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Topology of hypersurface singularities with 3-dimensional critical set

In this paper, we prove that the Milnor fibre of a singularity over an i.c.i.s. of dimension 3 has the homotopy type of a bouquet of spheres, provided that the function that defines the singularity has finite extended codimension with respect to the ideal that defines the i.c.i.s.


Introduction
In [12] Milnor introduced the Milnor fibration for any holomorphic germ f : (C n , O) → C and proved that the Milnor fibre is always a CW-complex of dimension at most (n−1). In the case in which f has an isolated singularity at the origin he also proved that the Milnor fibre is homotopy equivalent to a bouquet of (n − 1)-spheres. The number of spheres is equal to the Milnor number µ, which can be easily computed from the equation. If f has non-isolated singularities at the origin the situation is much more complicated. Up to now, the only general result is Kato and Matsumoto bound [9] which asserts that the Milnor fibre is (s − 2)-connected, where s is the codimension of the singular locus in C n . The homotopy type of the Milnor fibre of a general function germ can be very complicated. In fact, by a recent result of the first author [5] for any local analytic set in C m there is a function whose Milnor fibre is homotopy equivalent to the complement of the set in a sufficiently small ball. The class of such spaces is very rich (contains for example the class of complements of hyperplane and line arrangements) and there is a whole theory dedicated to its study. Hence we may not expect to find a simple description of the homotopy type of the Milnor fibre of a general function germ.
It is very interesting to find classes of non-isolated hypersurface singularities for which the homotopy type of the Milnor fibre admits an understandable description from the equation. This paper contributes to a program in this direction. Let I ⊂ O C n ,O be an ideal defining a 3-dimensional i.c.i.s. Σ 0 and let f be a function of finite extended codimension with respect to I (see Section 2 for a definition). Our main results are the following: (1) We prove that the Milnor fibre of f is homotopic to a bouquet of spheres of different dimensions (see Theorem 29). (2) We also compute the number of spheres appearing in terms of the equation (see Theorem 27). Simmilar results for the cases in which Σ 0 is of dimension 1 and 2 were produced by the work of Siersma (see [17] and [18]), Zaharia [22] and Nemethi [14]. If Σ is a hypersurface the result was proved by Shubladze [16] and Nemethi [13]. Actually we formulate the following: Conjecture. The Milnor fibre of a function of finite extended codimension with respect to an i.c.i.s. has the homotopy type of a bouquet of spheres.
Functions of finite extended codimension with respect to an i.c.i.s. are a particular case of I-isolated singularities as defined and studied in [3]. There it was given a bouquet theorem decomposing homotopically the Milnor fibre in a bouquet of several (n − 1)-spheres and an unknown space (Theorem 9.3 of [3]). The results of this paper identify the homotopy type of that space. It would be interesting to generalise this paper to other I-isolated singularities.
Other bouquet theorems in the context of singular ambient spaces were proved by Siersma [20] and Tibar [21].
Let us end with a description of some aplications of this kind of results. The class of singularities studied in this paper shows very surprising phenomena from the equisingularity viewpoint. It has been used in [3] in order to disprove several old equisingularity questions. At the moment of writing the paper [3] some of the Betti number formula contained in this paper were known to the first author. It was this knowledge which lead him to guess the counterexamples contained in [3] (see Section 12 for more detais). We hope that a systematic solution to our conjecture would lead to interesting examples showing other topological phenomena in nonisolated singularities as yet unknown to us.
The structure of this paper is inspired in the classical Picard-Lefschetz theory of isolated singularities and Sierma's generalisation for non-isolated singularities. In this theory, a function is perturbed to split a singular point into several Morse-type singularities (this process is usually refered to as Morsification). Then it is shown that the homology of the Milnor fibre of the original function can be recovered from the Milnor fibres of each Morse-type singularity. Finally, these homologies are computed by a local study of the Morse-type singularities. In Section 2 we use the results of [2] to prove that in our case we can do a process analogous to the Morsification, but instead of obtaining only Morse-type singularities, we will also obtain a non-isolated singularity over the Milnor fibre of the i.c.i.s. Σ. In Section 3 we show that, as in the isolated case, the homology of the original Milnor fibre can be recovered from the pieces of the Milnor fibres contained in small neighbourhoods of the singularities obtained after the deformation. Having done that, the hardest part of the argument is to study the Milnor fibre around the non-isolated singularity obtained after the deformation. This study is done by taking a suitable decomposition of the Milnor fibre of Σ in such a way that the space we want to study fibres naturally over each stratum of this decomposition. Section 5 describes this decomposition, and the following ones study the parts that fibre over the different strata. Sections 8 and 9 show how to glue these pieces to obtain the homology of the Milnor fibre around the deformation of Σ. Finally, we use all these data to recover the homotopy type of the whole Milnor fibre in Sections 10 and 11. The last section describes a distinguished family of functions belonging to the class studied in this paper which already had striking aplications in topological equisingularity.
1.1. Terminology. If X is a subspace of a topological space Y we denote byẊ the interior points of X, and by ∂X the boundary points of X in Y . Given two topological spaces X any Y we denote that they have the same homotopy type by X ≃ Y . We will denote by D δ the closed disc of radius δ in the complex plane and by B ǫ the closed ball of radius ǫ in a complex affine space. The centers of the discs and balls will be clear from the context unless they are explicitly mentioned in the text or in the notation (by B(x, ǫ)). Denote by S k an sphere of dimension k.

