Smooth perfectness for the group of diffeomorphisms

0.17 MB·English

by  Stefan Haller

SMOOTH PERFECTNESS FOR THE GROUP OF
4 DIFFEOMORPHISMS
0
0
2 STEFANHALLERANDJOSEFTEICHMANN

p
e
Abstract. Given the result of Herman, we provide a new elementary proof
S
of the fact that the group of diffeomorphisms is a perfect and hence simple

0 group. Moreover,weshowthateverydiffeomorphismcanberepresentedasa
3 product of commutators f =[g1,h1]···[gm,hm] where the factors gi and hi
canbechosentodependsmoothlyonf. Theelegance ofourapproachisdue

] to the fact that we prove smooth perfectness rather than perfectness. If the
G underlying manifold is of dimension n ≥ 2 then one can take m = 9(n+1)
D providedf issufficientlyclosetotheidentity.
.
h
t
a
m 1. Introduction and statement of the result
[
Definition1. ApossiblyinfinitedimensionalLiegroupGiscalledlocallysmoothly

1 perfect of order at most m if the smooth map K :G2m →G
v
5 K(g1,h1,...,gm,hm):=[g1,h1]◦···◦[gm,hm]
0
has a local smooth right inverse at the identity. More precisely, we ask for an
6
9 open neighborhood V of e ∈ Go and a smooth map σ : V → (Go)2m such that
0 K ◦σ = idV. Here Go denotes the connected component of e ∈ G. Be aware that
4 this concept is slightly different from the one used in [HT03].
0
∞
/ For a smooth manifold M without boundary we denote by Diff (M) the group
h c
ofcompactly supporteddiffeomorphisms ofM. This is aLie groupmodeled onthe
t
a convenient vector space X (M) of compactly supported vector fields, see [KM97]
m c
for all necessary details. For compact M this smooth structure coincides with the
: well known Fr´echet–Lie group structure on Diff∞(M).
v
i The aim of this paper is to establish the following
X
r Theorem 1. Let M be a smooth manifold without boundary of dimension n ≥ 2.
a Then Diff∞(M) is locally smoothly perfect of order at most 9(n+1).
c
∞
Since anyopenneighborhoodofe∈Diff (M) generatesthe group,Theorem1
c o
∞
implies that Diff (M) is a perfect group. This was already proved by Epstein
c o
[E84] using ideas of Mather, see [M74] and [M75], who dealt with the Cr–case,
1 ≤ r < ∞, r 6= n+1. The Epstein–Mather proof is elementary, but tricky. It is
basedontheSchauder–Tychonovfixedpointtheorem,andguarantiestheexistence
Date:DraftfromFebruary1,2008.
2000 Mathematics Subject Classification. 58D05.
Key words and phrases. diffeomorphism group; perfect group; simple group; fragmentation;
convenient calculus;foliation.
The first author is supported by the Fonds zur F¨orderung der wissenschaftlichen Forschung
(AustrianScienceFund),projectnumberP14195-MAT.
1
2 STEFANHALLERANDJOSEFTEICHMANN
of a presentation f = [g ,h ]···[g ,h ], but without any further control on the
1 1 m m
factors g and h .
i i
Theorem 1 actually implies the stronger statement that the universal covering
∞
of Diff (M) is a perfect group. This result is known too, see [T74] and [M84].
c o
Thurston’s proof is based on a result of Herman for the torus, see [H71], [H73]. It
also does not imply that the factors g and h can be chosen to depend smoothly
i i
on f.
∞
Note that the perfectness of Diff (M) implies that this group is simple, see
c o
[E70]. The methods used in [E70] are elementary and actually work for a rather
large class of groups of homeomorphisms.
Perfectnessandsimplicity results havebeenobtainedfor severalother groupsof
diffeomorphisms. For instance, the group of volume preserving diffeomorphisms is
notperfect. However,thereisawellunderstoodnormalsubgroup,thekernelofthe
flux homomorphism, which is simple according to unpublished work of Thurston,
see [B97] for a write-up, and the series of papers [MD82], [MD83], [MD84].
