_{ρ}(n), S^{n−1}), and the homology of F is given by the formula (3.5.a) (as it is noticed already in [18]).

3.7. Now, we are able to give the plan of the proof of the main theorem. If n < 3, then we give a direct proof. If n ≥ 3, then by (2.2–2.3) and (3.5), it is enough to verify the following facts:

a) The Hurewicz map Hur_{m−1} is onto, provided that D(0) ≠ 0; and b) H_{n}(F, Z) is free, provided that D(0) = 0.

4. The proof in the case D(0) ≠ 0.

Actually, we will prove a more general result: we verify the assertion for arbitrary k = m − n ≥ 1 and n ≥ 1. More precisely, let g = (g_{1}, …, g_{n}) : (C^{m}, 0) → (C^{n}, 0) be an icis, and consider the ideal I = (g_{1}, …, g_{n}) ∈ O as usual.

4.1. Theorem. Let f ∈ I^{2} be a germ so that c_{e}(f) < ∞ and D(0) ≠ 0. Then, the first two assertions of the main theorem (related to the case D(0) ≠ 0) are true for arbitrary k = m − n ≥ 1.

4.2. Proof of (4.1)

If n = 1, then the proof is trivial. If k = 1 then the result follows from [12].

In the sequel, the following notation appears frequently. If i_{1} : C → A and i_{2} : C → B are continuous maps, then their cofibered sum A ∐_{C} B is the quotient of the disjoint union A ∐ B obtained by identifying i_{1}(c) with i_{2}(c) for all c ∈ C (cf. [14], p. 99).

4.3. The case n = 2.

In this case, we can assume that f = g_{1}^{2} + g_{2}^{2}, hence by [9] the Milnor fiber of f is a bouquet of spheres S^{1} ⋁(⋁ S^{m−1}). Since we already know the Betti–numbers (cf. 3.5-3.6), the result follows.

For the reader’s convenience, we sketch the proof. The details are left to the reader, or they can be found in [9].

Write f = P ∘ g as above and we replace g by its good representative g : (X , 0) → (S, 0) in such a way that S is also a Milnor ball of P. D denotes the discriminant locus of g. Then for sufficiently small t (t ≠ 0), the space g^{−1}(P^{−1}(t)∩S) is homeomorphic to the Milnor fiber F of f ([9], 3.2.4). Since the critical locus of f is exactly g^{−1}(0), one has D ∩ P^{−1}(0) = {0}. The pair (P^{−1}(t) ∩ S, P^{−1}(t) ∩ D) is equivalent to (S^{1} × [0, 1], (D, P^{−1}(0))_{0} points), where (·, ·)_{0} denotes the intersection multiplicity. Since g over S − D is a fibration with fiber ⋁_{μ(g)} S^{m−2}, and any fiber over D can be obtained by attaching some (m − 1)–cells to the generic fiber, we obtain that F is homotopically equivalent to E ∐_{F (g)}(e_{m−1}^{1} ∪ … ∪ e_{m−1}^{N}), where E is the total space of a fibration over S^{1} with fiber the Milnor fiber of g, and F (g) is one of its fixed fibers, and the cells e_{m−1}^{i} (i = 1, …, N) are attached by maps ∂e_{m−1}^{i} ↦ F (g).

The main point is that we can assume that the fibration E is trivial. To see this, consider the deformation P_{u_{1},u_{2}}(c, d) = (c + i · d − u_{1})(c − i · d − u_{2}) of P. Consider the parameter (u_{1}, u_{2}) sufficiently small and such that: P_{u_{1},u_{2}}(c, d)^{−1}(0) intersects ∂S and D transversally, in particular {(c_{0}, d_{0})} = {c + i · d − u_{1} = c − i · d − u_{2} = 0} ∉ D. Then for sufficiently small t (t ≠ 0), the space g^{−1}[P_{u_{1},u_{2}}^{−1}(t) ∩ S] is equivalent to the Milnor fiber F. Let B_{0} be a small 2–disc with center (c_{0}, d_{0}) so that B_{0} ∩ D = ∅. Then for t ≠ 0 sufficiently small, there is a circle in B_{0} ∩ P_{u_{1},u_{2}}^{−1}(t) which generates π_{1}(P_{u_{1},u_{2}}^{−1}(t)). Therefore, the bundle over this circle is trivial. Now, in the above construction, we can assume that E is exactly this bundle over this circle. In particular, it is trivial.

Now, since P^{−1}(0) ∩ D = {0}, the space g^{−1}({c + i · d = 0}) has an isolated singularity, in particular, g^{−1}({c + i · d = u_{1}}) is (m − 2)–connected. The space g^{−1}({c + i · d = u_{1}}) is obtained from F (g) by attaching some (m − 1)–cells, and this set of cells is a subset of the set of cells which are used in construction of F. Therefore, by the maps ∂e_{m−1}^{i} ↦ F (g) one kills the homotopy group π_{m−2}(F (g)).

