On isomorphic classical diffeomorphism groups, III

Annals of Global Analysis and Geometry 13: 117-127, 1995. 117 (~) 1995 Kluwer Academic Publishers. Printed in the Netherlands.

On Isomorphic Classical Diffeomorphism

AUGUSTIN BANYAGA AND ANDREW MCINERNEY

Groups, III

Abstract: We prove that contact structures (in the restricted sense) are determined by their automorphism groups.

Key words: Contact structures, contact diffeomorphisms, Erlangen Program, Epstein axioms

M S C 1 9 9 1 : 5 3 C 1 5 ; 5 3 C 1 2 ; 5 8 D 0 5

1. I n t r o d u c t i o n

The goal of this paper is to prove tha t contact structures (in the restricted sense, see [10], [12]) are determined by their automorphism groups: a contribution to the Erlangen Program of Klein [11].

Filipkiewicz proved tha t C r structures on manifolds are determined by their automorphism groups, i.e. the group Diffr(M) of C r diffeomorphisms of the C~-mani - fold M: two manifolds are Cr-diffeomorphic if and only if their respective diffeomorphism groups are isomorphic [9]. The previous two papers [2], [3] in this series have extended this result to unimodular and symplectic structures. A topological version of some of these results is given in [4].

A contact form on a smooth manifold M of dimension 2n + 1 > 3 is a 1-form a such tha t a A (da) n is everywhere non zero. The contact s tructure (in the restricted sense) defined by a is the hyperplane field E(a ) C T M of kernels of a, also called the contact distribution. A contact structure in the wider sense is a hyperplane field E C T M such tha t there exists an open cover {(U~)} and contact forms ai on Ui such tha t Eu~ = kera i ([10], [12]). In this paper we consider only contact structures in the restricted sense.

The automorphism group of the contact s tructure E(a ) is the group Diff(M, a) = {~o e D i f f ~ ( M ) ; ~ * a -- As} for some smooth nowhere zero function ;~. Let DiffK(M, a) be the subgroup of Diff(M, a ) made of elements which have supports in a compact subset K, endowed with the compact-open topology, and let Diffe(M, a ) = lira Di f fg (M, a) , K running over all compact subsets, with the direct limit topology. We denote by G a ( M ) the identity component in Diffc(M, a) . Our main result is the following

T h e o r e m 1. Let (Mi, a~), (i = 1, 2) be paracompact, connected, smooth manifolds of dimension 2n + 1 equipped with contact forms at. I f • : Gal (M1) -+ Ga 2 (M2) is

118 A. BANYAGA AND A. MCINEKNEY

a group isomorphism, then there exists a unique C c¢ di~eomorphism w : .~I1 --~ M2 such that for all h • Gal (M1), O(h) = whw -1 and w*(~2 -- l a l for some nowhere zero function )~.

Theorem 1 was proved in [2] under the restrictive condition that • is a homeomorphism for the point-open topology. On the other hand, the referee has asked whether contact structures in the wider sense are determined by their automorphism groups as well. We are not in position to answer this question at this time, but we observe that the methods of this paper rely haevily on the existence of a globally defined contact form.

In the second paper of this series, it was noted that the methods outlined by Filipkiewicz had to be sidestepped in the volume and symplectic case because of the failure of a key lemma (the ~shrinking lemma") and the violation of Epstein's axioms [7]. We show here that in the contact case, the shrinking lemma holds and we establish Epstein's axioms. As a consequence of Epstein's theorem we obtain the following result:

T h e o r e m 2. Let (M,o~) be a contact manifold. Then [Ga(M),Ga(M)] , the commutator subgroup of Ga(M)~ is a simple group.

It is still an open problem whether G~(M) is a perfect group. The only perfectness result known for groups of contact diffeomorphisms is for (non compactly supported) contact diffeomorphisms of the Euclidean space R 2"+1 equipped with its standard contact form [5].

2. P r e l i m i n a r i e s

Let 7"{M denote the Lie algebra of smooth vector fields on a contact manifold (M, a) and by l :~(M) the Lie subalgebra of contact vector fields: a vector field X is a contact vector field if L x a = #a for some function #. The mapping X , > i ( X ) a is an isomorphism ~ between £ a ( M ) and the space C°°(M) of smooth functions on M (see for instance [13]). Therefore we can associate to any vector field V on M the contact vector field C(V) = g- l ( i (V)a) . Hence we have:

P r o p o s i t i o n 1. The Lie algebra Ca(M) is a retract of 74M.

