Connected components of partition preserving diffeomorphisms
aa r X i v : . [ m a t h . D S ] D ec CONNECTED COMPONENTS OF PARTITION PRESERVINGDIFFEOMORPHISMS
SERGIY MAKSYMENKO
Abstract.
Let f : R → R be a homogeneous polynomial and S ( f ) bethe group of diffeomorphisms h of R preserving f , i.e. f ◦ h = f . Denoteby S id ( f ) r , (0 ≤ r ≤ ∞ ), the identity component of S ( f ) with respect tothe weak Whitney C rW -topology. We prove that S id ( f ) ∞ = · · · = S id ( f ) for all f and that S id ( f ) = S id ( f ) if and only if f is a product of at leasttwo distinct irreducible over R quadratic forms. Introduction
Let f : R → R be a homogeneous polynomial of degree p ≥
1. Thus up toa sign we can write(1.1) f ( x, y ) = ± l Y i =1 L α i i ( x, y ) · k Y j =1 Q β j j ( x, y ) , where every L i is a linear function, Q j is a positive definite quadratic form, α i , β j ≥
1, and L i L i ′ = const for i = i ′ , Q j Q j ′ = const for j = j ′ . Denote by S ( f ) = { h ∈ D ( R ) : f ◦ h = f } the stabilizer of f with respectto the right action of the group D ( R ) of C ∞ -diffeomorphisms of R on thespace C ∞ ( R , R ). It consists of diffeomorphisms of R preserving every level-set f − ( c ) of f , ( c ∈ R ).Let S id ( f ) r , (0 ≤ r ≤ ∞ ) be the identity component of S ( f ) with respectto weak Whitney C rW -topology. Thus S id ( f ) r consists of diffeomorphisms h ∈S ( f ) isotopic in S ( f ) to id R via (an f -preserving isotopy) H : R × I → R whose partial derivatives in ( x, y ) ∈ R up to order r continuously depend on( x, y, t ), see Section 2 for a precise definition. Then it is easy to see that S id ( f ) ∞ ⊂ · · · S id ( f ) r ⊂ · · · ⊂ S id ( f ) ⊂ S id ( f ) . Date : 30.11.2008.This research is partially supported by Grant of President of Ukraine.
It follows from results [12, 13] that S id ( f ) ∞ = S id ( f ) . Moreover, it isactually proved in [9] that S id ( f ) ∞ = S id ( f ) for p ≤
2, see also [11]. The aimof this note is to prove the following theorem describing the relation between S id ( f ) r for all p ≥ Theorem 1.1.
Let f : R → R be a homogeneous polynomial of degree p ≥ .Then S id ( f ) ∞ = · · · = S id ( f ) . Moreover, S id ( f ) = S id ( f ) if and only if f isa product of at least two distinct definite quadratic forms, i.e. f = Q β · · · Q β k k for k ≥ . This theorem is based on a rather general result about partition preservingdiffeomorphisms, see Theorem 4.7. The applications of Theorem 1.1 will begiven in another paper concerning smooth functions on surfaces with isolatedsingularities.
Structure of the paper.
In Section 2 we describe homotopies which inducecontinuous paths into functional spaces with weak Whitney C rW -topologies.Section 3 introduces the so called singular partitions of manifolds being themain object of the paper. Section 4 contains the main result, Theorem 4.7,about invariant contractions of singular partitions. In Section 5 an applicationof this theorem to local extremes of smooth functions is given. Section 6 con-tains a description of the group of linear symmetries of f . Finally in Section 7we prove Theorem 1.1. 2. r -homotopies Denote N = N ∪ { } and N = N ∪ { , ∞} .Let M, N be two smooth manifolds of dimensions m and n respectively.Then for every r ∈ N the space C r ( M, N ) admits the so called weak
Whitneytopology denoted by C rW , see e.g. [8, 5].Recall, e.g. [7, § C ( I, C ( M, N )) ≈ C ( M × I, N )with respect to the corresponding C W -topologies (also called compact open ones) associating to every (continuous) path w : I → C ( M, N ) a homotopy H : M × I → N defined by H ( x, t ) = w ( t )( x ).We will now describe homotopies inducing continuous paths w : I → C r ( M, N )with respect to C rW -topologies. Definition 2.1.
Let H : M × I → N be a homotopy and r ∈ N . We say that H is an r -homotopy if (1) H t : M → N is C r for every t ∈ I ; (2) partial derivatives of H ( x, t ) by x up to order r continuously depend on ( x, t ) . ARTITION PRESERVING DIFFEOMORPHISMS 3
More precisely, let z ∈ M × I . Then in some local coordinates at z we canregard H as a map H = ( H , . . . , H n ) : R m × I → R n such that for every fixed t and i the function H i ( x, t ) is C r . Condition (2) requires that for every i = 1 , . . . , n and every non-negative integer vector k =( k , . . . , k m ) of norm | k | = P mj =1 k i ≤ r the function ∂ | k | H i ∂x k · · · ∂x k m m ( x , . . . , x m , t ) continuously depend on ( x, t ) . Thus a 0-homotopy H is just a usual homotopy.It easily follows from the definition of C rW -topologies that a path w : I → C r ( M, N ) is continuous from the standard topology of I to C rW -topology of C r ( M, N ) if and only if the corresponding homotopy H : M × I → N is an r -homotopy.We can also define a C r -homotopy as a C r -map M × I → N . Evidently,every C r -homotopy is an r -homotopy as well, but the converse statement isnot true. Example 2.2.
Let H : R × I → R be given by H ( x, t ) = (cid:26) t ln( x + t ) , ( x, t ) = (0 , , , ( x, t ) = (0 , . Then H is continuous, while ∂H∂x = txx + y is C ∞ for every fixed t as a functionin x but discontinuous at (0 , as a function in ( x, t ) . In other words H is a -homotopy but not a -homotopy.Moreover, define G : R × I → R by G ( x, t ) = R x H ( y, t ) dy . Then G a -homotopy but not a -homotopy. Singular partitions of manifolds
Let M be a smooth manifold equipped with a partition P = { ω i } i ∈ Λ , i.e. afamily of subsets ω i such that M = ∪ i ∈ Λ ω i , ω i ∩ ω j = ∅ ( i = j ) . In general Λ may be even uncountable and ω i are not necessarily closed in M .Let also Λ ′ be a (possibly empty) subset of Λ and Σ = { ω i } i ∈ Λ ′ be a subfamilyof P thought as a set of “singular” elements. Then the pair Θ = ( P , Σ) willbe called a singular partition of M . SERGIY MAKSYMENKO
Example 3.1.
