Cohomological connectivity of perturbations of map-germs

Let $f\colon (\mathbb C^n,S)\to (\mathbb C^p,0)$ be a finite map-germ with $n<p$ and $Y_\delta$ the image of a small perturbation $f_\delta$. We show that the reduced cohomology of $Y_\delta$ is concentrated in a range of degrees determined by the dimension of the instability locus of $f$. In the case $n\geq p$ we obtain an analogous result, replacing finiteness by $\mathcal K$-finiteness and $Y_\delta$ by the discriminant $\Delta(f_\delta)$. We also study the monodromy associated to the perturbation $f_\delta$.


Introduction
In this paper we establish bounds for the vanishing cohomology for images and discriminants of map-germs with non-isolated instabilities.As will become clear, these results are parallel to the classical bounds of Kato and Matsumoto on the cohomology of the Milnor fiber of non-isolated hypersurface singularities [KM75].
It is also our intention to illustrate that perverse sheaves are a powerful tool in the study of singular mappings.For the reader who is unfamiliar with the machinery of perverse sheaves, [Dim04] provides a quick introduction.Here we have included only the information that is necessary for our applications, and have avoided technical definitions.For those who are already well-versed in the topic, some non-trivially perverse sheaves related to the alternating cohomology of multiple points are introduced in Section 5.The perversity of these sheaves was discovered by Houston [Hou00] .
In this paper, all homology and cohomology groups are assumed to have complex coefficients, unless otherwise specified.origin in C and of radius δ > 0 with δ sufficiently small with respect to r.This gives rise to a locally trivial fibration g : B r ∩ g −1 (D * ) → D * , called the Milnor fibration, whose generic fibre is known as the Milnor fibre of g at 0 ∈ C n+1 .Milnor showed [Mil68] that if g −1 (0) has an isolated singularity at 0, then M is homotopy equivalent to a bouquet of n-dimensional spheres.In particular, the reduced cohomology of M -that is the vanishing cohomology of g at 0-is concentrated in the middle degree.Later, Kato and Matsumoto [KM75] established their results for functions with non-isolated singularities: If the critical locus Σ(g) has dimension d, then the Milnor fiber M is at least (n − d − 1)-connected.Consequently, the reduced cohomology Hp (M ) is concentrated in the range of degrees n − d ≤ p ≤ n.This concentration of reduced cohomology is what we refer to as cohomological connectivity.
Besides the cohomological version of Kato's and Matsumoto's connectivity result, we wish to also study the monodromy transformations.Since our setting will be slightly different from that of the Milnor fibration, we briefly review the general notion of monodromy for a topologically locally trivial fibration with fiber F E π −→ D * over a punctured disk D * .Let exp : S → D * be the universal cover of D * by an infinite strip S. Since S is simply connected, we may choose a global trivialization of the the fiber product and deliberately identify the fiber F δ = π −1 (δ) over a point δ ∈ D * with any fiber of E over a point δ mapping to δ ∈ D. The chosen global trivialization of E over S furnishes a notion of parallel transport of the fiber F over D * which is unique up to homotopy and independent of the choices made.The monodromy operator is defined to be the homeomorphism h : F → F obtained from parallel transport along a closed loop in D * passing once counterclockwise around the origin.It follows that the induced maps on cohomology are well defined automorphisms.
Let us recall the classical monodromy theorem in the Milnor fibration setting, see [Dim92, Theorem 3.1.20]: Theorem 1.1.Let h i : H i (M ) → H i (M ) be the i-th monodromy operator on the Milnor fibre associated to a germ g : (C n+1 , 0) → (C, 0) (not necessarily with isolated singularities).Then (1) the eigenvalues of h i are roots of unity.
(2) the size of the Jordan blocks of h i is at most i + 1.
It is worth noting that the first item is valid in the more general setting of the fibration induced by any analytic map-germ g : (X, 0) → (C, 0), where X is an arbitrary analytic space [Lê78].
1.2.The fibration associated to an unfolding.Rather than looking at germs of hypersurfaces, we will be looking at singularities that arise from multigerms of mappings f : (C n , S) → (C p , 0).
Here S is a finite subset of C n , mapped to 0 by f .Moreover, we always assume f to be K-finite (for its definition, see Definition 2.1).
Whenever p > n, our attention will be directed towards the image of f , denoted by (Y, 0) = (Im f, 0) ⊂ (C p , 0).For obvious reasons, the analytic space (Y, 0) is sometimes called a parametrizable singularity.
Whenever n ≥ p, K-finite mappings are surjective and there is no interest in studying the image of f in this case.The attention is directed towards the discriminant instead: the critical locus of f is the germ and the discriminant of f is defined to be the image Observe that the study of discriminants comprises the case of parametrizable singularities in the sense that, for p > n, the differential cannot be surjective and hence Σ(f ) = C n and Y = ∆(f ).
Let us summarize how the classical Milnor fibration is replaced by the fibration determined by a one-parameter family in the context of map germs: A one-parameter unfolding F = (f t , t) of f is a germ of the form F (x, t) = (f t (x), t) and such that f 0 = f .In the case n < p, the projection from the image Y = Im F to the parameter space gives a fibration, the generic fiber of which is Y δ , the image of a perturbation f δ .In the case n ≥ p we consider the fibration defined on the discriminant ∆(F ) whose generic fibre is ∆(f δ ).These fibrations depend not only on the map germ f , but also on the chosen unfolding F .Special attention is paid to the case where the perturbations f δ are stable.In this case, the generic fibre Y δ (or ∆(f δ )) plays a role closer to that of the classical Milnor fibre.In the case n < p, the image Y δ of a stable perturbation is called a disentanglement of f .For the precise definition of these terms see Section 3.1.
Unfoldings play the role of deformations and are still local objects.To define perturbations-the nearby objects-we need a well chosen representative.It is customary to absorb all associated technicalities in the definition of a good representative.Definition 1.2.Let V ⊂ C p and T ⊂ C be open neighborhoods of the origin and a representative of a one parameter unfolding defined on an open subset W ⊂ C n × C. We call F a good representative if it is a K-finite map and moreover satisfies the following conditions: (1) the family π : ∆(F ) → T is a locally trivial fibration over T \ {0}, (2) the central fiber π −1 (0) is contractible, (3) the space ∆(F ) retracts onto the central fiber.For any fixed nonzero value δ ∈ T \ {0} in the parameter space of a good representative the map With no risk of confusion, we also write f for the representative f 0 : W 0 → V .
Remark 1.3.A good representative of a K-finite germ can be obtained from an arbitrary representative F : U × T → V × T as follows: The discriminant ∆(F ) in V ×T is a closed complex analytic set and the projection π : V ×T → T a holomorphic function on it.The Milnor-Lê fibration asserts that for sufficiently small ball B r ⊂ V around the origin and a subsequently chosen disc D δ ⊂ T with r δ > 0 the restriction is a map satisfying the three properties mentioned above.Now the good representative is furnished by choosing B r × D δ small enough such that the restriction