This implies that the group C~(( I ,O) , (Diffc(M),idM)) of isotopies with compact supports retracts on the group C~(( I ,O) , (Diffc(M, a) , idM)) of contact isotopies. Recall that an isotopy is a mapping t , ~ ht • Diffc(M) such that the map H : M x [0, 1] --+ M, H(x , t) = hi(x) is smooth and H(x , O) = x, i.e. h0 = idM. An isotopy ht defines a family of vector fields ]~t :

ht(x) = ~ t (h;-l(x)).

From the isotopy ht we get the contact isotopy ht by integrating the time dependent differential equations described by the family of contact vector fields C(ht), with initial conditions h0 = idM. Using this fact, Banyaga and Pulido [5] proved the following contact version of a Palais-Cerf lemma:

P r o p o s i t i o n 2. Let (M, ol) be a contact manifold and let ht be a contact isotopy. Let F be any closed subset, and let U, W be open subsets of M such that U C ~f C W

ON ISOMORPHIC CLASSICAL DIFFEOMORPHISM GROUPS, III 119

and Ute[0,1] ht(F) C U. Then there is a contact isotopy ht such that htiF = ht and

supp ht C W.

The identity component G~(M) in Diffc(M, a) consists of diffeomorphisms h E Diffc(M, a) which are isotopic to the identity through compactly supported contact isotopies. This is a consequence of the following

P r o p o s i t i o n 3. The group Diffc(M, a) is locally connected by smooth arcs.

This in turn is a consequence of the existence of a smooth chart identifying a neighborhood of the identity in Diff,(M, a) with a neighborhood of zero in L:~(M) constructed by Lychagin [15]. For the convenience of the reader, we review Ly- chagin's construction, which parallels Weinstein's in the symplectic case [16]. Let j I M be the manifold of 1-jets of smooth functions on a smooth manifold M, i.e. J1M = UxeMJI(M), where J~(M) is the quotient C°°(M)/ I~ of the ring of smooth functions C °o (M) on M by the ideal Ix of functions vanishing at x and all of whose first partial derivatives (in some coordinate chart and hence in any chart) likewise vanish at x. J1M is naturally isomorphic with T*(M) × R and hence carries a natural contact form 0 which is equal to the pullback of the Liouville 1-form of T*(M) minus the pullback of the canonical 1-form on R.

A smooth map ¢ : L -+ M from an n-dimensional manifold L into a (2n -b 1)- dimensional contact manifold (M, a) is said to be a Legendre map if ¢*(~ --- 0. If ¢ is an embedding, L is called a Legendre submanifold. Legendre submanifolds play the role in contact geometry that Lagrange submanifolds play in symplectic geometry. The following results (Propositions 4 and 5) are due to Lychagin [14], [15]:

P r o p o s i t i o n 4. Let a : M ~ J I ( M ) be a section of the 1-jet bundle. Then ~(M) is a Legendre submanifold if and only if there is some f E Coo(M) such that a(x) = j l ( f ) ( x ) for all x E M.

The following proposition classifies neighborhoods of Legendre submanifolds in a similar way that neighborhoods of Lag-range submanifolds are classified [18].

P r o p o s i t i o n 5. Suppose L is a Legendre submanifold of the contact manifold (M, a). Then there is a diffeomorphism ~2 : U --~ V, where U C M is an open neighborhood of L and V C JI(L) is an open neighborhood of the zero section (iden- tified with L) such that ~[L = id and ~*0 = c~.

We apply the previous two propositions to the following situation: let (M, ~) be a contact manifold and define M = M x M × ( R \ 0 ) . Call 7rl, r2, t the projections onto the first, second and third factors respectively. Then the form ~ = tTr~'c~ - 7r~c~ is a contact form on M. Now let ¢ E Dif f (M,a) be such that ¢*c~ = ,~c~ for A E C ~ ( M ) . Define the "graph" of ¢ to be the map

re : M -~ M, x ~-~ (x, ¢(x), A(x)).

Note that F¢ is a Legendre embedding since F ~ = A(id*a) - ¢*a = As - As = 0.

Consider the Legendre submanifolds L¢ -- F¢(M) C 2~r and L0 = Fid(M) C which we identify with the zero section of J I ( M ) over M; we write L0 to represent the submanifolds both in M and in j1 (M).