UsingThurston’smethodsBanyagaclarifiedthealgebraicstructureofthegroup
ofcompactlysupportedsymplectomorphisms,see[B78]and[B97]. Again,thegroup
itself is not perfect but the normal subgroup of Hamiltonian diffeomorphisms is
simple for compact M. For non-compact M the group of compactly supported
Hamiltoniandiffeomorphisms containsanormalsubgroup,the kernelofthe Calabi
homomorphism,whichturnsouttobeasimplegroup. In[HR99]Banyaga’sresults
have been generalized to the slightly more general situation of locally conformal
symplectic manifolds.
To our knowledge the case of contact diffeomorphisms is still open, though it
appears reasonable to expect that the group of compactly supported contact dif-
feomorphisms is perfect and hence [E70] simple. Thurston’s approachshould work
in this case. However, a model case playing the role of Herman’s torus is lacking.
The Lie algebraof compactly supportedcontactvectorfields certainly is perfect in
view of [O97].
Rybicki [R95] proved that the group of compactly supported diffeomorphisms,
which preserve the leaves of a foliation of constant rank, is a perfect group —
though not simple of course.
Note, however, that none of the results above guaranties that the factors in a
presentationf =[g ,h ]···[g ,h ]canbechosentodependsmoothlyonf,except
1 1 m m
Herman’s result for the torus. Nor are there any estimates available on how many
commutatorsoneactuallyneeds. In[HT03]weprovidedaclassofclosedmanifolds,
including compact Lie groups and odd dimensional spheres, for which the factors
canbechosensmoothly,andestimatesonthenumberofcommutatorsareavailable.
OurproofofTheorem1restsonHerman’sresulttoobutisotherwiseelementary
and completely different fromThurston’s approach. The drawbackof our methods
is that they do not cover the manifold M =R.
The idea of our approach is to consider the foliation F on R × Tn−1 with
leaves {pt}×Tn−1. Parametrizing, see Section 4, Herman’s result for the torus,
see Section 5, one obtains that the group of leave preserving diffeomorphisms
Diff∞(R×Tn−1,F) is locally smoothly perfect. Perturbing F, we obtain finitely
c
manydiffeomorphicfoliationsF ,...,F onR×Tn−1whichspanthetangentbun-
1 k
dle. Actuallyk =3suffices,seeSection6. Decomposingadiffeomorphisminto leaf
preserving ones, see Section 3, we show that Diff∞(R×Tn−1) is locally smoothly
c
SMOOTH PERFECTNESS FOR THE GROUP OF DIFFEOMORPHISMS 3
perfect, see Proposition 4. Since R×Tn−1 embeds into Rn, we can cover M with
opensetsdiffeomorphictoR×Tn−1. Usingaclassicaltoolknownasfragmentation,
∞
see Section 2, we conclude that Diff (M) is locally smoothly perfect.
c
2. Smooth fragmentation
Let M be a smooth manifold without boundary. For an open subset U ⊆ M
∞ ∞
we denote by Diff (M) = Diff (U) the group of diffeomorphisms with compact
U c
supportcontainedinU. ThisisaLiegroupmodeledontheconvenientvectorspace
X (M)=X (U) of vector fields with compact support contained in U.
U c
The following is a folklore statement which can be found all over the literature
on diffeomorphism groups.
Proposition 1. Let M be a smooth manifold without boundary, and suppose U =
{U ,...,U } is a finite open covering of M. Then the smooth map
1 k
∞ ∞ ∞
P :Diff (M)×···×Diff (M)→Diff (M),
U1 Uk c
P(g ,...,g ):=g ◦···◦g
1 k 1 k
has a local smooth right inverse at the identity. More precisely, there exist an open
∞ ∞
neighborhood V of e ∈ Diff (M) and a smooth map σ : V → Diff (M) ×···×
∞ c o U1 o
DiffUk(M)o with σ(e)=(e,...,e) such that P ◦σ =idV.
Proof. Let π : TM → M denote the projection of the tangent bundle. Choose a
Riemannian metric on M and let exp denote the corresponding exponential map.