Let us summarize the facts: F ∼ S^{1} × F (g) ∐_{F (g)} E(g), where E(g) is obtained from F (g) by attaching N cells of dimension (m − 1) so that F (g) → E(g) is homotopically trivial. In particular, E(g) ∼ ⋁_{N−μ(g)} S^{m−1} and F ∼ [S^{1}×F (g)/F (g)] ⋁[⋁_{N−μ(g)} S^{m−1}] (cf. the topological lemma 4.7.b). But S^{1} × F (g)/F (g) ∼ S^{1} ⋁[ suspension of F (g)], i.e. F ∼ S^{1} ⋁(⋁_{N} S^{m−1}). 4.4. Remark. Using a similar argument as above, the proof of Theorem A,b in [9] can be simplified too. If in (3.2.11) [loc.cit.], we consider, similarly as above, a deformation P_{u_{1},u_{2}}(c, d) = P (c − u_{1}, d − u_{2}) with the transversal properties as above, then the monodromies {h_{i}}_{i} in (3.2.11) [loc.cit.] can be assumed to be trivial.

4.5. The particular case, when g defines an A_{1}–singularity (n ≥ 3 and k ≥ 2).

Assume that the Milnor number μ(g) of g is equal to 1. Let g : (X , 0) → (S, 0) be a good representative of g with discriminant D. By the general theory (see, for example, [7]) D is a smooth analytic space.

The fact that the critical locus of f is exactly g^{−1}(0) implies the following fact: 4.6. Lemma. Let P be the germ defined as in (3.6). Then:

a) The intersection D ∩ P^{−1}(0) is an (n − 2)–dimensional icis.

b) The intersection D ∩ P^{−1}(ϵ) is smooth, provided that ϵ is sufficiently small and ϵ ≠ 0. Therefore, this intersection can be identified with the Milnor fiber of the icis D ∩ P^{−1}(0). Proof. We have to prove that the origin is the only singular point of D ∩ P^{−1}(0) (respectively D ∩ P^{−1}(ϵ) has no singular point). Let t = 0 or = ϵ. Assume that w^{0} ≠ 0 is a singular point of D ∩ P^{−1}(t). By [7], there exists exactly one point x^{0} in the critical locus of g so that g(x^{0}) = w^{0}. By [loc. cit.], there are local coordinates x in (C^{m}, x^{0}) and w in (C^{n}, w^{0}) so that g(x) = (w_{1}, …, w_{n}) = (x_{1} − x_{1}^{0}, …, x_{n−1} − x_{n−1}^{0}, ∑_{i=n}^{m}(x_{i} − x_{i}^{0})^{2}). Moreover, the space–germ (P^{−1}(t), w^{0}) is smooth. Since D = {w_{n} = 0} and D ∩ P^{−1}(t) is singular at w^{0} it follows that P (w_{1}, …, w_{n}) = t + aw_{n} + P′(w), where a ≠ 0 and P′ ∈ m_{w}^{2} (the square of the maximal ideal (w_{1}, …, w_{n})). Therefore f(x) = t + a ∑_{i=n}^{m}(x_{i} − x_{i}^{0})^{2} + P′(g(x)) is singular at x^{0}. This contradicts the fact that the singular locus of f^{−1}(0) is exactly g^{−1}(0) (respectively f^{−1}(ϵ) is smooth). □

Now, by a similar argument as in [9] (3.2.4), we can identify the Milnor fiber of f with \overline{F} = g^{−1}(P^{−1}(ϵ) ∩ S) for ϵ ≠ 0 sufficiently small.

Fix a small ϵ (ϵ ≠ 0) and consider the restriction g : \overline{F} → F_{P}, where F_{P} = P^{−1}(ϵ) ∩ S is the Milnor fiber of P. Above F_{P} − D the map g is a fibration whose fiber is the Milnor fiber of g (which is homotopically a sphere S^{k}, k = m − n). Above the points P^{−1}(ϵ) ∩ D, the fiber is contractible. Choose Q : (C^{n}, 0) → (C, 0) so that D = Q^{−1}(0). Let p : (S, 0) → (B, 0) be a good representative of the icis (Q, P). Here B = {(c, d) ∈C^{2} : |(c, d)| < δ} and, obviously, we can assume that the source space of the good representative p is the same as the target space of the representative of g. Then we can identify the pairs (F_{P} , P^{−1}(ϵ) ∩ D) and (p^{−1}({d = ϵ}), p^{−1}((0, ϵ))). Let D_{p} be the discriminant locus of p. By (4.6.b), (0, ϵ) ∉ D_{p}. Let u : D_{2} → B ∩{d = ϵ} be an embedding of the real 2–disc D_{2} in B ∩ {d = ϵ} so that D_{p} ∩ {d = ϵ} ⊂ u(D_{2}) and (0, ϵ) ∉ u(D_{2}). Let v : [0, 1] → B ∩ {d = ϵ} be an embedding of the segment [0, 1] in B ∩ {d = ϵ} so that v(1) = (0, ϵ), v(0) ∈ ∂u(D_{2}), and v^{−1}(u(D_{2})) = {0}. Then, the inclusion (u(D_{2}) ∪ v([0, 1]), (0, ϵ)) → (B ∩ {d = ϵ}, (0, ϵ)) admits a strong deformation retract which has a p–lifting and a g–lifting. Therefore, with the notation T′ = p^{−1}(u(D_{2}) ∪ v([0, 1])) one has the homotopy equivalences: F_{P} ∼ T′ and F ∼ g^{−1}(T′).

Moreover, the inclusion u(D_{2}) ↪ u(D_{2}) ∪ v([0, 1]) also admits a strong deformation retract which has a p–lifting (but not a g–lifting). In particular, the inclusion T ↪ T′ is a homotopy equivalence, where T = p^{−1}(u(D_{2})).