and finally setting
Parametrizable singularities and even their disentanglements are usually highly singular.This is due to the fact that the image of a map f may be singular even if f is stable, that is when f does not admit any non-trivial unfoldings; see the examples below.Thus, the singular locus of (Y, 0) is not well suited to be the analogue of the critical locus of g from the classical Kato-Matsumoto result.Instead, we will be considering the instability locus which is an analytic subset of the discriminant of f (see Section 3.1).The bound on the vanishing cohomology of the disentanglement will be given in terms of the dimension d of Inst(f ).We give some examples to illustrate the situation.
Example 1.4.Consider the function g : C 3 → C, (x, y, z) → xy and the associated hypersurface X = g −1 (0).Since z is a dummy coordinate, the Milnor fiber M g (0) has the homotopy type of a circle.This is consistent with Kato and Matsumoto's theorem, because the critical locus of g, and hence also the singular locus of X, has dimension one, see Figure 1.
The situation changes when we think of (X, 0) as a parametrized singularity given by a bi-germ  with the two obvious branches, see Figure 2.
This map germ is known as a "transverse double point".It is stable, meaning that it cannot be perturbed by any unfolding, up to analytic isomorphism.Therefore, a sufficiently small representative of the image of f coincides with its disentanglement.We see that, unlike the Milnor fiber M g (0) of g, the disentanglement is a singular space with two smooth branches crossing transversally and it has trivial reduced cohomology.
Example 1.5.We will now consider map germs f with isolated instability, i.e. those for which d = dim Inst(f ) = 0.This property turns out to be equivalent to f being finitely determined [Mat68].The families S k , B k and H k of Mond [Mon85] are examples of such finitely determined germs (C 2 , 0) → (C 3 , 0).Here we consider the germ (x, y) → (x 2 , y 2 , x 3 + y 3 + xy), depicted in Figure 3.
Mond showed that the disentanglement of a finitely determined germ (C n , 0) → (C n+1 , 0) has the homotopy type of a bouquet of n-dimensional Figure 3.A germ f with isolated instability and its perturbation f δ .The red thick dot at the vertex of the cone represents the instability locus Inst(f ).
A similar connectivity result holds for discriminants of perturbations of finitely determined map-germs f : (C n , 0) → (C p , 0), when n ≥ p [DM91].The discriminant Σ(f δ ) has the homotopy type of a bouquet of spheres of dimension p − 1.
It is not finitely determined, because it has a line of instabilities.Maps with non-isolated instability locus may admit more than one disentanglement: The cuspidal edge can, for example, be perturbed into a cuspidal node (x, y) → (x, y 2 , y 3 − δy) which is stable and has the homotopy type of a circle.Another perturbation of f is (x, y) → (x, y 2 , y 3 + δy(x 2 − δ)).The image of this last one has the homotopy type of a bouquet of twodimensional spheres.
The main point of this article is to show that despite the fact that different unfoldings may lead to different disentanglements with possibly distinct homotopy types, there is always a bound on the degrees of the nontrivial reduced cohomology groups of any given disentanglement.As the three examples above suggest, this bound is not related to the dimension of the singular locus of the image (equal to one for all the germs (C 2 , 0) → (C 3 , 0) 1 The pairs (n, p) of dimensions where every map-germ (C n , S) → (C p , 0) admits a stable perturbation are called Mather's nice dimensions.These comprise all pairs (n, n + 1) with n ≤ 14, cf.[MN20, Section 5.2.2]). in our examples), but it is controlled by the dimension of the instability locus (empty, zero-dimensional and one-dimensional, respectively in the examples).
Before stating our results, we shall introduce a construction that connects the Milnor fibration to disentanglement fibration.Indeed, we will show that the study of the Milnor fibration of germs of hypersurfaces in C n is equivalent to that of disentanglements of bi-germs of immersions (C n , {p, q}) → (C n+1 , 0).
Example 1.7.To any hypersurface X = V (g) ⊆ (C n , 0), not necessarily with isolated singularity, we are going to associate a bi-germ of immersion (C n , 0) (C n , 0) → C n+1 .For convenience, label the two copies of (C n , 0) as U 1 and U 2 , then let f : U 1 U 2 → (C n+1 , 0) be the map of the form A different choice g of a generator of the ideal g gives rise to a different map f , but there is a change of coordinates in (C n+1 , 0) turning f into f .In other words, f and f are A-equivalent and, consequently, the study of their disentanglements is equivalent.Conversely, every bi-germ of immersion (C n , S) → (C n+1 , 0) arises-up to A-equivalence-by this construction: Given such a bi-germ, we can take a change of coordinates turning the second branch into x → (x, 0), and so that the normal vector to the first branch at the origin has a nonzero last coordinate in C n+1 .This makes the first branch locally into a graph, as desired.A more direct way to invert the process is as follows: The two branches of a bi-germ f of immersion are two germs f (1) , f (2) : (C n , 0) → Figure 5.The singularity {xy = 0} ⊆ C 2 and its Milnor fibre can be realized as the intersection of the branches of a bi-germ and its stable perturbation.The singular locus of {xy = 0} corresponds to the instability locus of the bi-germ.
(C n+1 , 0).Take a reduced equation L = 0 for the image of the first branch.Applying the above contruction to the function germ g = L • f (2) : (C n , 0) → (C, 0) gives rise to a bi-germ which is A-equivalent to the original immersion f .
Having explained the construction and its inverse, let us describe the relation between the hypersurface X and the immersion f : First and most obvious, the intersection of the two branches is Moreover, the two branches cross transversally, except on the singularities of X.It is well known that the instabilities of an immersion are located precisely at points where the branches do not intersect in general position (this is a particular case of Theorem 3.3 in [MN20], taking into account that monogerms of immersions are stable).In particular, it follows that The Milnor fibre of X has the form M = g −1 ({δ})∩B r , for suitable choices of r δ > 0. The same discussion about singularities and stabilities shows that a stabilization of f is given by the bi-germ F = (f t , t), with A stable perturbation of f is given by f δ : B r B r → C n+1 and the Milnor fibre M of g is recovered as the intersection of the two branches of f δ .
Our claim that the study of the Milnor fibration of hypersurfaces is the same as the study of disentanglements of bi-germs of immersions also carries over to the vanishing topology in the following sense: Remark 1.8.Let M be the Milnor fibre of a hypersurface X = V (g) and Y δ = Im f δ be the disentanglement of the bi-germ associated to g, as in Example 1.7.Then, there is an isomorphism compatible with the monodromy actions on M and Y δ .In fact, Y δ is homotopy equivalent to the suspension of M and we will come back to this isomorphism on Example 5.13 and Remark 5.14 to illustrate the more general approach to the computation of the cohomology of mappings.