Unfoldings
Let I := (g 1 , ..., g n−3 ) define a 3-dimensional i.c.i.s. Σ 0 in C n . Denote by Θ I,e the germs of vector fields tangent to the i.c.i.s. A function f : C n → C is singular at Σ if and only if it belongs to I 2 . As in [15] we define the extended codimension of f with respect to I as c I,e := dim C (I 2 /Θ I,e (f )).
From the deformation viewpoint, functions with finite extended codimension play the same role in the space of functions singular in Σ 0 than isolated singularities in the space of all funtion-germs. A geometric characterisation of germs of finite extended codimension was given in [22] (see [3] for another proof and generalisations): these are germs in I 2 which outside the origin only have either isolated A 1 singularities or singularities of type D(3, p), with p ∈ {0, 1, 2}.
The singularity D(k, p) has the following normal form (see [15]): Given a germ f ∈ I 2 we can express it as a matrix product f = (g 1 , ..., g n−3 )(h i,j )(g 1 , ..., g n−3 ) t with (h i,j ) a symmetric matrix of holomorphic germs of size n − 3. An easy computation shows that the restriction (h i,j )| Σ0 only depends on f . Let G 1 , ..., G n−3 : C n × B → C n−3 be the semiuniversal unfolding of the i.c.i.s. (g 1 , ..., g n−3 ). Its base B is a germ of complex manifold [10]. Given any b ∈ B denote by the mapping corresponding to the parameter value b. In the space SM (n − 3) of symmetric matrices with complex entries we consider the stratification where SM (n − 3, i) is the set of matrices of corank equal to i. Notice that SM (n − 3, i) consists of the set of matrices defined by the vanishing of the minors of size n − 2 − i. It is easy to check that SM (n − 3, i) is of codimension i(i + 1)/2 in SM (n − 3). We consider the unfolding of the function f defined by: .., G n−3 ) ⊂ C n × S. Given any s = (b, (c i,j )) ∈ S we denote by f s : C n → C the function corresponding to the parameter value s, by Σ s the locus V (G 1,b , ..., G n−3,b ) and by Define Σ[i] := H(F ) −1 (SM (n − 3, i)) and notice that we have the obvious equality Figure 1 shows a schematic view of these sets. The function F 0 coincides with f , where 0 ∈ S is the origin of the base of the unfolding.
Let ǫ and δ be radii for a Milnor fibration of f , that is radii such that (1) the central fibre f −1 (0) meets ∂B ǫ ′ transversely in the stratified sense for any ǫ ′ ≤ ǫ, (2) for any t ∈ D δ \ {0}, the fibre f −1 (t) meets ∂B ǫ transversely, (3) the only critical value of f | Bǫ is 0. From [2] and [3] we obtain: There exists a proper closed analytic subset ∆ of S, and a ball B η centred at 0 ∈ S such that for any s ∈ B η ∩ (D \ ∆) we have (1) for any t ∈ D δ the intersection of f −1 s (t) with ∂B ǫ is transversal (in the stratified sense if t = 0).
(2) the critical set of the function f s | Bǫ is the union of Σ s ∩ B ǫ with a finite number of Morse type singularities, whose critical values are pairwise different and different from 0. (3) the set Σ s ∩ B ǫ is smooth (a Milnor fibre of the i.c.i.s. (Σ 0 , O)) and the mapping is transversal to the stratification of SM (n − 3) by corank. In particular Σ[i] s is a manifold of codimension i(i + 1)/2 in the 3-dimensional manifold Σ s ∩B ǫ . Therefore the critical points of f s in Σ s are of type D(3, 0), D(3, 1) or D (3,2). Denote by C and D the critical set and the discriminant of the mapping where pr 2 denotes the projection of C n × B η to the second factor. Then the restriction is a locally trivial fibration with fibre diffeomorphic to the Milnor fibre of f .
Proof. For any s ∈ S the set of points of Σ s ∩ B ǫ where H(f s ) has corank at least 1 coincides with V (det(H(f s )). Denote by c Ix,e ((f s ) x ) the extended codimension of the germ f s at x with respect to the ideal I x defining the germ (Σ s , x). In [2] it is shown that the set of points such that H(f s ) is not transversal to the stratification of SM (n − 3) by corank coincides precisely with the set of points at which c Ix,e ((f s ) x ) is non-zero. Therefore the only point at which H(f ) is not transversal to the corank stratification is the origin if we take ǫ small enough. Thus at any Inspecting the normal form of the D(3, 1) singularity we find that V (det(H(f )), g 1 , ..., g n−3 ) has an isolated singularity at the origin. Now we study the deformation Σ[1] s := V (det(H(f s )), G 1,s , ..., G n−3,s ) ∩ B ǫ as we move in S. Choose s ∈ S \ ∆. Since H(f s ) is transversal to the stratification by corank there is a finite set of points Σ[2] s in Σ[1] s of type D(3, 2) and the rest of the points are of type D(3, 1). The normal form of the D(3, 2) singularity gives that Σ [1] s has an A 1 -type singularity at any point in Σ [2] s . Let a denote the cardinality of Σ s [2]. In the next Lemma we show that a is independent on s.

Lemma 4. The restriction
The topological triviality statements are easy after the normal forms of the D(3, p) singularities.
The space Σ is smooth since it is the product of the total space V of the versal deformation of the i.c.i.s.
We summarise for further reference the main invariants introduced for the function f . Definition 5. Define µ 0 and µ 1 to be the Milnor numbers at the origin of the i.c.i.s. Σ 0 and Σ[1] 0 . For s ∈ S \ ∆ close to the origin we define a to be the cardinality of Σ [2] s and #A 1 to be the number of Morse points of f s .

2.1.
The corank = 2 case. We will need an slightly larger unfolding of f in the particular case in which corank(H(f )(O)) is precisely equal to 2. In that case, after possibly changing the generators of the i.c.i.s we can assume that f is of the form When c I,e (f ) is finite the mapping is transverse to the corank stratification outside the origin. Therefore the origin is an isolated point of the locus where corank(H(f )) is at least 2. Since in this case the corank 2 locus is defined by the vanishing of the n functions h 1,1 , h 1,2 , h 2,2 , g 1 , ..., g n−3 , we have that Σ [2] 0 is a 0-dimensional i.c.i.s. concentrated at the origin. Let a be its length as 0-dimensional scheme. Let Given any finite set K denote by Aut(K) its permutation group. The monodromy action ρ : induces a monodromy action where Sym 2 (Σ[2] s0 ) \ Diag is the second symmetric product of Σ [2] s0 minus its diagonal (that is the set of subsets of cardinality precisely 2). The set of vanishing cycles is a subset of Sym 2 (Σ[2] s0 ) \ Diag. We enlarge the unfolding of f defined in (1) by considering the following one instead of it: F : C n × S → C given by The statements of Theorem 2 and Lemma 3 clearly remain true for this unfolding.

Homology splitting
We will compute the homology of the Milnor fibre using a general method of Siersma [17], [18], [19].
We have chosen radii ǫ and δ for a Milnor fibration of f . In that situation the total space Consider the versal unfolding F : C n × S → C defined in the previous section. Choose a direction in S such that the line l through the origin O of S in this direction has O as an isolated point of ∆. Let D ξ be a disc in l around O only meeting ∆ in O. Consider the associated 1 parameter unfolding .., v r } is diffeomorphic to the Milnor fibre of f . Therefore we are interested in the homology H k (f −1 s (w); Z), which is isomorphic to H k+1 (X s , f −1 s (w); Z) by the contractibility of X s . Consider D 0 , D 1 , . . . , D r a system of disjoint small disks inside D δ centered in 0, v 1 , . . . v k respectively. Choose points t i ∈ ∂D i , and disjoint paths α i joining t 0 with t i . We can take w = t 0 . Define It is clear that G is a deformation retract of D ∆ , and since f s is a locally trivial fibration outside G, we have that ). By excision: ). It is classical from Picard-Lefschetz theory that for any i > 0 we have Now let T be a tubular neighbourhood of Σ s . We can take for example T := (G 1,s , ..., G n−3,s ) −1 (B) for B a small ball around the origin in C n−3 . Taking T small enough and D 0 small in comparison with T we can assume that f −1 s (t) meets the boundary ∂T transversely for any t ∈ D 0 . We define (7) M := f −1 s (t 0 ) ∩ T. By this tranversality, and because of the fact that f s has no critical points in f −1 s (D 0 ) \ T , the part of the space f −1 s (D 0 ) that lives outside T can be retracted to f −1 s (t 0 ). This means that the pair (f −1 . We summarise what we have obtained: Proposition 9. Denote the Milnor fibre of f by F f . Let r be the number of A 1 points that f s has in B ǫ for s ∈ S \ ∆ close to the origin of S. Then By construction, T is homotopic to Σ s , which is the Milnor fibre of Σ 0 . We will spend a large part of this paper computing the homology of M.