Let F be a vector field on M , P F be the set of orbits of F ,and Σ F be the set of singular points of F . Then the pair Θ F = ( P F , Σ F ) willbe called the singular partition of F . Example 3.2.
Let f : R m → R n be a smooth map, x ∈ R m be a point, and J ( f, x ) be the Jacobi matrix of f at x . Then x ∈ R m is called critical for f if rank J ( f, x ) < min { m, n } . Otherwise x is regular . This definition naturallyextends to maps between manifolds.Let M , N be smooth manifolds and f : M → N a smooth map. Denoteby Σ f the set of critical points of f . Consider the following partition P f of M : a subset ω ⊂ M belongs to P f iff ω is either a critical point of f , or aconnected component of the set of the form f − ( y ) \ Σ f for some y ∈ N . Thenthe pair Θ f = ( P f , Σ f ) will be called the singular partition of f . Evidently,every ω ∈ P f \ Σ f is a submanifold of M . Example 3.3.
Assume that in Example 3.2 dim M = dim N + 1 and both M and N are orientable. Then every element of P f \ Σ f is one-dimensional andorientations of M and N allow to coherently orient all the elements of P f \ Σ f .Moreover, it is even possible to construct a vector field F on M such that thesingular partitions Θ f and Θ F coincide.In particular, let M be an orientable surface and f : M → R be a smoothfunction. Then M admits a symplectic structure, and in this case we canassume that F is the corresponding Hamiltonian vector field of f . Example 3.4.
Let F be a foliation on M with singular leaves, P be the setof leaves of F , and Σ be the set of its singular leaves (having non-maximaldimension). Then the pair Θ F = ( P F , Σ F ) will be called the singular partitionof F . This example generalizes all previous ones.Let Θ = ( P , Σ) be a singular partition on M . For every open subset V ⊂ M denote by E (Θ , V ) the subset of C ∞ ( V, M ) consisting of maps f : V → M such that(1) f ( ω i ∩ V ) ⊂ ω i for all ω i ∈ P and(2) f is a local diffeomorphism at every point z belonging to some singularelement ω ∈ Σ.Let also D (Θ , V ) be the subset of E (Θ , V ) consisting of immersions , i.e. local diffeomorphisms . For V = M we abbreviate E (Θ) = E (Θ , M ) , D (Θ) = D (Θ , M ) . For every r ∈ N denote by E id (Θ , V ) r , resp. D id (Θ , V ) r , the path-componentof the identity inclusion i V : V ⊂ M in E (Θ , V ), resp. in D (Θ , V ), with respectto the induced C rW -topology, see Section 2. ARTITION PRESERVING DIFFEOMORPHISMS 5
Thus E id (Θ , V ) r (resp. D id (Θ , V ) r ) consists of maps (resp. immersions) V ⊂ M which are r -homotopic ( r -isotopic) to i V : V ⊂ M in E (Θ , V ) (resp.in D (Θ , V )).Evidently,(3.1) E id (Θ , V ) ∞ ⊂ · · · ⊂ E id (Θ , V ) ⊂ E id (Θ , V ) , and similar relations hold for D id (Θ , V ) r .The following notion turns out to be useful for studying singular partitionsof vector fields.3.5. Shift-map of a vector field.
Let F be a vector field on M andΦ : M × R ⊃ dom(Φ) → M be the local flow of F defined on some open neighbourhood dom(Φ) of M × M × R . For every open subset V ⊂ M let alsofunc(Φ , V ) = { α ∈ C ∞ ( V, R ) : Γ α ⊂ dom(Φ) } , where Γ α = { ( x, α ( x )) : x ∈ V } ⊂ M × R is the graph of α . Then func(Φ , V )is the largest subset of C ∞ ( V, R ) on which the following shift-map is defined: ϕ V : func(Φ , V ) → C ∞ ( V, M ) , ϕ V ( α )( x ) = Φ( x, α ( x )) , for α ∈ func(Φ , V ), x ∈ V . Lemma 3.6.
Let Θ F be the singular partition of M by orbits of F . Then (3.2) im( ϕ V ) ⊂ E id (Θ F , V ) ∞ . Moreover, if D id (Θ F , V ) r ⊂ im( ϕ ) for some r ∈ N , then D id (Θ F , V ) ∞ = · · · = D id (Θ F , V ) r +1 = D id (Θ F , V ) r . Proof.
Let α ∈ func(Φ , V ) and f = ϕ ( α ), i.e. f ( x ) = Φ( x, α ( x )). Then f ( ω ∩ V ) ⊂ ω for every orbit of F . Moreover by [9, Lemma 20] f is a localdiffeomorphism at a point x ∈ V iff dα ( F )( x ) = −
1, where dα ( F )( x ) is theLie derivative of α along F at x . Hence f is so at every singular point z of F , since dα ( F )( z ) = 0 = −
1, [9, Corollary 21]. Therefore f ∈ E (Θ F , V ).Moreover an ∞ -homotopy of f to i V : V ⊂ M in E (Θ F , V ) can be given by f t ( x ) = Φ( x, tα ( x )). Thus f ∈ E id (Θ F , V ) ∞ .Finally, suppose that f ∈ D id (Θ F , V ) r . Then the restriction of f to anynon-constant orbit ω of F is an orientation preserving local diffeomorphism.Therefore dα ( F )( z ) > − V . Hence d ( tα )( F )( z ) > − t ∈ I as well, i.e. f t ∈ D (Θ F , V ). This implies that f ∈ D id (Θ F , V ) ∞ . (cid:3) SERGIY MAKSYMENKO
Example 3.7.
Let A be a real non-zero ( m × m )-matrix, F ( x ) = Ax be thecorresponding linear vector field on R m , and V be a neighbourhood of theorigin 0. Then the shift-map ϕ V is given by ϕ ( α )( x ) = Φ( x, α ( x )) = e Aα ( x ) x. It is shown in [9] that in this case im( ϕ V ) = E id (Θ F , V ) . Hence for all r ∈ N we have im( ϕ V ) = E id (Θ F , V ) ∞ = · · · = E id (Θ F , V ) r , D id (Θ F , V ) ∞ = · · · = D id (Θ F , V ) r . Invariant contractions
Let Θ = ( P , Σ) be a singular partition on a manifold M . We will say that asubset V ⊂ M is Θ -invariant , if it consists of full elements of Θ, i.e. if ω ∈ P and ω ∩ V = ∅ , then ω ⊂ V . Definition 4.1.