Main Results
In this paper, we always assume that f : This condition arises naturally in the study of contact equivalence of singular map-germs [MN20, Section 4.4] and it is generally considered mild.What appears here as its definition is usually regarded as a geometric characterization [MN20, Proposition 4.3].Observe that, for p > n, the condition for a germ f : (C n , S) → (C p , 0) to be K-finite is just that f is finite.
The K-finiteness condition is helpful in at least three ways: First of all, it entails that for dimensions p ≤ n + 1 the discriminant ∆(f ) is a hypersurface in C p (Proposition 3.3).Second, the instability locus of K-finite mappings is analytic (Proposition 3.2).Lastly, K-finiteness ensures the existence of stable unfoldings which are used in the definition of multiple point spaces in Section 5.
Theorem 2.2.Let f : (C n , S) → (C p , 0) be a K-finite map-germ, with p ≤ n, and let d = dim(Inst(f )).The reduced cohomology of the discriminant of any perturbation f δ satisfies The previous result also holds for disentanglements, as they are particular cases of perturbations.
Theorem 2.3.Let f : (C n , S) → (C n+1 , 0) be a finite map-germ and let d = dim(Inst(f )).The reduced cohomology of the image of any perturbation In the case of germs f : (C n , S) → (C n+1 , 0), apart from the image Y , it is common to study the source double point space D(f ) ⊆ C n .For a map f : X → Z between complex manifolds, it consists of the set of points x ∈ X such that the germ f : (X, Details on the analytic structure of D(f ) are found in Section 4. For now, it is enough to know the following: Whenever f : X → Z is a finite mapping between complex manifolds with dim Z = dim X + 1 and Theorem 2.4.Let f : (C n , S) → (C n+1 , 0) be a finite map-germ and let d = dim(Inst(f )).The reduced cohomology of the source double point space of any perturbation f δ satisfies Remark 2.5.Just as the disentanglements and discriminants of Example 1.5, the source double points space D(f δ ) of a finitely determined map germ f : (C n , S) → (C n+1 , 0) has the homotopy type of a bouquet of spheres of dimension n − 1.This observation is due to R. Giménez Conejero and J.J. Nuño-Ballesteros [GN20].
Example 2.6.Let us come back to the bi-germ of immersion associated to a hypersurface X ⊆ C n , described in Example 1.7.Since each of the branches f | U i is injective and immersive, the space D(f ) is the preimage by f of the intersection of the the two branches Therefore, the source double point space consist of two disjoint copies The same argument shows that the double point space of the stable perturbation of f is D(f δ ) = M M where M is the Milnor fibre of X. From this point of view the cohomological version of Kato's and Matsumoto's connectivity result is a particular case of Theorem 2.4.The classical connectivity result due to Milnor appears as a particular case of Remark 2.5.
Corollary 2.7.Let f : (C n , 0) → (C n+1 , 0) be a finite map-germ and assume that dim(Inst(f )) < n − 1.Then, the singular locus Sing Y δ of the image of any perturbation of f is connected.
Proof.Theorem 2.4 implies immediately that D(f δ ) is connected.Moreover, from the set-theoretical description of the source double point space given above, it follows that the singular locus of the image Y of a finite and generically one-to-one map f : X → Z between complex manifolds is, as a set, the image of the double point space.In other words, finite and generically oneto-one maps satisfy Sing Y = f (D(f )), which clearly implies our claim.Now observe that, for dimensions dim Y > dim X, stable mappings are generically one-to-one.Therefore, mappings satisfying dim(Inst(f )) < n−1 (hence generically stable) are necessarily generically one-to-one as well.
In order to obtain cohomological connectivity results for germs f : (C n , 0) → (C p , 0) with arbitrary p > n + 1, something stronger than K-finiteness is required: A map f : U → V between complex manifolds of dimensions n = dim U and p = dim V is dimensionally correct if its multiple point spaces D k (f ) are empty or have the minimal dimension kn − (k − 1)p (see Section 5 for the definition of multiple points).
Theorem 2.8.Let f : (C n , 0) → (C p , 0) be a dimensionally correct mapgerm and p > n + 1.Then, for any perturbation f δ , all possibly non-trivial reduced cohomologies Hq (Y δ ) are concentrated in the degrees q satisfying As the following example shows, Theorem 2.8 does not hold if the dimensionally correct hypothesis is omitted.
Similarly, adding a zero coordinate to the perturbation of Example 1.6 gives a perturbation f δ , given by (x, y) → (x, y 2 , y 3 + δy(x 2 − δ), 0), whose image is, obviously, isomorphic to that of the mentioned example.This image has non-trivial cohomology in degree two.If f was dimensionally correct, then Theorem 2.8 would allow for nontrivial vanishing cohomology only in degrees 0 and 1.Indeed, this condition on f is violated since the source double point space D(f ) contains the whole line {y = 0} (f is not immersive there) whereas, for germs (C 2 , 0) → (C 4 , 0), the minimal dimension for double points is 2 Remark 2.10.A first version of Theorems 2.3 and 2.8 was proven in [SZ18], under the extra assumptions that f δ is stable, that f has corank one and-in the case of p = n + 1-that f is a dimensionally correct monogerm.Two examples of map-germs that only the new version can handle are (x, y) → (x 2 , y 2 , x 3 + y 3 + xy) (which has corank two) and (x, y) → (x, y 3 , y 4 ) (which is not dimensionally correct, since it has a line of triple points, but satisfies the hypotheses of Theorem 2.3).
Finally, let us return to the study of monodromy.Let F be a oneparameter unfolding of a finite map-germ f : (C n , 0) → (C p , 0), with p > n.The projection from the image Y of a good representative of F to the deformation parameter restricts to a locally trivial fibration over the punctured disk.Since the disentanglement Y δ is the fibre of the previous fibration, for each i there is corresponding monodromy automorphism On one hand, the fact that f is finite implies that Y is analytic, and thus it is clear that the eigenvalues of h i are roots of the unity, cf.Remark 7.3.On the other hand Y is not smooth, (in fact Y need not be a complete intersection when p > n + 1) and thus the classical result about the size of Jordan blocks does not apply in our setting.As we will see, under the right hypothesis the classical bound for the Jordan blocks and a spectral sequence argument can be combined to bound the size of the Jordan blocks of the monodromy of perturbations.
Theorem 2.11.Let F be a stable one-parameter unfolding of a corank one map germ f : (C n , 0) → (C p , 0) with p > n.Then the size of the Jordan blocks of h i are at most i(i + 1)/2.Moreover, if f has only isolated instability locus and p > n + 1, the possibly non-trivial reduced cohomologies are Hkn−(k−1)(p−1) (Y δ ), for 1 < k ≤ p p−n , and the size of the corresponding Jordan block is at most kn − (k − 1)p + 1.
2.1.Some remarks on generality.In writing this paper, we decided to sacrifice some generality for the sake of clarity.The following remarks list some improvements that ended up being left behind: Remark 2.12.Throughout the text, the instability locus can be replaced by the topological instability locus.Topological stability is defined by using homeomorphisms instead of diffeomorphisms in the definition of trivial unfolding.In general, the topological instability locus is smaller, giving rise to sharper cohomological connectivity bounds.
Remark 2.13.Theorem 2.8 is stated for monogerms, but it applies to multigerms as well.However, the involved representations of the symmetric group made the exposition more complicated and seemed off-topic.
Remark 2.14.The "dimensionally correct" condition of Theorem 2.8 may be relaxed by introducing strict dimensional correctness and adding some extra hypotheses (counterintuitively, strict dimensional correctness is less demanding than dimensional correctness, see Definition 5.2).The cohomological connectivity predicted by Theorem 2.8 also holds in the following situations: (1) The unfolding F underlying f δ is strictly dimensionally correct.
(2) The germ f is strictly dimensionally correct, and Y δ is a disentanglement (i.e.f δ is a stable perturbation).The second situation is, in fact, a particular case of the first, because stable perturbations of strictly dimensionally correct germs arise only from strictly dimensionally correct unfoldings.The reason not to include this improvement is that it depends on the unfolding F , not only on f .Example 2.9 shows that perturbations f δ of a strictly dimensionally correct map-germ f need not satisfy the cohomological connectivity of Theorem 2.8.We thank David Massey, who inspired us to use perverse sheaves as a tool for understanding perturbations of map-germs.We also thank Juan José Nuño Ballesteros for useful discussions about K-finiteness.The third author would like to thank the Singularity group at BCAM in Bilbao for their kind hospitality.