Choose an open neighborhood of the zero section W ⊆ TM such that (π,exp) :
W →M×M is a diffeomorphismonto an open neighborhoodof the diagonal. Let
W′ ⊆T∗M ⊗TM be an open neighborhood of the zero section, and set
W := X ∈X (M) img(X)⊆W,img(∇X)⊆W′ ,
c
(cid:8) (cid:12) (cid:9)
(cid:12)
where ∇ denotes the Levi–Civita connection associated to the Riemannian met-
ric. This is a C1–open neighborhood of 0 ∈ X (M). For X ∈ W define f :=
c X
exp◦X ∈C∞(M,M). We choose W and W′ sufficiently small such that every f
c X
is a diffeomorphism. The map
∞
W →Diff (M), X 7→f (1)
c X
∞
providesa chartofDiff (M) centered atthe identity. This is the standardway to
c
∞
put a smooth structure on Diff (M), see [KM97].
c
Chooseasmoothpartitionofunityλ ,...,λ withsupp(λ )⊆U forall1≤i≤
1 k i i
k. Define
′
W := X ∈W (λ +···+λ )X ∈W for all 1≤i≤k .
(cid:8) (cid:12) 1 i (cid:9)
This is a C1–open neighborh(cid:12)ood of 0 ∈ X (M), and W′ ⊆ W. For X ∈ W′ and
c
∞
1 ≤ i ≤ k set X := (λ +···+λ )X, and note that f ∈ Diff (M). Clearly
i 1 i Xi c
supp(fX1) ⊆ supp(λ1) ⊆ U1. Moreover, for 1 < i ≤ k we have Xi−1 = Xi on
M \ supp(λ ), and hence f = f on M \supp(λ ). We conclude that the
i Xi−1 Xi i
diffeomorphism (f )−1 ◦f has compact support contained in supp(λ ) ⊆ U
Xi−1 Xi i i
for all 1<i≤k.
∞ ∞
Let V ⊆Diff (M) denote the open neighborhood of e∈Diff (M) correspond-
c c
ing to W′ via (1). Define a map
∞ ∞
σ :V →Diff (M)×···×Diff (M)
U1 Uk
4 STEFANHALLERANDJOSEFTEICHMANN
σ(f ):= f ,(f )−1◦f ,(f )−1◦f ,...,(f )−1◦f
X (cid:16) X1 X1 X2 X2 X3 Xk−1 Xk(cid:17)
andnoticethatitissmoothbecauseX dependssmoothlyonf inviewofourchart
X
(1). Obviously we have P(σ(fX))=fXk =fX and thus P ◦σ =idV. Moreover,it
is immediate from the construction that σ(e)=σ(f )=(e,...,e). (cid:3)
0
3. Smooth Decomposition into foliation preserving diffeomorphisms
Let M be a smooth manifold without boundary and suppose F is a foliation on
∞
M. We will always assume that foliations are of constant rank. Let Diff (M,F)
c
denotethegroupofcompactlysupporteddiffeomorphismswhichpreservetheleaves
of F. This is a Lie group modeled on the convenient vector space X (M,F) of
c
compactly supported vector fields tangential to F, see [KM97].
A weaker version of the following proposition was proved in [HT03] using the
Nash–Moser implicit function theorem. However, Proposition 2 applies to non-
compact manifolds as well, and its proof is elementary.
Proposition2. LetM beamanifold without boundaryandsuppose F ,...,F are
1 k
smooth foliations on M such that the corresponding distributions span the tangent
bundle of M. Then the smooth map
∞ ∞ ∞
F :Diff (M,F )×···×Diff (M,F )→Diff (M)
c 1 c k c
F(g ,...,g ):=g ◦···◦g
1 k 1 k
has a local smooth right inverse at the identity. More precisely, there exist an open
∞ ∞
neighborhood V of e ∈ Diff (M) and a smooth map σ : V → Diff (M,F ) ×
c o c 1 o
∞
···×Diffc (M,Fk)o with σ(e)=(e,...,e), such that F ◦σ =idV.
Proof. Let us write TF for the subbundle of TM consisting of vectors tangent
i
to F . Let Σ : TF ⊕···⊕TF → TM denote the vector bundle map given by
i 1 k
addition. Choose a vector bundle map s:TM →TF ⊕···⊕TF rightinverse to
1 k
Σ. This is possible since the distributions F span the tangent bundle.
i
Choose a Riemannian metric on M, and note that this provides Riemannian
metrics on the leaves of F . We obtain smooth exponential maps expFi :U →M,
i i
definedonanopenneighborhoodofthezerosectionU ⊆TF ,whichmapthefibers
i i
of TF into the leaves of F . Choose any linear connection on TF ⊕···⊕TF .