Let Z ⊂ M be a closed subset such that every point z ∈ Z is a singular element of Θ , i.e. { z } ∈ Σ . Say that Θ has an invariant r -contraction to Z if there exists a closed Θ -invariant neighbourhood V of Z being a smooth submanifold of M and a homotopy r : V × I → V such that: (i) r = id V ; (ii) r is a proper retraction of V to Z , i.e. r ( V ) = Z , r ( z ) = z for z ∈ Z , and r − ( K ) is compact for every compact K ⊂ Z ; (iii) for every t ∈ (0 , the map r t is a closed C r -embedding of V into V such that for each ω ∈ P (resp. ω ∈ Σ ) its image r t ( ω ) is also anelement of P (resp. Σ ); (iv) for each z ∈ Z the set V z = r − ( z ) is Θ -invariant, and r t ( V z ) ⊂ V z forall t ∈ I . Since V is Θ-invariant, it follows from (iii) that so is its image r t ( V ). Example 4.2.
Define f : R m → R by f ( x , . . . , x m ) = P mi =1 x i . Evidently, thesingular partition Θ f consists of the origin 0 and concentric spheres centeredat 0. For every s > V s = f − [0 , s ] be a closed m -disk of radius s . Then V s is Θ f -invariant and its invariant contraction of V s to Z = { } can be givenby r ( x, t ) = tx . Example 4.3.
The previous example can be parameterized as follows. Let p : M → Z be an m -dimensional vector bundle over a connected, smoothmanifold Z . We will identify Z with the image Z ⊂ M of the correspondingzero-section of p . Suppose that we are given a norm k · k on fibers such thatthe following function f : M → R is smooth: f ( ξ, z ) = k ξ k , z ∈ Z, ξ ∈ p − ( z ) . ARTITION PRESERVING DIFFEOMORPHISMS 7
Define the following singular partition Θ = ( P , Σ) on M , where P consists ofsubsets ω s,z = f − ( s ) ∩ p − ( z ) for s ≥ z ∈ Z , and Σ = { ω ,z = { z } : z ∈ Z } consists of points of Z . Thus every fiber p − ( z ) is Θ-invariant, and therestriction of Θ to p − ( z ) is the same as the one in Example 4.2.Fix s > V = f − [0 , s ]. Then V is Θ-invariant and a Θ-invariantcontraction of V to Z can be given by r ( ξ, z, t ) = ( tξ, z ).We will now generalize these examples. Let f : R m → R be a smoothfunction and suppose that there exists a neighbourhood V of 0 and smoothfunctions α , . . . , α m : V → R such that(4.1) f = α · f ′ x + · · · + α m · f ′ x m . Equivalently, let ∆( f,
0) be the
Jacobi ideal of f in C ∞ ( V, R ) generated bypartial derivatives of f . Then (4.1) means that f ∈ ∆( f, f be quasi-homogeneous of degree d with weights s , . . . , s m ,i.e. f ( t s x , . . . , t s m x m ) = t d f ( x , . . . , x m ) for t >
0, see e.g [1, § g ( x , . . . , x m ) = f ( x s , . . . , x s m m )is homogeneous of degree d . Then the following Euler identity holds true: f = x s f ′ x + · · · + x s m f ′ x m . In particular, f satisfies (4.1). Moreover in the complex analytical case theidentity (4.1) characterizes quasi-homogeneous functions, see [21]. Lemma 4.4.
Let f : M → R be a smooth function and z ∈ M be an isolatedlocal minimum of f . Suppose that f satisfies condition (4.1) at z , i.e. f ∈ ∆( f, z ) . Then the singular partition Θ f admits an invariant contraction to z .Proof. Since the situation is local, we may assume that M = R m , z = 0 is aunique critical point of f being its global minimum, f (0) = 0, and there existsan ε > V = f − [0 , ε ] is a smooth compact m -dimensional manifoldwith boundary L = f − ( ε ).First we give a precise description of the partition P f on V . Let F be anygradient like vector field on V for f , i.e. df ( F )( x ) > x = 0. Thenfollowing [15, Th. 3.1] we can construct a diffeomorphism η : V \ { } → L × (0 , ε ]such that f ◦ η − ( y, t ) = t for all ( y, t ) ∈ L × (0 , ε ], see Figure 4.1.Let CL = L × [0 , ε ] / { L × } be the cone over L and L t = f − ( t ) for t ∈ (0 , ε ]. Since diam( L t ) → t →
0, we obtain that η extends to ahomeomorphism η : V → CL by η (0) = { L × } . I would like to thank V. A. Vasilyev for referring me to the paper [21] by K. Saito.
SERGIY MAKSYMENKO
For every t ∈ (0 , ε ] put L t = f − ( t ). Then η diffeomorphically maps L t onto L × { t } . Figure 4.1.
