Transversality in families of mappings

Transcription

1 Transversality in families of mappings C. T. C. Wall November 8, 2005 Introduction By a classical result of Thom [20], if N and P are smooth manifolds, and Q is a smooth submanifold of P, the set of maps f : N P transverse to Q is residual in C (N, P ) (if Q is a closed subset of P, the set of transverse maps is also open). However if U is a further smooth manifold, and {f u } a family of smooth mappings, we cannot expect to be able to deform the family to make each f u transverse to Q. This situation was considered by Bruce [2], who showed that, for a residual set of maps F : N P U, we have (a) for each u U, the set of points in N at which f u is not transverse to Q is discrete; and (b) at each such point, the class of the contact of f u (N) with Q has finite codimension. (This is the result he established: the formal statement given was more explicit but weaker.) This result is not entirely satisfactory: we may hope also, for example, that each contact class presented is universally unfolded in the family. We address this problem, and show that this hope is justified under a hypothesis of nice dimensions. After recalling Mather s theory of topological stability, we establish a result valid in all dimensions with topologically versal unfoldings. Since the proofs of these results involve numerous translations between equivalent versality conditions, we begin with a section developing a notation and approach to facilitate such proofs. We then consider the corresponding question with the submanifold Q replaced by a stratified subset of P. This requires an extensive study of theories of contact equivalence relative to a subset of the target. Good algebraic conditions and criteria are only known under an analyticity hypothesis; for some results this needs to be strengthened to a holonomic condition. With these preliminaries, we obtain a parallel to the theory of the earlier section. However a full account of a topological analogue remains beyond our reach. 1

2 Preliminaries By manifold below, we will understand C manifold; similarly, all maps will be assumed smooth, i.e. C. Given a manifold N and a point x N, E N,x denotes the ring of germs at x of C functions on N. We write m N,x for the ideal in this ring of functions vanishing at the point x and more generally, for any subset Z N, m N,Z,x for the ideal in E N,x of functions vanishing on Z; we omit N from the subscript if it is clear from the context. If R is any commutative ring, M an R module, and A M, R.A denotes the R submodule of M generated by A; in particular, if M = R this is the ideal generated by A. For f : N P a smooth map, the critical set Σ(f) is the set of x N such that the tangent map T x f : T x N T f(x) P is not surjective. For y P, we set Σ(f, y) := Σ(f) f 1 (y). For singularity theory terminology, notation and source references, I mainly follow [11]; see also the expository article [23], and book [19]. However, the notation here for the space of tangent vectors along the germ at x N of f : N P is θ f rather than θ(f); in the case f is the identity θ N here (in the cited references B, θ x, θ n respectively), but θ n if (N, x) = (R n, O). Many of the key ideas will however be recalled below in some detail to facilitate the exposition. Indeed, the author has found that extending the original theory has required a closer understanding thereof than he originally possessed. Following Mather, we define an unfolding (of maps or of germs) to be a pullback diagram f N P i j N F P such that j is transverse to F ; usually we suppose j an embedding. We often write (i, j) : f F for short. We sometimes refer to these as non-parametrised unfoldings, to distinguish from the case where we write N = N U, P = P U, u 0 is a base point in U defining inclusions i, j, and we require that F be compatible with projection on U. Various categories of (parametrised) unfoldings have been defined, with corresponding theories of versality. We will refer below to our survey in [22] of early results and to Damon s axiomatic treatment [6]. We denote by π N and π U the projections of N U on its factors, and correspondingly for other products. Given F : N U P, we write F (x, u) = f u (x). We follow the convention of [9] and refer to F as a deformation of the maps f u and the map (F, π U ) : N U P U as the corresponding unfolding. In most of the text we are concerned with map-germs, either at a single point (usually denoted x 0 N) or at a finite set (usually S = {x i } N). We will state explicitly when we come to global considerations. We will need reference to topologies on function spaces. To avoid undue complications, we will use the Mather topology throughout: see [10] for its basic properties and [19, 3.4] for notation, definitions of several topologies, and a collection of results with acknowledgements to earlier references. The topology 2

3 used here is denoted τw in [19]. A residual subset is a countable intersection of dense open sets, hence (by Baire s theorem) is itself dense; these are also called G δ subsets. In particular, a countable intersection of residual subsets is residual. Thus we can impose a countable family of transversality conditions and still have a dense set of maps. This fails for an uncountable family, e.g. we cannot deform F : N U P to make each f u transverse to Q. Thom s original paper on transversality is [20]. This was quickly followed by improvements (the reference to residual subsets is not in the original) and generalisations, notably to jet space. A result general enough to include numerous extensions was given by Abraham [1]. Next, Mather proved a multitransversality theorem. The concept of multi-transversality was introduced in [13, 3]: however, Mather does not use this term. He proves that the set of maps f such that r j k f is transverse to a given submanifold V of a product of r copies of J k (N, P ) is residual. The submanifolds V of most interest are as follows: given submanifolds V i J k (N, P ) (1 i r), take the intersection of the product Π i V i with the pre-image of the diagonal P P r. Transversality at S in this case means that, first, f is transverse at x i to V i, with pre-image X i, say, where f(x i ) = y for each i, and second, that the images B i T y P of the T xi X i satisfy B i T y P = (T y P ) r, or equivalently, that T y P projects onto (T y P/B i ). An slightly stronger formulation was given by Looijenga [8, p 127]: he says that f is multi-transverse at S = {x i } to the submanifolds V i if (a) at each x i, j k f is transverse to V i, so each f 1 (V i ) is the germ of a manifold, and (b) these immerse in P and their images have transverse intersection. We see that, apart from the immersion requirement, these conditions are equivalent. 1 K equivalence Since much of this paper is devoted to detailed properties of K equivalence and variants and generalisations thereof, we begin with a rather detailed review of its definition and a number of its properties. Consider map-germs N P, and fix a source point x 0 N and a target point y 0 P. Define i : N N P by i(x) = (x, y 0 ). An element of K consists of a pair (H, h) where H is a diffeomorphism-germ of N P and h a diffeomorphism-germ of N such that H i = i h and π N H = h π N, i.e. the following diagram commutes. 1 y 0 (N, x 0 ) (N P, (x0, y 0 )) π N (N, x0 ) h H h (N, x 0 ) 1 y 0 (N P, (x0, y 0 )) π N (N, x0 ) Then K acts on the set of map-germs 1 (N, x 0 ) P via its action on their graphs: thus (H, h)f = g: if, for each x N (in some neighbourhood of x 0 ), H(x, f(x)) = (h(x), g(h(x))). 1 We do not always insist that f(x 0 ) = y 0. 3