Preliminaries
Here we include definitions related to stability, properties involving Kfiniteness and the basic notions of the theory of perverse sheaves which are used to show our results.
3.1.Stability and K-finiteness.An unfolding F : (C n × C r , S × {0}) → (C p , 0) of a germ f is trivial if there exist an unfolding Φ of (id C n , S) and an unfolding Ψ of (id C p , 0) such that the following diagram commutes: Note that these conditions make Φ and Ψ into germs of biholomorphism.
We say that the map f is stable if it is K-finite and it is stable at every q ∈ N .
A one-parameter unfolding F is called a stabilisation of f if it admits a good representative such that for every non-zero δ ∈ T the perturbations f δ are stable.In this case, f δ is called a stable perturbation of f , and The instability locus of f : M → N is the support , where θ(f ) is the sheaf of vector fields along f , and T A(f ) is the extended A-tangent space of f , see [Wal81].What gives its name to the instability locus is the following result from [MN20, Theorem 3.2]: Proposition 3.1.Let f : M → N be a K-finite map, let q ∈ N and S = f −1 (q) ∩ Σ(f ).If q / ∈ Inst(f ), then the germ of f at S is stable.
Introducing the instability locus as the support of a sheaf allows us to justify that it is an analytic space.The proof of the following proposition can be extracted from that of [MN20, Proposition 4.2]: We finish this subsection with a result which is well known, but whose proof we include for lack of a reference.
Proof.The case p = n + 1 is clear, because then Σ(f ) = X, the map is finite and ∆(f ) = Im f .We may thus assume q ≤ n.Since f is K-finite, ∆(f ) is an analytic space of the same dimension as Σ(f ).On the one hand, Sard's Theorem implies that ∆(f ) is not all of Z, hence its dimension is at most p−1.On the other hand, Σ(f ) is defined as the vanishing locus of p-minors of an n × p-matrix with p ≤ n.The results in [EN62] imply that the dimension of any component of ∆(f ) is greater or equal to n − (n − p + 1) = p − 1.

Perverse sheaves.
In this subsection, we will summarize those parts of the machinery of perverse sheaves on complex analytic spaces which we shall need for the proof of our theorems.We follow the standard reference [Dim04] and all the details can be found there.
Throughout this section, X stands for a complex analytic variety.For a sheaf F on X its sheaf cohomology groups will be denoted by We write C X for the constant sheaf on X associated to the field C. Recall that the cohomology of C X is isomorphic to the singular cohomology with complex coefficients: . By C(X) we denote the category of complexes of sheaves of C X -modules on X: From any such complex of sheaves we obtain the collection of cohomology sheaves, denoted by we will denote the shift by d of the complex F which is given by the terms for every k together with the appropriately shifted differentials from F.
By D(X) we denote the derived category of sheaves of C X -modules on X which is obtained from C(X) by localizing at the set of quasi-isomorphisms.In particular this construction entails that two complexes of sheaves F • and G • are isomorphic in D(X) if and only if they are quasi-isomorphic in C(X).
A complex of sheaves F • is in the bounded derived category D b (X) if its nontrivial cohomology sheaves H i (F • ) are confined to a bounded range of indices i.For any such complex of sheaves F • ∈ D b (X) we can define the support as where we denote by H i (F • ) x the stalk of the cohomology sheaf H i (F • ) at the point x ∈ X.
A sheaf F on X can be regarded as a complex of sheaves concentrated in degree 0. This is a fully faithful embedding of the category of sheaves on X into the bounded derived category, cf.[Dim04, Proposition 1.3.3iii].This allows us to simplify notation and write F instead of F • for the complex of sheaves associated to a sheaf F .
Under this embedding into the bounded derived category, classical sheaf cohomology reappears as hypercohomology.More generally, for a continuous map f : X → Y of topological spaces and a sheaf F on X one has where the left hand side denotes the i-th derived pushforward of a single sheaf and denotes the derived pushforward in the derived categories, cf.[Dim04, Section 2.3].For the special case of a projection p : X → {pt} to a point, one obtains Again for a map f : X → Y and a complex of sheaves F on X the hypercohomology functors satisfy the following fundamental property: Similar translations of the above identifications between the category of C X -modules and the derived category exist for relative cohomology for pairs of spaces; see [Dim04] for details.
We shall say that a complex of sheaves F ∈ D b (X) on a complex analytic space X is constructible if there exists a complex analytic stratification of X with locally closed complex analytic strata S α ⊂ X such that (1) The restriction of F to each one of the S α has locally constant cohomology sheaves, (2) All stalks are finite dimensional C-vector spaces.
The category of bounded constructible complexes of sheaves on X will be denoted by D b c (X).From now on, we will mostly be interested in a very special type of bounded complexes of constructible sheaves: the perverse sheaves for the middle perversity.Note that a perverse sheaf need not be a sheaf, but a complex of sheaves.It is our intention for the reader to accept "being perverse" as a good property which can be used without knowing its details.We proceed by stating the necessary properties and results for the intended usage.
Proposition 3.4.[Dim04, Remark 5.1.19]Let F be a perverse sheaf on a complex analytic variety X, and let d be the dimension of the support of F .For any point x ∈ X, the stalk cohomology groups satisfy Note that since by definition any perverse sheaf is constructible, its support is always a closed complex analytic subset and it makes sense to speak of its (complex) dimension.Proposition 3.5.[Dim04, Theorem 5.1.20]If X is a locally complete intersection of complex dimension d, then the shifted constant sheaf C X [d] is perverse.
Let g : X → C be a holomorphic function defined on a complex analytic variety and set X = g −1 (0).Associated to g, there is the vanishing cycle functor φ g : D b c (X ) → D b c (X).For any point x ∈ X and any F ∈ D b c (X ), the stalk cohomology of φ g F can be computed as follows (see e.g.[Dim04, page 106 (4.1)]): (1) where B r is a sufficiently small open ball in X centered at x and δ ∈ C \ {0} is small enough with respect to r.The last result we need is essential, as it allows to produce new perverse sheaves from old: Theorem 3.6.[Dim04, Theorem 5.2.21]The vanishing cycle functor shifted by one φ g [−1] : D b c (X ) → D b c (X) takes perverse sheaves on X to perverse sheaves on X.
The ingredients we have introduced can be put together to give a simple proof of the cohomological connectivity of Milnor fibers: Let g be a suitable representative of a holomorphic map germ (C n+1 , 0) → (C, 0) defined on some open subset X ⊂ C n+1 .Take the perverse sheaf C X [n + 1] and apply the shifted vanishing cycle functor associated to g to obtain a perverse sheaf on X = g −1 (0).The stalk cohomology at a point x ∈ X recovers the reduced cohomology of the Milnor fibre of the germ of X at x as follows:

