Smooth perfectness for the group of diffeomorphisms

Given a result of Herman, we provide a new elementary proof of the fact that the connected component of the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism $g$, which is sufficiently close to the identity, can be represented as a product of four commutators, $g=[h_1,k_1]\circ...\circ[h_4,k_4]$, where the factors $h_i$ and $k_i$ can be chosen to depend smoothly on $g$.


Introduction and statement of the result
Let M be a smooth manifold of dimension n. By Diff ∞ c (M ) we will denote the group of compactly supported diffeomorphisms of M . This is a regular Lie group in the sense of Kriegl-Michor, modelled on the convenient vector space X c (M ) of compactly supported smooth vector fields on M , see [9,Section 43.1]. A curve c : R → Diff ∞ c (M ) is smooth iff the associated mapĉ : R × M → M is smooth and its support satisfies the following condition: for every compact interval I ⊆ R there exists a compact set C ⊆ M so that supp(c(t)) ⊆ C, for all t ∈ I, see [9,Section 42.5]. If M is compact, this smooth structure coincides with the well known Fréchet-Lie group structure on Diff ∞ (M ).
Here exp(X) denotes the flow of a complete vector field X at time 1, and [k, h] := k • h • k −1 • h −1 denotes the commutator of two diffeomorphisms k and h. It is readily checked that K is smooth. Indeed, one only has to observe that K maps smooth curves to smooth curves in view of the characterization of smooth curves from the previous paragraph, cf. [9,Section 27.2]. Writing id for the identity in Diff ∞ c (M ), we clearly have K(id, . . . , id) = id.
The c ∞ -topology [9,Section 4] is the final topology with respect to all smooth curves, its open subsets are the natural domains for locally defined smooth maps between infinite dimensional manifolds [9,Section 27]. For compact M the c ∞topology on Diff ∞ (M ) coincides with the Whitney C ∞ -topology, cf. [9, Theorem 4.11 (1)]. In general the c ∞ -topology on Diff ∞ c (M ) is strictly finer than the one induced from the Whitney C ∞ -topology, cf. [9,Section 4.26]. The latter coincides with the inductive limit topology lim K Diff ∞ K (M ) where K runs through all compact subsets of M , see [9,Section 41.13].
The aim of this paper is to establish the following two results.
Theorem 1. Suppose M is a smooth manifold of dimension n ≥ 2. Then there exist four smooth complete vector fields X 1 , . . . , X 4 on M so that the map K, see (1), admits a smooth local right inverse at the identity, N = 4. Moreover, the vector fields X i may be chosen arbitrarily close to zero with respect to the strong Whitney C 0 -topology. If M admits a proper (circle valued) Morse function whose critical points all have index 0 or n, then the same statement remains true with three vector fields.
Particularly, on the manifolds M = R n , S n , T n , n ≥ 2, or the total space of a compact smooth fiber bundle M → S 1 , three commutators are sufficient. Circlevalued Morse theory was initiated by Novikov [14], see also Pajitnov's monograph [15] and the references therein. At the expense of more commutators, it is possible to gain further control on the vector fields. More precisely, we have: Theorem 2. Suppose M is a smooth manifold of dimension n ≥ 2 and set N := 6(n + 1). Then there exist smooth complete vector fields X 1 , . . . , X N on M so that the map K, see (1), admits a smooth local right inverse at the identity. Moreover, the vector fields X i may be chosen arbitrarily close to zero with respect to the strong Whitney C ∞ -topology.
Either of the two theorems implies that Diff ∞ c (M ) o , the connected component of the identity, is a perfect group. This was already proved by Epstein [4] using ideas of Mather [10,11] who dealt with the C r -case, 1 ≤ r < ∞, r = n + 1. The Epstein-Mather proof is based on a sophisticated construction, and uses the Schauder-Tychonov fixed point theorem. The existence of a presentation g = [h 1 , k 1 ] • · · · • [h N , k N ] is guarantied, but without any further control on the factors h i and k i . Rough estimates on the number of necessary factors are well known, too. In view of a result due to Tsuboi [19,Theorem 8.1], one can assume N = 4 n (n + 1), provided g is sufficiently close to the identity. More refined estimates for certain classes of manifolds can be found in [20] and [21]. That the factors h i and k i can be chosen to depend smoothly on g seems to be folklore as well.
