Vanishing cycles of matrix singularities

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $\mu=\tau$ conjecture about the families of a special type, and make first steps towards understading of the monodromy of matrix singularities. We also prove a Lyashko-Looijenga type theorem for simple matrix families, and point out a surprising relationship between certain Shephard-Todd groups, simple odd functions and a sporadic part of the Bruce-Tari simple matrix classification.

This paper is on the vanishing topology of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices, that is, of mappings of smooth manifolds to the corresponding matrix spaces. Complete classifications of simple families of the first two kinds were obtained more than 15 years ago by Bruce and Tari [5,7], and a partial simple classification of skew-symmetric matrices was done by Hasliger about the same time [19]. Topological and algebraic analysis of their results for a small number of parameters -when a generically perturbed family avoids singularities of the set ∆ of all degenerate matriceswas carried out in [6,17,16]. In particular, the monodromy of the Milnor fibre of the inverse image of the discriminant ∆ was described in these papers, and a relation between the rank of the vanishing homology of such a fibre and the relevant Tjurina number of the matrix family was obtained. The last relation was actually proved in [16] not just for the matrix singularities but for sections of a wide class of singular hypersurfaces.
Moving to a higher number of parameters in the matrix families brings a difference between two types of related Milnor fibres which have been diffeomorphic previously: the smooth Milnor fibre of a family which is the inverse image of the Milnor fibre of the discriminant, and the singular Milnor fibre which is the inverse image of ∆ itself under a generic perturbation of the family.
Like Milnor fibres of isolated function singularities, singular Milnor fibres of the matrix families are wedges of spheres of the middle dimension. This follows from [20,26]. However, being highly singular, they have been promising complicated calculations to yield any quantitative information, for example, the numbers µ ∆ of the spheres in their wedges (called the singular Milnor numbers). One of possible approaches to understand their topology was developed by Damon and Pike in [10,11] where they introduced a rather involved machinery to calculate the Euler characteristic of such fibres via sections corresponding to fixed flags in the configuration spaces. The method is based on a choice of an appropriate sequence of free divisors and extends the Lê-Greuel inductive procedure to calculate the Milnor number of an isolated complete intersection singularity [21]. Nevertheless, numerous questions related to the geometry of bifurcation diagrams of the matrix families, to the monodromy of these families, to their relations with other types of singularities and so on have stayed untouched.
The aim of this paper is to initiate a monodromy study of the singular Milnor fibres of the matrix families. The structure of the paper is as follows.
Section 1 reminds the equivalences of matrix families we are going to work with. In Section 2 we show that the complex link of the discriminant ∆ in the space of all arbitrary square, symmetric or skew-symmetric matrices is homotopic to a sphere of the middle dimension, and describe this sphere in geometric terms. This allows us to define in Section 3.1 vanishing cycles on singular Milnor fibres of our matrix singularities, and prove in Section 3.2 the Damon-Pike conjecture on the equality of the Tjurina and singular Milnor numbers for matrix families of a special type. Section 3.3 recalls the classification of simple matrix families from [5,7,19]. It also extends the above µ ∆ = τ result to a wider class of matrix singularities, namely, to those corresponding to Arnold's functions on manifolds with boundaries. The latter, in particular, includes nearly all simple matrix singularities.
In Section 4 we define two natural versions of bifurcation diagrams for matrix families, and prove a Lyashko-Looijenga type theorem for simple matrices.
Section 5 introduces a way to understand the monodromy of singular Milnor fibres which provides us with a complete description of such a monodromy for corank 2 symmetric matrices. Our approach may be considered as a generalisation of a modification of Arnold's construction of an intersection form for boundary function singularities (cf. [1]).
In Section 6, our attention is on corank 3 simple symmetric matrix families, the only part of the simple classification not covered in earlier sections. A higher-corank adjustment of the methods of the previous section reveals a surprising relationship between these simple matrix singularities, odd functions and certain Shephard-Todd groups. In Section 6 we also notice that for these families the µ ∆ = τ equality holds as well, which extends the validity of the equality to all simple matrix singularities.
We conclude the paper with a general µ ∆ = τ conjecture for all three types of our matrix families. Of course, another intriguing theme emerging from the paper is a problem of description of the monodromy of singular Milnor fibres of matrix families to go beyond the results of Section 5.