The space g^{−1}(p^{−1}(u(D_{2})∪v([0, 1)))) is the total space of a fibration with fiber S^{k} and base space T ∐_{R} R × [0, 1), where R = p^{−1}(v(0)), R × [0, 1) is (a trivialization of) p^{−1}(v[0, 1)), R ↪ T is the natural inclusion, and the map R ↪ R × [0, 1) is given by x ↦ (x, 0). Since T ∼ F_{P} ∼ S^{n−1} and R ∼ ⋁ S^{n−2}, the inclusion R ↪ T is homotopically trivial. Therefore, the the fibration over R is trivial, hence over R × [0, 1) is trivial too.

Actually, we have the following general result: 4.7. Lemma.

a) Let π : E → T be a fibration with fiber G, and connected base–space T with basepoint ∗ ∈ T. Assume that R is a subspace of T such that the inclusion R ↪ T is homotopically trivial. Then there is a homotopy equivalence of pair of spaces:

(E, π^{−1}(R)) ∼ (E ∐_{{∗}×G}(Cone(R) × G), (R × {1}) × G).

Here Cone(R) = R × [0, 1]/R × {0}, the inclusion {∗} × G ↪ E is via the identification of {∗} × G with π^{−1}(∗), the inclusion {∗} × G ↪ Cone(R) × G is given by (∗, x) ↦ ([R × {0}], x).

b) Assume that S and T are connected topological spaces, R ⊂ S, R ⊂ T, and the inclusion R ↪ T is homotopically trivial. Then T ∐_{R} S ∼ T ⋁(S/R). Proof. a) Since the inclusion j : R ↪ T is homotopically trivial, there exists a map h : Cone(R) → T such that the inclusion R → Cone(R), x ↦ [x, 1] composed with h is the map j. Obviously, we can assume that h([R×{0}]) = ∗. Consider the space T ∐_{{∗}} Cone(R), where {∗} ∈ T is identified with the vertex of the cone. Define the map k : T ∐_{{∗}} Cone(R) → T by k|T = id_{T} and k|Cone(R) = h. Then k : (T ∐_{{∗}} Cone(R), R × {1}) → (T, R) is a homotopy equivalence, which induces another homotopy equivalence:

(k^{∗}(E), k^{∗}(E)|R × {1}) → (E, π^{−1}(R)).

Since Cone(R) is contractible, the result follows. For the second part use a) with G = {a point}, namely T ∐_{R} S ∼ T ⋁ Cone(R) ∐_{R} S ∼ T ⋁(S/R). □

Now, consider the space g^{−1}(p^{−1}(v([0, 1]))) = g^{−1}(R × [0, 1]). The map g over R × [0, 1) is a trivial fibration with fiber (homotopically equivalent to) S^{k}, and the fiber of g over R×{1} = F_{P} ∩D is contractible. Therefore, there is a homotopy equivalence of pairs of spaces:

(g^{−1}(p^{−1}(v([0, 1]))), g^{−1}(p^{−1}(v(0)))) ∼ (R × Cone(S^{k}), R × S^{k}),

where the inclusion R × S^{k} ↪ R × Cone(S^{k}) is given by (r, s) → (r, [s, 1]). The above equivalences give:

g^{−1}(T′) = g^{−1}(T) ∐_{g^{−1}(R)} g^{−1}(p^{−1}(v([0, 1]))) ∼

∼ g^{−1}(T) ∐_{{∗}×S^{k}} Cone(R) × S^{k} ∐_{R×S^{k}} R × Cone(S^{k}).

Notice that for two spaces A and B, the cofibered sum Cone(A)×B ∐_{A×B} A× Cone(B) is the join space A ∗ B. In particular,

g^{−1}(T′) ∼ g^{−1}(T) ∐_{{∗}×S^{k}}(R ∗ S^{k}).

It is easy to verify that the inclusion (vertex of the Cone(A))×B ↪ Cone(A)× B ↪ A ∗ B is homotopically trivial. In particular, {∗} × S^{k} ↪ R ∗ S^{k} is homotopically trivial, hence (see (4.7.b))

g^{−1}(T′) ∼ (g^{−1}(T)/g^{−1}(∗)) ⋁(R ∗ S^{k}).

Here g^{−1}(T) is the total space of a fibration over T (∼ F_{P} ∼ S^{n−1}) with fiber ∼ S^{k}; and g^{−1}(∗) is one of its fibers. 4.8. Lemma. Let g : E → S^{n−1} be a locally trivial fibration with fiber S^{k} and choose a basepoint ∗ ∈ S^{n−1}. Then E/g^{−1}(∗) ∼ S^{n−1} ⋁ S^{k+n−1}.

Proof. First notice that quotient space E/g^{−1}(∗) is independent of the isomorphism type of the fibration g. Indeed, write S^{n−1} as the union D_{+} ∪D_{−}of two closed hemispheres D_{+} and D_{−} such that ∗ ∈ D_{+} and g over D_{−} is trivial. Then E/g^{−1}(∗) ∼ E/g^{−1}(D_{+}) ∼ D_{−} × S^{k}/∂D_{−} × S^{k}.