Theorem 1 or 2 actually implies that the universal covering of Diff ∞ c (M ) o is a perfect group. This result is known, too, see [18,12]. Thurston's proof is based on a result of Herman for the torus [6,7].
Our proof rests on Herman's result, too, but is otherwise elementary and different from Thurston's approach. In fact we only need Herman's result in dimension 1, which is a more structured situation, see also [22].
Note that the perfectness of Diff ∞ c (M ) o implies that this group is simple, see [3]. The methods used in [3] are elementary and actually work for a rather large class of homeomorphism groups.
Let us mention that some analogues of Theorems 1 and 2 for the homeomorphism groups in the category of topological manifolds have been obtained in [17] by using completely different arguments.
The remaining part of this note is organized as follows: In Section 2 we recall the above mentioned result of Herman and derive a corollary, see Proposition 1, which asserts that the statement of Theorem 2 holds true for the torus, M = T n , with N = 3. Using the exponential law we then establish a similar statement for diffeomorphisms on open subsets of R n , see Proposition 2 in Section 3. This construction allows us to circumvent Thurston's deformation construction, and at the same time restricts the approach to dimensions n ≥ 2. In Section 4 we formulate and prove a smooth version of the fragmentation lemma, see Proposition 3, and give a proof of Theorem 2. Finally, in Section 5 we discuss a technique to reduce the number of commutators which will eventually lead to a proof of Theorem 1.

Herman's theorem revisited
Let T n := R n /Z n denote the torus. For λ ∈ T n we let R λ ∈ Diff ∞ (T n ) denote the corresponding rotation. The main ingredient in the proof of Theorems 1 and 2 is the following result of Herman [7,6].
Theorem 3 (Herman). There exist γ ∈ T n so that the smooth map admits a smooth local right inverse at the identity. Moreover, γ may be chosen arbitrarily close to the identity in T n .
Herman's result is an application of the Nash-Moser inverse function theorem. When inverting the derivative one is quickly led to solve the linear equation Y = X − (R γ ) * X for given Y ∈ C ∞ (T n , R n ). This is accomplished using Fourier transformation. Here one has to choose γ sufficiently irrational so that tame estimates 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.
Below we will make use of the following corollary of Herman's result: There exist smooth vector fields X 1 , X 2 , X 3 on T n so that the smooth map Diff ∞ (T n ) 3 → Diff ∞ (T n ), admits a smooth local right inverse at the identity. Moreover, the vector fields X i may be chosen arbitrarily close to zero with respect to the Whitney C ∞ -topology.
Proof. This is an immediate consequence of Theorem 3 and the following observation, cf. [7]. The finite dimensional Lie group PSL 2 (R) acts effectively on the circle admits a smooth local right inverse at the identity. Moreover, Y i may be chosen arbitrarily close to 0 in sl 2 (R). Taking the product of n copies of such local right inverses and using the smooth embeddings the statement follows readily from Theorem 3 with

The exponential law
If F is a smooth foliation of M we let Diff ∞ c (M ; F ) denote the group of compactly supported diffeomorphisms preserving the leaves of F . This is a regular Lie group modelled on the convenient vector space of compactly supported smooth vector fields tangential to F . The group of foliation preserving diffeomorphisms has been studied in [16].
Proof. We proceed by constructing an inverse. To this end let π : M → M 2 denote the canonical projection and consider the smooth map . Note that the diffeomorphism σ 1 (g) := g • σ 2 (g) −1 preserves the leaves of the foliation F 1 , and this provides a smooth map σ 1 : U → Diff ∞ c (M ; F 1 ). We thus obtain a smooth map Clearly, F restricts to a bijection, and hence diffeomorphism, F : V ∼ = U, with inverse σ. Lemma 2. Suppose B and T are finite dimensional smooth manifolds, assume T compact, and let F denote the foliation with leaves {pt} × T on B × T . Then the canonical bijection Proof. This follows from the exponential law, see [9, Section 42.14].