Lemma 4.5. L is homotopy equivalent to S m − . We will prove this lemma below. Then it will follow from the generalizedPoincar´e conjecture that L is homeomorphic with the sphere S m − , and evendiffeomorphic to S m − for m = 4. For k = 1 , k = 3 this follows from a recent work of G. Perelman [18, 19],for k = 4 from M. Freedman [3], and for k ≥ L t is connected, whence the partition P f on V consistsof a unique singular element { } ∈ Σ f and sets L t , t ∈ (0 , ε ].Let us recall the definition of η . Notice that every orbit of F starts at 0and transversely intersect every L t . For each x ∈ V \ { } denote by q ( x ) aunique point of the intersection of the orbit of x with L = L ε = ∂V . Then η : V \ { } → L × (0 , ε ] can be given by the following formula: η ( x ) = ( q ( x ) , f ( x )) . Also notice that if φ : [0 , ε ] → [0 , ε ] is a (not necessarily surjective) C ∞ embedding such that φ (0) = 0, then we can define the embedding w φ : CL → CL, w φ ( y, t ) = ( y, φ ( t ))and therefore the embedding r φ = η ◦ w φ ◦ η − : V → V . Then r φ is C ∞ on V \ L t onto L q ( t ) for all t ∈ (0 , ε ]. Moreover r φ (0) = 0, but in general r φ is not even smooth at 0.Suppose now that f ∈ ∆( f, F = α ∂∂x + · · · + α m ∂∂x m . Then (4.1) means that f = df ( F ). Since f ( x ) > x = 0, it follows that F isa gradient like vector field for f . Therefore we can construct a homeomorphism η : V → CL using F as above. It follows from [10] that in our case this η hasthe following feature: ARTITION PRESERVING DIFFEOMORPHISMS 9 • if φ : [0 , ε ] → [0 , ε ] is a C ∞ embedding such that φ (0) = 0, thenthe corresponding embedding r φ : V → V is a diffeomorphism onto itsimage . Moreover, if φ s , ( s ∈ I ), is a C ∞ isotopy, then so is r φ s : V → V .In particular, consider the following homotopy φ : [0 , ε ] × I → [0 , ε ] , φ ( t, s ) = t (1 − s )which contracts [0 , ε ] to a point and being an isotopy for t >
0. Then theinduced homotopy r : V × I → V is an invariant ∞ -contraction of Θ f to0. (cid:3) Proof of Lemma 4.5.
It suffices to establish that(4.2) π k L ≈ π k S m − = (cid:26) , k = 0 , . . . , m − , Z , k = m − . Then the generator µ : S m − → L of π m − L ≈ Z will yield isomorphisms ofthe homotopy groups π k S m − ≈ π k L for all k ≤ m − S m − = dim L .Now by the well-known Whitehead’s theorem µ will be a homotopy equivalencebetween S m − and L .For the calculation of homotopy groups of L consider the exact sequence ofhomotopy groups of the pair ( V, L ):(4.3) · · · → π k +1 ( V, L ) → π k L → π k V → · · · Since V is homeomorphic with the cone CL , V is contractible, whence π k V = 0for all k ≥ π k +1 ( V, L ) = 0 for all k = 0 , . . . , m −
2. Indeed, let ξ : ( D k +1 , S k ) → ( V, L )be a continuous map. We have to show that ξ is homotopic (as a map of pairs)to a map into L . Since k + 1 ≤ m − < dim V , ξ is homotopic to a map into V \ { } . But L is a deformation retract of V \ { } , therefore ξ is homotopicto a map into L .Now it follows from (4.3) that π k L = 0 for k = 0 , . . . , m −
2. Hence fromHurewicz’s theorem we obtain that π m − L ≈ H m − ( L, Z ). It remains to notethat L is a connected closed orientable ( m − H m − ( L, Z ) ≈ Z . This proves (4.2). (cid:3) Remark 4.6. If f is a quasi-homogeneous function of degree d with weights s , . . . , s m , then we can define an invariant contraction by r ( x , . . . , x m , t ) = ( t s x , . . . , t s m x m ) . If f is homogeneous , then we can even put r ( x, t ) = tx . Theorem 4.7.
Let
Θ = ( P , Σ) be a singular partition on a manifold M , and Z ⊂ M be a closed subset such that every z ∈ Z is a singular element of P ,i.e. { z } ∈ Σ . Suppose that Θ has an invariant ∞ -contraction to Z definedon a Θ -invariant neighbourhood V of Z . Let also h ∈ E (Θ , V ) be a map fixedoutside some neighbourhood U of z such that U ⊂ Int V . Then h ∈ E id (Θ , V ) ,though h not necessarily belongs to E id (Θ , V ) r for some r ≥ . Remark 4.8.
Let Θ f be the singular partition of f ( x ) = k x k as in Ex-ample 4.2, V be the unit n -disk centered at 0 and r : V × I → V be theinvariant contraction of Θ f to a point Z = { } defined by r ( x, t ) = tx . Thena 0-homotopy between h and id V can be defined by H t ( x ) ( th ( xt ) , t > , , t = 0 . c.f. [5, Ch.4, Theorems 5.3 & 6.7]. Theorem 4.7 generalizes this example. Proof of Theorem 4.7.
Let r : V × I → V be an invariant ∞ -contraction ofΘ F to Z . Define the following map H : V × I → M by H ( x, t ) = ( r t ◦ h ◦ r − t ( x ) , if t > x ∈ r t ( V ) ,x, otherwise . We claim that H is a 0-homotopy (i.e. just a homotopy) between h and theidentity inclusion i V : V ⊂ M in E (Θ , V ). To make this more obvious werewrite the formulas for H in another way.The homotopy r can be regarded as the composition r = p ◦ e r : V × I e r −→ V × I p −−→ V, where e r is the following level-preserving map e r : V × I → V × I, e r ( x, t ) = ( r ( x, t ) , t ) , and p : V × I → V is the projection to the first coordinate. It follows from Figure 4.2.
ARTITION PRESERVING DIFFEOMORPHISMS 11 the definition that r yields a level-preserving embedding V × (0 ,
1] to V × I ,see Figure 4.2. Denote R ′ = e r ( V × (0 , , R = e r ( V × I ) . Then R \ R ′ = Z ×
0. Define also the following map e h : V × I → V × I, e h ( x, t ) = ( h ( x ) , t ) . In these terms, the homotopy H is defined by H = p ◦ e H : V × I e H −−→ V × I p −−→ V, where e H : V × I → V × I is a level-preserving map given by e H ( x, t ) = ( e r ◦ e h ◦ e r − ( x, t ) , ( x, t ) ∈ R ′ , ( x, t ) , ( x, t ) ∈ ( V × I ) \ R ′ . Now we can prove that H has the desired properties.Since r = id V , we have H = h . Moreover H = id V .
1. Continuity of e H on V × (0 , . Notice that e r ◦ e h ◦ e r − is well-defined andcontinuous on R ′ . Moreover, since h is fixed on V \ U , it follows that e h is fixedon ( V \ U ) × I , whence e r ◦ e h ◦ e r − is fixed on the subset e r (cid:0) ( V \ U ) × (0 , (cid:1) ⊂ R ′ .This implies that e H is continuous on V × (0 ,
2. Continuity of e H when t → . Let z ∈ V . Then e H ( z,
0) = ( z, z ∈ V \ Z . Since Z is closed in V , e H is also fixed and thereforecontinuous on some neighbourhood of ( z,
0) in ( V × I ) \ R .Let z ∈ Z and let W be a neighbourhood of ( z,
0) in V × I . We have to findanother neighbourhood W ′ of ( z,
0) such that e H ( W ′ ) ⊂ W .Recall that for every y ∈ Z we denoted V y = r − ( y ). Then V y is compactand Θ-invariant. Claim.