Matrix equivalences
A matrix family in this paper will be a holomorphic mapping M : C s → Mat n , where Mat n is one of three spaces of complex n × n matrices: Sq n , arbitrary square matrices, Sym n , symmetric matrices, Sk n , n = 2k, skew-symmetric matrices.
In the Sq n case, we say that two matrix families, M 1 and M 2 , are GL-equivalent if there exist a biholomorphism ϕ of the source and two holomorphic maps A, B : C s → GL n (C) such that For the GL-equivalence in the symmetric and skew-symmetric cases, we require existence of only one holomorphic s-parameter family A of invertible n × n matrices such that Restriction to mappings A, B : C s → SL n (C) in all three cases provides notions of the SL-equivalences.
Relation between the matrix GL-and SL-equivalences is similar to that between the contact (K) and right (R) equivalences of holomorphic functions.
Germ versions of the above definitions are straightforward. A relevant equivalence class of germs of matrix families will be called a matrix singularity. We denote by O s the space of all holomorphic function germs on (C s , 0), and identify the space of all holomorphic map germs M : (C s , 0) → Mat n with the O s -module O N s where N = dim C Mat n . Let E jℓ be the n × n matrix with the jℓ-entry 1, and all other entries zero. The extended tangent spaces to the GL-equivalence classes of germs M : (C s , 0) → Mat n , with the source coordinates x 1 , . . . , x s , are T GL,Sqn M = O s ∂M/∂x i , E jℓ M, ME pq i=1,...,s; j,ℓ,p,q=1,...,n , and ..,s; j,ℓ=1,...,n for Mat n = Sym n , Sk n .
For the SL-equivalences, all the diagonal matrices E jj and E pp in these expressions should be replaced, for example, by the differences E jj − E 11 and E pp − E 11 . We will denote by τ GL,M atn (M) and τ SL,M atn (M) the corresponding Tjurina numbers of M, that is, the codimensions of the above extended tangent spaces in O N s . All our groups of matrix equivalences are Damon's geometric subgroups of the K-equivalence group (see [8]), and therefore GL-and SL-miniversal deformations of a matrix family germ M may be written as where τ is the relevant Tjurina number of M, and the ϕ i ∈ O N s form a basis of the quotient of O N s by the corresponding extended tangent space.
By the matrix corank of a germ M : (C s , 0) → Mat n we will understand the corank of the matrix M(0). In our symmetric and arbitrary square settings, a matrix corank c family is equivalent to a family where M ′ is a germ of a c × c matrix family of the same type and M ′ (0) is the zero matrix. Similar reduction exists in the skew-symmetric case, with the only difference that the identity corner should be replaced by the block-diagonal matrix J n−c with the elementary blocks J 2 = 0 1 −1 0 along the diagonal.
Two germs of matrix families M i : (C s , 0) → Mat n i , i = 1, 2, will be called stably GLor SL-equivalent if there exists n such that the two 'extended' families In what follows we will need convenient skew-symmetric analogs of the standard trace and eigenvalues of a square matrix. So, we set • the skew trace of A ∈ Sk 2k to be sktr(A) = k i=1 a 2i−1,2i , and • the skew eigenvalues of such an A to be the solutions λ 1 , . . . , λ k of the characteristic equation P f (A − λJ 2k ) = 0.
We will also need a special kind of skew-symmetric matrices which we call quaternionic.
These are matrices A ∈ Sk 2k in which all the 2 × 2 cells a 2i−1,2j−1 a 2i−1,2j a 2i,2j−1 a 2i,2j are additionally required to be of the form z w −w z , z, w ∈ C. In particular, all diagonal 2 × 2 blocks 0 a 2i−1,2i −a 2i−1,2i 0 of a quaternionic matrix are real. Proof. For a natural number i, Then the signed product a i 1 j 1 · · · · · a i k j k participating in the Pfaffian of a quaternionic A ∈ Sk 2k is the complex conjugate of the signed product a i ′ 1 j ′ 1 · · · · · a i ′ k j ′ k also participating in P f (A).

Complex links of matrix discriminants
We denote by ∆ ⊂ Mat n the set of all degenerate matrices, and call this set the discriminant. Under the discriminant ∆(M) of a particular matrix family M we understand the zero set of the function det • M or P f • M, that is, the inverse image of ∆ under the mapping M. We use the notation T r α for the hyperplane in Mat n of all matrices with the (skew) trace α. We write ∆ sym , T r sq α , etc. if we want to emphasise which of the three cases we are considering.