120 A. BANYAGA AND A. MCINERNEY

By PropAosition 5, there is a diffeomorphism • : U --+ V, where U is a neighborhood of L0 in M and V is a neighborhood of L0 = M in J I ( M ) , so that ~]Lo = id and • *0 = ~, where as above 0 is the s tandard contact form on J I ( M ) . Note that if ¢ is CLclose to the identity, then L¢ C U. Hence, in this case, ~(L¢) is a Legendre submanifold of j I ( M ) , Cl-close to L0. The Cl-closeness to L0 implies tha t ~(L¢) is the graph of some section. Hence Proposition 4 implies tha t • (L¢) = j l (re) for some f¢ E C ~ ( M ) , C2-close to 0. The contact vector field 12(¢) = ¢-1(f¢) depends only on ¢ and belongs to a small C 1 neighborhood l) of 0 in £~(M). Conversely, given Z E 12, there is a unique ¢ e Di f f (M,a ) such that ~(L¢) = j l (¢ (X) ) . Therefore,we have:

L y c h a g i n ' s T h e o r e m . There is diffeomorphism ~ : )A] --+ )2 where 14~ is a neighborhood of the identity in Diffc(M, a), and )2 is a neighborhood of 0 in £~(M) . Here "diffeomorphism " refers to the natural smooth structure of Diffc(M, a) .

3. E p s t e i n ' s A x i o m s

The goal of this section is to show tha t unlike in the symplectic and volume preserving situations, Epstein 's axioms hold in the contact case.

Let G be a group of homeomorphisms of a paracompact Hansdorff topological space M, and let H be a basis of the topology of M. The Epstein axioms for the triple (G,/4, M) are:

A x i o m 1. If U E/4 and g E G, then gUE/4 .

A x i o m 2. G acts transitively on/4.

A x i o m 3. Let g E G , U E /4 and B an open cover of M. Then there exist an integer n, and Hi,g2, .-,Ha E G and V1, V2, ...Vn E B such that:

(i) g = gngn-1 .... gl; (ii) supp(gi) C_ ~ , (iii) supp(gl) U (Hi - l - . . gl& r) # M for 1 < i < n.

E p s t e i n ' s t h e o r e m . Let (G,/4, M) as above satisfying the 3 axioms, then the commutator subgroup [G, G] of G is a simple group.

Epstein showed tha t if G is the group of C ~ diffeomorphisms of a smooth manifold M which have compact support and are isotopic to the identity through C ~ compactly supported isotopies, and i f /4 is the basis of the topology consisting of embedded euclidean balls, then the triple (G,/4, M) satisfies the 3 axioms above.

I t is clear that if G is a group of diffeomorphisms preserving a volume on a smooth manifold M, there is no basis /4 of the topology of M so tha t the triple (G,/4, M) satisfy the 3 axioms. Indeed, the very definition of a basis of a topology requires tha t some elements of/4 must be contained in others, thus having smaller volume. This is impossible if a group of volume preserving diffeomorphisms was to act transitively on/4,

Consider now a contact manifold ( M , a ) and let G = Ga(M) be the identity component in Diffc(M, or). Fix a point p E M and let (V, ¢) be a Darboux chart of an open neighborhood V of p such that ¢ (V) = W C R 2~+1, ¢(p) ---- 0. Let ~- be a positive number such tha t the open ball D4r of radius 4~- centered at 0 in R 2n+l is contained in W. Let B = (¢)-1 (Dr) C M, we will call the reference contact ball.


L e m m a 1. The subsets L/ = {~B, ~ E Ga(M)} form a basis for the topology of M .

Denote by ( x , y , z ) a point of R 2n+1 where x , y E R n , z E R and let

wo = x ldy l + .... + xndyn + dz

be the canonical contact form on R 2~+1. For each t E R, let

Rt : R 2n+1 --4 R 2n+l (x, y, z) v-+ (tx, ty, t2z)

be the contact homothety, which is a contraction for any t smaller than 1 and a dilatation for t larger than 1. Proposition 2 applied with F = Dr yields the following:

L e m m a 2. For each arbitrarily small real number a and T such that 3"r <_ 1, there is a contact isotopy 7-¢~ o f R 2n+1 which is equal to Rat on Dr and supported in D3r. Moreover the set Fix(T¢~) of fixed points of T¢~ is exactly ( R 2n+l - D3r) U {0}.