Now, we prove that S^{n−1} × S^{k}/{∗} × S^{k} ∼ S^{n−1} ⋁ S^{k+n−1}. Write S^{l} = e^{0} ∪ e^{l}, where e^{i} denotes a i–dimensional cell. Then S^{n−1} × S^{k}/{∗} × S^{k} = e^{0} ∪ e^{n−1} ∪ e^{n−1} × e^{k} = S^{n−1} ∪_{α} e^{n−1} × e^{k}. The gluing is given by a map α : ∂(e^{n−1} × e^{k}) → S^{n−1}, provided by the composition

∂(e^{n−1} × e^{k}) → e^{n−1} × e^{k} →^{pr_{2}} e^{n−1} → e^{n−1}/∂e^{n−1} = S^{n−1}.

Since α is the restriction of a map defined on e^{n−1} × e^{k}, it is homotopically trivial. Therefore, S^{n−1} ∪_{α} e^{n+k−1} ∼ S^{n−1} ⋁ S^{n+k−1}. □

This last lemma implies that

F ∼ g^{−1}(T′) ∼ (S^{n−1} ⋁ S^{n+k−1}) ⋁(R ∗ S^{k}).

Since R ∼ ⋁_{μ} S^{n−2} (where μ is the Milnor number of the icis p) one has R ∗ S^{k} ∼ ⋁_{μ} S^{n+k−1}. Therefore,

F ∼ S^{n−1} ⋁( ⋁_{1+μ} S^{n+k−1}),

and we finished the proof for the case when g defines an A_{1}–singularity. 4.9. Remark. In this case, when g defines an A_{1}–singularity, by an Euler– number computation (see 3.5-3.6): μ_{Σ} + σ = 1 + μ, i.e. σ = μ(P^{−1}(0) ∩ D).

4.10. The general case, n ≥ 3 and k ≥ 2.

Now, we return to the general case: f = ∑_{i=1}^{n} g_{i}^{2}, where g = (g_{1}, …, g_{n}) defines an arbitrary icis. We assume that c_{e}(f) < ∞, or equivalently, that the singular locus of f is exactly g^{−1}(0).

First notice that π_{1}(F) = 0. Indeed, π_{1}(F) = π_{2}(X_{s}, X_{s,t}) = π_{2}(B_{s}^{0}, F_{s}^{0}) (because the dimension of the Lefschetz thimbles is m ≥ 5). By the fibration

S^{n−1} → F_{s}^{0} →^{π} Σ_{s} we obtain that π_{1}(F_{s}^{0}) = 0, therefore π_{2}(B_{s}^{0}) → π_{2}(B_{s}^{0}, F_{s}^{0})

is onto. In particular, π_{1}(F) is abelian, hence π_{1}(F) = H_{1}(F) = 0.

Now, by the criterion (2.2) and (3.5-3.6), it is sufficient to prove that the Hurewicz map Hur_{m−1} is onto (since Hur_{n−1} is an isomorphism by the result of Hurewicz mentioned in 2.3).

Let us identify a set of generators in H_{m−1}(F) ≃ Z^{σ+μ_{Σ}} (cf. 3.5-3.6). By section 3, H_{m−1}(F) ≃ H_{m}(X_{s}, X_{s,t}) ≃ Z^{σ} ⊕ H_{m}(B_{s}^{0}, F_{s}^{0}) ≃ Z^{σ} ⊕ H_{m−1}(F_{s}^{0}). By the generalised Gysin exact sequence of the fibration π ([14], p. 483) the map G̃ : H_{k}(Σ_{s}) → H_{m−1}(F_{s}^{0}), which appears in this sequence, is an isomorphism.

The subgroup Z^{σ} of H_{m−1}(F) is generated by the vanishing cycles ∂e_{m−1} associated with the A_{1}–singular points. In particular, Z^{σ} is in the image imHur_{m−1} of the Hurewicz map.

If μ_{Σ} = 1, then H_{m−1}(F_{s}^{0}) = Z is generated by G̃([c]), where [c] is the generator of H_{k}(Σ_{s}) = Z(cf. 4.9). By (4.5-4.9), F ∼ S^{n−1} ⋁(⋁_{σ+1} S^{m−1}), hence Hur_{m−1} is onto. In particular, G̃([c]) ∈ imHur_{m−1}.

If μ_{Σ} > 1, the group H_{k}(Σ_{s}) is generated by the vanishing cycles [δ_{i}]_{i} of Σ_{s}. Therefore, we can reduce this general case to the case μ_{Σ} = 1 as follows.

Consider the miniversal deformation G : (C^{m} × C^{k}, 0) → (C^{n} × C^{k}, 0), G(x, v) = (G_{1}(x, v), …, G_{n}(x, v), v) of g ([7], 6.7), and the deformation f_{s}(x) = ∑_{i=1}^{n}(G_{i}(x, v) − u_{i})^{2} of f, where s = (u, v). One has the following lemma (cf. (3.1)): 4.12. Lemma. There exist (S, 0) ⊂ (C^{n} × C^{k}, 0), a small ϵ–ball B_{ϵ} ⊂ C^{m}, and a disc Λ = {z : |z| < δ} such that:

a) for any s ∈ S and t ∈ Λ the fiber f_{s}^{−1}(t) (as a stratified set) intersects ∂B_{ϵ} transversely; and