4 Since the definition of K depends crucially on the target point y 0 we should, and occasionally will, emphasise this by writing K y0. The group K is the semi-direct product of the group R of diffeomorphismgerms of (N, x 0 ), acting on map-germs by composition, and the subgroup C of pairs (H, h) with h the identity, which it is often convenient to consider separately. We also consider the set of diffeomorphism-germs of N with source x 0 and variable target point, which we regard as an extended version of R and denote R e2, and write K e := R e.c. The tangent space at f to the set of map-germs (at x 0 ) N P is the set of germs θ f of vector fields ξ : N T P making the following diagram commute: T f T N T P π N π P N f P This, and other similar expressions below, is to be interpreted (following Mather [11], e.g. Proposition 7.4) as giving genuine calculations at the level of tangent spaces to jet spaces: the tangent space to J k (N, P ) at the germ of f at x 0 is identified with θ f /m k+1 x 0.θ f. In particular, we can identify θ f /m x0.θ f with T y0 P ; for ξ θ f, we denote by ξ its image in T y0 P. We will usually abstain from further such details. In particular, taking f to be the identity, the tangent space to the set of self-maps, or to the group of diffeomorphisms, is the set θ N of germs at x 0 of vector fields tangent to N; we can identify this with the tangent space T R e. The tangent space T R of R consists of germs at x 0 of tangent vector fields on N vanishing at x 0, which is equal to m x0.θ N. The action of R e on the set of map-germs N P corresponds to a tangent map tf : θ N θ f. In local coordinates, we can identify θ N, θ f with the free E N,x0 modules with the symbols / x i, respectively / y j as bases, and tf as the E N,x0 linear map with matrix (y j f)/ x i. The tangent space of C consists of germs of vector fields on N P pointing in the P direction (i.e. with zero projection on N), and vanishing on i(n). In local coordinates, the N projections vanish; the P projections lie in m y0.e p N P,(x 0,y 0) (where y 0 is the target point). Pulling back by the graph Γf : N N P, this gives f m y0.θ f. We can thus write the tangent spaces to the orbits of f under these groups and sets of transformations as T R e f = tf(θ N ), T Rf = tf(m x0 θ N ), T Cf = f m y0.θ f, T Kf = T Rf + T Cf = tf(m x0 θ N ) + f m y0.θ f, T K e f = T R e f + T Cf = tf(θ N ) + f m y0.θ f. We introduce notation for the quotients Nf := m x0.θ f /T Kf, N e f := θ f /T K e f. 2 We thus write T R e f for what was denoted T erf in [22]. and 4

5 We also need to refer to left- and right-left-equivalence. The group L consists of diffeomorphism-germs of (P, y 0 ) and acts on the set of maps f by composition on the left. Correspondingly, we have L e, T L e = θ P, T L = m y0.θ P ; the action on the space of maps f : N P induced by composition inducing the map ωf : θ P θ f of tangent spaces, the group A := R L; T L e f = ωf(θ P ), T Lf = ωf(m y0.θ P ), and T Af = T Rf + T Lf. When we consider germs at a finite set S rather than at a single point x 0, each of θ f, T R(f), T R e (f), T C(f), T K(f) and T K e (f) is a direct sum over points of S, but T L(f) is the diagonal pre-image of θ P, so this and T A(f) are not direct sums. Mather defines germs to be contact equivalent if they are diffeomorphic to K equivalent germs: it suffices to compose a K equivalence with a diffeomorphism of the target. The tangent space to the contact class is thus T Kf +T L e f = tf(m x0.θ N )+f m y0.θ f +ωf(θ P ). Since ωf(m y0.θ P ) f m y0.θ f, ωf induces a surjection of T P onto (T Kf + T L e f)/t Kf. The same definition applies to germs at a finite set S: note that here it is still T P (not a direct sum of copies thereof) that maps onto the quotient of the tangent space to the contact class by that to the K orbit. We recall that if a k-jet z is such that all germs f with this k-jet are K equivalent, then z is said to be k-sufficient, and the germs f to be k K determined, and in particular, finitely K determined. Corresponding definitions hold with A etc. replacing K. An important property (of which more anon) is that if the germ of f at x 0 is finitely K determined and critical, x 0 is an isolated point of Σ(f, f(x 0 )). Finite determinacy is necessary and sufficient for the existence of unfoldings which are versal in a corresponding sense: we will recall a general formulation in Theorem 4.1 below. It is also convenient to use Mather s expression finite singularity type for the condition of K finite determinacy of germs. There is also a global notion which we will recall below. Mather also gives a characterisation [12, Theorem 2.1]: Lemma 1.1 Finitely K determined germs are contact equivalent if and only if the corresponding local algebras Q(f) := E x0 /f m y0.e x0 are isomorphic. We now give a new presentation of some properties of germs f, which will serve as a model for other accounts below (MT-theory, theory relative to a subvariety of the target). Let C(f) denote the set of jets contact equivalent to j k f (where we assume this jet to be K sufficient), then C(f) is a submanifold of J k (N, P ). Write C s (f) for the (germ of) the set of points x in the source N with j k f(x) C(f), and C t (f) for its image in the target P. Lemma 1.2 If f is finitely-k determined and not a submersion germ, ξ θ N and tf(ξ) f m y0.θ f, then ξ m x0.θ N. Proof We recall the proof of [22, Theorem 4.5.1]. If ξ m x0.θ N, then in suitable local coordinates (with each of x 0 and y 0 at the origin) we can take 5

