Liftable vector fields, unfoldings and augmentations

We study liftable vector fields of smooth map-germs. We show how to obtain the module of liftable vector fields of any map-germ of finite singularity type from the module of liftable vector fields of a stable unfolding of it. As an application, we obtain the liftable vector fields for the family $H_k$ in Mond's list. We then show the relation between the liftable vector fields of a stable germ and its augmentations.


Introduction
Since the beginning of the study of singularities of differentiable maps with Whitney and Thom, classification of singularities has been one of the main research subjects. Classical techniques, even with the help of computers, have, in a way, exhausted their potential and forced the appearance of new methods in order to attend the growing need of harder classifications. Examples of these new methods can be found in [3,12]. In [1] Arnol'd defined liftable vector fields and showed how they can be used for classification purposes. In fact, the new methods developed rely in many aspects in the computation of liftable vector fields. However, it is not easy at all to compute liftable vector fields in general.
Here we consider A -classification of map-germs f : (K n , S) → (K p , 0), where K = R or K = C, S ⊂ K is a finite subset and the map-germ is either smooth when K = R or holomorphic when K = C. In [11], the authors gave a method to construct liftable vector fields for a certain class of map-germs f which includes stable germs. They then showed how to obtain liftable vector fields of a finite singularity type germ f from liftable vector fields of F , a stable unfolding of it. By finite singularity type, we mean a germ which admits a stable unfolding with a finite number of parameters. Furthermore, if F is a one-parameter stable unfolding, they showed that any liftable vector field of f comes from a liftable vector fields of F by the previous construction. Although the class of map-germs which admit one-parameter stable unfoldings seems to be quite big amongst simple germs, it is still not fully understood when this is the case.
In this paper we generalize the result in [11], that is, we prove that all liftable vector fields of any finite singularity type map-germ can be obtained from the liftable vector fields of a stable unfolding of it, with no restriction on the number of parameters. The main result is that if F is an r-parameter stable unfolding of f , then where π 1 is the projection onto the first p components, i * is the morphism induced by i(X) = (X, 0) and G is the submodule of θ p+r generated by ∂/∂X k , with 1 ≤ k ≤ p and Λ i ∂/∂Λ j , 1 ≤ i, j ≤ r (here we denote by (X, Λ) the coordinates in C p × C r ). As an application, we obtain the liftable vector fields for the family H k in Mond's list of simple germs f : (K 2 , 0) → (K 3 , 0) [9]. In the last section, we turn our attention to families of singularities which can be obtained by the method of augmentation. We show the relation between the liftable vector fields of a one-parameter stable unfolding of a map-germ and of its augmentations. We then use this result to prove the following: let F be a oneparameter stable unfolding of f : (C n , S) → (C p , 0) with p ≤ n + 1 and let Af be the augmentation of f by F and a quasihomogeneous function φ, then π 2 (i * (Lift(Af ))) = π 2 (i * (Lift(F ))), where now π 2 is the projection onto the last component. This property allows us to use some results in [2,12,13], where this condition is required.

Notation
We consider map-germs f : (K n , S) → (K p , 0), where K = R or K = C, and S ⊂ K n a finite subset. For simplicity, we will say that f is smooth if it is smooth (i.e. C ∞ ) when K = R or holomorphic when K = C. We denote by O n = O K n ,S and O p = O K p ,0 the rings of smooth function germs in the source and target respectively and by θ n = θ K n ,S and θ p = θ K p ,0 the corresponding modules of vector field germs. The module of vector fields along f will be denoted by θ(f ). Associated with θ(f ) we have two morphisms tf : θ n → θ(f ), given by tf (χ) = df •χ, and wf : θ p → θ(f ), given by wf (η) = η • f . The A e -tangent space and the A e -codimension of f are defined respectively as It follows from Mather's infinitesimal stability criterion [8] that a germ is stable if and only if its A e -codimension is 0. We refer to Wall's survey paper [14] for general background on the theory of singularities of mappings.
We will say that f has finite singularity type if where m p is the maximal ideal of O p and f * : O p → O n is the induced map. Another remarkable result of Mather is that f has finite singularity type if and only if it admits an r-parameter stable unfolding (see [14]). We recall that an r-parameter unfolding of f is another map-germ and such that f 0 = f . Along the paper, we use the notation of small letters x 1 , . . . , x n , λ 1 , . . . , λ r for the coordinates in K n ×K r and capital letters X 1 , . . . , X p , Λ 1 , . . . , Λ r for the coordinates in K p × K r . Definition 2.1. Let f : (K n , S) → (K p , 0) be a smooth map-germ.
(1) A vector field germ η ∈ θ p is called liftable over f , if there exists ξ ∈ θ n such that df • ξ = η • f (i.e., tf (ξ) = wf (η)). The set of vector field germs liftable over f is denoted by Lift(f ) and is an O p -submodule of θ p . (2) We also define τ (f ) = ev 0 (Lift(f )), the subspace of T 0 K p given by the evaluation at the origin of elements of Lift(f ).
When f has finite A e -codimension, the space τ (f ) can be interpreted geometrically as the tangent space to the isosingular locus (or analytic stratum in Mather's terminology) of f . This is the well defined manifold in the target along which the map f is trivial. See [3,6] for some basic properties of τ (f ).
When K = C and f has finite singularity type, we always have the inclusion Lift(f ) ⊆ Derlog(∆(f )), where ∆(f ) is the discriminant of f (i.e., the image of non submersive points of f ) and Derlog(∆(f )) is the submodule of θ p of vector fields which are tangent to ∆(f ). Moreover, we have the equality Lift(f ) = Derlog(∆(f )) in case f has finite A e -codimension (see [4,10]). Definition 2.2. Let h : (K n , S) → (K p , 0) be a map-germ with a 1-parameter stable unfolding H(x, λ) = (h λ (x), λ). Let g : (K q , 0) → (K, 0) be a functiongerm. Then, the augmentation of h by H and g is the map A H,g (h) given by (x, z) → (h g(z) (x), z).