The Milnor fibre of the D(k, p) singularity
We collect and reprove the following proposition, which follows from [4] and [6].
Proposition 10. The Milnor fibre of the D(k, p) singularity in C n is homotopyequivalent to the sphere S n+p−k−1 .
Proof. Since the D(k, p) in C n singularity is quasi-homogeneous its Milnor fibre is diffeomophic to the global hypersurface X ⊂ C n defined by Hence X is homotopic to the n − p − k-suspension of the hypersurface Y ⊂ C , and hence contractible. We conclude that Y is homotopy equivalent to the unit sphere S 2k−1 in C k . Consequently X is homotopic to the sphere S n+k−p−1 .
is homotopy equivalent to the pair (S n−1 , S n−2 ) with S n−2 embedded in S n−1 as the equator.
Proof. By suspension it is enough to consider n = 2, and in this case it is obvious, since the Milnor fibre of x 1 x 2 2 projects to the double cover of C \ {O} by the projection pr(x, y) = x and the Milnor fibre of f | H1 is a fibre of this projection.
Consider the D(3, 2) singularity Recall from Notation 1 the definition of Σ[i]. As here we are considering no unfolding we have the equality Σ = Σ 0 . The restriction of f to any (n − 3)dimensional transversal to Σ[0] is a Morse type singularity in C n−3 . The restriction of f to any (n − 2)-dimensional transversal to Σ[1] is a D(1, 1) singularity in C n−2 .
The stratum Σ [1] is equal to V (det(H(f )) \ {0} and hence is homotopic to its link L ǫ := Σ[1] ∩ S ǫ at the origin. Since the singularity is homogeneous we can take ǫ = 1 and denote L ǫ by L. This link is diffeomorphic to RP 3 , since the surface be the holomorphic normal bundles of Σ and Σ [1] in C n respectively. We have the inclussion of restrictions compatible with the bundle maps.
Observe that, since in this case Σ is a 3-dimensional coordinate subspace, the first bundle is trivial with fibre C n−3 . Notice that, as L is compact, there is a positive ρ such that the ρ-neighbourhood of the zero section of the restriction embedds in C n holomorphically on each fibre. We denote by For any y ∈ L the restriction of f to the fibre of the embedded normal bundle Since the restriction of the function det(H(f )) to Crit(y) is non-singular at the point y for any y ∈ L, for u small enough the intersection Crit(y) ∩ det(H(f )) −1 (u) is a unique point for any y ∈ L, and hence defines a cross-section of the embedded normal bundle, and the restriction be an holomorphic embedding of a ρ-neighbourhood of the zero section of the restriction to Ξ u of the normal bundle of Σ in C n . It is a trivial bundle with fibre a complex n − 3 dimensional ball. For any x ∈ Ξ u the restriction of f to the fibre (N ρ (C n , Σ)| Ξu ) x is a Morse type singularity. If ρ and u are chosen small enough we may assume that, for any x ∈ Ξ u , we have an inclussion of fibres . Now we study the restrictions of the bundle maps κ ρ and κ ρ 1 to a fibre of f −1 (δ) for small δ. Define is a locally trivial fibration of pairs with fibre homotopic to (S n−3 , S n−4 ), with S n−4 embedded in S n−3 as an equator and whose monodromy is isotopic to the identity in S n−4 and is the reflection over the equator in S n−3 .
Proof. The statement of the homotopy type of the fibre is just Lemma 11.