6 ξ = / x 1. The hypothesis is equivalent to stating that there exist functions a i,j (x) such that, for each i, f i / x 1 = j a i,j (x)f j (x). But now it follows in turn that f takes the x 1 axis to 0, that the ideal f m y0.e x0 is constant along the x 1 axis, and that so is the contact class. Since, by hypothesis, f is singular at x 0, the whole x 1 axis is contained in Σ(f), contradicting the fact that the origin is isolated in Σ(f, y 0 ). An alternative argument goes: since tf(ξ) is tangent to the point y 0, ξ is tangent to the pre-image of y 0. Since x 0 is an isolated singular point of f 1 (y 0 ), the assertion follows. We can now compare the normal spaces N(f) and N e (f). Corollary 1.3 The kernel of the natural map N(f) N e (f) is isomorphic to (T f) 1 (0)/{α Ker (T f) tf(α) f m y0.θ f }; its cokernel to T P/T f(t N). If f is finitely K determined and not a submersion germ, the kernel simplifies to (T f) 1 (0). P Q 1 Proof We first note that given the commutative diagram of Q 2 R inclusions of abelian groups, the maps Q 1 /P R/Q 2 and Q 2 /P R/Q 1 both have kernels isomorphic to Q 1 Q 2 /P and cokernels isomorphic to R/Q 1 + Q 2. We may thus replace the map N(f) N e (f) by the map from (tf(θ N ) + f m y0.θ f )/(tf(m x0 θ N ) + f m y0.θ f ) to θ f /(m x0.θ f ). We can identify θ f /m x0.θ f with the tangent space (at y 0 ) T P. There is a natural map from T N = θ N /m x0.θ N onto the first quotient; its kernel consists of the classes ξ T N of the ξ θ N with tf(ξ) (tf(m x0.θ N ) + f m y0.θ f ). We can add an element of m x0.θ N to ξ without changing ξ, and so suppose tf(ξ) f m y0.θ f. Finally, if f is finitely K determined and not a submersion germ, then by Lemma 1.2 this condition implies that ξ = 0, so we have replaced N(f) N e (f) by T f : T N T P. Proposition 1.4 Exclude the case of submersion-germs, where Nf = N e f = 0. Then f is a stable germ if and only if it is transverse to C(f). For f stable, (i) The preimage C s (f) of C(f) is a submanifold of N, and its normal space can be identified with Nf. (ii) The restriction of f to C s (f) is an immersion. (iii) The set C t (f) is a submanifold of P ; its normal space can be identified with N e f. Proof Mather s criterion for stability of a germ [13, Theorem 4.1] is tf(θ N )+ f m y0.θ f + ωf(θ P ) = θ f, which expresses precisely that f is transverse to C(f). 6

7 (i) It follows that the preimage C s (f) of C(f) is a submanifold of N, and its normal space can be identified with that of C(f) in J k (N, P ), viz. θ f /T Kf + ωf(θ P ). The natural map Nf = mx 0.θ f θ T Kf f T Kf+ωf(θ P ) is surjective, since ωf(θ P ) maps onto θ f /m x0.θ f = T P ; it is injective since if α mx0.θ f can be expressed as β + γ + δ with β tf(m x.θ N ), γ f m y.θ f and δ ωf(θ P ), then we must have δ m x0.θ f, hence δ ωf(m y.θ P ) f m y.θ f. Thus α T Kf. (ii) Suppose ξ θ N projects to a vector ξ θ N /m x0.θ N = Tx0 N which is tangent to C s (f) and lies in the kernel of T f : T x0 N T y0 P. Then on one hand, for some η θ P, tf(ξ) ωf(η) T Kf; on the other, tf(ξ) m x0.θ f. It follows that ωf(η) m x0.θ f, and hence that η m y0.θ P, so that ωf(η) T Kf, and hence tf(ξ) T Kf. Adding a suitable element of m x0.θ N to ξ, we can now suppose that tf(ξ) f m y0.θ f. But now by (i), ξ m x0.θ N, so ξ = 0. (iii) It now follows that indeed C t (f) is (locally) a submanifold. Its tangent vectors lift to those η θ P which correspond to some ξ lifting a tangent vector of C s (f), i.e. (arguing as above) with tf(ξ) ωf(η) T Kf. We thus have the η with ωf(η) T K e. Since, by Mather s criterion for stability, ωf maps θ P onto θ f /T K e f = N e f, the result follows. We proceed to germs of f at a finite set S = {x i } of points of N, with a common target y 0 P. Lemma 1.5 (a) The following are equivalent: (i) the germ of f at S is stable, (ii) i tf(θ N,xi ) + i f m y0.θ fxi + ωf(θ P ) = i θ fxi, (iii) T y0 P maps onto the direct sum i N e f xi, (iv) r j k f is transverse at S to the contact class of f at S, (v) f is multi-transverse at S to the contact classes of f at the x i. (b) When these conditions hold, C t (f, y 0 ) := i C t(f xi ) is a smooth submanifold of P with normal space i N e f xi. Proof The equivalence of (i) and (ii) is the criterion of Mather [13, Theorem 4.1]; equivalence of (ii) with (iv) follows from the calculation of the tangent space to the (multi-)contact class. By Proposition 1.4, the germs of f at the x i are stable if and only of f is transverse there to the contact classes; moreover, when this holds, the C s (f xi ) map immersively to P, with images the C t (f xi ). For multi-transversality we require also that these images have transverse intersection, which is condition (iii). Expanding (iii) as an equation gives (ii). This proves (a), and (b) follows. A useful property of the spaces Nf and N e f is their stability under unfoldings. Lemma 1.6 Let (i, j) : f F be an unfolding of a map-germ. Then there are natural isomorphisms Nf NF and N e f N e F. 7

8 Proof We can identify N e f with the cokernel of the map of free modules over Q(f) whose matrix is the Jacobian matrix of f, with entries reduced modulo f m y0.e N. Now the equations of an unfolding can be given in the form F i (x 1,..., x n, x n+1,..., x n+a ) = f i (x 1,..., x n ) + g i (x 1,..., x n ) (1 i p), where the g i belong to the ideal p 1 x n+i.e N U, and F p+j (x 1,..., x n, x n+1,..., x n+a ) = x p+j (1 j a). We deduce first, that in the quotient Q(F ) we must set x p+j = 0 for each j; modulo these we have f i (x 1 (,..., x n ) = 0; ) thus Q(f) = Q(F ). Now the Jacobian J(f) matrix of F has the form, where I 0 I a denotes the identity a a a matrix. Thus N e f = N e F. We can describe Nf analogously as the cokernel of the map of modules which are direct sums of copies of the maximal ideal m x0.q(f) of Q(f), induced by the same matrix. The same argument thus goes through. Lemma 1.7 Let (i, j) : f F be an unfolding. (i) Let x 0 N be such that the germ of F at i(x 0 ) is stable. Then the germ of f at x 0 is stable if and only if i is transverse at x 0 to C s (F i(x0)). (ii) Let S = {x i } be a finite set of points of N, with a common target y 0 P ; suppose the germ of F at i(s) is stable. Then the germ of f at S is stable if and only if j is transverse at y 0 to i C t(f i(xi)). Proof (i) The germ of f at x 0 is stable if and only if T x0 N maps onto the normal space Nf of C(f x0 ). By Lemma 1.6, Nf x0 = NFi(x0). By Proposition 1.4, NF i(x0) is the normal space of C s (F i(x0)). The assertion follows. (ii) By Lemma 1.5, the germ of f at S is stable if and only if T y0 P maps onto i N e f xi. As in (i), this is isomorphic to i N e F i(xi), which is the normal space to i C t(f i(xi)). This proves the result. 2 Transversality to a submanifold of the target Now consider a deformation F : N U P and the corresponding unfolding (F, π U ). We define the partial jet extension j1 k F : N U J k (N, P ) by j1 k F (x, u) := j k f u (x). We require a relative notion of multi-transversality. Adapting the notation of [13, 2], define r ju k F : N (r) U r J k (N, P ) by rju k F (x 1,..., x r, u) := (j k f u (x 1 ),..., j k f u (x r )). Then for any submanifold V of J k (N, P ) r (e.g. of the form (V 1... V r ) with V i J k (N, P )) and finite subset S = {x i } of N, we say F is multi-transverse at (S, u) to V rel π U if r ju k F is transverse to V at (S, u). The arguments of [13, 3] adapt easily to show Lemma 2.1 For any submanifold V of J k (N, P ) r, the set of those deformations F : N U P which are everywhere multi-transverse to V rel π U is residual in C (N U, P ). 8