b) for any s ∈ S and point x_{0} ∈ B_{ϵ} ∩ f_{s}^{−1}(0) the germ of f_{s} at x_{0} has finite extended codimension. In particular, its singular locus is exactly (Σ_{s}, x_{0}), where Σ_{s} = {x ∈ B_{ϵ} : G_{i}(x, v) = u_{i}, i = 1, …, n; s = (u, v)}. If x_{0} is a smooth point of Σ_{s} then the germ of f_{s} at x_{0} is of type D(2, 0). Proof. The first part is clear. For the second part consider the following sheaf. For any open set U ⊂ B_{ϵ} define F(U) by I_{U}^{2}/I_{U}^{2} ∩J(f)_{U}, where I_{U} and J(f)_{U} are O(U)–modules, I_{U} is generated by the restrictions g_{i}|U, and J(f)_{U} is the Jacobian ideal of f on U. Then F is a coherent sheaf of O–modules, is concentrated in a point and the dimension of its sections is exactly the extended codimension of f (cf. [18], 5.2.2). Therefore, for any s ∈ S, the corresponding sheaf is concentrated in a finite number of points. □

Now, consider an affine line L in C^{n} ×C^{k}, sufficiently closed to the origin, and such that it intersects the discriminant D_{G} of G transversally. Fix a point s ∈ L−D_{G} and embedded pathes [0, 1] → L which join s with the intersection points L ∩ D_{G} = {s_{i}}_{i}. For each point s_{i} (which is a smooth point of the discriminant space D_{G}), consider the unique singular point x_{s_{i}} of Σ_{s_{i}} (which is of type A_{1}) (see [7], section 4).

Then, for each point s_{i} of the intersection L ∩ D_{G}, one has a well–defined vanishing cycle [δ_{i}] in H_{k}(Σ_{s}) (see, for example, [1]). Moreover, there is a natural embedding of the local Milnor fiber F_{s_{i}} of the germ f_{s_{i}} (at x_{s_{i}}) in the Milnor fiber F of f (use 4.12.a).

Since the vanishing cycles [δ_{i}]_{i} generate H_{k}(Σ_{s}), the cycles of type G̃([δ_{i}]) generate H_{m−1}(F_{s}^{0}) ⊂ H_{m−1}(F). Consider the maps j_{i} : H_{m−1}(F_{s_{i}}) → H_{m−1}(F) and π_{i} : π_{m−1}(F_{s_{i}}) → π_{m−1}(F) induced by the above embeddings. Since G([δ_{i}]) ∈ imj_{i} one has Z^{σ} + ∑_{i} imj_{i} = H_{m−1}(F). Moreover, j_{i} ∘ Hur_{m−1}^{i} = Hur_{m−1} ∘ π_{i}, where the Hurewicz map Hur_{m−1}^{i} of F_{s_{i}} is onto (by the case μ_{Σ} = 1). In particular, imj_{i} ⊂ imHur_{m−1}. Recall that Z^{σ} ⊂ imHur_{m−1}. Therefore, H_{m−1}(F) = imHur_{m−1}. Hence the proof follows from (2.2-2.3). □.

5. The proof of the case D(0) = 0.

5.1. Assume that n = 1. In this case f = h_{11}g_{1}^{2}, where g_{1} defines an isolated singularity. Therefore, the result follows from [9], Corollary 2.5.5. Actually, a possible direct proof follows the following line. By [9] (2.5.4) (or by using the fact that the singular locus of f is exactly g_{1}^{−1}(0)), one has that c_{e}(f) < ∞ if and only if h_{11}^{−1}(0) has an isolated singularity and the pair (h_{11}, g_{1}) defines an icis. In particular, the discriminant locus of the pair (h_{11}, g_{1}) intersects the zero set of P (c, d) = c · d^{2} only at the origin. Now, repeat the arguments of (4.3) of this paper. 5.2. In the sequel we will assume that n ≥ 2. In this section we start the proof of Proposition 3.5.b, and of the main theorem (case D(0) = 0).

Recall, that the definition and the topology of the space F_{s}^{0} is given in (3.4). For any Y ⊂ Σ_{s}, denote the space π^{−1}(Y) by F_{Y}. Recall that π is a fiber bundle over Σ_{s} − Δ_{s} with fiber (homotopically equivalent to) S^{n−1}.

Assume that n = 2. Consider the covering F_{T_{s}} ∪ F_{Σ_{s}−Δ_{s}} of F_{s}^{0}, with F_{T_{s}} ∩ F_{Σ_{s}−Δ_{s}} = F_{T_{s}−Δ_{s}}. Since T_{s} − Δ_{s} ∼ S^{1} × Δ_{s}, and Σ_{s} − Δ_{s} ∼ S_{1} ∨ (∨S^{2}) (see e.g. 5.5), using the homotopy exact sequences of the fibrations (3.4.ab) and Van Kampen’s Theorem, one has π_{1}(F_{s}^{0}) = 1. This shows that π_{2}(B_{s}^{0}, F_{s}^{0}) is abelian. But (X_{Λ,s}, X_{t,s}) and (B_{s}^{0}, F_{s}^{0}) differs by some 4–cells only, and X_{Λ,s} is contractible, hence π_{2}(B_{s}^{0}, F_{s}^{0}) = π_{1}(X_{t,s}). Therefore π_{1}(F) (where F = X_{t,s}, cf. 3.1.e) is abelian, in particular, π_{1}(F) = H_{1}(F).