Lemma 3. Let B be a precompact open subset in a finite dimensional smooth manifold M . Then there exist compactly supported smooth vector fields X 1 , X 2 , X 3 on M × T n , tangential to the foliation F with leaves {pt} × T n , so that the map ] admits a smooth local right inverse at the identity. Moreover, the vector fields X i may be chosen arbitrarily close to zero with respect to the strong Whitney C ∞topology.
Proof. According to Proposition 1 there exist smooth vector fields 3 at the identity. We extend the vector fields Y i in a constant manner to smooth vector fields Z i on M × T n , tangential to F . Multiplying Z i with a compactly supported smooth function which equals 1 onB × T n , we obtain compactly supported smooth vector fields Moreover, X i depends continuously on Y i with respect to the Whitney C ∞ -topologies. Consequently, X i may be assumed arbitrarily close to zero with respect to the Whitney Hence σ is the desired smooth local right inverse.
Lemma 4. Suppose p ≥ 1, q ≥ 0, set n := p+q, and let F denote the foliation with leaves {pt} × R q on R n . Moreover, let B be a precompact open subset in R n . Then there exist compactly supported smooth vector fields X 1 , X 2 , X 3 on R n , a c ∞ -open neighborhood U of the identity in Diff ∞ c (B; F ) and smooth maps σ 1 , σ 2 , σ 3 : U → Diff ∞ c (R n ) so that σ i (id) = id and, for all g ∈ U, Moreover, the vector fields X i may be chosen arbitrarily close to zero with respect to the strong Whitney C ∞ -topology on R n .
Proof. Since p ≥ 1, there exists a smooth embedding ϕ : admits a smooth local right inverse ρ = (ρ 1 , ρ 2 , ρ 3 ) : V → Diff ∞ c (U ; G) at the identity. We extend the vector fields X i by zero to compactly supported smooth vector fields on R n , and observe that these extensions may be assumed arbitrarily close to zero with respect to the Whitney C ∞ -topology on R n . Let U denote the preimage of V under the canonical inclusion Diff ∞ c (B; F ) → Diff ∞ c (U ; G). Restricting ρ i , we obtain smooth maps σ 1 , σ 2 , σ 3 : U → Diff ∞ c (U ; G) ⊆ Diff ∞ c (R n ) with the desired property.
Moreover, the vector fields X i may be chosen arbitrarily close to zero with respect to the strong Whitney C ∞ -topology.
Proof. Fix p, q ≥ 1 so that n = p + q. Without loss of generality we may assume B = B p × B q . Let F 1 and F 2 denote the foliations with leaves R p × {pt} and {pt} × R q on R n , respectively. According to Lemma 4, there exist compactly supported smooth vector fields X 4 , X 5 , X 6 on R n , a c ∞ -open neighborhood W of the identity in Diff ∞ c (B; F 2 ), and smooth mapsσ 4 ,σ 5 ,σ 6 : W → Diff ∞ c (R n ) so that σ i (id) = id and, for all g ∈ W, Similarly, there exist compactly supported smooth vector fields X 1 , X 2 , X 3 on R n , a c ∞ -open neighborhood V of the identity in Diff ∞ c (B; F 1 ), and smooth maps In view of Lemma 1, the composition Diff ∞ c (B; F 1 ) × Diff ∞ c (B; F 2 ) → Diff ∞ c (R n ) admits a smooth local right inverse ρ = (ρ 1 , ρ 2 ) : U → V × W at the identity. Hence the smooth maps σ i :=σ i • ρ 1 , 1 ≤ i ≤ 3, and σ i :=σ i−3 • ρ 2 , 4 ≤ i ≤ 6, will have the desired property.

Smooth fragmentation
Suppose U ⊆ M is an open subset. Every compactly supported diffeomorphism of U can be regarded as a compactly supported diffeomorphism of M by extending it identically outside U . The resulting injective homomorphism Diff ∞ c (U ) → Diff ∞ c (M ) is clearly smooth. Note, however, that a curve in Diff ∞ c (U ), which is smooth when considered as a curve in Diff ∞ c (M ), need not be smooth as a curve into Diff ∞ c (U ). Nevertheless, if there exists a closed subset A of M with A ⊆ U and if the curve has support contained in A, then one can conclude that the curve is smooth in Diff ∞ c (U ), too. This follows immediately from the characterization of smooth curves given at the beginning of Section 1.