9 For F : N U P a deformation and y 0 P, write X := F 1 (y 0 ) and denote the projection X N U U by Π F. The next result may be regarded as a version of Lemma 1.5 for deformations. The equivalence in (b) is central to the paper, and motivates much of what follows. Proposition 2.2 (a) The family F induces a K e versal unfolding of f u at (x 0, u) if and only if, for k large enough, j1 k F is transverse at that point to the corresponding K orbit. If this holds, F is transverse at (x 0, u) to y 0, so that X is a manifold. (b) For S a finite set with common target y under f u, the following are equivalent: (i) the germ of f u at S is K e versally unfolded by the family F, (ii) F is multi-transverse rel π U at (S, u) to the corresponding K orbit, (iii) the germ of Π F : X U at (S, u) is stable. Proof (a) The transversality condition is that j1 k F induces a surjective map of the tangent space of N U to the quotient θ f /(tf(m x θ N )+f m y.θ f ). Equivalently, taking out the tangent space of N, we need a surjection of T u U to θ f /{tf(θ N ) + f m y.θ f } = N e f. The versality theorem in the form of [22, Theorem 3.3] states that the unfolding (dim U, F ) of f is K e versal if and only if t F (T u0 U) + T K e (f) = θ f, i.e. t F induces a surjective map of T u0 U to the quotient θ f /T K e (f) = N e f. It remains to observe that the maps of T u0 U induced by j1 k F and t F agree: indeed, both are eventually induced by tf. The final assertion follows. (b) The condition for versality is that the induced map of T u U to the direct sum of the N e f at the points of S is surjective. The transversality condition is that the tangent space of N (r) U maps onto the direct sum of the θ f /(tf(m xi θ N ) + f m y.θ f ). Thus (i) and (ii) are equivalent. In the diagram i 0 N N U X P i 0 X f u0 (F, π U ) (id, π U ) Π F, (1) P i 0 P U = P U i 0 U each of the pairs of arrows labelled i 0 is the inclusion in an unfolding, and X P N U is an open inclusion. Thus by Lemma 1.6, we have isomorphisms N e f (u0,x i) = N e (F, π U ) (u0,x i) = N e (id, π U ) (u0,x i) = N e Π F(u0,x i. Hence both ) conditions are equivalent to T u0 U mapping onto i N e f u0,x i. We now study the contact of the deformation F : N U P with a smooth submanifold Q of P. The contact of f at x N with Q is measured (compare Lemma 1.1) by the isomorphism class of the algebra A := E N,x /f m P,Q,y.E N,x (if f(x) = y Q). Transversality holds if and only if A = E R n p+q,o. We can also express this in terms of F. Consider the unfolding (F, π U ) : N U P U. Then A is naturally isomorphic to E N U,(x,u) /F m P,Q,y.E N U,(x,u) + πu m U,u.E N U,(x,u). 9

10 It is convenient to express P locally as a product Q R, with Q = Q O. Write h := π R f, h t := π R f t and H := π R F. Then A is isomorphic to E N,x /h m R,O.E N,x (and hence to E N U,(x,u) /(H, π U ) m R U,(O,u).E N U,(x,u) ), and the contact is also measured by the K equivalence class of the map-germ h. Proposition 2.3 For each u and finite S fu 1 (Q), the following are equivalent: (i) the contact classes of f u at points of S are simultaneously versally unfolded by the family F, (ii) the germ of h u at S is KO e versally unfolded by the family F, (iii) H is multi-transverse rel π U at (S, u) to the corresponding K O orbit, (iv) F is multi-transverse rel π U at (S, u) to the pre-image by the projection J k (N, P ) J k (N, R) of the K O orbit, (v) the germ of Π F := Π H : F 1 (Q) = H 1 (O) = X U at (S, u) is stable. Proof Equivalence of (ii), (iii) and (v) follows by applying Proposition 2.2 to H. Now (ii) is a reformulation of (i). Equivalence of (iii) and (iv) follows since the pre-image (by a submersion) of a submanifold has isomorphic normal spaces to the original normal space. It is not clear from this statement that the pre-image in J k (N, P ) of the K O orbit in J k (N, R) is independent of the choice of local splitting P = Q R. We will establish this directly below (in Proposition 5.7), and re-interpret these manifolds. Applying this result to all subsets S gives Corollary 2.4 The following are equivalent: (i) for each u and finite S fu 1 (Q), the contact classes of f u at points of S are simultaneously versally unfolded by F, (ii) for each u and finite S h 1 u (O), the germ of h u at S is KO e versally unfolded by F, (iii) H is multi-transverse rel π U to all K O orbits, (iv) F is multi-transverse rel π U to the pre-images by the projection J k (N, P ) J k (N, R) of all K O orbits, (v) Π F : X U is locally stable. Observe that the defining condition refers only to (multi-)jets of F at points of F 1 (Q). Deformations F satisfying these conditions (for some, hence all large enough k) may be regarded as being as transverse to Q as possible ; we will refer to them as C -d-transverse to Q. It follows, in particular, that F is transverse to Q as a map. We began this paper by noting that trying to make each f u transverse to Q imposed an uncountable family of transversality conditions; we have replaced this by seeking to make H (multi-)transverse to the uncountable family of K O orbits. However, progress was made in this situation by Mather in [13]. 10