A decomposition of M
In Proposition 9 it has become clear that in order to understand the homology of the Milnor fibre of f we need to compute the homology of the intersection M = f −1 s (t 0 ) ∩ T , with s ∈ S \ ∆ small enough and t 0 = 0 small enough. The tubular neighbourhood T is the total space of a trivial fibration (9) π : By Theorem 2, for a generic parameter s close to the origin of the base S of the unfolding of f , the maximal corank of H(f s )(x) is two for any x ∈ Σ s . Recall that the set of points where the corank is at least 1 is the surface Σ For each point p i let B i (ζ) be a ball of radius ζ around p i in C n such that f s | Bi(ζ) is biholomorphic to the restriction of the singularity D(3, 2) to the unit ball of C n . Taking ζ small enough we can assume that the balls are mutually disjoint and that the intersections A i (ζ) := B i (ζ) ∩ Σ s are balls in Σ s centered in each of the points p i . Taking T , ζ, s, and t 0 small enough the space (10) A ) is diffeomeorphic to the Milnor fibre of f s at p i for any i, and hence homotopy equivalent to S n−2 .
We take ζ small enough so that ∂A i (ζ ′ ) is transverse to Σ[1] s for any 0 < ζ ′ < ζ. We choose ζ 0 < ζ sufficiently close to ζ so that the inclussion A schematic picture of this decomposition can be seen in Figure 2.
is a locally trivial fibration with fibre diffeomorphic to the Milnor fibre of the D(1, 1) in C n−1 , and hence homotopy equivalent to S n−3 . For any ξ ′ > 0 we define the complement of a tube around Σ[1] s in Σ s , and the piece of M lying over it. For T and t 0 small enough the restiction is a locally trivial fibration with fibre diffeomorphic to the Milnor fibre of the Morse singularity in C n−3 , and hence homotopic to S n−4 . We fix a positive ξ 0 smaller and close to ξ and define: The restriction The mapping is a diffeomorphism. Hence the mapping is a locally trivial fibration of pairs with fibre the pair (S n−3 , S n−4 ), being S n−4 embedded as an equator of S n−3 .
6. The topology of B 0 6.1. The fundamental group of B 0 . The space SM (k) of symmetric matrices of size k with complex coefficients is a complex vector space of dimension k(k + 1)/2. The smooth locally closed algebraic subset SM (k, l) has codimension l(l + 1)/2, and we have seen that its Zariski closure SM (k, l) is defined by the vanishing of all l × l minors. It is easy to check that SM (k, l) is far to be, in general, a complete intersection.
Proof. The space of matrices k × (k − 1) which are not of maximal rank is of codimension at least 2 in a complex vector space.
The mapping Lemma 14. The fiber F k has two connected components.
Proof. We will work by induction over k ≥ 2. For k = 2, it is a direct computation. Now let us compute the fibre F k . Consider the following matricial equation: Let v i be the vector in C k−1 given by the i-th row of (m i,j ). Denote by Re(v i ) and Im(v i ) its real and imaginary parts respectively. Now the previous matricial equation becomes the following system of vector equations: if (i, j) = (k, k) then Obviously B k is the set of vectors v 1 satisfying the above system of equations for i = j = 1. The vector v 1 belongs to B k if and only if ||Re(v 1 )|| 2 is one unit longer than ||Im(v 1 )|| 2 and both vectors are orthogonal. That is, the vector Re(v 1 ) can be anywhere except in the interior of the unit sphere in R k−1 . If ||Re(v 1 )|| 2 equals 1 then the vector Im(v 1 ) is zero. In any other case, the vector Im(v 1 ) lies in the k − 2-sphere of radius 1 − ||Re(v 1 )|| 2 in the hyperplane orthogonal to Re(v 1 ). It is easy to show that B k admits the unit sphere in R k−1 embedded in the real part of C k−1 as a deformation retract.
The fiber τ −1 k ((1, 0, . . . , 0)) is equal to the fiber F k−1 of α k−1 over A ′ where A ′ is the result of deleting the first row and the first column in A.
We have constructed a fibration of F k over a space with the homotpy type of S k−2 whose fibre is F k−1 . If F k−1 has two connected components and k ≥ 4, the homotopy exact sequence of the fibrations gives the result. For k = 3 we have to check that the monodromy of the fibration does not interchange the two connected components of F 2 , but this is direct computation.
Proof. This is just the homotopy exact sequence of the fibration α k .
Proof. The unfolding with s ∈ C n−3 × SM (n − 3) can be obtained by pullback from the unfoldings of f that we considederd in Section 2 both in the corank(H(f )(O) = 2 and corank(H(f )(O) = 2 cases. In both cases a generic parameter s ∈ C n−3 ×SM (n−3) maps to a parameter outside the discriminant ∆. Thus we can use this unfolding in order to compute the topology of B 0 .
The mapping is obviously a submersion wherever it is defined. Define Since SM (n − 3, i) is a cone for any i, and the fundamental group of SM (n−3, 1) is isomorphic to Z 2 , we have that the local fundamental group of the germ SM (n−3, 1) at the origin is Z 2 . Since the mapping α is a submersion, the local fundamental group of (Z 1 \ Z 2 ) at the origin is Z 2 . Fix a positive ǫ and a generic sufficiently close to the origin. Consider set of functions . Applying Hamm-Lê Theorem (Main Theorem II.1.4 of [7]) repeatedly for the above set of functions, and using the relative homotopy exact sequence we get that the fundamental group of is isomorphic to Z 2 . But it is clear that the above space is homotopic to B 0 .
6.2. Homology of B 0 . We will now compute the homology of B 0 , which is the same as B. Given the function we use the Mayer-Vietoris sequence of the decomposition of det(H(f s )) −1 (D ξ ) as the union of ∪A i (ζ) and B given in Section 5.
, which is homotopic to a bouquet of (µ 1 − a) 2-spheres (see Definition 5). This is because , which is an i.c.i.s. with Milnor number µ 1 , and det(H(f s )) −1 (0) has only a Morse-points as singularities.
On the other hand, the intersection of each space A i (ζ) with B is the link RP 3 of a Morse type singularity, and the spaces A i (ζ) are contractible.
This facts, together with the computation of π 1 (B 0 ) allows us to compute the following Mayer-Vietoris sequence: The homology of B with coefficients in Z 2 can be computed analogously, or by using the universal coefficient theorem. We obtain  There are several sphere fibrations involved in the computation of the homology of the Milnor fibre, and we will need to deal with the corresponding Gysin sequences. These are greatly simplified if we are in the case n ≥ 8. The homology of the Milnor fibre can be always deduced (by suspension) from the homology of the Milnor fibre of a function f : (C n , O) → C with n ≥ 8. We will assume in n ≥ 8 whenever is needed.
Consider the fibration B u → B u . As we have seen previously, it is a fibration, with fiber homotopically equivalent to S n−4 . This fibration can be extended to h −1 s (u), which is simply conected, and hence, the fibration is orientable. Its Gysin exact sequence leads to the isomorphisms: for i = 0, 1, 2, 3. The rest of the homology groups of B u vanish. Consider the projection B → B u . As we have seen before, it is a fibration with fibre homotopically equivalent to S n−3 , and the monodromy reverses the orientation. Since the fibration is not orientable, we can only consider its Gysin sequence with coefficients in Z 2 , which gives the following isomorphisms: for i = 0, 1, 2, 3. The rest of the homology groups of B with coefficients in Z 2 vanish. The fibration of pairs (B, B u ) → B u has as fibre the pair (S n−3 , S n−4 ) with S n−4 embedded as the equator of S n−3 . Its monodromy acts trivally on S n−4 and reverses the hemispheres of S n−3 along the only non-trivial class of π 1 (B u ) ∼ = Z 2 .
In order to compute the homology of the pair (B, B u ) we can simultaneously thicken the equator S n−4 of each fibre to a small collar S n−4 × [−η, η)] in S n−3 . By excision we can remove fibrewise the interior of the collar. We obtain a fibration over B u with fibre two n − 3-disks relative to their boundary, such that the monodromy interchanges them.
Since π 1 (B u ) ∼ = Z 2 , its universal cover is the only connected double cover. The fibration of pairs is then homologically equivalent to the composition of an orientable fibration ϕ : Y →B u of (n − 3)-spheres overB u with the covering map σ. The Gysin sequence of the fibration ϕ gives if k ≥ n − 3 and zero otherwise. The spaceB u is homotopically equivalent to the double coverB 0 of B 0 branched over its a singular points, minus the preimage of these a points.B 0 is a 2dimensional Stein space (for being a branched cover of the 2-dimensional Stein space B 0 ), and hence it has the homotopy type of a 2-dimensional CW-complex. Therefore, H 2 (B 0 ; Z) is free and H 3 (B 0 ; Z) vanishes. Since the singularities of B 0 are of Morse type, and the 2-dimensional Morse singularity is the quotient of C 2 by the action of the group of two elements, the spaceB 0 is smooth. HenceB u is the result of deleting fromB 0 small balls around the a preimages by the double cover of the singular points of B 0 . Using the Mayer-Vietoris sequence we see that such deletion leaves unchanged the homology except in dimension 3, where we obtain a copy of Z for each deleted point. Sumarizing, we get that Since the Euler characteristic ofB u is twice the one of B u , k must be equal to 2µ 1 − 3a + 1.
Its is easy to check that the following diagram is commutative and hence also δ n , is multiplication by 2.
For being σ a covering there is a well defined pull-back mapping in homology It is clear that the map σ * σ * : is multiplication by 2 (the degree of the covering). This, together with the previous commutative diagram, implies that 2H i−1 (B u ; Z) is always in the image of δ i for any i. In view of this and of the long exact sequence of the pair (B, B u ) we obtain that H n−2 (B, Z) can not have p-torsion for p = 2. By the above diagram and connectedness ofB u we have that δ n−3 is an isomorphism.
Using these facts, together with the previous computations of H • (B u ; Z), H • (B; Z 2 ) and H • (B, B u ; Z), plus the universal coefficients theorem allows us to completely determine the long integral homology exact sequence of the pair (B, B u ). : The non-zero lower homology groups are isomorphic to those of B u , which coincide with those of B.

Homology of X
Let X be the union of ∪ a i=1 A i and B. We will now consider the Mayer-Vietoris sequence of this union with coefficients in Z 2 . To do so, we need to compute the groups H • (A i ; Z 2 ) and H • (A i ∩B; Z 2 ), since H • (B; Z 2 ) has already been computed.
The space A i is the Milnor fiber of the singularity D(3, 2), and hence, it has the homotopy type of the sphere S n−2 .
To study the homology of A i ∩ B, we can use the Gysin sequence of the fibration with fibre S n−3 . The groups H i (RP 3 ; Z 2 ) are Z 2 for i = 0, 1, 2, 3, and zero otherwise. We obtain that H i (A i ∩ B, Z 2 ) = Z 2 for i = 0, 1, 2, 3, n − 3, n − 2, n − 1, n, and zero otherwise.
To study the maps ι k : i H k (A i ∩ B; Z 2 ) → H k (B; Z 2 ) induced by inclussion, we will see them as the Gysin lift of the maps i H j (A i ∩ B; Z 2 ) → H j (B; Z 2 ) for j = k or j = k − n + 3. Using the version of the Mayer Vietoris sequence (14) with coefficients in Z 2 , we get easily • ι n and ι 3 are isomorphisms.
• ι 2 is a monomorphism. We need also the following Lemma, whose proof we postpone:

by inclussion is an isomorphism.
With all this facts, we can compute the Mayer-Vietoris sequence: We omit the lower part of the sequence. The non-vanishing remaining homology groups of X are , H 0 (X ; Z 2 ) ∼ = Z 2 .