The following is a folklore statement which, in one form or the other, can be found all over the literature on diffeomorphism groups [1]. For the reader's convenience we include a version emphasizing the smoothness of the construction.

Proposition 3 (Fragmentation).
Let M be a smooth manifold of dimension n, and suppose U 1 , . . . , U k is an open covering of M , ie. M = U 1 ∪ · · · ∪ U k . Then the smooth map P (g 1 , . . . , g k ) := g 1 • · · · • g k , admits a smooth local right inverse at the identity.
. Choosing W and W ′ sufficiently small, we may assume that every f X , X ∈ W, is a diffeomorphism [8]. The map provides a chart of Diff ∞ c (M ) centered at the identity. This is the standard way to put a smooth structure on Diff ∞ c (M ), see [9, Section 42.1]. Choose a smooth partition of unity λ 1 , . . . , λ k with supp(λ i ) ⊆ U i , 1 ≤ i ≤ k, and define Let U ⊆ Diff ∞ c (M ) denote the c ∞ -open neighborhood of the identity in Diff ∞ c (M ) corresponding to V via (2), and define a map Since the support of (f Xi−1 ) −1 • f Xi is contained in supp(λ i ) ⊆ U i , it is clear that σ is a smooth map, cf. the remark at the beginning of this section. Obviously we have P (σ(f X )) = f X k = f X and thus P • σ = id U . Moreover, it is immediate from the construction that σ(id) = σ(f 0 ) = (id, . . . , id).
Combining Propositions 2 and 3 permits to show the following result. Moreover, the vector fields X i may be chosen arbitrarily close to zero with respect to the strong Whitney C ∞ -topology.
Proof. It is well known that there exist open subsets U 1 , . . . , U n+1 of M so that U ⊆ U 1 ∪ · · · ∪ U n+1 ⊆ V and such that each U i , 1 ≤ i ≤ n + 1, is diffeomorphic to a disjoint union of copies of the open unit ball B n . Moreover, we may assume that there exist embeddings Applying Proposition 2 to each connected component of U i , we find complete vector fields X i,1 , . . . , X i,6 on M with supp(X i,j ) ⊆ V , a c ∞ -open neighborhood U i of the identity in Diff ∞ c (U i ) and smooth maps σ i,1 , . . . , σ i, 6 : Moreover, the vector fields X i,j may be chosen arbitrarily close to zero with respect to the strong Whitney C ∞ -topology on M . In view of Proposition 3, the map admits a local smooth right inverse at the identity. Combining this with the σ i,j above, we immediately obtain the statement.
Specializing Proposition 4 to U = M , we obtain Theorem 2.

Reducing the number of commutators
Proceeding as in [2] permits to reduce the number of commutators considerably, see also [20] and [21]. Moreover, the vector field X may be chosen arbitrarily close to zero in the strong Whitney C ∞ -topology on M .

Proof. Clearly, there exists an open subset
If g, h ∈ Diff ∞ c (V ) and 0 ≤ i = j ≤ N , then the diffeomorphisms φ i • g • φ −i and φ j • h • φ −j commute as their supports are disjoint. Hence for all g 1 , . . . , g N ∈ Diff ∞ c (V ). According to Proposition 4 there exist smooth complete vector fields X 1 , . . . , X N on M with supp(X i ) ⊆ V , a c ∞ -open neighborhood U of the identity in Diff ∞ c (U ) and smooth maps σ 1 , . . . , σ N : U → Diff ∞ c (V ) so that σ i (id) = id and, for all g ∈ U, Combining this with (3) we obtain, for all g ∈ U, Since X depends continuously on X 1 , . . . , X N , we may assume X to be arbitrarily close to zero with respect to the strong Whitney C ∞ -topology. Clearly, ̺ 1 and ̺ 2 are smooth, and we have ̺ 1 (id) = ̺ 2 (id) = id.