i i 1 k
For1≤i≤k and(ξ ,...,ξ )∈TF ⊕···⊕TF let τ (ξ ,...,ξ )denote the point
1 k 1 k i 1 k
obtained by parallel transport in TF ⊕···⊕TF along the curve t 7→expFi(tξ )
1 k i
up to time t=1. It is evident that τ constitutes a smooth map:
i
τ :TF ⊕···⊕U ⊕···⊕TF →TF ⊕···⊕TF
i 1 i k 1 k
We certainly find an open neighborhood of the zero section W ⊆TM such that
the compositions
α :=π◦τ ◦τ ◦···◦τ ◦s:W →M
i i i+1 k
are well defined for all 1 ≤ i ≤ k. Here π : TF ⊕···⊕TF → M denotes the
1 k
vector bundle projection.
Set α := α , and consider (π,α) : W → M ×M. Its derivative at 0 ∈ T M is
1 x x
invertible. Using the inverse function theorem and possibly shrinking W we may
assume that (π,α):W →M ×M is a diffeomorphism onto an open neighborhood
ofthediagonal. LetW′ ⊆T∗M⊗TM beanopenneighborhoodofthezerosection,
and define
W := X ∈X (M) img(X)⊆W,img(∇X)⊆W′ ,
c
(cid:8) (cid:12) (cid:9)
(cid:12)
SMOOTH PERFECTNESS FOR THE GROUP OF DIFFEOMORPHISMS 5
where∇denotestheLevi–CivitaconnectiononM. ThisisaC1–openneighborhood
of 0 ∈ X (M). For X ∈ W define f := α◦X ∈ C∞(M,M). We choose W and
c X c
W′ sufficiently small so that every f is a diffeomorphism. It is now clear that
X
∞
W →Diff (M), X 7→f (2)
c X
∞
constitutes a chart for Diff (M) centered at the identity, cf. the concept of local
c
addition in [KM97, Section 42.4].
For 1 ≤ i ≤ k and X ∈ W define f := α ◦X. Again, we choose W and
X,i i
W′ sufficiently small so that f is a diffeomorphism for every X ∈ W and all
X,i
∞
1 ≤ i ≤ k. Obviously we have f = f . Also note that f ∈ Diff (M,F ).
X,1 X X,k c k
Similarly,itfollowsfromtheconstructionthatf ◦(f )−1 ∈Diff∞(M,F )for
X,i X,i+1 c i
all 1≤i<k.
∞
Let V denote the open neighborhood of e ∈ Diff (M) corresponding to W via
c
(2). Define a map
∞ ∞
σ :V →Diff (M,F )×···×Diff (M,F )
c 1 c k
σ(fX):=(cid:16)fX,1◦(fX,2)−1,fX,2◦(fX,3)−1,...,fX,k−1◦(fX,k)−1,fX,k(cid:17)
andnotethatitissmoothbecauseX depends smoothlyonf inview ofourchart
X
(2). Moreover we have F(σ(fX))=fX,1 =fX and thus F ◦σ =idV. Since the αi
restrict to the identity on the zero section we have σ(e)=σ(f )=(e,...,e). (cid:3)
0
4. The exponential law and smooth parametrization of
diffeomorphism
Suppose B is a smooth manifold without boundary and suppose G is a possibly
infinitedimensionalLiegroup. Considerthe spaceC∞(B,G)ofsmoothmapsfrom
c
B to G which are constant equal to e ∈G outside a compact subset of B. This is
a Lie group with point wise multiplication, see [KM97].
Lemma1. IfGislocallysmoothlyperfectoforderatmostm,thensoisC∞(B,G).
c
Proof. Let σ : V → G2m be a local right inverse as in Definition 1. Let V′ ⊆
C∞(B,G) denote the subset of maps which take values in V ⊆G. This is an open
c
neighborhood of e∈C∞(B,G). Consider the smooth map
c
σ′ :=σ∗ :V′ →Cc∞(B,G2m)=Cc∞(B,G)2m
given by composition with σ. Its restriction to the connected component of V′
clearly is a local smooth right inverse as required. (cid:3)
Remark 1. NotethatifGisonlyperfectthentheargumentintheproofofLemma1
breaks down, and one cannot conclude that C∞(M,G) is perfect.
c
Let S and B be smooth manifolds without boundary. Set M := B×S and let
F denote the foliation of M with leaves {pt}×S.