According to [5,7,19], a generic map germ (C N −1 , 0) → (Mat n , 0) is -up to any of our equivalences -an embedding whose image is the hyperplane germ (T r 0 , 0). Moreover, GLand SL-miniversal deformations of such maps are one-parameter families of embeddings of C N −1 into Mat n as the hyperplanes T r α , α ∈ (C, 0). Therefore, the complex links of the matrix discriminants are the sets of all degenerate matrices with a fixed non-zero (skew) trace.

Theorem 2.1
The complex link of the discriminant ∆ ⊂ Mat n is homotopy equivalent to a sphere S N −2 . If λ is a positive real number, then this sphere may be taken to be the set of respectively (Sym) all degenerate real matrices A ∈ Sym n with the trace nλ and all eigenvalues nonnegative; (Sq) all degenerate hermitian matrices A ∈ Sq n with the trace nλ and all eigenvalues nonnegative; (Sk) all degenerate quaternionic matrices A ∈ Sk n , n = 2k, with the skew trace kλ and all skew eigenvalues non-negative.
Proof. We consider the symmetric case in detail, and point out the adjustments required in the two other cases. Symmetric matrices. The only level with non-isolated critical points of the restriction of the determinantal function to the hyperplane T r sym nλ , λ = 0, is its zero level. If we show that this restriction of the determinantal function has exactly one Morse critical point outside the zero level then the claim about the homotopy type of the link will follow from [20,26].
So, we parametrise the hyperplane T r sym nλ ⊂ Sym n as a shift of T r sym 0 on which all the entries x ij , 1 ≤ i ≤ j ≤ n, except for x 11 , are taken as independent coordinates: x ∈ C N −1 , N = n(n + 1)/2, λ = 0.

Lemma 2.2
The only invertible matrix in the family (1) which is a critical point of the determinantal function is M λ (0) = λI n .
Proof of the lemma. Vanishing of the partial derivatives of det(M λ (x)) with respect to all the x ij , i < j, that is, vanishing of all the off-diagonal cofactors of M λ (x) implies that the inverse of an invertible critical matrix is diagonal. Hence an invertible critical matrix itself is also diagonal: x ij = 0 for all i < j. This reduces the determinant of (1) to the product of its diagonal entries subject to the constraints that their sum nλ is fixed and that all of them are non-zero. Such a product has only one critical point, when all its factors are equal.
We now look at the quadratic in x terms of det(M λ (x)). They are λ n−2 times the sum of all 2 × 2 diagonal minors of L sym . This sum reduces to This quadratic form is non-degenerate, and therefore the critical point M λ (0) = λI n is indeed a Morse critical point of the determinantal function.
For λ real and positive, the form (2) is negative definite on the real part of T r sym nλ . This implies the (Sym) claim of the Theorem. Indeed, taking small ε > 0, we see that the Morse vanishing cycle on the under-critical level det = λ n − ε is real, and it is sent by the negative gradient flow of the determinantal function on the real part of T r sym nλ to the vanishing sphere S N −2 in the complex link ∆ sym ∩ T r sym nλ . Square matrices. We parametrise T r sq nλ similar to (1) with the only difference that the over-and under-diagonal entries are now independent: x ∈ C n 2 −1 , λ = 0. This replaces the first term in (2) with − 1≤i<j≤n x ij x ji , and we set x ji =x ij for all i < j to treat it as a negative definite quadratic form on a real space.
Skew-symmetric matrices. For a parametrisation of T r sk kλ we take where Similar to the proof of Lemma 2.2, vanishing of the derivatives of det(M λ (x)) = P f 2 (M λ (x)) with respect to all the x-coordinates except for the x 2i−1,2i , i > 1, implies that a critical invertible matrix must be block-diagonal, with the multiples of the J 2 blocks along the diagonal, that is, its Pfiaffian is The only critical point of this product outside its zero level is M λ (0) = λJ 2k . The The determinants here become positive definite quadratic forms on R 4 when we set Taking also all the x 2i−1,2i real, we end up with our skew-symmetric matrices being quaternionic.

Vanishing cycles
From now on we assume that the dimension s of the domains of our matrix families C s → Mat n is such that in general their images meet the singular loci of the discriminants ∆ ⊂ Mat n , that is, s > 2, 3, 5 in respectively symmetric, arbitrary square and skew-symmetric cases.