There exist ε > and an open neighbourhood N of z in V such that N × [0 , ε ] ⊂ W and (4.4) e r ( V y × [0 , ε ]) ⊂ W, ( y ∈ N ∩ Z ) . Proof.
Let N be an open neighbourhood of z such that N is compact and N × ⊂ W . Denote Q = r − ( N ∩ Z ) = ∪ y ∈ N ∩ Z r − ( y ) = ∪ y ∈ N ∩ Z V y . Then Q is a compact subset of V , and e r − ( W ) is an open neighbourhood of Q × V × I . Hence there exists ε > Q × [0 , ε ] ⊂ e r − ( W ). Thisimplies (4.4). Decreasing ε is necessary we can also assume that N × [0 , ε ] ⊂ W as well. (cid:3) Denote W ′ = N × [0 , ε ). We claim that e H ( W ′ ) ⊂ W . Let ( x, t ) ∈ W ′ . If either ( x, t ) ∈ W ′ \ R ′ or t = 0, then e H ( x, t ) = ( x, t ) ∈ W ′ ⊂ W .Suppose that ( x, t ) ∈ W ′ ∩ R ′ . Then t >
0. Let also y = r ( x ) ∈ Z . Then e r − ( x, t ) ∈ V y × t for all t ∈ I . Hence e r ◦ e h ◦ e r − ( x, t ) ∈ e r ◦ e h ( V y × t ) ⊂ e r ( V y × t ) (4.4) ⊂ W. In the second inclusion we have used a Θ-invariantness of V y and the assump-tion that h ∈ E (Θ , V ).
3. Proof that H t ∈ E (Θ , V ) for t ∈ I . We have to show that (i) for every t ∈ I the mapping H t is C ∞ , (ii) H t ( ω ) ⊂ ω for every element ω ∈ P includedin V , and (iii) H t is a local diffeomorphism at every point z belonging to some ω ∈ Σ.(i) Since r t , ( t > C ∞ and h is fixed on V \ U , it follows that H t is C ∞ as well.(ii) Let ω ⊂ V be an element of P (resp. Σ).If ω ⊂ V \ r t ( V ), then H t is fixed on ω , whence H t ( ω ) = ω ∈ P (resp. Σ).Suppose that ω ⊂ r t ( V ). Since r t ( V ) is Θ-invariant, ω = r t ( ω ′ ) for someanother element ω ′ ∈ P (resp. Σ). Then h ( ω ′ ) ⊂ ω ′ , whence H t ( ω ) = r t ◦ h ◦ r − t ( ω ) = r t ◦ h ( ω ′ ) ⊂ r t ( ω ′ ) = ω. (iii) Suppose that ω ∈ Σ and let x ∈ ω .If x ∈ V \ r t ( U ), then H t is fixed in a neighbourhood of x , and therefore itis a local diffeomorphism at x .Suppose that x = r t ( x ′ ) ∈ r t ( U ) for some x ′ ∈ U and let ω ′ ∈ Σ be theelement containing x ′ . Then h is a local diffeomorphism at x ′ , whence H t = r t ◦ h ◦ r − t is a local diffeomorphism at x . (cid:3) Stabilizers of smooth functions
Let B m ⊂ R m be the unit disk centered at the origin 0, S m − = ∂B m beits boundary sphere, f : B m → R be a C ∞ function, and Θ f be the singularpartition of f .Let S ( f ) = { h ∈ D ( B m ) : f ◦ h = f } be the stabilizer of f with respect tothe right action of the group D ( B m ) of diffeomorphisms of B m on the space C ∞ ( B m , R ). Denote by S + ( f ) the subgroup of S ( f ) consisting of orienta-tion preserving diffeomorphisms. For r ∈ N let also S id ( f ) r be the identitycomponent of S ( f ) with respect to the C r -topology. Then S id ( f ) ∞ ⊂ · · · ⊂ S id ( f ) r +1 ⊂ S id ( f ) r ⊂ · · · ⊂ S id ( f ) ⊂ S + ( f ) . Theorem 5.1.
Let m ≥ , f : B m → [0 , be a C ∞ function such that is aunique critical point of f being its global minimum, f (0) = 0 , and f ( S m − ) = 1 .Denote by S the subgroup of S ( f ) consisting of diffeomorphisms h such that ARTITION PRESERVING DIFFEOMORPHISMS 13 h | S m − : S m − → S m − is C ∞ -isotopic to id S m − . Suppose also that the singularpartition Θ f of f has an invariant ∞ -contraction to . Then S ⊂ S id ( f ) .If m = 2 , , , then S = S id ( f ) = S + ( f ) . For the proof we need the following two simple standard statements con-cerning smoothing homotopies at the beginning and at the end, see e.g. [20,pp.74 & 118] and [16, p. 205]. Let M be a closed smooth manifold. Claim 5.2.
Let a, b, c ∈ R be numbers such that < a < b < c , and N = M × (0 , c ] . Then we have a foliation on N by submanifolds M × t , t ∈ (0 , c ] . Let also h : N → N be a C ∞ leaf preserving diffeomorphism, i.e., h ( x, t ) = ( φ ( x, t ) , t ) for some C ∞ map φ : M × (0 , c ] → M such that for every t ∈ (0 , c ] the map φ t : M → M is a diffeomorphism. Then there exists a leaf preserving isotopyrelatively to M × (0 , a ] of h to a diffeomorphism ˆ h ( x, t ) = ( ˆ φ ( x, t ) , t ) such that ˆ φ t = φ c for all t ∈ [ b, c ] .Proof. Let µ : (0 , c ] → (0 , c ] be a C ∞ function such that µ ( t ) = t for t ∈ (0 , a ]and µ ( t ) = c for t ∈ [ b, c ], see Figure 5.1a). Define the following lead preservingisotopy H : N × I → N by H s ( x, t ) = ( φ ( x, (1 − s ) t + sµ ( t )) , t ) . Then it easy to see that H = id N , H t = h on M × (0 , a ] and ˆ h = H satisfiesconditions of our claim. (cid:3) Claim 5.3.