Proof. Let ,k be a bump function which is positive on D2r, equal (r on Dr and zero outside D3r. Then the isotopy Tgg is the isotopy defined by the family of vector fields Xt = C(),7~t). []

Let T be the number considered above in the definition of B. Using the chart (V, ¢) above we obtain a contact isotopy p~ of M which is equal to ( ¢ ) - 1 . T~ 7 . ¢ on V and the identity outside of V. The contact diffeomorphism po = p~ has the proper ty of fixing the point p and shrinking the reference ball B into an arbitrari ly small neighborhood p~(B) = B~ E/4 ofp .

R e m a r k 1. Let (.9p = ¢ - l ( D 3 r ) , then

Fix(p 1) = (M - Op) U {p}.

Proof of Lemma 1. Pick an arbi trary point x E M and an arbitrari ly open neighborhood O of x. By Boothby's transit ivity theorem [6], there is h E Go(M) such tha t h(p) = x and for a small enough, U = h(Bo) is contained in (_9. Clearly x e U and V = (pah)(B) E/~. []

T h e o r e m 3. With the notations above, the triple (G = G~(M) , U, M) satisfy the Epstein axioms.

Proof. Axioms 1 and 2 are trivially verified. In order to verify Axiom 3, we need several results.

P r o p o s i t i o n 6. Let X E £ a ( M ) and let ~ E C ~ ( M ) be a compactly supported function. Denote by X v the contact vector field C(t/X) --- ~-l( i(vX)c~). Then [[ZVI]l _< muI]Zl]l for some my > O. Here we denoted by H . i]1 the e l -n o rm .

Proof. If f x = i(X)c~, then:

X~' "-= ( ' f x ) ~ q- o - l ( ( i (~ )d( t , ' f x ) )ce - d ( v f x ) )

= t / ( f x ~ q- { ~ - l ( ( i ( ~ ) d f x ) o t - d f x ) ) + fx (O-](( i (~)d~' )a - d~,))

-= tJX q- fx(f fP-l(( i(~)dp)~x- dr)) ,


where • is the isomorphism between horizontal vector fields and semi-basic one- forms. Hence

I lxq l l _< ][vXl[1 + Ilfxg lh, (1)

where H . is the horizontal part of the contact vector field ~- l (v) corresponding to u.

Let be such that ll lh < Since X i(X) is a linear map, llfxlh -< cllOXIII, where Cl = [Sail depends only on a. Furthermore, since • and • -1 are linear operators, we have IIHuII1 < c211X~lll ~ c2Au (where c2 depends only on 0, in turn depending only on d~). So (1) yields IiX"l]l <_ AvlIXI]I + (cl l iZlh)(c2A,) = A~,(1 -~- CLC2)HxII1. []

Properties (i) and (if) of Axiom 3 are called the fragmentation property in [1] and in [17]. It is well known that the group Diffc~(M) has the fragmentation property [16]. Fragmentation lemmas have been proved for some subgroups of volume preserving diffeomorphisms in [17], of measure preserving homeomorphisms in [8], and of symplectic diffeomorphisms in [1]. We prove here the fragmentation property for contact isotopies:

L e m m a 3. Let ¢ E G~(M) with supp ¢ C K compact, and let {Ui} be a (finite) open cover of K . Then ¢ = ~)1 O ' ' . 0 C n , where supp ¢i C Ui, ¢i E Ga(M) .

Proof. We may assume that ¢ is as close to the identity as we want: indeed, if Ct E D/fie(M, a) is an isotopy from the identity to ¢, then ¢1 = ¢ can be writ ten

- 1 ¢ = Cn. . . ¢1, where ¢i = (¢i/n o ¢(i-1)/n)' for n a suffiently large integer. Let ~ : 14 ~ --+ 1) be the Lychagin chart, and choose a neighborhood of the identity

O C )IV. Let ~ = sup{s I/~s(0) C fl(O)} where/~s(0) = { Z E £:~(M), HXII1 < s}. We choose ¢ close enough to the identity that if fl(¢) = X then IIXll] < ~/k, where k is a number specified as follows: Choose a parti t ion of unity {),i} subordinate to

i the open cover {Ui}, and define new functions #4 = ~ j = l )U" By Proposition 6, there is mi such that IlX,,lll < millxiI1 for each i. Then take k = max{m/}. Therefore each IIX'~I[1 < ?, hence t X ~ is in the image of the chart ~1.