Liftable vector fields and stable unfoldings
In this section we show that all the liftable vector fields of any map-germ of finite singularity type can be obtained from the liftable vector fields of a stable unfolding (not necessarily versal) of it.
Theorem 3.1. Let f : (K n , S) −→ (K p , 0) be a non-stable germ of finite singularity type and let F be an r-parameter stable unfolding. Then, where π 1 is the projection onto the first p components, i * is the morphism induced by i(X) = (X, 0) and G is the submodule of θ p+r generated by Proof. The proof is similar to the proof in Theorem 2 in [11]. We will prove it by a double inclusion. For simplicity, we set ϕ = π 1 • i * : θ p+r → θ p .
Consider η ∈ Lift(F ) ∩ G. There exists ξ ∈ θ n+r such that dF • ξ = η • F . Evaluating this system of equations in λ = 0 we have the following: is the map germ g(X, Λ) = (X, Λ 2 ) as was stated in Theorem 2 in [11]. However, if r > 1 and g : (K p × K r , S) → (K p × K r , 0), where S = {z 1 , . . . , z r } and the branch at z i is given by g i (X, Λ) = (X, Λ 1 , . . . , Λ 2 i , . . . , Λ r ), then 3.1. Liftable vector fields of H k . As an application of Theorem 3.1 we compute the liftable vector fields of a family of germs which do not admit a 1-parameter stable unfolding. Consider H k : For any k ≥ 2, H k admits a 2-parameter stable unfolding A -equivalent to F : which is versal for k = 2. In fact, the change of variables in the source In this case, In Example 3.4 in [7], the generators for Lift(F ) are calculated. We show these vector fields as matrices for simplicity: In [11,Lemma 6.1], it is shown that if two maps f and g are A -equivalent, i.e. there exist h and H diffeomorphisms in source and target such that f = H • g • h, then the map L f,g : Lift(g) → Lift(f ) such that L f,g (η) = dH •η•H −1 is a bijection. We therefore can obtain Lift(F k ) from Lift(F ) by composing its generators by G −1 k on the right and multiplying by dG k on the left.
The module Lift(F k ) ∩ G is formed by vector fields of type η =α 1 η ke +α 2 η 1 k1 +α 3 η 2 k1 +α 4 η 1 k2 +α 5 η 3 k1 +α 6 η 2 k2 +α 7 η 3 k2 , such that the first and third components are divisible by U 1 or V 2 , where thẽ α i are functions in U 1 , V 1 , V 2 , W 1 and W 2 . This means that the first and third components evaluated at U 1 = V 2 = 0 are equal to 0, which yields two conditions on The conditions can be rewritten as By Theorem 3.1, the liftable vector fields of H k are obtained from the second, fourth and fifth components vector fields of the type of η evaluated at U 1 = V 2 = 0. That is, where the α i are germs of functions.
We use conditions (1) and (2) to eliminate the functions α 4 , α 5 . Then, after some easy simplifications in the obtained generators, we arrive to the following: Theorem 3.3. The module of liftable vector fields over the map germs H k : (K 2 , 0) → (K 3 , 0), given by H k (x, y) = (x, y 3 , y 3k−1 + xy) is generated by the following vector fields  Remark 3.4. In fact, these vector fields can be obtained by applying ϕ to the following vector fields in Lift(F k ) ∩ G:

Liftable vector fields and augmentations
In this section, we restrict ourselves to the complex case K = C. Suppose f : (C n , S) −→ (C p , 0) admits a one-parameter stable unfolding F (x, λ) = (f λ (x), λ). Let φ : (C, 0) → (C, 0), and let Af (x, z) = (f φ(z) (x), z) = (X, Z) be the augmentation of f by F and φ. We shall establish a relation between the liftable vector fields of an augmentation Af and those of the stable unfolding F .
Proof. We put η p+1 (X, Z) = Zψ(X, Z) + β(X), for some functions ψ, β, hence For simplicity we suppose η ∈ Derlog(h) and comment on the general case later. So, we have Since φ(Z) = Z k and φ ′ (Z) = kZ k−1 , the right hand side of this equality can be seen as a series in Z of type On the other hand, for i = 1, . . . , p, Analogously for Zψ(X, Z) and ∂H ∂Λ (X, φ(Z)). By identifying the coefficients at both sides of the equality, the left hand side has a part equal to 0 (the sum of the coefficients whose terms do not appear in the right hand side) and the rest is used to rewrite the equality for some functions A i , i = 1, . . . , p + 1. The part which was equal to 0 corresponds to a certain ϕ ∈ Derlog(h). Dividing the previous equation by φ ′ (Z) and substituting Λ = φ(Z), we obtain and η = η − ϕ.
In [12] the operation of simultaneous augmentation and concatenation was defined as the multigerm defined by {Af, g} where g is a fold map or an immersion. The authors proved a formula relating the A e -codimensions of {Af, g} and of f under a certain condition, namely that π 2 (i * (Lift(Af ))) = π 2 (i * (Lift(F ))) where π 2 is the projection to the last component and i is the immersion i(X) = (X, 0). This condition appears again in [2] and in [13] where the authors say that all examples they have studied satisfy this condition but they were not able to prove it. We will use Theorem 3.1 and Lemmas 4.1, 4.2 in order to give a proof of this fact when the augmenting function φ is quasihomogeneous.