Let M : (C s , 0) → Mat n be a germ of a matrix family with a finite SL Tjurina number τ, and {M λ } λ∈Λ , M 0 = M, a representative of its SL-miniversal deformation. According to [20,26], for sufficiently small radii ε > 0 and η > 0, there exist closed 2s-and 2τ -dimensional balls B ε ⊂ C s and B η ⊂ Λ, centred at the origins, such that the set ∆(M λ ) ∩ B ε is homotopic to a wedge of a finite number of (s Due to the constraints on the dimension s, this is the only level of the function which is critical for all λ ∈ Λ if the corank of M(0) is higher than the lowest positive. Moreover, the singular locus of the zero level is of positive dimension if s is not the minimal we have allowed. Critical points p i , i = 1, . . . , r, on all other levels are isolated, and therefore the function has the corresponding ordinary Milnor numbers µ i at them. Due to [20,26], the number of the spheres in the wedge of the set For a generic λ ∈ B η , the number of the spheres in the wedge achieves its maximum. For convenience, we will use the same terminology even if the corank of M(0) is the lowest positive. Of course, in such cases the above ∆(M λ ) ∩ B ε is just the ordinary smooth Milnor fibre of the function det • M λ or P f • M λ , and µ ∆ (M) is its ordinary Milnor number.
In this section we calculate singular Milnor numbers of matrix families related to isolated singularities of functions on manifolds with and without boundaries. In particular, they include nearly all simple matrix singularities from [5,7]. The only remaining simple singularities, of 4-parameter corank 3 symmetric matrices and of their 7-parameter square versions, will wait till the final section of this paper.
Later on, to avoid reminders about the ball choices, we will refer to the sets ∆(M λ ) ∩ B ε with λ ∈ B η as local sets ∆(M λ ).

Matrix vanishing cycles
In what follows, it will be more convenient for us to write miniversal deformations of the parametrisations L : (C N −1 , 0) → (Mat n , 0) of the hyperplanes T r 0 used in (1), (3), (4) in slightly different forms than in those formulas, namely as respectively Now, any map germ M : (C s , 0) → (Mat n , 0), s = N − 1 + m, with a generic corank 1 linear part may be considered as an m-parameter deformation of the relevant family L, and therefore reduces to the form (5) with λ = g(z), where z = (z 1 , . . . , z m ) are the additional variables. In the simplest situation g has a Morse critical point at the origin, and the matrix families are SL-and GL-equivalent to For these families τ SL,M atn = τ GL,M atn = 1, and we have the miniversal deformations where we are back to using λ ∈ C as a deformation parameter. These are versal deformations of the only codimension 1 matrix singularities. According to Theorem 2.1, they provide the following description of all possible vanishing cycles appearing in our matrix singularities.
The singular Milnor fibre of each of the matrix families in (6) is homotopy equivalent to a sphere S s−1 , s = N − 1 + m, m ≥ 0. If λ is a fixed positive real number, then this sphere may be taken to be the set of respectively (Sym) all degenerate real matrices A ∈ Sym n with the trace λ − z 2 1 − · · · − z 2 m , z ∈ R m , and all eigenvalues non-negative; (Sq) all degenerate hermitian matrices in A ∈ Sq n with the trace λ − z 2 1 − · · · − z 2 m , z ∈ R m , and all eigenvalues non-negative; (Sk) all degenerate quaternionic matrices A ∈ Sk n , n = 2k, with the skew trace λ − z 2 1 − · · · − z 2 m , z ∈ R m , and all skew eigenvalues non-negative.
For a proof, we just need to notice that, for a fixed λ = 0, all families in (7) contain only one critical point of the determinantal function off its zero level. Indeed, for example in the symmetric case, the argument used in Lemma 2.2 shows that a critical matrix is diagonal. After that the vanishing of the derivative of the determinant with respect to any z i yields z i = 0 outside ∆.
Thus, all the vanishing cycles are the order m suspensions of the complex links of the matrix discriminants.

Definition 3.3
Vanishing cycles corresponding to the families (6) of the matrix corank n will be called corank n vahishing cycles.
In particular, corank 1 vanishing cycles in the symmetric and square cases, as well as corank 2 vanishing cycles in the skew-symmetric case are the ordinary Morse vanishing cycles of isolated hypersurface sungularities.

Damon-Pike conjecture
We will now prove Theorem 3.4 Consider matrix singularities M with a generic corank 1 linear part, that is, x ∈ C N −1 , g ∈ O m . Assume the SL Tjurina number of such a matrix family is finite. Then µ ∆ (M) = τ SL,M atn (M) = µ(g) .