Let d < e ∈ (0 , and G : M × I → M be a C ∞ homotopy(isotopy). Then there exists another C ∞ homotopy (isotopy) G ′ : M × I → M such that ˆ G t = G for t ∈ [0 , d ] and ˆ G t = G for t ∈ [ e, .Proof. Take any C ∞ function ν : I → I such that ν [0 , d ] = 0 and ν [ e,
1] = 1,and put ˆ G t = G ν ( t ) , see Figure 5.1b). (cid:3) a) b) Figure 5.1.
Proof of Theorem 5.1.
Since 0 is a unique critical point of f and f is constanton S m − , it follows from the arguments of the proof of Lemma 4.4 that there exists a diffeomorphism η : D m \ → S m − × (0 ,
1] such that f ◦ η − ( y, t ) = t .In particular, for every t ∈ [0 ,
1] the set f − ( t ) is diffeomorphic with S m − .Since S m − is connected, we obtain that D (Θ f ) = S ( f ). Therefore D id (Θ f ) r = S id ( f ) r for all r ∈ N .By assumption there exists an invariant ∞ -contraction of Θ f to 0 definedon some Θ f -invariant neighbourhood V of 0. Therefore we can assume that V = f − [0 , c ] for some c ∈ (0 , ). Lemma 5.4.
There exists a C ∞ -isotopy of h in S to a diffeomorphism ˆ h fixedon f − [ c, . Then is follows from Theorem 4.7 that ˆ h and therefore h belongto D id (Θ f ) = S id ( f ) .Proof. Since h preserves f , it follows that the following diffeomorphism g = η ◦ h ◦ η − : S m − × (0 , → S m − × (0 , h ( S m − × t ) = S m − × t for all t ∈ (0 , g | S m − × t = h | S m − for all t ∈ [0 . , a ∈ (0 , c ). It suffices to find a leaf preserving isotopy relatively to S m − × (0 , a ] of g to a diffeomorphism ˆ g fixed on S m − × [ c, f − [0 , a ] of h in S to a diffeomorphism ˆ h which is fixed on f − [ c, C ∞ isotopy G : S m − × [1 , → S m − such that G = h | S m − and G = id S m − . By Claim 5.3 we can assumethat G t = h | S m − for all t ∈ [1 , . g and G yield the following C ∞ leafpreserving diffeomorphism T : S m − × (0 , → S m − × (0 , , T ( y, s ) = (cid:26) g ( y, s ) , s ∈ (0 , G ( y, s ) , s ∈ [1 , . Notice that T ( y,
2) = G ( y,
2) = y . Then by Claim 5.3 T is isotopic via aleaf preserving isotopy relatively S m − × (0 , a ] to a diffeomorphism ˆ T which isfixed on S m − × ( c, g = ˆ T | S m − × (0 , . The restriction of this isotopyto S m − × (0 ,
1] gives a leaf preserving isotopy relatively S m − × (0 , a ] of g toa diffeomorphism ˆ g with desired properties. The construction of homotopy isschematically presented in Figure 5.2. (cid:3) Suppose now that m = 2 , ,
4. Then every orientation preserving diffeomor-phism of S m − is C ∞ -isotopic to id S m − , whence S = S + ( f ), and therefore S = S id ( f ) = S + ( f ). For m = 2 this is rather trivial, for m = 3 is proved byS. Smale [22], and for m = 4 by A. Hatcher [4].If m ≥
5, then D + ( S m − ) is not connected in general, see e.g. [17], andtherefore Theorem 4.7 is not applicable. (cid:3) ARTITION PRESERVING DIFFEOMORPHISMS 15
Figure 5.2. Linear symmetries of homogeneous polynomials
Let f : R → R be a homogeneous polynomial of degree p ≥ f ( x, y ) = ± l Y i =1 L α i i ( x, y ) · k Y j =1 Q β j j ( x, y ) . Denote LS ( f ) = S + ( f ) ∩ GL + (2 , R ) . Thus LS ( f ) consists of preserving orientation linear automorphisms h : R → R such that f ◦ h = f . Also notice that LS ( f ) is a closed subgroup ofGL + (2 , R ), and therefore it is a Lie group. Denote by LS ( f ) the connectedcomponent of the unit matrix id R in LS ( f ).In this section we recall the structure of LS ( f ). Notice that we may makelinear changes of coordinates to reduce f to a convenient form. Then LS ( f )will change to a conjugate subgroup in GL + (2 , R ). Lemma 6.1. If deg f is even, then f ( − z ) ≡ f ( z ) , i.e. − id R ∈ LS ( f ) .Therefore in this case LS ( f ) is a non-trivial group. (cid:3) We will distinguish the following five cases of f .(A) l = 1, k = 0, f = L α . By linear change of coordinates we can assumethat L ( x, y ) = y and thus f ( x, y ) = y α . Then LS ( y α ) = { ( a b ) : a > } . If α is odd then LS = LS ( f ), otherwise, LS ( f ) consists of two connectedcomponents LS and −LS .(B) l = 2, k = 0, f = L α L α . By linear change of coordinates we canassume that L ( x, y ) = x , L ( x, y ) = y and thus f ( x, y ) = x α y α . Then LS ( x α y α ) = (cid:8)(cid:0) e α t e − α t (cid:1) : t ∈ R (cid:9) . Moreover, LS ( f ) / LS is isomorphic with some subgroup of Z generated bythe rotation of R by π/ l = 0, k = 1, f = Q β . By linear change of coordinates we can assumethat Q ( x, y ) = x + y , whence f ( x, y ) = ( x + y ) β . Then LS ( x + y ) = SO (2 , R ) = { ( cos t sin t − sin t cos t ) : t ∈ [0 , π ) } . The above statements are elementary and we left them for the reader. Noticealso that in the cases (A)-(C) l + 2 k ≤
2. The remaining two cases are thefollowing:(D) l = 0, k ≥ f = Q β · · · Q β k k . In this case deg f is even, whence LS ( f )is non-trivial.(E) l ≥ l + 2 k ≥ Lemma 6.2.