If n ≥ 3, then by [3], F is (n − 2)–connected. Hence, for any n ≥ 2, the (n − 1)–connectedness of F follows from the vanishing H_{n−1}(F) = 0. 5.3. Since F is an (n + 1)—dimensional Stein space, H_{q}(F) = 0 if q ≥ m = n + 2, and H_{n+1}(F) is free. Therefore, if we once verify that H_{n−1}(F) = 0 (which implies the (n − 1)–connectedness of F, cf. 5.2), and H_{n}(F) is free (of rank 0 or 1), then by (2.2-2.3) the main theorem follows.

If z is a new variable, then the Milnor fiber F_{z} of f(x)+z^{2} is the suspension of the Milnor F of f, in particular, H̃_{∗}(F) = H̃_{∗+1}(F_{z}). Therefore, in the wanted homological assertions described above, we can assume that n ≥ 5.

Since B_{s}^{0} ∼ Σ_{s} has the homotopy type of a bouquet of 2–spheres, by (3.3) for q =∈ {n − 1, n} one has H_{q}(F) = H_{q}(F_{s}^{0}). Therefore, in order to prove Proposition 3.5.b and the main theorem (case D(0) = 0), we have to verify that H_{n−1}(F_{s}^{0}) = 0 and H_{n}(F_{s}^{0}) is free (of rank 0 or 1), provided that n ≥ 5. 5.4. In order to understand the fibration π given by (3.4.a), consider the case of a D(2, 1) singularity f(x_{1}, …, x_{m}) = x_{2}x_{3}^{2} + ∑_{i≥4} x_{i}^{2}. Then Σ = {x_{i} = 0; i ≥ 3} and Δ = {x_{i} = 0; i ≥ 2}. Let B^{0} be a tubular neighbourhood: {x : |x_{1}|^{2} + |x_{2}|^{2} = η} of Σ, and F^{0} = {f = ϵ} ∩ B^{0} (0 < ϵ << η). The projection π(x) = (x_{1}, x_{2}) induces a fibration π : F^{0} ∩π^{−1}(Σ−Δ) → Σ−Δ. Notice that π_{1}(Σ − Δ) = Z generated by the path γ(α) = (x_{1}, x_{2}e^{2πiα}), α ∈ [0, 1]. The fiber above γ(α) is given by {(x_{3}, …, x_{m}) : e^{2πiα}x_{2}x_{3}^{2} + x_{4}^{2} + ⋯ = ϵ} which has the homotopy type of S^{n−1}. We leave to the reader to verify (using the natural C^{∗}–action) that the monodromy action on H_{n−1}(S^{n−1}) corresponding to the closed path γ (and with fixed ϵ) is −1_{Z}. In particular, π is not an orientable fibration.

In general, in the context of the fibration (3.4.a), above any small neighbourhood of any point of Δ_{s}, we have similar local situation as described above. Hence π is not an orientable fibration, and the monodromy corresponding to a closed path around Δ_{s} is −1_{Z}. In particular, the computation of the homology groups needs some caution.

For this computation we need some preparation. 5.5. In this paragraph we collect some properties of Σ_{s} − Δ_{s} and T_{s} − Δ_{s}.

Recall that Δ_{s} ⊂ Σ_{s} is a smooth fiber of the analytic map D_{s} : Σ_{s} → C; therefore, the tubular neighbourhood T_{s} of Δ_{s} in Σ_{s} has a product structure T_{s} ≃ Δ_{s} × D (where D denotes a disc). Via this identification, the first projection is exactly pr : T_{s} → Δ_{s} introduced in (3.4). Obviously, T_{s} − Δ_{s} ≃ Δ_{s} × D^{∗}, where D^{∗} denotes the punctured disc.

It is not difficult to prove (see, for example, [17], 4.5), that Σ_{s} − Δ_{s} ∼ S^{1} ⋁(⋁ S^{2}). Here the circle S^{1} corresponds to an elementary closed path around Δ_{s}. Moreover, the map, induced by the inclusion α : π_{1}(T_{s} − Δ_{s}) → π_{1}(Σ_{s} − Δ_{s}) = Z can be identified with the map induced by the second projection pr_{2}: π_{1}(T_{s} − Δ_{s}) = π_{1}(Δ_{s} × D^{∗}) → π_{1}(D^{∗}) = Z.

In our homological computations, similarly as in [17] p. 165, it is convenient to replace the tubular neighbourhood T_{s} by two small tubular neighbourhoods U_{1} and U_{2} of Δ_{s} in Σ_{s} with the following properties: (a) U_{1} ⊂ U_{2} ⊂ Σ_{s}, (b) the restriction of D_{s} to U_{i} (i = 1, 2) is a trivial fibration over some small disc from C, (c) the restriction of π above Σ_{s} − U_{1} is a locally trivial fibration, (d) the composed map π^{−1}(U_{2}) →^{π} U_{2} → Δ_{s} is a locally trivial fibration (where the map U_{2} → Δ_{s} is obtained with the help of a trivializaton of the restriction of D_{s} to U_{2}, (cf. the properties of T_{s} above).

The fiber of the fibration π is homotopy S^{n−1}. A closed (n − 1)-cycle in a fiber, which generates its (n−1)^{th}-homology, will be called “the fundamental cycle of the fiber”.