= Hi+n
sing (B r ∩ g −1 (δ)).Since the Milnor fibre of g at a regular point of the function has trivial reduced cohomology, we deduce that the support of φ g C X [n] is contained in Sing X. Letting d = dim(Sing X), Proposition 3.4 gives a bound on the stalk cohomologies at the origin which turns out to be the desired cohomological connectivity result for the Milnor fiber of g.
As the reader will see, the proofs of our results follow the same pattern: Find an appropriate perverse sheaf and apply the vanishing cycle functor associated to the projection to the parameter space.By virtue of Proposition 3.3, an estimate on the dimension of the support of the sheaf will directly translate to a bound on the non-zero degrees of the reduced cohomology of a nearby object.

Proof of Theorems 2.2, 2.3 and 2.4
Recall that, for p = n + 1, finiteness and K-finiteness are the same and Y = ∆(f ).Consequently, Theorem 2.2 and Theorem 2.3 can be considered two instances of the same result for K-finite germs f : (C n , 0) → (C p , 0) with p ≤ n + 1.For a K-finite germ f , any unfolding F is K-finite and, since p ≤ n + 1, the discriminant ∆(F ) is a hypersurface in C p × C, by Proposition 3.3.In particular, the shifted constant sheaf C ∆(F ) [p] is perverse.Projecting on the unfolding parameter gives a family and one checks the fiber over δ ∈ T to be ∆(f δ ).Applying the associated vanishing cycle functor gives a perverse sheaf Proof.Given a point away from the instability locus y ∈ ∆(f ) \ Inst(f ), S := f −1 (y) ∩ Σ(f ) is a finite set because f is K-finite.By Proposition 3.1, the germ f : (U, S) → (V, y) of f at S is stable and thus the unfolding is a trivial unfolding.Consequently, there exist unfoldings Φ and Ψ of the identity mappings id U and id V , respectively, such that the following diagram commutes: The stalk at y of the sheaf of vanishing cycles of π on ∆(F ) is which finishes the proof.
Proof of Theorems 2.2 and 2.3.From the inclusion Supp(φ π C ∆(F ) [p − 1]) ⊆ Inst(f ) and the hypothesis that dim Inst(f ) ≤ d, applying Proposition 3.4 we obtain that Theorems 2.2 and 2.3 follow, because for a good representative the stalk at the origin is precisely where the last equality is due to the fact that ∆(F ) is contractible.
The proof of Theorem 2.4 is very similar and thus will only be sketched.Before that, we discuss some subtleties of the analytic structure of D(f ).In order to avoid pathologies related to unfoldings, the source double point space D(f ) is given an analytic structure which need not be reduced.For our purposes, we do not need to know the details of this construction [MNP12, Definition 2.2], but only the following two properties: (1) For any unfolding F = (f t , t) : U × T → V × T , the fiber over δ ∈ T of the family D(F ) π −→ T is the complex space D(f δ ).
(2) If f : (C n , S) → (C n+1 , 0) is finite and generically one-to-one, then the space D(f ) is a hypersurface.The first statement follows from [MNP12, Lemma 4.2].For the second statement, the proof of [MNP12, Lemma 2.3] shows that D(f ) is a Cohen Macaulay subspace of C n of codimension one, hence a hypersurface.
Proof of Theorem 2.4.First of all, observe that the statement of Theorem 2.4 is trivial unless the dimension d of the instability locus is smaller than n − 1.Consequently, we may assume f : (C n , S) → (C n+1 , 0) to be a finite map-germ with dim(Inst(f )) < n − 1.
The map f is generically one-to-one, because stable mappings are generically one-to-one, and the conditions of finiteness and dim(Inst(f )) < n − 1 imply that the preimage of the instability locus is nowhere dense.
Any unfolding F of f is also finite and generically one-to-one, hence D(F ) is a hypersurface in C n+1 and the sheaf C D(F ) [n] is perverse.The shifted vanishing cycle functor associated to the projection π : where the dimension of f −1 (Inst(f )) is equal to d, because f is finite.Then the result follows from the computation of the stalk

Multiple point spaces
For finite germs f : (C n , 0) → (C p , 0) with p > n + 1, the image Y is no longer a hypersurface, and may even fail to be a complete intersection, as shown by the well known twisted cubic: Since the constant sheaf on non complete intersections is not necessarily perverse, we cannot follow the same reasoning as in the proof of Theorems 2.3 and 2.2.Instead, we study the cohomology of the disentanglement via the image computing spectral sequence due to Goryunov and Mond [GM93, Proposition 2.3]: . and the perversity of the sheaves on the image Y , discovered by Houston [Hou00, Theorem 2.9].This involves the strict multiple point spaces D k (f ) and their alternating cohomology which we will now discuss.
Unfortunately, there is no common agreement on the definition of multiple point spaces and different notions are in circulation.What we will refer to as the strict multiple points D k (f ) is the definition used by Goryunov, Mond [GM93], and Houston [Hou97a], [Hou97b], [Hou00].As Example 5.4 shows, the strict multiple point spaces are badly behaved in deformations: They do not specialize to fibers.To remedy this fact, there is another, more subtle definition of multiple point spaces D k (f ) due to Gaffney [Ga83] which we will describe in Section 5.2.These are what we will refer to as simply the multiple point spaces of f .Fortunately, the results about D k (f ) we want to use can be adapted to the spaces D k (f ) without difficulties.
5.1.The strict multiple point spaces D k ).Let f : X → Z be a complex analytic map between holomorphic manifolds.For k ≥ 1, the k-th strict multiple point space of f is defined to be We recall the construction of the alternating complex due to Goryunov and Mond [GM93].D k (f ) are symmetric in the sense that group S k acts on them by permutating the points x 1 , . . ., x k .This action preserves the fibers of the maps As a consequence, there is an action of S k on the pushforward sheaf ε k * C D k (f ) on Z.For each σ ∈ S k , we write the associated automorphism as