Note that for x @ U/, #4 = #4-1. Also note #n = 1. So define h~ = ~ - l ( t X ' ~ ) . Then, h~ = h~_ 1 outside Ui, i.e. ¢~ = h~o(h~_l) -1 has support in U/. By construction,

= [] 1"

As in Epstein [7], property (i/i) in Axiom 3 is a consequence of the fragmentation lemma and the following

P r o p o s i t i o n 7. Let g E G where suppg C V E lg. Let U be a non-dense open subset of M. Then g = g2 o gl where g/ E G, suppg/ C V and suppgl U U ~ M, supp g2 U gl 0 ¢ M.

Proof. Let g, 11, U be as in the hypotheses. If suppg C /)', choose gl = id, g2 = g. Otherwise choose a point x E V - U" such that g(x) ~ x. Choose an isotopy gt E D/fie(M, c~) from the identity to gl = g. Let N1 be an open neighborhood of x small enough so that there are open sets N2, N3 satisfying:

U gt(/v1) c N2 c 192 c N3 c V. tEI

ON ISOMORPHIC CLASSICAL DIFFEOMORPHISM GROUPS, I I I 123

Choose a point y ~ (0" U N3) and let N4 be an open neighborhood of y such tha t N3 M/V4 = 0. By Proposition 2, there is a contact isotopy h t such tha t ht]f¢l = gt and supp h t C g3.

Now simply define gl = hi, g2 = g(hl) -1. Clearly, suppgi C V for i = 1,2. Since y ~ U and y ¢ supph, s u p p g l U U # M. Finally, suppg2Ug10 # M: indeed, gl(x) 01U since x ¢ U; moreover for any z E N1 we have g2(gl(z)) = g(z) = h(z) = gl(z). Therefore gl(x) ~ supp g2- []

End of proof of Theorem 3. We are now in position to finish the proof of Axiom 3: let g , /4 , B be as in the hypotheses of the axiom. Since 9 has compact support K , we can find a covering of K by finitely many elements of B, ( ~ } i = 1 , . . . , m.

Now apply the contact fragmentation lemma to g relative to the cover {Y~, i.e. express g = gmO" 'og l with gi E G, suppgi C ~ . We can then apply the proposition

2 1 to each of the (gi, Y~, gi-1 o - . . o glU). In this way we obtain 9i = gi gi, supp ~ C V~, suppg~U ¢ M, and suppgi 2 U g~U ¢ M. This proves the Axiom 3: n = 2m,

2 1 g = g r o g s v j = v , . []

4. P r o o f o f t h e M a i n T h e o r e m

Our main theorem will follow from the results of the previous two papers in this series. Before we review these results, we must make several definitions.

D e f i n i t i o n 1. Let G(M) be a group of diffeomorphisms of a manifold M. G (M) is said to satisfy Condition A if, given x, y E M, x ~ y, and a pa th c : I -+ M such tha t c(0) = x, c(1) = y, there exists an h E G(M) such tha t h(x) = y and supp(h) is contained in an arbitrari ly small open neighborhood of Utsle(t).

D e f i n i t i o n 2. Let G ( M ) be as above. G ( M ) satisfies Condition B if for any open ball U "centered" at x0, there is an h E G(M) such that Fix(h) = ( i - U) U (x0), where Fix(h) = {x E i [ h(x) = x}. An open ball in i is the image of an embedding preserving some given structure, of an open euclidean ball centered at the origin. The center of the open ball in M is the image point of the origin.

Let G(U) C G( M) be the subgroup of elements of G (M) which have support in an open set U.

D e f i n i t i o n 3. A subgroup F C G(M) is said to have Property L if the condition tha t for each x E M there is an open ball U~ E /4 containing x and such that [G(U~), G(Ux)] C F, implies that [ G ( / ) , G ( / ) ] C F.

If G ( M ) is a group of homeomorphisms of the manifold M, we denote by Sx (G (M)) the stabilizer of a point x of M. In the second paper of the series, the following is proved:

T h e o r e m . ([3]) Let ¢ : G(M) --> G(N) be a group isomorphism between two groups of diffeomorphisms of smooth manifolds i and N . I f G (M) and G(N) are nonabelian and satisfy conditions A and B, and if Fn = ¢ - I ( S n G ( N ) ) and Fm = ¢ (SmG(M)) have the property L for all m E M, n E N , then there exists a unique homeomorphism w : M ~ N with ¢ ( f ) ---- w f w -1 for all f E G(M) .