8.1.
A basis of H n−2 (X ; Z 2 ). Fix a base point x 1 ∈ A 1 (ζ) ∩ B u . Choose paths γ i : [0, 1] → B u such that γ 1 is a generator of the fundamental group of A 1 (ζ) ∩ B u , and γ i connects x 1 with some point x i ∈ A i (ζ) ∩ B u . We choose chains G i ⊂ B such that the natural projection π| Gi is a locally trivial fibration over γ i with fibre diffeomorphic to a S n−3 generating the homology of the corresponding fibre of (ρ| Bu ) −1 •ρ•π| B . Since γ 1 is closed, the chain G 1 is closed with coefficients in Z 2 .
For each i, we choose an (n − 2)-sphere generating H n−2 (A i ; Z). Take a hemisphere K i of such sphere; its boundary ∂K i is an (n − 3)-sphere in A i . The boundary ∂G i consists of two (n − 3)-spheres L 1 and L i , being L i contained in A i . Since A i is homotopic to S n−2 there exists a chain The generators of H n−2 (X ; Z 2 ) are represented by the Z 2 -closed chains Z 1 := G 1 and Z i := K 1 + W 1 + G i + W i + K i . Notice that since the coefficients are in Z 2 we have K 1 + W 1 + C 1 + W 1 + K 1 = G 1 , and so the way of defining the generators is homogeneous. To check that these are really generators we observe that Z 2 , ..., Z a are sent by the connecting homomorphism of the Mayer-Vietoris sequence 17 to the kernel of the first mapping of the (n − 3)-row, and that Z 1 generates the cokernel of the first mapping of the (n − 2)-row.
Lemma 20. Let γ ′ i : [0, 1] → B u be any other path joining x ′ 1 and x ′ i , being x ′ 1 and x ′ i points in A 1 (ζ) ∩ B and A i (ζ) ∩ B respectively. As above we can associate with γ ′ i an element [Z ′ i ] ∈ H n−2 (X ; Z 2 ). We have the equality for a certain m ∈ Z 2 .
Proof. Let α j be a path joining x j and x ′ j for j = 1, i. The product of paths Since the fundamental group π 1 (B u , x 1 ) is isomorphic to Z 2 and generated by γ 1 , the loop γ i α i (γ ′ i ) −1 (α 1 ) −1 is homotopic to mγ 1 for a certain m. After this, the above equality follows easily from the construction of the chains Z i .

8.2.
A system of generators of H n−2 (X i ; Z). To lift the computation to coefficients in Z, we need to compute the integer homology of A i ∩ B. We can do so by computing the long exact sequence of the pair (A i ∩ B, A i ∩ B u ) using the same arguments that we used to compute the long exact sequence of the pair (B, B u ). We obtain: and zero otherwise.
With these data, and the universal coefficients theorem, we can compute the Mayer-Vietoris sequence (17) with coefficients in Z: The non-zero lower homology groups are isomorphic to those of Σ[1] s . We give a sufficient system of generators of H n−2 (X ; Z). The restriction det(H(f s ))| Σs : Σ s → C has isolated critical points. Therefore, taking η so small that the disk D η only contains 0 as critical value of the restriction, the set Σ s is homotopy equivalent to the result of attaching to (det(H(f s ))| Σs ) −1 (D η ) the Lefschetz thimbles associated to the critical points of det(H(f s ))| Σs not contained in the zero level. There are exactly µ 0 + µ 1 − a such Lefschetz thimbles (see [10]). Since the Lefschetz thimbles are 3disks they are attached along 2-spheres to the boundary of (det(H(f s ))| Σs ) −1 (D η ), which is 5-dimensional. Hence, a transversality argument ensures that all the attaching spheres are disjoint. Denote by C 1 , ..., C µ0+µ1−a the Lefschetz thimbles.
We have found a homotopy equivalence which in fact (since we are working with CW -complexes) is a deformation retract.
Since we have a locally trivial fibration we can lift the deformation retract (18) to a deformation retract We will compute the homology of M ′ using a Mayer-Vietoris sequence. By the previous deformation retract we identify the homology of M ′ and M. Denote π −1 (C i ) by C i . Since C i is contractible the fibration over it is trivial, and, hence, C i and π −1 (∂C i ) are homotopy equivalent to C i × S n−4 , and The associated Mayer-Vietoris sequence (with coefficients in Z 2 ) is: for some e ∈ N. We will find out what are the possible values for e. We have given a basis of H n−2 (X ; Z 2 ) in 8.1.