Goryunov and Mond have already argued that one has an isomorphism
(2) where the term on the right hand side is the alternating cohomology a subspace of the singular cohomology of D k (f ).These considerations were taken to the derived category in [Hou00].Note that, since ε k is finite, the associated pushforward of sheaves is an exact functor and thus in particular as complexes of sheaves on Y in the derived category.The same holds for their respective alternating subsheaves.Definition 5.2.Let f : X → Z be a complex analytic mapping between two complex analytic manifolds with n = dim X and p = dim Z. Then f is called strictly dimensionally correct if for all k ≥ 2 the strict multiple point space D k (f ) is either empty, or has dimension kn − (k − 1)p.
With these notations gathered we may cite the key result due to Houston, cf.[Hou00, Theorem 2.9], slightly adapted to our setup:

The multiple point spaces D k (f ).
As we already mentioned before, the strict multiple point spaces discussed in the previous section are not well behaved in families.This is illustrated by the following example.
Example 5.4.The cuspidal edge f : (x, y) → (x, y 2 , y 3 ) of Example 1.6 and the similar germ f : (x, y) → (x, y 2 , y 3 , 0) of Example 2.9 are both strictly dimensionally correct because the maps are injective and therefore the multiple point spaces are empty.However, the map F : (x, y, t) → (x, y 2 , y 3 + ty(x 2 − 1), 0, t) from the unfolding in Example 2.9 is not strictly dimensionally correct: Its multiple point spaces D 1 (F ) have dimension two while the expected dimension is one.
Without the dummy variable, i.e. for the map f as in Example 1.6, not only f , but also the unfolding is strictly dimensionally correct with expected dimension two for the double point spaces.In this example we encounter another problem: Observe that the strict double points of f are empty while those of F are not.This shows the failure of specialization and thereby illustrates the necessity to use Gaffney's multiple point spaces.
The pathological behaviour exhibited in this example can be avoided by taking a different definition of multiple points due to Gaffney [Ga83].Our exposition follows [NP17], where the reader can find proofs and details omitted here.
Let f : (C n , S) → (C p , 0) be a finite multi-germ.Since f is finite, hence K-finite, it admits a stable unfolding [MN20, Theorem 7.2] where t is the coordinate of the parameter space The multiple point spaces of f are defined as The space D k (f ) does not depend on the chosen stable unfolding F .Multiple point spaces of finite mappings between complex manifolds are defined by patching the multiple point spaces of their corresponding multi-germs.
To compare D k (f ) to D k (f ), we write the the k-th fat diagonal of X as Proposition 5.5.(Properties of the multiple point spaces) Let f : X → Z be a finite mapping between complex manifolds.Then (1) the spaces D k (f ) and D k (f ) satisfy the relation (3) unlike D k (f ), the spaces D k (f ) behave well under deformations in the sense that, for any unfolding F : X × T → Z × T , the family π : Just as in the case of D k (f ), the spaces D k (f ) are S k -invariant spaces endowed with finite maps ε k,j : While the spaces D k (f ) and D k (f ) and hence also the sheaves may differ, the corresponding alternating sheaves do not: Lemma 5.6.For any finite map f between complex manifolds, the complexes Note that with this lemma we may replace Alt The idea behind the proof of Lemma 5.6 is not new and appears already in [GM93], as well as [Hou00] and [Hou97b].Due to the lack of a citeable reference, we give a self contained argument.
Proof.We verify that the stalks of both sheaves coincide at an arbitrary point y ∈ Y .Fix one such point and let be its preimage under the finite map ε k : D k (f ) → Y .Set P := D k (f ) ∩ P to be the preimage in the subspace D k (f ).According to Proposition 5.5, every point x ∈ P \ P has to be contained in the fat diagonal ∆ k .Now the pushforward of C D k (f ) along ε k decomposes as The alternating operator clearly respects this decomposition since P is itself an S k -invariant subspace.Note that the first summand is nothing but the stalk of ε k * C D k (f ) by definition of P .Thus, we may conclude the proof by showing that the second summand has no alternating part.
To see the latter, let x = (x 1 , . . ., x k ) ∈ P \ P be an arbitrary point and c ∈ C D k (f ),x an element of its associated stalk.Since x has to be a diagonal point, there exists one pair of indices 0 < i < j ≤ k such that x i = x j .Let τ ∈ S k be the corresponding transposition.Then we have because Alt c is alternating and τ takes the point x to itself.Thus, Alt c = 0 for every element c of any stalk of C D k (f ) at points outside D k (f ).
Remark 5.7.Parallel to (2) we have the equality This shows indirectly that D k (f ) and D k (f ) have the same alternating cohomology, thanks to the equality Alt ε As in the case of D k (f ), if f : X → Z is a mapping between two complex analytic manifolds with n = dim X and p = dim Z, then every irreducible component of D k (f ) has dimension at least kn − (k − 1)p.The use of D k (f ) rather than D k (f ) calls for the following adaptation: It is clear that every unfolding F of a finite map-germ f must be finite as well.Moreover, any stable unfolding of F is a stable unfolding of f as well.From this observation, one concludes easily the following result: Proposition 5.9.If a map-germ f is dimensionally correct, then every unfolding F of f is dimensionally correct.
Remark 5.10.The previous assertion does not hold if one replaces the property of being dimensionally correct by that of being strictly dimensionally correct (see Example 5.4).Being dimensionally correct implies being strictly dimensionally correct, but the converse is not true, as shown again by the germ (x, y) → (x, y 2 , y 3 , 0).