Lemma 5. Let M be a smooth manifold of dimension n. Then there exists an open covering M = U 1 ∪ U 2 ∪ U 3 and smooth complete vector fields X 1 , X 2 , X 3 on M so that exp(X 1 )(U 1 ) ⊆ U 2 , exp(X 2 )(U 2 ) ⊆ U 3 , and such that the closures of the sets are mutually disjoint. Moreover, the vector fields X 1 , X 2 , X 3 may be chosen arbitrarily close to zero with respect to the strong Whitney C 0 -topology. If M admits a proper (circle valued) Morse function whose critical points all have index 0 or n, then we may, moreover, choose U 1 = ∅ and X 1 = 0.
Proof. Fix a proper Morse function f : M → R, see [8], and let X denote the set of critical points of f . Since X is a discrete subset of M , the properness of f implies that the critical values form a discrete subset of R. We may, moreover, assume that each critical level contains precisely one critical point, see [13]. For notational brevity we will write ind(t) for the Morse index of the unique critical point corresponding to a critical value t of f . For any subset X ⊆ R, we introduce the notation M X := f −1 (X) ⊆ M .
Let U be a zero neighborhood in the space of vector fields on M with respect to the strong Whitney C 0 -topology. A neighborhood basis in this topology is obtained by taking neighborhoods of the zero section in T M and considering vector fields which take values in this subset. We will assume that U is of this form, for a neighborhood of the zero section which is fiberwise convex. Particularly, U is invariant with respect to multiplication by functions whose modulus does not exceed 1. Below we will show that the following assertions hold true: (a) Suppose t is a regular value of f . Then, for sufficiently small ε > 0 and all η > 0 there exists a vector field Y ε,η 2 ∈ U so that supp(Y ε,η 2 ) ⊆ M (t,t+ε+η) and exp(Y ε,η 2 )(M (t,t+ε) ) ⊆ M (t,t+η) .
(b) Suppose t is a regular value of f . Then, for sufficiently small η > 0 there exists a vector field Y η 3 ∈ U with supp(Y η 3 ) ⊆ M (t−2η,t+2η) such that the closures of the subsets Suppose t is a critical value of f with ind(t) = n, i.e. a local maximum. Then, for sufficiently small ε > 0 and all η > 0, there exists a vector field Y ε,η . (e) Suppose t is a critical value of f with ind(t) = 0, i.e. a local minimum. Then, for sufficiently small ε > 0 and all η > 0, there exists a vector field Y ε,η . We postpone the proof of these statements and first indicate how the lemma can be derived from them. For each critical value t of f we choose ε t > 0 and η t > 0 as in (c), (d) or (e), respectively, depending on ind(t). Shrinking ε t and η t we may assume that the intervals [t − ε t − η t , t + ε t + η t ] are mutually disjoint. We let where t runs through all critical values of f corresponding to critical points of index different from 0 and n. Moreover, V εt , denote the open subsets and vector fields constructed in (c), respectively. Furthermore, we set where t runs through all critical values t of f corresponding to critical points of index 0 or n, and the vector fields Y εt,ηt 2 are the ones constructed in (d) or (e), respectively. By construction we have ,t+εt) , where the latter union is over all critical values, and M (t+εt−ηt,t+εt) .
Note that we are still free to shrink all η t without affecting these properties. Using It thus remains to verify assertions (a) through (e) above. If t is a regular value of f , then there exists an open interval I containing t and a diffeomorphisms M I ∼ = M t × I intertwining the map f : M I → I with the standard projection M t × I → I, see [13]. In order to prove statements (a) and (b) it thus suffices to write down appropriate vector fields on I which is straightforward. Now suppose t is a critical value of f with corresponding critical point y of index k. Choose a Morse chart [13] centered at y, i.e. (x 1 , . . . , x n ) : W → R n are local coordinates such that n on the open neighborhood W of y. We fix a Riemannian metric g on M such that its restriction to W is the standard Euclidean metric, i.e.
We are now in a position to complete the proof of Theorem 1. Fix an open covering M = U 1 ∪ U 2 ∪ U 3 and smooth complete vector fields X 1 , X 2 , X 3 as in Lemma 5 above. According to Proposition 3 there exists a c ∞ -open neighborhood V of the identity in Diff ∞ c (M ) and smooth mapsσ i : V → Diff ∞ c (U i ) such that σ i (id) = id and, for all g ∈ V, g =σ 1 (g) •σ 2 (g) •σ 3 (g).