11 For any n, r and k, there is a closed algebraic subset Y k (n, r) J k (n, r) whose complement is a finite union of J k K orbits. Its codimension is denoted σ k (n, r); and σ k (n, r) increases with k till it attains a constant value σ(n, r). From now on, we assume k large enough so that this bound is attained. Write n = dim N, p = dim P, q = dim Q, but b = dim U; then x := n + b p + q is the expected dimension of X. Then if x < σ k (n, p q), the set of maps F such that j k 1 H(X) avoids Y k (n, p q) is residual. As there are only finitely many) K O orbits outside Y k (n, p q), we can now apply Lemma 2.1 to make the jets of H at points of X transverse (rel π U ) to them. Hence Theorem 2.5 Suppose x < σ(n, p q). Then the deformations C -d-transverse to Q are residual in C (N U, P ); for such deformations F, the associated map Π F : X U is locally C stable. A trivial case is when x < 0; here a deformation is C -d-transverse to Q only if it avoids Q, so X =. Almost as trivial is the case x = 0: again C - d-transversality to Q is generic, and for the corresponding F, X is discrete and Π F an embedding. We recall from [14] that σ(n, p) depends only on n p, except that if n min(p, 2) or n = 3 p 4 we have σ(n, p) > σ(n + 1, p + 1). Write τ(e) = σ(n, n e) for any n min(4, e + 1). Then (n, p) are nice dimensions in the sense of Mather if and only if n < σ(n, p) or equivalently, n < τ(n p). Thus if (x, b) are nice dimensions, x < τ(x b) = τ(n p + q) σ(n, p q) and the Theorem applies. It also applies if n = 1 p (when σ(n, p) = ) and in a few other cases with n = 2 or 3. The above theory is purely local. Globally, there are two natural questions: (i) is the set of deformations C -d-transverse to Q open as well as dense in C (N U, P )? (ii) is the map Π F actually, not just locally stable? To pass from local to actual stability we must impose some global condition, so will suppose that f : N P is proper. For future convenience, we will say that F is a proper deformation when (F, π U ) is a proper unfolding. If this holds, the restriction of (F, π U ) to X Q U is also proper. If, in addition, Q is compact, the composite Π F : X U is a proper map. The same also follows if Q is closed and N is compact, so X is closed and π U : N U U proper. Since (for any N and P ) the space C pr (N, P ) of proper maps N P is open in C (N, P ) (see e.g. [19, ]), its intersection with any residual subset of C (N, P ) is residual in C pr (N, P ). For proper maps, local and global notions of C stability are equivalent; indeed, according to [19, Theorem 4.3.7], they are also equivalent to a large number of alternative forms of stability. We will also require Mather s global notion of finite singularity type. For a map f : N P, consider the space Θ(f) of smooth vector fields over f, and factor out the image by tf of the space Θ(N) of smooth tangent vector fields to N. The result can be regarded as a module over the ring E(P ) of smooth 11

12 functions (not germs) on P, and we say that f has finite singularity type if this module is finitely generated. A geometrical consequence of this is that the restriction f Σ(f) of f to its critical set is a finite map. A basic theorem states that a map has a (finite dimensional) unfolding which is an (infinitesimally) stable map if and only if it has finite singularity type. We recall the Right Stability Theorem. Theorem 2.6 [19, Theorem 3.6.3] Let Q be a closed smooth submanifold of P, and let f : N P be a proper smooth map, transverse to Q. Then there exist a neighbourhood W of f in C (N, P ) and a continuous map Φ : W Diff(N) such that for g W, Φ(g)(g 1 Q) = f 1 Q. We can now state our global result. Theorem 2.7 Suppose (x, b) are nice dimensions, that Q is closed, and that Q or N is compact. Then the set of proper deformations F : N U P C - d-transverse to Q is open and dense in the space C 1,pr(N U, P ) of all proper deformations. For any such F, the associated map Π F : X U is C stable. Proof Apply Theorem 2.6, with N U replacing N and F replacing f. Thus W is a neighbourhood of f in C (N U, P ) and Φ : W Diff(N U) a continuous map, such that, for g W, Φ(g) takes g 1 Q to X. By [19, Lemma 3.4.4], the map Ψ with Ψ(g) = Φ(g) 1 is also continuous. Since the inclusion X N U is proper, it follows from [19, Lemma 3.4.4] that the family π U (Ψ(g) X) : X U depends continuously on g, so we have a continuous map W C (X, U). Then the preimage of the dense open subset of C (X, U) consisting of C stable maps is also open. The formulation in Theorem 2.7 suggests that the natural result to seek outside the nice dimensions should be C 0 stability of the map Π F. In the next section we recall the theory of C 0 stability developed by Thom and Mather, which we will refer to as MT-stability. 3 Topological stability theory Fix source and target dimensions n and p and a degree k. Write W k (n, p) for the subset of the jet space J k (n, p) consisting of jets which are not sufficient for K equivalence. Then W k (n, p) is a closed algebraic subset of J k (n, p), and it can be shown (we will recall the proof in 4) that its codimension tends to infinity with k. In each context below, we pick k large enough to make this codimension large enough (e.g. > dim(n U)). The complement J k (n, p)\w k (n, p) supports a well defined stratification A k (n, p). This is due to Mather: a quick sketch is given in [15], with details in [16]; alternative accounts are given in [8, IV, 2] and [18]. We now recall its construction and main properties: first we need some results on stratifications. 12

13 Let f : N P be a smooth map, Σ(f) the set of x N such that df x is not surjective (the critical set) and (f) := f(σ(f)) the discriminant, or critical value set. A Whitney stratification C of (f) is called a critical value stratification (c.v.s.) if, for any strata S i, S j of C, (c.v.s. i) Σ(f, S i ) := f 1 (S i ) Σ(f) is a smooth submanifold of N and f Σ(f, S i ) S i is a local diffeomorphism, (c.v.s. ii) Σ(f, S j ) is Whitney regular over Σ(f, S i ), and (c.v.s. iii) f 1 (S j ) \ Σ(f, S j ) is Whitney regular over Σ(f, S i ). A c.v.s. is canonical if, when the strata are constructed inductively starting from those of highest dimension, each stratum is taken as large as possible subject to the regularity conditions involving it with larger strata, i.e. we remove inductively from (f) all strata already constructed, together with the set of points at which one of these conditions fails. Then [8, I,3.5] if U, V are semialgebraic open sets in R n, R p respectively, and f : U V a polynomial map with f Σ(f) proper and finite, then f admits a canonical c.v.s. having only finitely many semialgebraic strata. Also [8, I,3.1] if f : N P is a smooth mapping with Σ(f) closed, which admits a c.v.s., then we obtain a Thom stratification (X, X ) of f by taking X to have strata those of C together with P \ (f) and X to have strata Σ(f, S j ) and f 1 (S j ) \ Σ(f, S j ) for each S j C, together with N \f 1 ( (f)). Such stratifications have the property that each stratum in the source is either a subset of Σf which is mapped by a local diffeomorphism to a target stratum, or is mapped submersively (with fibre dimension (n p)) to a target stratum. It follows from the definition that canonical c.v.s. are preserved under diffeomorphisms of source and target. More generally, let us regard two stratifications of P as the same if they define the same filtration F r P = {S i dim S i r}. Then the canonical c.v.s. localises if we restrict f to U = f 1 (V ), where V is an open subset of P, or indeed to any neighbourhood U of Σ(f, V ) in f 1 (V ). It does not localise when we restrict to an arbitrary open subset U of N: the reason is that for a point y P, the critical preimage Σ(f, y) depends on the choice of domain U. If f : N P is a stable map-germ, then since (by [12, Proposition 3.5]), stable map-germs are finitely A determined, there exist local coordinates with respect to which f is polynomial. For a suitable representative, f Σ(f) is proper and finite, so the above conditions hold, and we obtain a canonical c.v.s. and a canonical Thom stratification. We see from the above remark that some care is needed in choosing a representative. The safest way is to choose a conelike representative: the (non-trivial) existence was established by Fukuda [7]. It suffices to work with representatives which are special in the sense that F 1 (y 0 ) contains only the one critical point x 0, and admits a canonical stratification such that x 0 lies in the closure of any connected component of a stratum in the source. We conclude that a stable germ determines uniquely the germ of a canonical c.v.s. We can apply the same arguments to a stable germ at a finite subset S of points of N with a common image y 0 P. In this situation, it follows from stability that the c.v.s. corresponding to the several x i S are transverse to 13