5.3.
The image computing spectral sequence.Let f : (C n , 0) → (C p , 0) be a finite map-germ with p > n and let F be a one-parameter unfolding of f .We wish to compute the reduced cohomology of the disentanglement Y δ of a good representative of F .Goryunov and Mond have shown [GM93] how to do this in terms of the alternating complex.We are basically going to rephrase their argument in the derived category so that we can bring the perversity of the alternating sheaves, Theorem 5.3, into the picture.
In the derived category, Proposition 5.1 and Lemma 5.6 assert that the constant sheaf C Y on the image Y of the unfolding F is quasi-isomorphic to the alternating complex Alt ε • * C D k (F ) , ∂ • and we may thus replace C Y by this complex.
If we let π : Y → T be the projection onto the unfolding parameter, then the reduced cohomology of the disentanglement Y δ are the vanishing cycles of π at the origin: The nearby and the vanishing cycles of π : Y → T are computed using the well known diagram Lemma 5.11.For a good representative F of a 1-parameter unfolding of a K-finite map f : (C n , 0) → (C p , 0) with n < p there is a spectral sequence with first page Here π : Y → T is the projection of the image Y of F to the deformation parameter.
Proof.The standard procedure to obtain a quasi-isomorphism of the complex Alt ε • * C D • (F ) with a complex of injectives is to construct a double complex of injectives I •,• where each column I •,j resolves the sheaf Alt ε j+1 * C D j+1 (F ) , cf. [GM93, Section 2].Then the original complex is quasi-isomorphic to the total complex Tot(I •,• ) of this double complex I •,• .
The nearby cycles of π are defined as This is a composition of three functors.The first one, (ρ•j) −1 , takes injective sheaves on Y to injective sheaves on the open subset Y \ Y 0 , see [KS94, Proposition 2.4.1] and the third one, i −1 , is an exact functor.To compute the nearby cycles of Alt ε • * C D • (F ) -which involves the derived pushforward in the middle-we may therefore apply the functor i −1 (ρ • j) * (ρ • j) −1 to Tot(I •,• ).Note that applying this functor to a seperate column I •,j yields the nearby cycles of the single sheaf Alt ε j+1 * C D j+1 (F ) considered as a complex of sheaves concentrated in a single degree.
The vanishing cycles φ π C Y of π are defined as the mapping cone of the comparison morphism Following Definition 1.2), the image Y of a good representative is connected, hence the stalk of i −1 C Y at the origin is simply the vector space C, concentrated in degree zero.It is easy to check that on this single nontrivial term the comparison morphism c is always an inclusion which makes the vanishing cycles coincide with the reduced cohomology of the disentanglement, cf.(1).We may again replace C Y by the alternating complex Alt ε • * C D • (F ) and subsequently by Tot(I •,• ) in the distinguished triangle of the comparison morphism: Now note that the functor i −1 (ρ • j) * (ρ • j) −1 commutes with taking the total complex in a natural way.Thus we may identify the distinguished triangle of the comparison morphism c with the total complex of the following "comparison morphism of double complexes" which turns the vanishing cycles φ π Tot(I •,• ) into the total complex of the double complex Cone(c).The spectral sequence in question is now the spectral sequence of this double complex Cone(c).A column-wise inspection of this double complex reveals that we indeed find the vanishing cycles of the alternating sheaves on the first page.
Remark 5.12.From the fact that the spaces D k (F ) are empty for k big enough, it follows that the spectral sequence collapses at a certain page.Since we chose a field for the coefficients in cohomology, the infinity page determines the cohomology of the total complex.To be precise, there is an isomorphism of vector spaces Example 5.13.Here we come back to the immersion bi-germ f : (C n , S) → (C n+1 , 0) associated to a hypersurface X = V (g), introduced in Example 2.6, where we find a particularly simple spectral sequence.First of all, there are no triple or higher multiplicity points.This is because the stable unfolding F of f (or any unfolding of f for that matter) is also a bi-germ of an immersion, hence it cannot map more than two points to one.This gives D k (F ) = ∅, and thus D k (f ) = ∅, for all k ≥ 3 .Consequently, the nonzero terms of the first page E •,• 1 are concentrated in the second column, corresponding to the alternating cohomologies H i Alt (D 2 (f δ )).The fact that there are no triple points implies also that the maps D 2 (F ) → D(F ) and D 2 (f δ ) → D(f δ ) are homeomorphisms.Recall also from Example 2.6 that D(f δ ) = M 1 M 2 , where M 1 and M 2 are copies of the Milnor fibre of X = V (g).The homeomorphism D 2 (f δ ) → D(f δ ) transforms the action of the generator σ ∈ S 2 into the map taking a point x ∈ M 1 to the same point in M 2 , and vice-versa.In particular, after identifying D 2 (f δ ) with M 1 M 2 , we can choose a system of generators of the cohomology H i (D 2 (f δ )) consisting of some cocycles c i generating the cohomology of M 1 and their corresponding cocycles σ•c i , which generate the cohomology of M 2 .It now follows immediately that the alternating part is generated by the co- as Remark 1.8 claimed.
Remark 5.14.The previous example has a more visual version in homology (Figure 6 depicts the case of g = x 2 +y 2 which, after a change of coordinates, is the same as in Figure 5, but with a real picture better suited for the present discussion).We know that D(f δ ) consists of two copies M 1 and M 2 of M , contained in two copies U 1 and U 2 of an open ball U ∈ C n .Now let c be a cycle in M and let c i be the corresponding copies in U i (the two circles one the left side in Figure 5).Since U is contractible, there are chains a i in U i with boundary δa i = c i (the green and blue disks).Observe that ∂a i is supported on M i , which is the double point space of f δ , and that f δ glues a 1 and a 2 along the boundaries ∂a i .After changing the sign of a 2 if necessary, we observe that f * (a 1 + a 2 ) is an (i + 1)-dimensional cycle on Y δ (the blue and green cycle on the right side).The desired isomorphism is determined by c → f * (a 1 + a 2 ).

Proof of Theorem 2.8
In this section we will use the notation of Theorem 2.8 and impose the hypothesis found there, that is, f : (C n , 0) → (C p , 0) is a dimensionally correct map-germ with p > n and instability locus Inst(f ) of dimension d, F : W → V × T is a good representative of a one-parameter unfolding of f and Y = Im F is the image of F .
We intend to control the stalk cohomology of φ π C Y , i.e. the reduced cohomology of the disentanglement, by means of the first page of the spectral sequence in Lemma 5.11.Thus, we have to bound the vanishing cohomology of the alternating sheaves Alt ε j+1 * C D j+1 (F ) on the image and we will do so by exploiting the perversity of their respective shifts and estimating the dimension of the support of their vanishing cycles.
Lemma 6.1.The support of Proof.Take a point away from the instability locus y ∈ Y \ Inst(f ) and let S = f −1 (y).Since f is stable at y, the unfolding (F, S × {0}) of the germ (f, S) is trivial.Therefore, there exist an unfolding Φ of (id C n , S) and an unfolding Ψ of (id C p , 0), such that We choose representatives F, f, Φ and Ψ satisfying the equality As one may easily check, this makes the multiple points D k (F ) and D k (f × id T ) isomorphic via Φ −1 × k . . .×Φ −1 .Moreover, the multiple points in D k (f × id T ) are of the form ((x 1 , t), . . ., (x k , t)).Forgetting all but one of the copies of t gives an isomorphism [NP17, Proposition 3.8] In turn, the spaces Y and Y × T are isomorphic via Ψ.
There are two geometric considerations where the k-fold product structure of Φ −1 × k . . .×Φ −1 plays a role: First of all, the isomorphisms and the ε k mappings are compatible in the sense that there is a commutative diagram ) is compatible with the action of the symmetric group S k on both D k (F ) and on D k (f ) × T (the latter induced by the action of S k on D k (f )).This makes the construction of the Alt operators on both sides equivalent and the pushforward Ψ * takes the sheaf Alt We can now show that y is not in the support of φ π Alt ε k * C D k (F ) by checking the vanishing of the stalk cohomology: Take a suitable ball B r in V around y.Then, for δ sufficiently small we find The vanishing in the last line follows from the fact that due to the product structure clearly (Y × T ) ∩ B r retracts onto the fiber (Y × {δ}) ∩ B r for r δ > 0.
Now we can determine where the non-trivial entries of the first page of the spectral sequence in Lemma 5.11 are concentrated.Proposition 6.2.Let F be a dimensionally correct one-parameter unfolding of a germ (C n , 0) → (C p , 0) with p > n.For every integer k ≥ 2, the nonzero cohomologies Shifting everything back by kn − (k − 1)p, we obtain the desired bounds for k ≥ 2. Note that, since the alternating cohomology ) is a subspace of the singular reduced cohomology of the disentanglement, there can be no contributions in negative degrees.
For k = 1 the disentanglement always provides a product structure (D 1 (F ), (0, 0)) ∼ = (X × T, (0, 0)) and therefore In this case, the sequence necessarily collapses on this first page.In general, the differentials between the nonzero entries on the first page can lead to cancellations for the following ones and on the limit page.In either case, the positions of the nonzero entries on the first page give a bound on the nonzero entries of i E i,k−i ∞ ∼ = H k (φ π C Y ) 0 .For this particular example we see that the non-trivial cohomologies of a disentanglement Y δ are concentrated on the degrees ∈ {3, 4} ∪ {6, 7, 8} ∪ {10, 11, 12}.