Lemma 21. The composition
is surjective.
Proof. For each of the a singular points p i of det(H(f s )) −1 (0) there is a vanishing cycle E i which is a embedded 2-sphere in det(H(f s )) −1 (u). The parameters s, u, ζ (see Section 5) can be choosen so that The sphere ∂C k can be choosen to be embedded in det(H(f s )) −1 (u) and, after a perturbation, transverse to E i for any i. Let would be a non-zero homology class. We claim the following equality Let us finish the proof assuming this claim. Any Lefschetz thimble C k gives rise to a class [∂C k ] ∈ H 2 (det(H(f s )) −1 (D ξ ); Z). It is easy to check that its image by the connecting homomorphism δ 2 is equal to where ψ i is a generator of H 1 (∂A i (ζ) ∩ B; Z) for any i. The first equality is by connstruction of the connecting homomorphism and the second is true because i b k,i is even and, hence we have the equality b k,1 = a i=2 b k,i in Z 2 . Let α 1 be the first mapping of the 1-row of the sequence (14). Define the isomorphism θ : H n−2 (X , The concidence of the coefficients in the last terms of equations (22) and (23) give the equality τ ([Y ]) = [Z ′ ]. Now we prove the claim. Choose a point x 0 ∈ ∂C k \ ∪ i,j D k,i,j and choose a disk D 0 around it in ∂C k disjoint to the disks D k,i,j . Deform the immersion ι k | D0 so that the embedding of its boundary remains fixed, it meets E 1 transversely precisely at b k points, all different from x 0 , and it is disjoint from E j for any j = 1. After this deformation the intersection ι k | −1 D0 (A 1 (ζ)) consists of b k disjoint disks {D ′ k,i,j } i∈{1,..,a},j∈{1,...,b k,i } (we choose the indexing to make it easy to make a bijection with the disks D k,i,j ).
Choose non-intersecting paths α k,i,j in D 0 \ (∪ i,jḊ ′ k,i,j ) joining x 0 with a point y k,i,j ∈ ∂D ′ k,i,j . Choose non-intersecting paths joining ∂D ′ k,i,j with ∂D k,i,j . For a schematic picture, see Figure 3.  Let Z ′ k,i,j be the chain associated to β k,i,j by the procedure given in 8.1. Adding and substracting S n−2 -hemispheres K k,i,j and K ′ k,i,j for any i, j (see the procedure in 8.1), the chain ∂C k + ∂ψ is shown to be equal to a sum i,j where Y i is a closed chain contained in A i . The (n−2)-row of the sequence (17) shows that, for any i, any class in H n−2 (X ; Z 2 ) supported by a chain contained in A i is a multiple of [Z 1 ]. On the other hand, by Lemma 20 there exists ]. This proves the claim. This means that the only possible values for e are 0 and 1. We will now characterize the cases in which each value is obtained.
Proof. Consider the unfolding given in (1). If corank(H[f 0 ](0)) ≥ 3 there exists a parameter s 0 ∈ S and a point x ∈ Σ s0 such that the germ f s0 at x is right-equivalent to a germ of the form: and this one can be computed by projecting to the variables (y 1 , y 2 , y 3 ). This projection is a fibration over C 3 \ {0} ≈ S 5 whose fibre is contractible for being given by the solutions of a system of linear equations. So M has the homotopy type of S 5 and H 4 (M ) = 0. The general case is a suspension of this one. Hence the Milnor fibre of the germ f s0 at x is homotopic to S n−1 and its (n − 2)-homology vanishes.
with (c i,j ) ∈ SM (n − 3). Observe that this unfolding can be obtained by pullback and localising near x from the unfolding (24). Let B(x, ǫ 0 ) be a Milnor ball for (f s0 ) x contained in the Milnor ball B ǫ of f . If s is generic and very close to s 0 and t is small enough then an inclussion of the Milnor fibre of (f s0 ) x into the Milnor fibre of f .
Since corank(H(f s0 ))(x) ≥ 2, if s is close to s 0 there exists at least a point p i of Σ [2] s contained in B(x, ǫ 0 ). If e = 0, that is, if H n−2 (M; Z 2 ) = 0 then, by Proposition 9 with coefficients in Z 2 there is a S n−2 in A i ⊂ f −1 s (t)∩B ǫ representing a non-trivial homology class in the Milnor fibre of f . But this is impossible because A i is already contained in f −1 s (t) ∩ B(x, ǫ 0 ) and in this space there is no non-trivial (n − 2)-homology. Now we will see that, in the case where corank(H[f 0 ](0)) = 2, the number e turns out to be 1. Recall that A i is homotopic to S n−2 , denote by Z the generator of H n−2 (A i ; Z). Since we have an inclusion i : A i ֒→ M, we need to check that i * (Z) = 0.
Lemma 23. If f is of the form Proof. Let ω be a closed differential form defined in C 2 \ {0} such that S 3 ω = 0 for a sphere S 3 around the origin in C 2 . Consider the map φ : C n → C 2 defined by φ := (g 1 , g 2 ).
Then Ω := φ ⋆ ω is a closed differential form defined over the Milnor fibre of f . The change of variables formula gives the inequality i⋆(Z) φ ⋆ ω = 0 Now let's generalize this argument for the case where the corank is two, but the dimension is higher: Lemma 24. If corank(H[f 0 ](0)) = 2, then e = 1.
We consider the unfolding F defined in (6). Clearly there are parameters s such that the functions (h 1,1,s , h 1,2,s , h 2,2,s , G 1,s , . . . , G n−3,s ) vanish at the origin and form a holomorphic coordinate system around it. In this case the local Milnor fibre of the deformed function F s at the origin has the homotopy type of a (n−2)-sphere.
Let ǫ and δ be radii for the Milnor fibration of f . Let ǫ ′ and δ ′ be radii for the Milnor fibration of f s at the origin. By Theorem 2 we have that F −1 s (δ ′ ) ∩ B ǫ is diffeomorphic to the Milnor fibre of f . Let Z be a cycle in the local Milnor fibre F −1 s (δ ′ ) ∩ B ǫ ′ generating the (n − 2)-homology group. In order to show that e = 1 it is enough to show that the homology class [Z] is nonzero considered in the bigger space F −1 s (δ ′ ) ∩ B ǫ . For this it suffices to find a closed (n − 2)-differential form Ω, defined in F −1 s (δ ′ ) ∩ B ǫ such that Z Ω = 0. In order to define such a form, choose a positive function β : C → R ⊆ C such that β| D(0,η/2) ≡ 0 and β| C\D(0,η) ≡ 1 for a sufficiently small radius η. Now take ω a closed 3-form in C 2 \ {0} that generates the de Rham cohomology in degree 3.
The cycle Z admits the following parametrisation: since the functions (h 1,1,s , h 1,2,s , h 2,2,s , G 1,s , . . . , G n−3,s ) form a holomorphic coordinate system at the origin the vanishing of the first 5 of them defines a germ (M, O) of (n − 5)-dimensional complex manifold at the origin. Let B(0, √ δ ′ ) denote the ball of radius √ δ ′ centered at the origin of R n−5 , being R n−5 the real locus of (M, O). The parametrisation is given by . Now we compute Z Ω using Fubini's theorem: which is nonzero.
Now we can prove easily Lemma 19 using an example: Proof of Lemma 19. Since we are with coefficients in Z 2 , if the mapping is not an isomorphism then it is identically zero. The function f : C 5 → C given by has finite extended codimension with respect to (x 1 , x 2 ). By a procedure simmilar to the one we have used to compute the homotopy type of the Milnor fibre of the D(3, 2) singularity, in [4] we have shown that the Milnor fibre of f at the origin is homotopy equivalent to S 3 . If we take a generic parameter s of the unfolding F associated to f in Section 2 we see that F s has no Morse points outside Σ s = Σ 0 , there are precisely 2 points of type D (3, 2), and the Milnor number of the i.c.i.s. Σ 0 ∩ {det(H(f )) = 0} is equal to 3. Let us assume that the mapping in the statement of Lemma 19 is identically zero. In this case the previous long exact sequences can be used to compute the homology of the Milnor fibre of f , and they give that the 4-homology group is non-zero. This gives a contradiction. 9.2. Integral coefficients. From the integer homology of X , it is easy to see by the Mayer-Vietoris sequence that H k (M; Z) = H k (X ; Z) for k = n − 1, n − 2.
On the other hand, the group H n−1 (M; Z) is torsion free since M is a (n − 1)dimensional Stein space. By the Universal Coefficients Theorem and our computation of homology with coeffficients in Z 2 , it is easily obtained that H n−2 (M; Z) has no 2-torsion: as we have seen before, when e = 1, the Z 2 component of H n−2 (M; Z 2 ) is represented by a torsion free class (its integral against a closed form is non-zero), and hence it comes from a Z component in H n−2 (M; Z).
Summarising, the Mayer-Vietoris sequence with coefficients in Z is as follows: where T is a torsion group without 2-torsion. We prove now that T = 0. We have to deal separatedly with the cases corank(H(f )) ≥ 3 and corank(H(f )) = 2.
Proof. Let F : C n × S → C be the unfolding associated with f in Section 2. By Theorem 2 there is a monodromy representation ρ : π 1 (S \ ∆) → Aut(H n−2 (F −1 s (δ) ∩ B ǫ ; Z)). By Lemma 4, if one of the generators of the form S i of H n−2 (X ; Z) maps to zero in H n−2 (F −1 s (δ) ∩ B ǫ ; Z), then every other generator of the form S j maps to zero too. In the proof Lemma 22 we have seen that this is the case. By Homology Splitting we conclude that any S i is zero in H n−2 (M; Z). Now let z ∈ H n−2 (M; Z) be a p-torsion element with p = 2. Then pz = 0, which means that, considered with coefficients in Z 2 its class [z] ∈ H n−2 (M; Z 2 ) must be also zero. This means that z is homologous to z ′ = i 2a i Z i . But from the exactness of the sequence we get that 2Z i can be expressed as a linear combination of the S i 's (recall that the S i are the images of the generators of H n−2 (A i ; Z), and that the Z i corresponds to the generators of the kernel of ⊕ i H n−3 (A i ∩ B; Z) → H n−3 (B; Z), which is isomorphic to Z a−1 2 ). We can finally conclude that z ′ can be expressed as a sum of some S i 's, but as we have seen before, all of them are zero in H n−2 (M; Z). Proof. Let z ∈ H n−2 (M; Z) be a p-torsion element with p = 2. Then we have the following equality with coefficients in Z 2 : As before, this means that homologically, z can be expressed as z = i a i S i . Assume that all S i are equal in H n−2 (M; Z). We would have that, integrating against the form Ω of Lemma 23 and Lemma 24 (normalizing it if necessary) we get which, by the hypothesis of z being of p-torsion, means that i a i = 0, and, hence, that [z] = 0.
We only need to prove that S i and S j represent the same class in H n−2 (M; Z) for any i, j.