The first paper of the series ensures us tha t the homeomorphism above is in fact a structure-preserving diffeomorphism.

124 A. BANYAGA AND A. I~/[CINERNEY

T h e o r e m . ([2]) Let ( Mi, c~i), i = 1, 2 be two manifolds equipped with volume forms, symplectie forms, or contact forms al. Let w : M1 -~ M2 be a bijective map such that for any map f : M1 -~ M2, w f w -1 E Ga2(M2 ) if and only if f E Ga l (MI ). Then w is a C c¢ diffeomorphism and w*c~2 = ~oL1 for some function )% which is constant if o~i are symplectic or volume forms.

We must show, then, tha t the group Ga(M) of contact diffeomorphisms isotopic to the identity through contact diffeomorphisms with compact support satisfy Con- ditions A and B, and tha t the stabilizers of points have Proper ty L.

P r o p o s i t i o n 8. Ga(M) satisfies Condition A.

Proof. This is a consequence of the proof of Boothby's theorem on the k-transit ivity of the group e ~ ( i ) ([6], see also [12]). []

P r o p o s i t i o n 9. Ga(M) satisfies Condition B.

Proof. This follows from Remark 1 and Lemma 2. []

In order to check Proper ty L, we will need the following contact modification of a "shrinking lemma" of Filipkiewicz [9].

Let Da n be the closed ball in R n of radius a (when the dimension is understood we will suppress the superscript). Also let B(x , e) be the ball of radius e centered at x.

L e m m a 4. Let C be a covering of D~ n+l by open subsets of R 2n+l. Then if a E (0, 1], there are fi ,gi E Gwo(R2n+l), 1 < i < r such that

(i) For 1 < i < r, 3Ci E C such that fi ,gi E G~o(Ci);

(ii) [h , gr] o . . . o [fl,gl](D1) C Da. Here wo is the standard contact form ~ i n l xi dyi + dz on R 2n+1.

Proof. Following Filipkiewicz, we define the set A -- {a • (0, 1] : the lemma is true for a}. Note A is not empty since 1 • A; if a -- 1 let f -- g -- id, and the lemma is satisfied.

Now let ao = inf A; we wish to show tha t ao = 0. Suppose ao > 0. Let {l~} be a finite cover of the boundary ODao of Dao, and

choose e less than the Lebesgue number of the covering {~}. For each x • OD, o choose pairs (gx, [Ix) with gx • Gwo(B(x,e)) such that:

(i) Ux C Dao+E/2 -- Dao-e/2 and Ux C B(x , e);

(ii) gx(U,) C D~o_~/2 and g z I ( u x ) C R 2n+1 - Dao+~/2.

To do this, choose points p , • B(x , e) n (0D~o_3~/4), qx • B(x , e) M (0D~o+3~/4). By Boothby 's t ransi t ivi ty theorem, there exist u, v • G~ o (B(x, e)) such that

u(x) = px; u(qx) = qx, v(qx) = x, v(p~) = p~.

Setting g~ = vu, then gx(x) = p, (gx)- l (x) = q~:. It follows that if U~ is a sufficiently small neighborhood of x, then g~(U~) C Dao-c/2 and g~- l (u~) C R 2~+1 - D~o+~/2.

Prom this covering {U~} of OD,~ o choose a finite covering {Ui} and corresponding {gd-


Choose e' < e/2 such t ha t

(Dao+~, - Dao-~,) C 0 U~. i=1

Now choose some f 6 G~ o ( I n t ( V ~ o + d - V . o - d ) ) such tha t f (D .o+d/c ) C Vao-d /c for some c > 1.

To cons t ruc t such an f , consider contac t h o m o t h e t y #~ (xi, Yl, z) = (Axi, Aye, A2z),

where A = ( ~ ~ ~ao+d/cs < 1, and c > 1 is chosen so t ha t

(a0 -- c ' /c)A 2 > a0 -- e'.