Monodromy for disentanglements
Our study of the monodromy for disentanglements in Theorem 2.11 will be based on the monodromy on the multiple point spaces and their alternating cohomology.This requires an adaptation of the definitions to fibrations with group actions.
Note that any stable one parameter unfolding As already noted by K. Houston in [Hou97a, Lemma 2.14], there exist equivariant local trivializations.We may therefore suppose that the trivialization of the pullback of the fibration (4) to the universal cover exp : S → D \ {0} is S k -equivariant and that the parallel transport of the fiber along a closed, counterclockwise loop in D * around the origin commutes with the S k -action on D k (f δ ).It is easy to see that this implies that also the maps induced by h on cohomology restrict to operators on the alternating cohomology H i Alt (D k (f δ )).We therefore have a well defined monodromy action on the vanishing cycles of the alternating sheaves.As we shall see now, this gives the vanishing cycles of the alternating sheaves the structure of a C[s, s −1 ]-module where s is a formal variable standing for the monodromy automorphism h.
To relate monodromy to our usual spectral sequence, it is useful to introduce a point of view about eigenvalues and Jordan blocks slightly different from that of the introduction: Consider the monodromy of a fibration E π − → D * over the punctured disk with fibre F , and assume that the fibration was obtained as a small representative of an analytic germ (E, 0) → (D, 0), as in [Lê78].Consider the one-variable Laurent polynomial ring R := C[s, s −1 ].The monodromy action gives each cohomology H i (F ) an R-module structure, by declaring that multiplication by s acts on H i (F ) by the monodromy automorphism h i : H i (F ) → H i (F ).It is well known that F is a homotopy equivalent to a finite CW-complex, and that each monodromy automorphism h i is idempotent.As a consequence, each H i (F ) is a finitely generated torsion R-module.Note that R is a principal ideal domain so that according to the structure theorem, every finitely generated torsion R-module A can be uniquely decomposed as A ∼ = ⊕ m j=1 R/(s − λ j ) a j , where λ j are non-zero complex numbers and a j are positive integers.It turns out that the set of eigenvalues of h i can be recovered as Supp(A) := {λ j } 1≤j≤m ,

1. 1 .
The Milnor fibration.Consider a germ of a non-constant holomorphic function g : (C n+1 , 0) → (C, 0) and a representative of g thereof, defined on some open neighborhood of a closed ball B r of sufficiently small radius r centered at the origin.Consider a punctured open disc D * , centered at the 2010 Mathematics Subject Classification.58K15, 58K60, 32S30, 57R45.The first author is partially supported by the starting grant KY2340000123 from University of Science and Technology of China, National Natural Science Funds of China (Grant No. 12001511) and the project "Analysis and Geometry on Bundles" of Ministry of Science and Technology of the People's Republic of China.The second author is partially supported by the ERCEA 615655 NMST Consolidator Grant and by the Basque Government through the BERC 2014-2017 program, the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and by Programa de Becas Posdoctorales en la UNAM, DGAPA, Instituto de Matemáticas, UNAM.

Figure 2 .
Figure 2. The same singularity as in Figure 1, now regarded as the image of a transverse double point bi-germ.

Figure 4 .
Figure 4.The cuspidal edge and two different perturbations.The red dashed line represents the instability locus Inst(f ).
representative F of a 1-parameter family and its image Y. Here, exp : S → T * is the universal cover of the punctured unit disc and Y δ × S the trivialized fiber product over the infinite strip S, cf.[Dim04, Chapter 4.2].

Figure 6 .
Figure 6.The disentanglement of a bi-germ of immersion has the homology of a two-point suspension.
0 are concentrated in degrees i ≥ 0 withkn − (k − 1)p − d ≤ i ≤ kn − (k − 1)p.In the case k = 1 the above vanishing cycles are all zero.Proof.Since F is dimensionally correct, the sheavesAlt ε k * C D k (F ) [kn − (k − 1)p + 1]are perverse, and, by virtue of Theorem 3.6, so are the sheavesφ π Alt ε k * C D k (F ) [kn − (k − 1)p] on Y 0 .Since these sheaves are supported on a space of dimension at most d, it follows from Proposition 3.4 that the non trivial cohomologies of their stalks at the origin are concentrated in degrees i with −d ≤ i ≤ 0.
T, X × δ) = 0.This finishes the proof.Proof of Theorem 2.8.The proof follows by the same spectral sequence argument as in [SZ18, Theorem 1.1]:The concentration of cohomology follows immediately from the isomorphismH (φ π C Y ) 0 ∼ = i E i, −i∞ from Remark 5.12 and the vanishing of the entries of the first page coming from Proposition 6.2.Let us give an example which illustrates the general situation: Example 6.3.Let f : (C 16 , 0) → (C 21 , 0) be a dimensionally correct map whose instability locus has dimension d = 2.According to Proposition 6.2, the possibly nonzero entries in the first page of the spectral sequence E i,j 1 from Lemma 5.11 are the following: i ↑ 11 H 11 Alt (D 2 (f δ )) 10 H 10 Alt (D 2 (f δ ))

F
: (C n , 0) × (C, 0) → (C p , 0) × (C, 0) of a finite map germ f with p > n as in Theorem 2.11 gives rise to analytic fibrations(4) π • ε k : D k (F ) ∩ (ε k ) −1 (B × D \ {0}) → D \ {0}for suitable choices of a representative F , a ball B ⊂ C p in the target of f , and a disc D ⊂ C in the parameter space.By construction, the symmetric group S k acts fiberwise on the multiple point space D k (F ) ∩ (ε k ) −1 (B × D), for each k > 0.