If the functions The fact that the functions (27) have Milnor number 1 at the origin implies that at least n − 1 of them must be linearly independent variables (after a suitable change of coordinates). After this it is easy to see that we can restrict ourselves to one of the following cases that we will list and analyse below. In this analysis we will use repeatedly the following fundamental fact, which is clear from Homology Splitting and from Section 5: The list of cases is the following: Case 1. Suppose f = (g 1 , g 2 ) · g 3 g 4 g 4 g 5 · g 1 g 2 with g 1 , g 2 , g 3 , g 4 independent variables. In this case, we can take coordinates such that g i = x i for i = 1, . . . , 4, and g 5 = ax 3 + bx 2 5 + φ, being φ a sum of higher order terms. Consider the following family of functions: It is clear that f 1 = f . For any t the singular set Σ is smooth, the set Σ [1] is the surface given by the suspension of two smooth branches with intersection multiplicity equal to 2, and the set Σ [2] is just the origin. After a perturbation the triple (Σ, Σ [1], Σ [2]) becomes a triple which has the topology of (C 3 , V (z 1 (z 1 + z 2 2 − 1) + z 2 3 , V (z 1 , z 2 2 − 1, z 3 )) independently of t. Moreover in the generic perturbation there are no A 1 points appearin outside Σ for any t. Therefore, by Fact 1 in order to compute the homology of the Milnor fibre we may assume t = 0. Write Since it is quasi-homogenous, we can take infinite Milnor radius and we are reduced to compute the homology of: , we see that there exists a preimage if and only if (x 2 1 + ax 2 2 , x 1 x 2 , bx 2 2 ) = (0, 0, 0), that is, everywehere except in the point (x 1 , x 2 ) = (0, 0). It can be easily checked that the fibre over each point is contractible, and hence the Milnor fibre F f0 has the homotopy type of C 2 \ {0} ≈ S 3 .
Case 2. Suppose f = (g 1 , g 2 ) · g 3 g 4 g 4 g 5 · g 1 g 2 with g 1 , g 2 , g 3 , g 5 independent variables. We can write f = ( being φ again a sum of higher order terms. After an appropriate change of basis in x 1 and x 2 we get which falls into the previous case. Case 3. Suppose f = (g 1 , g 2 ) · g 3 g 4 g 4 g 5 · g 1 g 2 with g 1 and g 2 are not linearly independent variables. After a change of base, we may assume that f is of the form x 1 q where q has a Taylor development starting by a generic cuadric. Like in Case 1, using Fact 1 and an apropiate family f t , we may reduce the to the case in which The triple (Σ, Σ [1], Σ [2]) and its deformations (Σ s , Σ[1] s , Σ[2] s ) when we move s in the base S of the unfolding are always contained in the hyperplane x 1 = 0. We restrict to this hyperplane and forget the variable x 1 for the rest of the analysis of this case.
In this hyperplane, the i.c.i.s. Σ is given the hypersurface q = 0, and the singular locus of det = x 3 ·x 5 − x 2 4 is the x 2 -axis. When we consider the Milnor Fibre q −1 (s), it intersects the x 2 axis in two points. This two points correspond to two vanishing cyles S 1 , S 2 in the Milnor Fibre F of Σ[1] = V (x 1 , q, det). Each vanishing cycle S i corresponds to a point p i ∈ Σ [2] s , which gives a class S i in H 3 (M; Z). Wee need to prove that these two classes are equal. Running in this particular case the general considerations previously made in order to compute the homology of M, we observe that if we find a vanishing cycle S 3 in F meeting each S 1 and S 2 transversely at a point, we can use it and the fibrations above it, in order to express the chain S 1 − S 2 as a boundary.
Case 4. If f is of the form with the linear part of g 6 lineally dependent with x 1 , x 2 , x 3 , x 4 , • If corank(H(f )(0) = 0: Proof. Our computations work for the case corank(H(f )(0)) > 0, if n ≥ 8. In order to remove this restrictions we notice that by Thom-Sebastiani the Milnor fibre of f + z 2 with z a new variable is the suspension of the original Milnor fibre, and that the case corank(H(f )(0)) = 0 was proved by Nemethi in [14]. 11. The homotopy type of the Milnor fibre Proposition 28. The Milnor fibre F f is simply connected if corank(h i,j (0)) = 0.
Proof. For n ≥ 6, the Kato-Matsumoto bound [9] tells us that F f is simply connected. For the case where n = 5, we will need the following reasoning. Let Z 1 , ..., Z #A1 be representatives of the vanishing cycles of F f corrsponding to the A 1 points that appear outside Σ s in a generic deformation. Let C(Z i ) denote the cone over Z i . Let C(π) be the cylinder of the mapping π : M → Σ s . The space C(π) is simply connected because it admits the simply connected space Σ s as a deformation retract.
By construction we have that is homotopy equivalent to the contractible space X s (see Section 3). Since each Z i is homeomorphic to S 4 , by Seifert-Van Kampen theorem, the gluing of the C(Z i ) has no efect over the fundamental group, since both π 1 (C(S 4 )) and π 1 (S 4 ) are trivial. The same reasoning tells us that, if π 1 (M) is trivial, so must be π 1 (F f ). The space M is obtained from X by gluing the preimage by π of several Lefschetz thimbles. These pieces are topologically D 3 ×S 1 glued along S 2 ×S 1 . By Seifert-Van Kampen theorem, if π 1 (X ) is trivial, the adition of these pieces does not change the fundamental group. So, to prove that π 1 (M) = 0 it is enough to prove that π 1 (X ) = 0.
We may compute π 1 (X ) using Seifert-Van Kampen with the decomposition In Section 5 it is shown how the mapping π allows to express each of the pieces of the decomposition as fibrations with fibres homotopy spheres of dimension at leat 2 over the corresponding piece of the decomposition Using this it is easy to see that the computation of π 1 (X ) by Seifert van Kampen mimics the computation of π 1 (Σ s ∩ det(H(F s )) −1 (0)), but this space is simply connected (in fact a bouquet of 2-spheres).
We now have all the necessary ingredients to prove our Bouquet Theorem.
Theorem 29. The Milnor fibre of a singularity over a 3-dimensional i.c.i.s. with finite extended codimension has the homotopy type of a bouquet of spheres of different dimensions.
Proof. For the previous Proposition, we know that the Milnor fibre is simply connected.
In the case where corankH(f )(O) ≥ 2 (that is, a = 0) we have computed the integer homology, getting that H n−1 (F f ; Z)) and H n−2 (F f ; Z) are free and finitely generated and H i (F f ; Z) ∼ = 0 otherwise. In this situation, since the Milnor Fibre has the homotopy type of a (n−1)-complex, we can apply Criterion 2.2 and Remark 2.3 of [14] and we get the result.
We are left with the case in which corank(H(f )(O) = 1. By Criterion 2.2 in [14], we only need to represent each generator of the non-zero homology groups by a chain modelled in a sphere. When corank(H(f )(O) = 1, in the decomposition of M given in Section 5 we have that B coincides with X , that B u is diffeomorphic to B 0 , which are Milnor fibres of the 2-dimensional i.c.i.s. Σ 0 ∩ V (det(H(f ))) and that the fibration (11) becomes a homotopy S n−3 -fibration The generator of H n−3 (F f ; Z) is the Gysin lift of the generator of H 0 (B; Z), and hence it is represented by a sphere. By Homology Splitting, the generators of H n−1 (F f ; Z) come from two different places: the ones comming from the A 1singularities of f s outside Σ s and those coming from H n−1 (M; Z). The first generators are clearly represented by spheres (the vanshing cycles of the A 1 -singularities). The generators of H n−1 (M; Z) come in turn from two different places: the ones comming from the image of H n−1 (X ; Z) in H n−1 (M; Z), and those coming from the addition to X of the spaces C i (see the decomposition formula (21)). Recall that each C i is the product of a Lefschetz thimble associated to a vanishing cycle of B u = {det(H(f s ) = u} ∩ Σ s with the homotopy-sphere S n−4 , which is the fibre of the fibration (19). The ones coming from H n−1 (X ; Z) are Gysin-liftings over the vanishing cycles of B u of the fibration (28).
We claim that the fibration of (n − 3)-spheres over B 0 is trivial. Since B 0 is a bouquet of 2-spheres given by vanishing cycles it is enough to prove that the fibration, restricted to each of the vanishing cycles of B 0 is trivial. Choose a vanising cycle C i . Move the parameter s so that that s is very close to a parameter s 0 in which Σ s ∩ V (det((f s ))) adquires an A 1 singularity to which the vanishing cycle C i collapses. In this situation a local change of coordinates shows that to prove that the fibration is trivial over C i is equivalent to prove that the fibration of (n − 3)-spheres associated to the function x 2 i is trivial over the vanishing cycle of the restriction of x 2 1 + x 2 2 + x 2 3 to V (x 4 , ..., x n ). Proving this is an easy local computation. Now we represent each of the two kinds of generators of H n−1 (M; Z) by spheres. Let us start by the first kind. By the claim the group H n−1 (X ; Z) is generated by chains of the form where τ (S 2 ×S n−3 ) is a Gysin lift of a canishing cycle C i of B 0 by the fibration (28).
Choose a section s of this fibration such that s(C i ) is inside τ (S 2 × S n−3 ). For n = 5, the sphere s(S 2 ) is trivial in H 2 (X ; Z), since this group is generated by the fibre. This implies that it is also zero in H 2 (F f ; Z), and, by Hurewitz's Theorem, it is also trivial in π 2 (F f ). For n > 5 the triviality of s(S 2 ) in π 2 (F t ) holds by the connectivity of the Milnor fibre. This means that s(S 2 × {point}) can be killed by a 3-disc inside F f . By Lemma 4.5 in [14], we have that the homology class [τ (S 2 × S n−3 )] can be represented by a sphere.
We study now the homology classes in H n−1 (F 1 ; Z) coming from a the addition of an space C i . The space C i is the product of a Lefschetz thimble L i associated to a vanishing cycle C i of B u with the sphere S n−4 , which is the homotopy-fibre of the fibration (19). Recall that over B u we have in fact a fibration of pairs with fibre homotopic to (S n−3 , S n−4 ) being S n−4 embedded as the equator of S n−3 . Consider a collar K ∼ = ∂L i × [0, 1] of ∂L i in the 3-cell L i . We deform continuously the chain given by the embedding of L × S n−4 in M so that fibrewise S n−4 is the equator of S n−3 over any point of the internal boundary of the collar and so that S n−4 is collapsed to the north pole of S n−3 at the external boundary ∂L i of the collar. The obtained chain is called ϕ : L i × S n−4 → F f .