14 each other in a neighbourhood of y 0, and from [8, I,3.10] that their intersection gives the canonical c.v.s. of the germ of f at S. Let f be a non-submersive stable germ with source x 0 and target y 0. Take the canonical c.v.s. of f, and write C t,mt (f) for the stratum (in the target) containing y 0 and C s,mt (f) for the connected component of f 1 (C t,mt (f)) Σ(f) containing x 0. By (c.v.s. i), f induces a local diffeomorphism of C s,mt (f) on C t,mt (f). Choose k such that j k f is a K sufficient jet. Define C MT (f) to be the saturation of j k f(c s,mt (f)) under contact equivalence. Since f is stable, j k f is transverse to contact classes, so since f 1 (C MT (f)) = C s,mt (f) is smooth, it follows that C MT (f) is smooth. It also follows that all nearby singular contact classes have non-empty pre-images under f, so each lies in some stratum of the above c.v.s.. Thus the partition of the set of singular strata in a neighbourhood of z into the C MT for germs of f at points near x 0 is locally finite (since the strata of the c.v.s. are), and is Whitney regular (as a consequence of (c.v.s. ii)). I believe that this outline could be elaborated into a new proof, but can refer to [8, IV, 2] for justification of all the statements made. Now for any z J k (n, p)\w k (n, p) pick a representative map-germ f. Then f has finite singularity type, so has a finite dimensional A stable unfolding F : (R n+a, 0) (R p+a, 0). Since F is a stable germ, it has a c.v.s.. It follows from Lemma 1.6 that the map J k (n, p) J k (n+a, p+a) induced by suspension is transverse to contact classes, and hence to C MT (F ). We can thus define C MT (f) to be the pre-image of C MT (F ). This contains z, so the various C MT (f) partition J k (n, p) \ W k (n, p). We denote this partition by A k (n, p): it follows from the preceding paragraph that this is locally finite and Whitney regular. It is shown in [18] (proving a claim of Mather) that A k (n, p) has only finitely many strata. Since these strata are invariant under diffeomorphisms of source and target, we can define a stratification A k (N, P ) (for any n-manifold N and p-manifold P ) inducing A k (n, p) on the fibres. We now proceed along the same lines as in 1 and 2, but with the strata of A k (N, P ) replacing the contact classes. We say that a (K finite) germ f at a point x 0 is MT-stable if (for some k) j k f is transverse to C MT (f). A germ f at a finite set S with common target y 0 is MT-stable if j k f is multi-transverse at S to the strata of A k (N, P ). Write N MT (f) for the normal space to C MT (f) and C s,mt (f), and NMT e (f) for the normal space to C t,mt (f). We now have analogues of the results of 1. Lemma 3.1 (a) Exclude the case of submersion-germs. Then f is an MTstable germ if and only if it is transverse to C MT (f), i.e. T x0 N N MT (f) is surjective. For f MT-stable, (i) The preimage C s,mt (f) of C MT (f) is a submanifold of N, and its normal space can be identified with N MT f. (ii) The restriction of f to C s,mt (f) is an immersion. (iii) The set C t,mt (f) is a submanifold of P ; its normal space can be identified with NMT e f. (b) Let f : (N, S) (P, y 0 ) be a map-germ, then the following are equivalent: 14

15 (i) the germ of f at S is MT-stable, (ii) T y0 P maps onto the direct sum i N MT e f x i, (iii) f is multi-transverse at S to A k (N, P ). When these conditions hold, i C t,mt (f xi ) is a smooth submanifold of P with normal space i N MT e f x i. (c) Now let (i, j) : f F be an unfolding of a map-germ. Then there are natural isomorphisms N MT f N MT F and NMT e f N MT e F. (d)(i) Let x 0 N be such that the germ of F at i(x 0 ) is MT-stable. Then the germ of f at x 0 is stable if and only if i is transverse at x 0 to C s,mt (F i(x0)). (ii) Let S = {x i } be a finite set of points of N, with a common target y 0 P ; suppose the germ of F at i(s) is MT-stable. Then the germ of f at S is MTstable if and only if j is transverse at y 0 to i C t,mt (F i(xi)). Proof The proof of (a) follows that of Proposition 1.4. The crucial fact is establishing (ii): here this follows since it holds for f stable, as a consequence of the condition in (c.v.s. i) that f is a local diffeomorphism. Next, (b) follows, as for Lemma 1.5. (c) corresponds to Lemma 1.6, but the proof is different. In fact as we defined C MT (f) to be the pre-image of C MT (F ) by a map transverse to it, we have an isomorphism of the normal spaces. This proves the result for N MT ; that for NMT e follows. For (d), compare [8, IV,3.1]. The proof is essentially the same as for Lemma 1.7. Globally, we define f : N P to be MT-transverse if each multi-germ is MT-stable, or equivalently, if j k f avoids W k (n, p) and is multi-transverse to A k (n, p). This condition is independent of k, provided k is large enough. It follows from the multi-transversality theorem that for any N, P the set of MT-transverse maps N P is a residual subset of C (N, P ). According to Mather [15] and [16], a proper MT-transverse smooth map is topologically stable. Such maps and their germs we will call MT-stable. We turn to deformations N U P R, and parallel the development in 2. We have defined a stratification A k (n, p q) of J k (n, p q) \ W k (n, p q), and have seen that the strata play a role parallel to that of the contact classes in the C theory. We now define WQ k (n, p) := π 1 R W k (n, p q) and A k Q (n, p) to consist of the preimages of the strata of A k (n, p q); again we defer (to Corollary 5.8) the proof that this is independent of the local splitting P = Q R. Here the open stratum consists of all jets with target Q. The collection of jets transverse to Q is open in the complement to this. In Proposition 2.3, we required j1 k H to be transverse rel π U to K classes, which are the intersections of the contact classes with fibres over points of P. We thus define MT K classes to be the intersections of the strata of A k (n, p q) with fibres over points of P, and MT K Q classes to be their pre-images in J k (N, P ). 15