The µ(g) here is the standard Milnor number of the isolated function singularity g.
This property was conjectured by Damon and Pike in [11], with quasi-homogeneity of g requested for the last equality.
Proof. We consider only the symmetric case, the other two being its word-to-word repetitions.
So, take a small generic deformation {M t } t∈(C,0) of Following the approach used in the previous subsection, we can set where {g t } t∈(C,0) is a small generic deformation of g 0 = g. To prove the equality of µ ∆ (M) and µ(g), we need to show that, for a fixed t = 0, the function det • M t has exactly µ(g) Morse critical points close to the origin and outside its zero level ∆(M t ).
Following the argument used in Lemma 2.2, we see that a critical matrix in the M t family off ∆ must be diagonal and hence has determinant Vanishing of the derivatives of this expression with respect to all the x ii outside zeros of the expression itself implies x 22 = x 33 = · · · = x nn = g t (z)/n .
Therefore the determinant of a critical matrix is (g t (z)/n) n . Critical points of the last function outside its zeros are exactly critical points of g t . Due to the genericity of the deformation {g t } of g 0 = g, these points are all Morse and their number is µ(g).
The remaining part of the theorem, that τ SL,M atn (M) = µ(g), follows from a calculation exercise (cf. [5]) showing that The last isomorphism holds also for the square matrices in (8), and -with replacement of E 11 by (E 12 −E 21 ) -in the skew-symmetric case too. Due to that, SL-miniversal deformations of all matrix families (8) may be obtained by replacing the function g in them by its Rminiversal deformation.
Remark 3.5 In the GL situation, for the matrix families (8) we have τ GL,M atn (M) = τ (g), where τ (g) is the Tjurina number for the K-equivalence of functions. Respectively, for GL-miniversal deformations of these families, we replace the function g by its K-miniversal deformation.

µ = τ theorem for a matrix version of boundary function singularities
The type of matrix families we are studying in this subsection generalises that from the Damon-Pike conjecture, and contains nearly all simple matrix classes, at least in the symmetric and square settings. The families we are considering now have a more general top left corner: Here again x ∈ C N −1 and z ∈ C m , while h ∈ O m+1 . Note that the summations in the matrices are one term shorter now comparing with the matrices L from the previous subsection. Of course, we are back to the situation of that subsection if the derivative of h at the origin by its only x argument is not zero.
We now recall the GL-simple classification within our earlier assumption that the number s = N −1+m of the matrix parameters should be high enough for the singular Milnor fibre to be indeed singular. The above function germs h ∈ O m+1 are considered in this classification up to the R ∂ -equivalence of functions on the manifold C m+1 with the boundary C m given by the vanishing of its only x coordinate [1]. Theorem 3.6 (a) [5,7,19] GL-simple matrix singularities of the forms (9), (10) and (11) are classified by the R ∂ -simple types X τ = A τ , D τ , E τ , B τ , C τ , F 4 of the function germs h.
(b) [5] Up to the stable GL-equivalence, the only other simple classes in the symmetric case are the following corank 3 matrix families in four variables: Thus, matrix singularities of the forms (9-11) are playing a special role in the classification, and our aim now is to extend the Damon-Pike conjecture to such families.
So, let us find the singular Milnor numbers for matrices (9)(10)(11). According to [5,7] and similar skew-symmetric calculations, all possible deformations of these matrix families may be obtained by deforming the functions h. To make some difference with the previous subsection, we shall now consider the skew-symmetric case. Therefore, let {h t } t∈(C,0) be a generic small deformation of h 0 = h in (11) and {M t } t∈(C,0) the corresponding matrix deformation. Vanishing of the derivatives of the P f • M t outside ∆(M t ) with respect to all the x j,ℓ except for the x 2i−1,2i , i > 1, reduces the Pfaffian of a critical matrix in the family M t , t = 0, to the product Vanishing of the derivatives of this product with respect to all the x 2i−1,2i , i > 2, implies that the values of all these co-ordinates must be the same and equal to h t (x 3,4 , z)/(k − 1). This leaves us with the Pfaffian equal to Vanishing of the derivatives with respect to all the still remaining co-ordinates gives us the final conditions ∂h t /∂z 1 = ∂h t /∂z 2 = · · · = ∂h t /∂z m = h t + (k − 1)x 3,4 ∂h t /∂x 3,4 = 0 . Therefore, The expression for the local algebra Q h in the symmetric and square cases has n instead of k, and x 22 instead of x 3,4 .