The mapping
s : ∂L i → X ⊂ F f which assigns to a point of ∂L i the north pole of the fibre S n−3 has been seen before to be a trivial element in π 2 (F f ). Therefore there exists a 3-disk L ′ bounding ∂L i and an extension s ′ : L ′ → F t of s. The identification L ∪ ∂Li L ′ along their common boundary is a 3-sphere. A representative of our homology class is given by the chain ψ : (L i ∪ ∂Li L ′ ) × S n−4 → F t defined by ψ| L×S n−4 := ϕ and ψ| L ′ ×S n−4 := s ′ •pr 1 , where pr 1 is the projection of L ′ × S n−4 to the first factor. Notice that the source of ψ is a product of spheres, which we view as a trivial fibration of S n−4 over L∪ ∂Li L ′ ∼ = S 3 , and that ψ fatorises through the result of collapsing to a point the fibre over any point of L ′ . Again Lemma 4.5 in [14] represents the homology class by a sphere.

Examples
Despite the apparent simplicity of the homotopy type of the Milnor fibre of the class singularities considered in this paper, it is possible to find among them unexpected topological behaviours which at the moment have not been observed in singularities with smaller critical set. As an illustration of this we summarise here the properties of a family of examples, which fall in the general class studied in this paper, and which was used in [4] to produce counterexamples to several old equisingularity questions.
If ϕ is not identical to 0 the function f ϕ is of finite codimension with respect to the ideal I = (y 1 , y 2 ). The critical set Σ = V (y 1 , y 2 ) is 3-dimensional and smooth. It is easily checked that the I-unfolding where ord(ϕ) denotes the order of the series ϕ in x 1 , is the versal I-unfolding of f ϕ in the sense of [15] and [3]. Hence we can obtain all I-unfoldings of f ϕ by considering deformations of the form ϕ + ord(ϕ)−2 i=0 t i x i 1 .
Notice that the determinant detH(f ϕ ) = x 3 (ϕ(x 1 ) − x 3 ) − x 2 2 : (Σ, O) → C has a singularity at the origin of type A 2ord(ϕ)−1 . An easy computation shows that if (f ϕ ) s is a generic deformation of f ϕ in its versal I-unfolding, the cardinality of the set Σ[2] s of points where H(f 2 ) has corank precisely 2 is equal to ord(ϕ).
It is also easy to check that for any s in the base of the versal I unfolding the critical set of f s is equal to Σ = V (y 1 , y 2 ). Hence there are no A 1 points popping out of Σ in a generic I-unfolding of f ϕ .
Noticing that corank(H(f ϕ ))(O) = 2 we may apply Theorem 27 to show that the Milnor fibre is 2-connected, with third Betti number equal to 1 and fourth Betti number equal to: b 4 = µ 0 + 2µ 1 − 4a + 2 + #A 1 = 0 + 2(2ord(ϕ) − 1) − 4ord(ϕ) + 2 + 0 = 0, which, surprisingly, is independent of ϕ. By Theorem 29 we conclude that the Milnor fibre of f ϕ is homotopy equivalent to a 3-sphere. The remarkable fact is that the homotopy type of the Milnor fibre is independent on ϕ and at the same time the topology of the pair of germs In [4] it is shown that in fact the diffeomerphism type of the Milnor fibration of the germ f ϕ and the generic Lê-numbers are independent of ϕ. Using that the topology of the pair (30) depends on s it is also proven that the topology of the abstract link of f ϕ does depend on ϕ. This kind of examples and their stabilisations are at the moment the only known families of examples with constant Lê numbers and constant Milnor fibration and changing topological type. They answer negatively a question of D. Massey in [11]. In [4] modifications of these examples are also used to give the first known counterexample of Zariski's Question B of [23]. Also in [4] these examples were used to construct a family of reduced projective hypersurfaces with constant homotopy type and changing topological type (therefore most classical algebro-topological invariants can not detect the change in topology).