In the cases (D) and (E) LS ( f ) is a finite cyclic subgroup of GL + (2 , R ) . Moreover, in the case (E) LS ( f ) is a subgroup of Z l .Proof. In fact the cyclicity of LS ( f ) for the case l +2 k − ≥ f : C → C as a complex polynomial withreal coefficients. Then by [6] the subgroup LS C ( f ) of GL (2 , C ) consisting of complex symmetries of f turned out to be of one of the following types: cyclic,dihedral, tetrahedral, octahedral, and icosahedral. Notice LS ( f ) is the sub-group of LS C ( f ) consisting of preserving orientation real symmetries of f , i.e.automorphisms which also leave invariant 2-plane R ⊂ C of real coordinatesand preserve its orientation. Then it follows from the structure of symmetriesof regular polyhedrons, that LS ( f ) must be cyclic.Nevertheless, since we need a very particular case of [6] and for the sakeof completeness, we will present a short elementary proof. It suffices to showthat LS ( f ) is finite, see 6.6. This will imply that LS ( f ) is isomorphic witha finite subgroup of SO (2), and therefore is cyclic. Also notice that the factthat LS ( f ) is discrete also follows from [12]. First we establish the followingthree statements: Claim 6.3.
Let h ∈ GL + (2 , R ) and Q be a positive definite quadratic formsuch that Q ◦ h = t Q for some t > . Then t = det( h ) .Proof. By linear change of coordinates we can assume that Q ( z ) = | z | . Then h ( z ) = √ te iψ z for some ψ ∈ R , hence det( h ) = t . (cid:3) Claim 6.4.
Let Q , Q be a positive definite quadratic form such that Q Q const . Let also h ∈ GL + (2 , R ) be such that Q i ◦ h = tQ i for i = 1 , , where t = det( h ) . Then h ( z ) = ±√ t z .Proof. We can assume that Q ( x, y ) = x + y and Q ( x, y ) = ax + by , where a, b > a = 1 or b = 1. Denote g ( z ) = h ( z ) / √ t . Then Q i ◦ g = Q i , ARTITION PRESERVING DIFFEOMORPHISMS 17 i.e. g preserves every circle x + y = const and every ellipse ax + by = const.Therefore g = ± id R , and h ( z ) = ±√ tz . (cid:3) Claim 6.5. If t · id R ∈ LS ( f ) for some t ∈ R , then t = ± .Proof. Let z ∈ R be such that f ( z ) = 0. Since f is homogeneous, we have f ( z ) = f ( tz ) = t deg f f ( z ) , whence t = ± (cid:3) Let h ∈ LS ( f ). Since L i and Q j are irreducible over R , so are L i ◦ h and Q j ◦ h . Therefore the identity f ◦ h = f implies that “ h permutes L i and Q j up to non-zero multiples”. This means that for every i there exist i ′ and s i ∈ R \ { } , and for every j there exist j ′ and t j > L i ( h ( z )) = s i L i ′ ( z ) , Q j ( h ( z )) = t j Q j ′ ( z ) . Denote by Sym( r ) the group of permutations of r symbols. Then we have awell-defined homomorphism µ : LS ( f ) → Sym( l ) × Sym( k )associating to every h ∈ LS its permutations of L i and Q j . Claim 6.6. If l + 2 k ≥ , then ker µ ⊂ {± id R } , whence LS ( f ) is a finitegroup.Proof. Let h ∈ ker µ . Thus L i ◦ h = s i L i and Q j ◦ h = t j Q j for all i, j . Wewill show that h = t · id R for some t = 0. Then it will follow from Claim 6.5 h = ± id.Notice that h preserves every line { L i = 0 } and thus has l distinct eigendirections.a) Therefore if l ≥
3, then h = t · id R for some t ∈ R .b) Moreover, if k ≥
2, then by Claim 6.4, we also have h = ± id R .c) Suppose that 1 ≤ l ≤ k = 1. We can assume that Q ( z ) = | z | and that h ( z ) = t e iψ z for some t > ψ ∈ R . Since h has l ≥ h ( z ) = ±√ tz . (cid:3) Thus LS ( f ) ≈ Z n for some n ∈ N . Let h be a generator of LS ( f ). Thenwe can assume that h ( z ) = e πi/n z . It remains to prove the latter statement. Claim 6.7.
Suppose that l ≥ . Then n divides l , whence LS ( f ) is isomor-phic with a subgroup of Z l .Proof. Since f ◦ h = f , it follows that h ( f − (0)) = f − (0). By assumption l ≥
1, whence f − (0) = l ∪ i =1 { L i = 0 } is the union of l lines passing through theorigin. This set can be viewed as the union of 2 l rays starting at the origin,and these rays are cyclically shifted by h . Moreover, if h t preserves at leastone of these rays, then h t = id R . Therefore n divides 2 l . (cid:3) Lemma 6.2 is completed. (cid:3) Proof of Theorem 1.1.
Let f : R → R be a homogeneous polynomial of degree p ≥ f ( x, y ) = ± l Y i =1 L α i i ( x, y ) · k Y j =1 Q β j j ( x, y ) . We will refer to the cases (A)-(E) of f considered in the previous section. Wehave to show that S id ( f ) ∞ = · · · = S id ( f ) and that S id ( f ) = S id ( f ) iff f isof the case (D).Our first aim is to identify S id ( f ) r with the group D id (Θ G ) r for some vectorfield G on R , see Lemma 7.1. Then we will use the shift map of G . Denote D = ± l Y i =1 L α i − i · k Y j =1 Q β j − j . Then f = L · · · L l · Q · · · Q q · D and it is easy to see that D is the greatest common divisor of f ′ x and f ′ y in thering R [ x, y ].Let F = − f ′ y ∂∂x + f ′ x ∂∂x be the Hamiltonian vector field of f on R and G = F/D = − ( f ′ y /D ) ∂∂x + ( f ′ x /D ) ∂∂x . We will call G the reduced Hamiltonian vector field of f . Notice thatdeg G = l + 2 k − G are relatively prime in R [ x, y ].As noted in Example 3.3 the singular partitions Θ f and Θ F coincide. Letus describe the singular partition Θ G = ( P G , Σ G ). Recall that elements of P G are the orbits of G and Σ G consists of zeros of G . Consider the following cases,see Figure 7.1.(A) f = y α . Then D = y α − and F ( x, y ) = α y α − ∂∂y . Hence G ( x, y ) = α ∂∂y is a constant vector field and the partition Θ G consists of horizontal lines { y = const } being non-singular elements of Θ G .(C) and (D) f = Q β · · · Q β k k . In this case Θ F = Θ G . The origin is a uniquesingular element of Θ G . All other elements of Θ G are level-sets f − ( c ) of f for c > l = 2 and k = 0 or l ≥ k ≥
1. In both cases theset of singular points of Θ F consist of the origin and the set D − (0) = ∪ i : α i ≥ { L i = 0 } ARTITION PRESERVING DIFFEOMORPHISMS 19 of zeros of D being the union of those lines { L i = 0 } for which L i is a multiplefactor of f . Since after division of F by D the coordinate functions of G = F/D are relatively prime, it follows that 0 is a unique singular element of Θ G . Hencenon-singular elements of Θ G are the connected components f − ( c ) for c = 0and the half-lines in f − (0) \
0, see Figure 7.2.Case (A) Case (C) Cases (D) Cases (B) and (E)
Figure 7.1. a) F ( x, y ) = − xy ∂∂x + y ∂∂y b) G ( x, y ) = − x ∂∂x + y ∂∂y Figure 7.2.