16 Proposition 3.2 For each u and finite S fu 1 (Q), the following are equivalent: (i) j1 k H avoids W k (n, p q), and is multi-transverse rel π U at (S, u) to the corresponding M T K class, (ii) j1 k F avoids WQ k(n, p), and is multi-transverse rel π U at (S, u) to the corresponding MT K Q class, (iii) The germ of Π F : X U at (S, u) is MT-stable. We can interpret (ii) as stating that the contact classes of f u at points of S are simultaneously topologically versally unfolded by the family F. Proof The equivalence of (i) and (ii) is immediate from the definitions. Equivalence of (i) and (iii) follows as before using the diagram (1), together with Lemma 3.1 (c). As before, the defining condition refers only to (multi-)jets of F at points of F 1 (Q), and implies that F is transverse to Q as a map. The condition is independent of k provided k is chosen large enough so that x < cod W k (n, p q). Again, by applying this result to all subsets S we obtain Corollary 3.3 The following are equivalent: (i) j1 k H avoids W k (n, p q), and is multi-transverse rel π U to all MT K classes, (ii) j1 k F avoids WQ k(n, p), and is multi-transverse rel π U to all MT K Q classes, (iii) Π F : X U is MT-transverse. Deformations F satisfying these conditions (for some, hence all large enough k) may be regarded as being as transverse to Q as possible ; we will refer to them as C 0 -d-transverse to Q. Note that in the nice dimensions x < σ(x, p q), this coincides with the notion of C -d-transversality to Q. It follows from Lemma 2.1 that maps F C 0 -d-transverse to Q form a residual set in C (N U, P ). Theorem 3.4 Suppose that Q is closed in P, and that Q or N is compact. Then the set of proper deformations F : N U P C 0 -d-transverse to Q is dense and open in the space C 1,pr(N U, P ) of all proper deformations. For such F, the associated map Π F is MT-stable. Proof As before, under this hypothesis, if F is a proper deformation, Π F is a proper map. By Corollary 3.3, F is C 0 -d-transverse to Q if and only if Π F : X U is MT-transverse. But for Π F proper, Π F is MT-transverse if and only if it is MT-stable. It follows as before from Theorem 2.6 that Π F depends continuously on F. Since (see e.g. [8, IV, 4.1]), MT-stable maps are open in the space of proper maps, the desired openness follows. Since f and the unfolding (F, π U ) have finite singularity type, if F is C 0 -dtransverse to Q, the restriction to the critical set is a finite map: each critical stratum is immersed. In particular, for our family {f u } : N P, transversality fails for only finitely many points in each fibre N {u}. Moreover, f u is transverse to Q for an open dense set of u; for such u, the only stratum that appears is the transverse stratum. 16

17 4 Generalisations of K equivalence We next seek to replace the single submanifold Q of P by a stratification of a subset. It is natural to hope that we can obtain a theory generalising that in the case when Q is smooth, and that a suitable genericity condition will be given by transversality conditions on F. It is more natural to recast the arguments of the preceding section by eliminating the splitting P = Q R and the auxiliary maps h, H and speaking instead of K equivalence of f and F relative to Q or, in the terminology of Damon [5], of K Q equivalence. Thus in this section we will study variants of K equivalence where a certain subset of the source or target is respected in some sense. Let X be a subset of N, x 0 N, and define R e X to be the set of diffeomorphismgerms at x 0 of N preserving X and R X the subgroup of those with target x 0. Recall that we denote by m X,x0 the ideal in E X,x0 of function-germs vanishing on X. The tangent space T R e X is the set of vector fields tangent to X, viz. θ X N := {ζ θ N ζ(g) m X,x0 for all g m X,x0 }. (2) This contains m X,x0.θ N,x0, but does not in general coincide with it. For X arbitrary, this is just a formal definition, but Damon [5] shows that, in favourable circumstances (e.g. if X is a complex analytic or real analytic coherent subset of N), this E N,x0 module is finitely generated and ties in with the geometry. The vector fields ξ θ X N are often called logarithmic. We call the image of θ X N in θ N /m x0.θ n = T x0 N the small tangent space to X at x 0, and denote it by Tx s 0 X. The tangent space to the orbit of f under R e X is tf(θ X N), and T R X f = tf(θ X N m x0 θ N ). Similarly, if Q is a subset of P and y 0 P, we have the set L e Q of diffeomorphismgerms with source y 0 preserving Q, its subgroup L Q of diffeomorphism-germs with target y 0 ; and T L e Q := θ Q P, T L e Q f = ωf(θ Q P ) and T L Q := θ Q P m y0.θ P. For Q P, we have two candidates for the analogue to K equivalence. We define KQ e to be the group of pairs (H, h), where H is a diffeomorphismgerm of N P with source (x 0, y 0 ) and h a diffeomorphism-germ of N with source x 0, such that π N H = h π N and H(N Q) N Q. Define KQ,y e 0 to be the subgroup satisfying the additional condition H i = i h defining K: thus KQ,y e 0 = KQ e Ke y 0. When we also restrict the target to be (x 0, y 0 ) we obtain the groups K Q,y0 and K Q,y0 = K Q K y0. These are semi-direct products K Q = R.C Q, K Q,y0 = R.C Q,y0, where C Q is the group of diffeomorphism-germs H over π N which preserve N Q, and C Q,y0 the subgroup which also fix N y 0. Correspondingly, KQ e = Re.C Q and KQ,y e 0 = R e.c Q,y0. Since E N P is flat over E P, the set of germs, with zero N projection, tangent to N Q is T C Q = E N θ Q P, and those vanishing along N y 0 form the tangent space T C Q,y0 = E N (θ Q P m y0 θ P ). Thus T C Q f = E N.ωf(θ Q P ) and T C Q,y0 f = E N.ωf(θ Q P m y0 θ P ). For X a subset of N, define X K to be the subgroup of pairs (H, h) K such that h(x) = X and H(X P ) = X P. We also consider a variant X K, 17