We check first t h a t #~(D~o+~,/~ ) C Dao-~,/~. Let p E Dao+~,/~,P = (2i, ~ , 2). T h e n

71

I.>,(p)l = [X2(E(~F + ~2) + x42211/2 i=1

< [~2(~(~ +~)+ ~2)]~12= ~lpl-< :,(ao+~'/c)----ao-c'/c. We now app ly Propos i t ion 2 to obta in a contac t d i f feomorphism f which agrees

wi th #x on D,~o+d/c - Dao_d/~ suppor t ed inside Int(D~o+~,/~ ). To verify t ha t this is possible, we need to show tha t if [p[ > ao - e'/c, then I#x(P)I > a0 - d. Bu t

n I.~(p)l = [~2(E(~ +~21 + ~4~]1/2 i=1

___ [~4(y](~ +~2)+ ~2)]1/2 = ~21p I > X2(a ° -?/c) > ao--?/~.

Finally, we mus t show tha t c can be chosen to sat isfy the necessary condit ion above. The condi t ion

# ao -- £1/C 2 (a0 - - e / c ) ( ~ 0 0 + - - - ~ ) > (a0 - e')

is equivalent (by cross mul t ip l ica t ion and simplification) to

c3(a0 - e'lc) 3 > c3(a0 - et)(a0 A- eric) 2.

This can be expanded into a polynomial inequali ty P(c) > 0; P(c) is cubic in c wi th posi t ive leading te rm, and hence P(c) > 0 can be satisfied by choosing c sufficiently large.

Now app ly L e m m a 3 to this f subjec t to the cover {Ui} to get fi wi th supp( f i ) C u~.

Define hl = [fi, gi]. T h e n

{ fi(~) • c ui hi(x) = g i f lg? l (x ) x C gi(Ui)

x otherwise, [supp(f i) CUi] .

The remainder of the p roof is identical to t ha t of Filipkiewicz: if

x E Dao+E' -- Dao-g


then

h~ o . - . o hi(x) = f~o . . , o f l(x)

and so

h~ o .. . o hl(Dao) C D~o_~,/2.

Further, for each i there is an xl E ODao such that

fi,gi e a~oo(B(xi, e));

since e is less than the Lebesgue number of the covering C, there is an open Ci E C with

G o(C ) for each 1 < i < r. But this contradicts the minimality of a0. []

We are now in a position to prove:

P r o p o s i t i o n 10. Every subgroup F of Ga(M) has property L.

Proof. We follow Filipkiewicz' method directly. Let H be the subgroup of [Ga(M), Ga(M)] generated by the groups [Ga(U), Ga(U)] as U ranges over/4. H is a normal subgroup of [G~(M), G~(M)]. By Theorem 2, g = [G~(M), G~(M)].

Now let F be a subgroup of G~(M) satisfying the hypotheses of property L, i.e. that for all x E M, there is a neighborhood U, E/4 of x such that [G(Uz), G(Uz)] C F. We need only to show then that if W e / 4 then [G~(W), Ga(W)] C F. So take W E/4, i.e. W = ¢¢(D1), where ¢ is a given Darboux chart with domain U such that D1 C U, and ¢ E Ga(M). By hypothesis we can cover lYV by U1, . . . , Us ELt so that [Ga(Ui),Ga(Ui)] C F. Then define ~ = (¢¢)-1(Ui); these define a cover of D1 C R 2n+l with each ~ C U (if not, choose Ui smaller). Assume 0 6 V1. For some a > 0 we have Da C V1, so apply the shrinking lemma to obtain commutators [fj,gj] (1 < j < r) with f j ,gj 6 Go, o(Vi,j) and [fr,g~] o . . . o [fl,gl](D1) C Da.

Now since each of the the f j ,gj 6 Gwo(U) we can define f j ,~j 6 Ga(M) as follows

= / ( ¢¢ ) f s (¢¢ ) - I x ¢ ¢ ( u )

( x otherwise

= (¢¢)gs(¢¢) -I x ¢ ¢ ( u ) j(x) L x otherwise.

So [~,~j] E [G~(Ui,j),G~(UI,j)]. Now define T ---- [f'~,~r] o . . . o [~,gl] . Then by hypotheses r E F; also ~-(W) C U1.

Hence [G~(W),G~(W)] C T-I[Gu,,Gu,]r C F. []

This completes the proof of the main theorem.

A k n o w l e d g e m e n t s . It is a pleasure to thank the referee for his/her remarks; in particular for having raised the question whether contact structures in the wider sense are determined by their automorphism groups. The first author (A.B.) was partially supported by NSF grants DMS 90-01861 and DMS 9403196 and the second author (A.M.) by a Penn State University graduate assistantship.


R e f e r e n c e s

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A. BANYAGA A. MCINERNEY Department of Mathematics The Pennsylvannia State University University Park, PA 16802 USA

(Receiced March 2, 1994; new version August 29, 1994)

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On isomorphic classical diffeomorphism groups, III