∞
Proposition 3. If Diff (S) is locally smoothly perfect of order at most m, then
c
∞
so is Diff (M,F).
c
Proof. It follows from the exponential law in [KM97] that the homomorphism
∞ ∞ ∞
C (B,Diff (S))→Diff (M,F)
c c c
α7→ (b,s)7→(b,α(b)(s))
(cid:0) (cid:1)
is an isomorphism of Lie groups. The proposition now follows from Lemma 1
above. (cid:3)
6 STEFANHALLERANDJOSEFTEICHMANN
5. Herman’s result for the torus
Let Tn := Rn/Zn denote the torus of dimension n. The group Diff∞(Tn) is
a Fr´echet–Lie group modeled on the Fr´echet space X(Tn) of vector fields on Tn.
For β ∈ Tn we let R ∈ Diff∞(Tn) denote the rotation by β. In this way we can
β
consider Tn as subgroup of Diff∞(Tn) .
o
Theorem 2 (Herman). There exist γ ∈Tn such that the map
H :Tn×Diff∞(Tn) →Diff∞(Tn) ,
o o
H(λ,h):=R ◦[R ,h]
λ γ
has a local smooth right inverse at the identity. More precisely, there exist an open
neighborhood V of e ∈ Diff∞(Tn) and a smooth map σ : V → Tn×Diff∞(Tn)
o o
with σ(e)=(0,e) such that H ◦σ =idV.
Herman’s result is an application of the Nash–Moser inverse function theo-
rem. When inverting the derivative one is quickly led to solve the linear equation
Y = X −(R )∗X for given Y ∈ C∞(Tn,Rn). This is accomplished using Fourier
γ
transformation. Here one has to choose γ sufficiently irrational so that tame es-
timates on the Sobolev norms of X in terms of the Sobolev norms of Y can be
obtained. The corresponding small denominator problem can be solved due to a
number theoretic result of Khintchine. For a proof of Herman’s theorem see [H73]
and [H71].
Using the fact that the perfect group of Mo¨bius transformations acts on T1,
Herman concluded that Diff∞(Tn) is a perfect group. In fact, one can estimate
o
the number of commutators necessary, see [HT03, Section 4], and obtains
Corollary 1. Diff∞(Tn) is locally smoothly perfect of order at most 3.
6. Proof of Theorem 1
We start proving the following
Proposition 4. Suppose n≥2. Then Diff∞(R×Tn−1) is locally smoothly perfect
c
of order at most 9.
Proof. Consider the foliation F on R×Tn−1 with leaves {pt}×Tn−1. In view of
Proposition3andCorollary1weknowthatDiff∞(R×Tn−1,F)islocallysmoothly
c
perfect of order at most 3.
One easily constructs diffeomorphisms ϕ of R×Tn−1 such that the foliations
i
F :=ϕ (F), 1≤i≤3,spanthe tangentbundle ofR×Tn−1. Indeed, choosefunc-
i i
tionsh :Tn−1 →Rsuchthath =0andthe criticalsetsofh andh aredisjoint.
i 1 2 3
Nowdefineϕ (x,t):=(x+h (t),t). Itisreadilycheckedthatthesediffeomorphisms
i i
havethe desiredproperty. ClearlyDiff∞(R×Tn−1,F )≃Diff∞(R×Tn−1,F)also
i
is locally smoothly perfect of order at most 3. It follows from Proposition 2 that
Diff∞(R×Tn−1) is locally smoothly perfect of order at most 3·3=9. (cid:3)
c
ForasmoothmanifoldM ofdimensionnletcov(M)denotetheminimalinteger
k such that there exists an open covering U = {U ,...,U } of M for which every
1 k
U is diffeomorphic toa disjointunionofcopies ofR×Tn−1. Using a triangulation
i
and the fact that R×Tn−1 embeds into Rn, one easily shows cov(M) ≤ n+1.
Therefore Theorem 3 below implies Theorem 1.
SMOOTH PERFECTNESS FOR THE GROUP OF DIFFEOMORPHISMS 7
∞
Theorem 3. Let M be a smooth manifold of dimension n≥2. Then Diff (M) is
c
locally smoothly perfect of order at most 9·cov(M).