In the following lemmas we will assume that n ≥ 5 (cf. 5.3). 5.6. Lemma. H_{q}(F_{s}^{0} − F_{U_{1}}) = 0 for q = n − 2 and q = n. Moreover, H_{n−1}(F_{s}^{0} − F_{U_{1}}) = Z_{2} generated by the “fundamental cycle of the fiber of π”. Proof. Consider the fibration π : F_{s}^{0} −F_{U_{1}} → Σ_{s} −U_{1}. The base space (resp. the fiber) has the homotopy type of S^{1} ∨ (∨S^{2}) (resp. S^{n−1}). Therefore F_{s}^{0} − F_{U_{1}} is homotopically equivalent to E′ ∪_{S^{n−1}} E′′, where E′ → S^{1} and E′′ → ∨S^{2} are fiber bundles with fiber S^{n−1} and in E′ ∪_{S^{n−1}} E′′ a fiber of E′ is identified with a fiber of E′′. By (5.4), E′ is not orientable (i.e. its monodromy is −1_{Z}), hence by the Wang exact sequence H_{q}(E′) = 0 if q ∈ {n − 2, n}, and H_{n−1}(E′) = Z_{2} (generated by the “fundamental class [S^{n−1}] of the fiber”). On the other hand, E′′ → ∨S^{2} is orientable fibration, hence by Gysin exact sequence H_{q}(E′′) = 0 for q ∈ {n − 2, n}, and the Gysin map G : H_{0}(∨S^{2}) = Z → H_{n−1}(E′′) is an isomorphism. Notice that G(1_{Z}) is again [S^{n−1}]. Now, the result follows by the Mayer–Vietoris theorem. □ 5.7. Lemma. H_{n−2}(F_{U_{2}} − F_{U_{1}}) = 0 and H_{n}(F_{U_{2}} − F_{U_{1}}) is a torsion group. H_{n−1}(F_{U_{2}} − F_{U_{1}}) = Z_{2} and it is generated by the “fundamental (n − 1)–cycle of the fiber of π”. In particular, the map H_{n−1}(F_{U_{2}} −F_{U_{1}}) → H_{n−1}(F_{s}^{0}−F_{U_{1}}), induced by inclusion, is an isomorphism. Proof. Consider the fibration π : F_{U_{2}} −F_{U_{1}} → U_{2} −U_{1} with base space (resp. fiber) U_{2} − U_{1} ∼ S^{1} × Δ_{s} (resp. S^{n−1}). Since the fiber is (n − 2)–connected H_{n−2}(F_{U_{2}} −F_{U_{1}}) = 0 (use e.g. the homotopy exact sequence of π). Moreover, H_{n−1}(F_{U_{2}} − F_{U_{1}}) is cyclic and it is generated by the fundamental cycle of the fiber S^{n−1}. Because of (5.4), π is not orientable, hence H_{n−1}(F_{U_{2}} −F_{U_{1}}) = Z_{2}. Since the base space has a product structure, one can write F_{U_{2}} − F_{U_{1}} as a fiber bundle over Δ_{s} with fiber Z, where Z is the total space of a fiber bundle with base S^{1} and fiber S^{n−1}. By (5.4) Z is not orientable, hence by the Wang exact sequence H̃_{∗}(Z, Q) = 0. Therefore, H_{∗}(F_{U_{2}}−F_{U_{1}}, Q) = H_{∗}(Δ_{s}, Q). □ 5.8. Lemma. H_{n−1}(F_{U_{2}}) = 0. Moreover, H_{n}(F_{U_{2}}) = Z if Ξ is trivial, and H_{n}(F_{U_{2}}) = Z_{2} if Ξ is not trivial. (Obviously, here U_{2} can be replaced by T_{s}.) Proof. This follows from the fibration pr ∘ π : F_{U_{2}} → Δ_{s} (cf. 3.4.b), whose fiber has the homotopy type of S^{n}. □ 5.9. Corollary. H_{q}(F_{s}^{0}, F_{U_{2}}) = 0 if q = n, n − 1. Proof. Use the long exact sequence of the pair (F_{s}^{0}−F_{U_{1}}, F_{U_{2}} −F_{U_{1}}), excision and (5.6-5.7). □ 9.10. Corollary.

(a) H_{n−1}(F_{s}^{0}) = 0, hence by (5.3) F is (n − 1)–connected;

(b) rank H_{n}(F_{s}^{0}) ≤ 1, hence by (5.3) rank H_{n}(F) ≤ 1. Proof. Use the long exact sequence of the pair (F_{s}^{0}, F_{U_{2}}), and (5.8-5.9). □ 5.11. In the sequel we want to show that H_{n}(F_{s}^{0}) is free. Consider the following diagram:

[figure: π_{1}(Σ_{s} − Δ_{s}) = Z; H_{n+1}(F_{s}^{0}, F_{T_{s}}); H_{n}(F_{T_{s}}); H_{n}(F_{s}^{0}); 0; ∂_{1}; H_{n+1}(F_{Σ_{s}−Δ_{s}}, F_{T_{s}−Δ_{s}}); H_{n}(F_{T_{s}−Δ_{s}}); H_{1}(T_{s} − Δ_{s}); H_{2}(Σ_{s} − Δ_{s}, T_{s} − Δ_{s}); π_{2}(Σ_{s} − Δ_{s}, T_{s} − Δ_{s}); π_{1}(T_{s} − Δ_{s}); ∂_{2}; δ; δ′; α; e; ≃; j; h_{1}; h_{2}; G; G′]