On the other hand, direct calculations (cf. [5,7]) of the extended tangent spaces to the SL-equivalence classes of the families (9), (10) and (11) show that O N N −1+m /T SL,M atn M is isomorphic to either Q h E 11 or respectively Q h (E 12 − E 21 ).
Thus we have proved  (9), (10) and (11): it has already been noticed in [5,7] that for these matrix singularities without any quasi-homogeneity requirements. Indeed, the GL-equivalence splits the last generator of the ideal in (13) into its two summands.  The diagram Σ splits into components Σ c numbered by the coranks c of the cycles vanishing on the fibres det • M λ = 0, λ / ∈ Σ, when λ approaches Σ.   Let Π d ≃ C d be the space of monic degree d polynomials in one variable. For a matrix family M, we have a Lyashko-Looijenga type map from Λ \ Θ to Π µ ∆ which sends a point λ to the polynomial whose roots are all non-zero critical values of the function det • M λ or P f • M λ . By the continuity, it extends to the holomorphic map L : Λ → Π µ ∆ which sends Θ to the set Ξ ⊂ Π µ ∆ of all polynomials with either multiple or zero roots.
For a simple singularity X M atn µ , the map L is a map between two spaces of the same dimension µ. Taking quasi-homogeneous SL-miniversal deformation of X M atn µ , we can assume Λ = C µ . Theorem 4.6 For a simple matrix singularity X M atn µ , the map L : Λ → Π µ is a proper holomorphic map. Its restriction to the complement Λ\Θ is a finite order unramified covering of the complement Π µ \ Ξ.
Proof. All the parameters in a quasi-homogeneous SL-miniversal deformation of X M atn µ have positive weights. The mapping L is in this case also quasi-homogeneous with all its components of positive degrees. Therefore, the properness follows from L −1 (0) = {0} which is guaranteed by the same property of the Lyashko-Looijenga map L Xµ of a simple boundary function singularity X µ (see [22,23,24]).
The local biholomorphicity of L outside Θ also follows from the local biholomorphicity of the map L Xµ outside the bifurcation diagram of the boundary function X µ (see [22,23,24] again).
The complement Π µ \ Ξ is a k(π, 1)-space for Brieskorn's generalised braid group BrB µ associated with the Weyl group B µ [4]. Therefore we have Corollary 4.7 For a simple matrix singularity X M atn µ , the complement Λ \ Θ to its full bifurcation diagram is a k(π, 1)-space, where π is a subgroup of a finite index in the braid group BrB µ . In the symmetric and square matrix cases, the index is Here C and |X µ | are the Coxeter number and the order of the Weyl group X µ , and the values of α are as follows: In the skew-symmetric case, n should be replaced in the formula by k = n/2.
The index has been calculated here as the degree of the quasi-homogeneous map L (see (12)).

Monodromy of corank 2 symmetric families
Let us go back to the miniversal deformations (7) of codimension 1 matrix singularities. Quasi-homogeneous considerations show that in all these cases the action of the only Picard-Lefschetz operator on the only vanishing cycle of the singular Milnor fibre is multiplication by (−1) m . Description of the monodromy representation of π 1 (Λ \ Σ) on the homology of singular Milnor fibres of more complicated matrix families desires a good definition of an intersection form on the homology. In this section we are introducing an approach which allows to understand the whole monodromy group of a corank 2 symmetric matrix singularity.
Coincidence of simple classifications of corank 2 symmetric matrix families and of functions on manifolds with boundary suggests introduction of an order 2 covering of the singular Milnor fibres. However, our covering will be different from the one used by Arnold for boundary functions.
Namely, since degenerate binary quadratic forms are squares of linear forms, we represent the cone ∆ ⊂ Sym 2 as a quotient of a plane C The isolatedness of the singularity of M follows from the transversality of the map M to the stratified variety ∆ on C s \ {0}. The latter holds due to τ GL,Sym 2 (M) < ∞ (see, for example, [10,11]). Doing the same for a whole SL-versal deformation of M, we lift the singular Milnor fibre V of M to the smooth σ-invariant Milnor fibre V of M . The 2-covering V → V is ramified over the singularities of V.