Case (B). Hamiltonian and reduced Hamiltonianvector fields for f ( x, y ) = x y .Since f is constant along orbits of F and G , it follows that(7.1) D (Θ F ) ⊂ D (Θ G ) ⊂ S ( f ) . Also notice that D (Θ F ) consists of those h ∈ D (Θ G ) which fixes every criticalpoint of f . Lemma 7.1. D id (Θ G ) r = S id ( f ) r for all r ∈ N .Proof. It follows from (7.1) that D id (Θ G ) r ⊂ S id ( f ) r .Conversely, let h ∈ S id ( f ) r , so there exists an r -isotopy h t : R → R between h = id R and h = h in S ( f ), i.e.(7.2) f ◦ h t = f, t ∈ I. We claim that every h t ∈ D (Θ G ), i.e. h t ( ω ) = ω for every element ω of Θ G .This will mean that { h t } is an r -isotopy in D (Θ G ), whence h ∈ D id (Θ G ) r .It follows from (7.2) that h t ( f − ( c )) = f − ( c ) for every c ∈ R and h t (Σ f ) =Σ f . Since h = id R preserves every connected component ω of f − ( c ) \ Σ f , sodoes h t , t ∈ I . If either c = 0, or c = 0 but f is of either the cases (A), (C), or(D), then by definition every such ω is an element of Θ G . Let c = 0. We claim that in the cases (B) and (E) h t (0) = 0 for all t ∈ I .Indeed, in these cases the origin is “the most degenerate point among all otherpoints of f − (0)”. This means the following.For every z ∈ f − (0) denote by p z the least number such that p z -jet of f at z does not vanish, i.e. j p z − ( f, z ) = 0 while j p z ( f, z ) = 0. In other words, theTaylor series of f at z starts with terms of order p z . It is easy to see that for theorigin p = deg f , while for all other points z ∈ f − (0) we have that p z < deg f .Also notice that this number p z is preserved by any diffeomorphism h ∈ S ( f ),i.e. p h ( z ) = p z . It follows that h t (0) = 0.It remains to note that by continuity every h t preserves connected compo-nents of D − (0) \ { } . Hence h t ∈ D (Θ G ) for all t ∈ I . (cid:3) Now we can complete Theorem 1.1. Let Φ be the local flow on R generatedby G , and ϕ be the shift map of G , see Section 3.5. The following statementwas established in [9]. Lemma 7.2.
In the cases (A)-(C) (i.e. when deg G ≤ ) for every h ∈E id (Θ G ) there exists a smooth function σ : R → R such that h ( z ) = Φ( x, σ ( x )) ,i.e. im( ϕ ) = E id (Θ G ) . It follows from Lemmas 3.6 and 7.1 that in the cases (A)-(C)(7.3) S id ( f ) ∞ = D id (Θ G ) ∞ = · · · = D id (Θ G ) = S id ( f ) . The following lemma is a consequence of results of [12, 13].
Lemma 7.3.
In the cases (D) and (E) (i.e. when deg G ≥ ) im( ϕ ) consistsof all h ∈ E (Θ G ) whose tangent map T h : T R → T R at is the identity. Corollary 7.4. E id (Θ G ) ⊂ im( ϕ ) , whence similarly to (7.3) we get S id ( f ) ∞ = · · · = S id ( f ) .Proof of Corollary. Let h ∈ E id (Θ G ) . Then there exists a 1-homotopy h t between h = id R and h = h in E (Θ G ). In particular, T h t is continuous in t .Since f is homogeneous, it follows from [9, Lemma 36], that T h t regardedas a linear automorphism of R also must preserve f , i.e. T h t ∈ LS ( f ).Therefore the family of maps T h t can be regarded as a homotopy in LS ( f ).But by Lemma 6.2 in the cases (D) and (E) the group LS ( f ) is discrete,whence all T h t coincide with the identity map T h = id R . In particular, T h = id R , whence by Lemma 7.3 h ∈ im( ϕ ). (cid:3) It remains to show that S id ( f ) and S id ( f ) coincide in the case (E) and aredistinct in the case (D).(D) Let f be a product of at least two distinct definite quadratic forms.Then by Theorem 5.1 S id ( f ) = S + ( f ). Moreover, since deg f is even, wehave that − id R ∈ S + ( f ) = S id ( f ) , see Lemma 6.1. On the other hand by ARTITION PRESERVING DIFFEOMORPHISMS 21
Lemma 7.3 for every h ∈ S id ( f ) its tangent map T h = id R = − id R . Hence S id ( f ) = S id ( f ) .(E) In this case f − (0) is a union of l ≥ { L i = 0 } passingthrough the origin. Let h ∈ D id (Θ G ) . Since there exists a homotopy between h and id R in D id (Θ G ) , it follows that h preserves every half-line of f − (0) \{ } .Therefore so does T h ∈ LS ( f ). Then it follows from Claim 6.7 that T h =id R , whence by Lemma 7.3 h ∈ im( ϕ ). Thus D id (Θ G ) ⊂ im( ϕ ).Now we get from Lemma 3.6 that S id ( f ) = D id (Θ G ) = D id (Θ G ) = S id ( f ) . Theorem 1.1 is completed.8.
Acknowledgement
I would like to thank V. V. Sharko and E. Polulyah for useful discussions. Ialso thank anonymous referee for careful reading of this manuscript and criticalremarks which allow to clarify many points.
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