18 defined by H X P = h id. Both these definitions appear in [6] (Examples 6, 8 on pp 57 58), but our terminology X K differs from Damon s. These also are semi-direct products X K = R X.C and X K = R X. X C, where X C := {H C H X P = id. The extended groups are X K e = R e X.C and X K e = R e X.X C. The tangent space of X C consists of vector fields which, moreover, vanish on X P. Hence we have T X C e = m X θ P and T X C = m X m y0.θ P, and the tangent spaces to the orbits are m X.θ f and m X.f m y0.θ f. We summarise in a table B R X C Q C Q,y0 X C T B e f tf(θ X N ) m X.θ f T Bf tf(θ X N m x θ N ) E N.ωf(θ Q P ) E N.ωf(θ Q P m y0 θ P ) m X.m y0.θ P (3) to be used recalling also the products K Q = R.C Q, K Q,y0 = R.C Q,y0, XK = R X.C, X K = R X. X C. We also have the group L Q and set L e Q of left equivalences preserving Q (with T L e Q f = ωf(θ Q P )), A Q := R L Q and X A := R X L: we will not study these, but will need to refer to A Q. Remark It would be interesting to have results corresponding to Lemma 1.1 here. Perhaps equivalence for K Q is characterised by having an isomorphism of pairs (E N, f m Q.E N ); for K Q,y0 by isomorphism of triples (E N, f m y0.e N, f m Q.E N ) and for X K by isomorphism of triples (E N, m N,X, f m y0.e N ). In [4], Damon gives a set of axioms for what he calls geometric subgroups of K. The main results of [4] state: Theorem 4.1 If B is a geometric group, the following are equivalent: dim(θ f0 /T B e f 0 ) <, f 0 is finitely B determined, f 0 has a (finite-dimensional) B versal unfolding. Moreover, any B versal unfolding is a trivial unfolding of a B miniversal unfolding, and any two B miniversal unfoldings are equivalent. In [5] Damon shows that K Q,y0 is a geometric subgroup of K if Q is either complex analytic or real analytic; in the C case if Q is real analytic and coherent. The arguments in the C case are more delicate; the calculations work only modulo flat fields. If Q is real analytic but not coherent, he obtains all these conclusions except that having a B versal unfolding need not imply the other conditions. A key point in this development is the requirement that the ideal m Q defining Q should be finitely generated: this is the source of the problem about coherence. The same ideas show that, under the same hypotheses, K Q is a geometric subgroup of K. In [6] he obtains corresponding results for X K and X K, and also shows that all three belong to what he calls the special class of geometric subgroups, which are those for which he establishes topological triviality and versality results. 18

19 As in [22], we call a germ B stable (for an equivalence relation B) if T B e f = θ f. We apply this for the above groups B. Since in each case T B e f is an E N,x0 submodule of θ f, it follows from Nakayama s Lemma that T B e f = θ f if and only if T B e f maps onto θ f /m x0.θ f, which can be identified with T f(x0)p. This reduces the calculations to easy remarks. The map-germ f at x 0 is R stable if and only if θ f = T R e f = tf(θ N ). This is equivalent to the condition that tf induces a surjection of θ N onto T f(x0)p, i.e. that the tangent map T x0 f : T x0 N T f(x0)p is surjective. The map-germ f at x 0 is R X stable if and only if θ f = T R e X f = tf(θ X N ). It follows as above that this is equivalent to having the tangent map T x0 f surjective when restricted to the small tangent space Tx s 0 X. The map-germ f at x 0 is C y0 stable if and only if θ f = T C e f = f m y0.θ f. If f(x 0 ) y 0, f m y0 contains functions not vanishing at x 0, so f m y0.θ f = θ f and the stability condition holds; otherwise, it does not. The map-germ f at x 0 is C Q,y0 stable if and only if θ f = T CQ e f = E N.(ωf(θ Q P ) m y0.θ P ). If f(x 0 ) = y 0, this cannot occur; if f(x 0 ) y 0, the condition reduces to θ f = E N.ωf(θ Q P ). Reducing modulo m y0, we see that this only happens if θ Q P maps onto T y0 P, i.e. Ty s 0 Q = T y0 P, hence if and only if f(x 0 ) Q. The map-germ f at x 0 is X C stable if and only if θ f = T X C e f = m X,x0.θ f, i.e. if and only if x 0 X. The map-germ f at x 0 is K y0 stable if and only if θ f = T K e f = tf(θ N ) + f m y0.θ f. This holds if either f(x 0 ) y 0 or T x0 f is surjective; otherwise it fails. The map-germ f at x 0 is K Q stable if and only if θ f = T e K Q (f) = tf(θ N )+ E N.ωf(θ Q P ). This holds if and only if either f(x 0 ) Q or T f(x0)p = T x0 f(t x0 N)+ Tf(x s Q, i.e. T 0) x 0 f is transverse to Q. The map-germ f at x 0 is K Q,y0 stable if and only if θ f = T KQ,y e 0 f = tf(θ N ) + E N.(ωf(θ Q P ) m y0.θ P ). If f(x 0 ) = y 0, the second term lies in m x0.θ f, so the condition holds only if T x0 f is surjective. Otherwise the equation reduces to θ f = T KQ e f = tf(θ N) + E N.ωf(θ Q P ). If f(x 0 ) Q, this is always satisfied. If f(x 0 ) Q, it reduces to T f(x0)p = T x0 f(t x0 N) + Tf(x s Q, i.e. T 0) x 0 f is transverse to Q. The map-germ f at x 0 is X K stable iff θ f = T X K e = tf(θ X N )+f m y0.θ f. As before, this holds if and only if one of the summands is equal to the whole of θ f, i.e. either f(x 0 ) y 0 or T x0 f maps Tx s 0 X onto T y0 P. The map-germ f at x 0 is X K stable iff θ f = T X K e = tf(θ X N ) + m X.θ f. Arguing as usual, we find that this holds if and only if either x 0 X or T x0 f : Tx s 0 X T f(x0)p is surjective. Although the small tangent space is the natural one algebraically, it is not convenient geometrically unless we impose further conditions. Also, unless we impose further restrictions, the analogue of Theorem 2.6 cannot hold for the present situation. Consider, for example, the case when Q contains four 2- dimensional sheets in R 3 meeting along a smooth curve C. Then the tangent planes define a cross-ratio at each point of C, which is a C invariant. For the case when F (N) meets C transversely at a varying point, the different pre- 19