This way the vanishing of a corank 2 cycle from Section 3.1 −x − z 2 1 − · · · − z 2 s−2 + λ y y x lifts to the standard vanishing a 2 + b 2 + z 2 1 + · · · + z 2 s−2 = λ of a Morse cycle in C s . We call this σ-invariant Morse cycle a short cycle.
On the other hand, the lift of a corank 1 vanishing cycle consists of two disjoint Morse cycles e and σ * (e). We call their sum e + σ * (e) a long cycle.
The self-intersections of short and long cycles are the same as in Arnold's boundary function situation: zeros if s is even, 2 and respectively 4 if s ≡ 1 mod4, and −2 and respectively −4 if s ≡ 3 mod4.  h(x, z) y y x , z ∈ C s−2 , the hypersurface ∆(M) lifts to the hypersurface h(b 2 , z)−a 2 = 0 in C s a,b,z . The latter is Arnold's lift h(x, z)| x=b 2 of the boundary function h followed by a one-variable stabilisation. The action of Arnold's boundary involution b → −b is extended here to the sign change of the stabilising variable (cf. [27]). We have already noted that replacement of the function h in M by its R ∂ -versal deformation provides a GL-versal deformation of M, with the bifurcation diagram of h becoming the matrix bifurcation diagram Σ. Therefore, the monodromy group of the matrix family M coincides in this case with the monodromy group of the one-variable stabilisation of the boundary function h.

Corank 3 simple symmetric matrices and simple odd functions
The singular Milnor fibre V of a corank n matrix family of any of the three types we have been considering in this paper has a filtration where V c is the set of matrices of corank at least c. The monodromy action of π 1 (Λ \ Σ) respects this filtration. The action of π 1 (Λ \ Σ) on H * (V c ) reduces to that of its quotient π 1 (Λ \ ∪ n i=c Σ i ). In the case of corank n symmetric matrix families M(u) = (m ij (u)) n i,j=1 , u ∈ C s , the approach developed in the previous subsection generalises to the description of the monodromy on V n−1 by introduction of an isolated complete intersection singularity M : (m ij (u)) = (a i a j ) in C s+n u,a (17) symmetric under the involution σ : a → −a. The subset V n−1 of the singular Milnor fibre of M is 2-covered by the σ-invariant Milnor fibre V n−1 of M. The definition of the intersection form on V n−1 follows Definition 5.1 with the only difference that we should now take the part H (−) n of the homology of V n−1 on which σ acts as multiplication by (−1) n . Respectively, long cycles change to e + (−1) n σ * (e). In the self-intersection numbers of short and long cycles and in the expressions (16) of the Picard-Lefschetz operators, the dimension s should be replaced by s + n − 2.
For symmetric matrix singularities of form (10), this construction produces on H * (V n−1 ) either odd or even (depending on the dimension) versions of the monodromy groups corresponding to the isolated boundary function singularities h.
We are going to use our double-cover construction to explain a relation between corank 3 simple symmetric matrix singularities in 4 variables (those from item (b) of Theorem 3.6) and simple singularities of odd functions. This relation was suggested by a coincidence between sets of degrees of basic invariants of certain Shephard-Todd groups and sets of weights of parameters of versal deformations of such matrix families. Table 1 shows these quasi-homogeneous versal matrix deformations along with the associated curve singularities.
We will now provide detailed information about the matrix singularities of Table 1 and about its last two columns. Some of this information does not contribute directly to revealing the relationship between the matrices and odd functions. Recall that the SL Tjurina number τ of a table family M is the subscript in its notation. (b) the bifurcation diagram Σ is just Σ 2 and has multiplicity τ + 1.
Proof. (a) For the I k+1 series, we notice that critical matrices off ∆ must have z = w = 0. After that it is easy to derive that, for a generic λ ∈ C k+1 , critical matrices are in one-to-one correspondence with the solutions of the degree k + 1 equation 2x 2 P ′ + xP + λ 2 k = 0 where P is the polynomial in x in the central entry of the deformed matrix.
For the II τ families, consideration of the adjoint matrix and of its inverse establishes a similar one-to-one correspondence between non-degenerate critical matrices and solutions of the degree τ equation 3(p + 2wp ′ ) 2 = 4wqq ′2 where p and q are the polynomials in w in the deformed entries m 12 and m 22 in Table 1.
(b) We have Σ 3 = ∅ due to the dimension of the domain. A Σ 1 = ∅ check has been done by separate calculations for the I k+1 and II τ cases, but we prefer to omit these calculations.