Above G and G′ are the corresponding Gysin maps, h_{i} (i = 1, 2) the Hurewicz maps, and e is the excision map. The horizontal lines are exact. 5.12. Assume that Ξ is trivial. Then H_{n}(F_{T_{s}}) = Z (cf. 5.8), and H_{n}(F_{T_{s}−Δ_{s}}) is a torsion group (cf. 5.7). Therefore, ∂_{1} ∘ e = j ∘ ∂_{2} = 0. Since e is an isomorphism, ∂_{1} = 0, hence H_{n}(F_{s}^{0}) = Z. By (5.3), H_{n}(F) = Z, hence the main theorem in this case follows. 5.13. Assume that Ξ is not trivial. Then H_{n}(F_{T_{s}}) = Z_{2} (cf. 5.8). We have to show that ∂_{1} is onto.

Let s_{i} : S^{1} → Δ_{s}, i = 1, …, μ_{Δ}, be embeddings such that [s_{i}]_{i} generates π_{1}(Δ_{s}). (In order to simplify the notations, we do not deal with the base– points.) Now, by the product structure of T_{s} (and the identification of α with (pr_{2})_{∗}, see (5.5)), there exist embeddings s′_{i} : S^{1} → T_{s} − Δ_{s} ( unique up to isotopy) such that pr ∘ s′_{i} = s_{i} and α[s′_{i}] = 0. Then the diagram gives that [s′_{i}] ∈ imδ′, i.e. there exist d_{i} : D_{2} = {real unit 2–disc} → Σ_{s} − Δ_{s} with δ′[d_{i}] = [s′_{i}] (i = 1, …, μ_{Δ}).

Since Ξ is not trivial, there exists at least one s_{i}, such that the monodromy of the fibration pr ∘ π : π^{−1}(T_{s}) ∩ F_{s}^{0} → Δ_{s} is not trivial. Fix such an s_{i}, and the corresponding s′_{i} and d_{i}. We claim that ∂_{1}eGh_{1}[d_{i}] is not trivial in H_{n}(F_{T_{s}}) = Z_{2}, which shows that ∂_{1} is onto. For this it is enough to show that jG′h_{2}[s′_{i}] ≠ 0.

Let S^{1} = im s_{i} and T^{1} = im s′_{i}. By construction pr : T^{1} → S^{1} is a homeomorphism. Denote E = π^{−1}pr^{−1}(S^{1}) and E′ = π^{−1}(T^{1}). Since T^{1} = ∂d_{i}(D_{2}) and D_{2} is contractible, the fibration π : E′ → T^{1} is trivial (this is the very reason why we used the relative homotopy group). Consider the composed map E′ →^{π} T′ →^{pr} S^{1} and denote q = pr ∘ π. Then q : (E, E′) → S^{1} is a fibration with fiber (S^{n}, S^{n−1}) such that q|E′ is trivial, and q : E → S^{1} is not trivial.

Let j_{1} : H_{n}(E′) → H_{n}(E) and j_{2} : H_{n}(E) → H_{n}(F_{T_{s}}) be the natural maps. By the Wang sequence H_{n}(E) ≃ Z_{2} (generated by the fiber S^{n}), therefore j_{2} is an isomorphism. Then (j ∘G′)(h_{2}[s′_{i}]) = (j_{2} ∘j_{1})(G̃[S^{1}]), where G̃ : H_{1}(S^{1}) → H_{n}(E′) is the Gysin map of the (trivial) fibration E′ → S^{1}, i.e. G̃[S^{1}] is the generator of H_{n}(E′) ≃ Z. In particular, it is enough to prove that j_{1} is onto.

5.14. Lemma. Let q : (E, E′) → S^{1} be a (homotopy) fibration with fiber (S^{n}, S^{n−1}) so that q|E′ : E′ → S^{1} is trivial and q : E → S^{1} is not trivial. Consider the homology groups H_{n}(E′) ≃ Z generated by the “fundamental class” [E′], and H_{n}(E) ≃ Z_{2} generated by the “fundamental class “ of a fiber [q^{−1}(∗)] of the fibration q : E → S^{1}. Then the map j_{1} : H_{n}(E′) → H_{n}(E), induced by the inclusion, is onto. Proof. Set S^{n} = {(x_{0}, …, x_{n}) ∈ R^{n+1}; ∑_{i} |x_{i}|^{2} = 1} and the symmetry i : S^{n} → S^{n}, i(x_{0}, …, x_{n}) = (x_{0}, …, x_{n−1}, −x_{n}). Define E = S^{n} × [0, 1]/ ∼, where the equivalence ∼ means: (x, 0) ∼ (i(x), 1) for x ∈ S^{n}. Moreover, set S^{1} = [0, 1]/0 ∼ 1 and define q : E → S^{1} by [x, s] ↦ [s]. Take E′ = {[x, s] : x_{n} = 0} = S^{n−1} × S^{1} ⊂ E. Now, it follows by an easy verification that [S^{n−1} × S^{1}] = [S^{n} × {1/2}] in H_{n}(E), which finishes the proof of the lemma. □

The above discussion shows that if Ξ is not trivial then H_{n}(F_{s}^{0}) = 0, hence by (5.3) F is n–connected. This finishes the proof of the main theorem.

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IMAR, Bucharest, Romania, Current address:

Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA. e-mail: nemethi@math.ohio-state.edu