Proof. Choose an open coveringU ={U ,...,U } of M such that every U is
1 cov(M) i
diffeomorphic to a disjoint union of copies of R×Tn−1. Using Proposition 4 we
seethatDiff∞(M)= Diff∞(R×Tn−1)is locallysmoothlyperfectoforder
Ui Lπ0(Ui) c ∞
at most 9. In view of Proposition1 this implies that Diff (M) is locally smoothly
c
perfect of order at most 9·cov(M). (cid:3)
It is easy to show that cov(Rn)=cov(Sn)=2. Hence we obtain
Corollary 2. Let n ≥ 2. Then Diff∞(Sn) and Diff∞(Rn) are locally smoothly
c
perfect of order at most 18.
Note thatthe caseM =S1 is coveredbyCorollary1. However,the caseM =R
is not.
References
[B78] A.Banyaga, Sur la structure du groupe des diff´eomorphismes qui pr´eservent une forme
symplectique.Comment.Math.Helv.53(1978), 174–227.
[B97] A. Banyaga, The structure of classical diffeomorphism groups. Kluwer Academic Pub-
lishersGroup,Dordrecht,1979.
[E70] D.B.A.Epstein,Thesimplicityofcertaingroupsofhomeomorphisms.CompositioMath.
22(1970), 165–173.
[E84] D.B.A.Epstein,CommutatorsofC∞–diffeomorphisms. Appendixto: ‘Acuriousremark
concerning the geometric transfer map” by John N. Mather. Comment. Math. Helv.
59(1984), 111–122.
[HR99] S.HallerandT.Rybicki,Onthegroupofdiffeomorphismspreservingalocallyconformal
symplectic structure.Ann.GlobalAnal.Geom.17(1999), 475–502.
[HT03] S. Haller and J. Teichmann, Smooth perfectness through decomposition of diffeomor-
phisms into fiberpreserving ones.Ann.GlobalAnal.Geom.23(2003),53–63.
[H71] M.R. Herman, Simplicit´e du groupe des diff´eomorphismes de classe C∞, isotopes `a
l’identit´e, du tore de dimension n.C.R.Acad.Sci.ParisSr.A273(1971),232–234.
[H73] M.R.Herman,Surlegroupe desdiffomorphismes dutore.Ann.Inst.Fourier(Grenoble)
23(1973), 75–86.
[KM97] A. Kriegl and P.W. Michor, The convenient setting of global analysis. Mathematical
SurveysandMonographs 53,AmericanMathematical Society,1997.
[M74] J.N. Mather, Commutators of diffeomorphisms. Comment. Math. Helv. 49(1974), 512–
528.
[M75] J.N. Mather, Commutators of diffeomorphisms. II. Comment. Math. Helv. 50(1975),
33–40.
[M84] J.N.Mather,Acuriousremarkconcerningthegeometrictransfermap.Comment.Math.
Helv.59(1984), 86–110.
[MD82] D.McDuff,Localhomology ofgroupsofvolumepreservingdiffeomorphisms. I.Ann.Sci.
E´coleNorm.Sup.(4)15(1982), 609–648.
[MD83] D. McDuff, Local homology of groups of volume-preserving diffeomorphisms. II. Com-
ment.Math.Helv.58(1983),135–165.
[MD84] D. McDuff, Local homology of groups of volume-preserving diffeomorphisms. III. Ann.
Sci.E´coleNorm.Sup.(4)16(1984),529–540.
[O97] H. Omori, Infinite-dimensional Lie groups. Translated from the 1979 Japanese original
and revised by the author. Translations of Mathematical Monographs 158. American
Mathematical Society, Providence,RI,1997.
[R95] T.Rybicki,Theidentitycomponentoftheleafpreservingdiffeomorphismgroupisperfect.
Monatsh.Math.120(1995),289–305.
[T74] W. Thurston, Foliations and groups of diffeomorphisms. Bull. Amer. Math. Soc.
80(1974), 304–307.
8 STEFANHALLERANDJOSEFTEICHMANN
StefanHaller,DepartmentofMathematics,UniversityofVienna,Nordbergstraße
15,A-1090,Vienna,Austria.
E-mail address: [email protected]
Josef Teichmann, Institute of MathematicalMethods in Economics, Technical Uni-
versity of Vienna,WiednerHauptstrasse 8–10,A-1040Vienna,Austria.
E-mail address: [email protected]