Part (a) of the Proposition completes an extension of the Damon-Pike conjecture to all simple matrix singularities of Theorem 3.6.
1. Third column of Table 1. In our dimensional settings, the singular locus V 2 = V n−1 of the singular Milnor fibre is a smooth curve. Following (17), we lift the SL-miniversal deformations to C 4+τ +3 x,y,z,w;λ;a.b.c by equating them  for I τ and to   a 2 ab ac ab b 2 bc ac bc c 2   for II τ .
Since the rank of our non-deformed matrix families (C 4 , 0) → (Sym 3 , 0) is 4, this gives us deformations of complete intersection curves M embeddable into C 3 a,b,c and symmetric under the involution σ : (a, b, c) → (−a, −b, −c). The lifted SL-miniversal deformations turn out to be miniversal in the class of σ-symmetric complete intersections.
The curve singularities M coming from the matrix families I k+1 and II 4 are the complete intersections D 2k+3 and U 9 from Giusti's list of simple curves in C 3 without any symmetry requirements [12,13]. They are the only centrally symmetric curves on that list.
The curves M corresponding to the families II 5 and II 6 become non-simple if the symmetry is removed. The way we denote them in the Table extends naturally Giusti's notations.
The subscripts in the notations of the complete intersections M are their Milnor numbers which all turn out to be equal to 2τ + 1. The map V 2 → V 2 is an unramified 2-covering, and therefore, in all table cases V 2 is a wedge of τ + 1 circles. Table 1. Exclude from the equations of V 2 ⊂ C 3 a,b,c the variable entering both of them linearly. This defines a projection p of V 2 onto a curve V ′ 2 ⊂ C 2 which is a deformation of the planar curve singularity M ′ given in the 4th column of the Table. The V ′ 2 passes through the origin 0 ∈ C 2 , and V 2 ≃ V ′ 2 \ {0}. This is why µ( M ′ ) = µ( M ) − 1 = 2τ. The involution σ of C 3 projects to the central symmetry σ ′ of C 2 , and the curve V ′ 2 is σ ′ -symmetric. Moreover, the whole σ-symmetric miniversal deformation of the complete intersection M ⊂ C 3 a,b,c (obtained earlier by the lift similar to (17)) becomes now a deformation miniversal in the class of odd functions on the plane. The notations of the odd functions used in the Table are taken from [14,15,18].

Last column of
Let Σ and Σ ′ be the natural bifurcation diagrams of respectively M and M ′ , that is, the sets of all points in the bases (C τ , 0) of their σ-symmetric and odd miniversal deformations for which the corresponding symmetric curves are singular. The effect of the transition M → M → M ′ on the bases of the corresponding miniversal deformations and bifurcation diagrams is described by Proposition 6.2 For all matrix singularities of Table 1, the three pairs (C τ , Σ), (C τ , Σ) and (C τ , Σ ′ ) are biholomorphic.
The variables in P λ are exactly those used in Table 1.
An alternative description of the G 31 discriminant which extends to the diagrams Σ of the other two exceptional matrix families of Table 1 is as follows. The SL-miniversal deformations of these families are of the form   x p(w) y p(w) −y + q(w) z y z w   .
Set here b = uc, divide the equation by c and set c 2 = w, to end up with In the three table cases this is a deformation of respectively A 5 , D 6 and E 7 . It extends to an R-miniversal deformation by addition of the term λ τ u 2 . Therefore, the bifurcation diagrams of the families II 4 , III 5 and IV 6 are non-generic sections of the discriminants of the Weyl groups A 5 , D 6 and E 7 by smooth hypersurfaces. The deformations (18) contain the constant and linear terms, which is consistent with the fact that the multiplicities of the diagrams Σ are τ + 1.
4. Coincidence of three simple classifications. The following fact is easy to prove. Proposition 6.3 The last column of Table 1 contains a complete classification of simple singularities of odd functions on C 2 .
One may notice that odd functions on C are all A even , and there are no simple odd functions on C >2 .
Together with part (c) of Theorem 3.6, the last Proposition provides a not very much expected Corollary 6.4 There exists a one-to-one correspondence between simple singularities of three classifications: • of odd functions on C 2 , • of corank 3 SL-or GL-simple symmetric matrix families in 4 variables, and • of corank 3 SL-or GL-simple families of arbitrary square matrices in 7 variables.
Each triplet of simple singularities has the same bifurcation diagram.