Annales de l’institut Fourier (AIF)

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ANNALES
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ÉDITEUR DES AIF
Erratum: “Weil representation and beta-extensions” by Corinne Blondel,
volume 62.4 (2012), pp. 1319-1366.
Tome 62, no 6 (2012), p. 2385-2385.
<http://aif.cedram.org/item?id=AIF_2012__62_6_2385_0>
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Ann. Inst. Fourier, Grenoble
62, 6 (2012) 2385
ERRATUM: “WEIL REPRESENTATION AND
BETA-EXTENSIONS” BY CORINNE BLONDEL,
VOLUME 62.4 (2012), PP. 1319-1366.
by ÉDITEUR DES AIF
AIF (2012) 62.4, page 1364, suite à une erreur de l’éditeur, à la page
1364, pour les références 5 et 6 il fallait lire :
AIF (2012) 62.4, page 1364, due to the publisher’s mistake, references 5
and 6 must be read :
[5] C. Bushnell & P. Kutzko, The admissible dual of GL(N) via compact
open subgroups, Annals of Mathematics Studies 129, Princeton, 1993.
[6] C. Bushnell & P. Kutzko, « Smooth representations of reductive p-adic
groups : structure theory via types », Proc. London Math. Soc. 77 (1998),
p. 582-634


Erratum: “Weil representation and beta-extensions” by Corinne Blondel, volume 62.4 (2012), pp. 1319-1366.

Böcherer, Siegfried

Ibukiyama, Tomoyoshi

Surjectivity of Siegel $\Phi $-operator for square free level and small weight

Olsson, Martin

Integral models for moduli spaces of $G$-torsors

Satriano, Matthew

de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities

Ishida, Tomohiko

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Tomohiko ISHIDA
Second cohomology classes of the group of C1-flat diffeomorphisms
Tome 62, no 1 (2012), p. 77-85.
<http://aif.cedram.org/item?id=AIF_2012__62_1_77_0>
© Association des Annales de l’institut Fourier, 2012, tous droits
réservés.
L’accès aux articles de la revue « Annales de l’institut Fourier »
(http://aif.cedram.org/), implique l’accord avec les conditions
générales d’utilisation (http://aif.cedram.org/legal/). Toute re-
production en tout ou partie cet article sous quelque forme que ce
soit pour tout usage autre que l’utilisation à fin strictement per-
sonnelle du copiste est constitutive d’une infraction pénale. Toute
copie ou impression de ce fichier doit contenir la présente mention
de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Ann. Inst. Fourier, Grenoble
62, 1 (2012) 77-85
SECOND COHOMOLOGY CLASSES OF THE GROUP
OF C1-FLAT DIFFEOMORPHISMS
by Tomohiko ISHIDA
Abstract. — We study the cohomology of the group consisting of all C∞-
diffeomorphisms of the line, which are C1-flat to the identity at the origin. We
construct non-trivial two second real cohomology classes and uncountably many
second integral homology classes of this group.
Résumé. — On étudie la cohomologie du groupe des C∞-difféomorphismes de
la droite, qui sout C1-tangents à l’identité à l’origine. On construit deux classes
non-triviales de cohomologie réelle de degré deux et un nombre non-dénombrable
de classes d’homologie de dimension deux de ce groupe.
1. Notations and main results
We denote by a1 the Lie algebra of all formal vector fields on R with the
Krull topology. For k > 0, we denote by ak1 the Lie subalgebra of a1 consist-
ing of formal vector fields which are Ck-flat at the origin. Let Diff∞0 (R) be
the group of orientation-preserving C∞-diffeomorphisms of R which fix the
origin. Let G(1) be the group of germs of local C∞-diffeomorphisms at the
origin of R. Let G∞(1) be the group of∞-jets of local C∞-diffeomorphisms
at the origin of R. For k > 1, we denote by Diff∞k (R), Gk(1) and G∞k (1)
the subgroup of Diff∞0 (R), G(1) and G∞(1) respectively, consisting of ele-
ments which are Ck-flat to the identity at the origin. The groups G∞(1)
and G∞k (1) can be considered as infinite-dimensional Lie groups, whose Lie
algebras are a01 and ak1 , respectively.
We define the Gel’fand-Fuks cohomology [2] of a11 in § 2. It is known
to be 2-dimensional for each degree [3][7]. Moreover, Millionschikov proved
its generators in degree greater than 1 can be described by the Massey
Keywords: cohomology of diffeomorphism groups, flat diffeomorphism, Massey product.
Math. classification: 58D05, 57S05.
78 Tomohiko ISHIDA
products [5]. We carried out the calculation of the Massey products on
Diff∞1 (R), and we give two 2-cocycles of Diff∞1 (R) in § 3.
For l > k, let αl and αl1 . . . αli be the 1-cochains of Diff
∞
k (R) defined by
αl(f) =
dl
dxl
f(0) for f ∈ Diff∞k (R),
and
αl1 . . . αli(f) = αl1(f) . . . αli(f) for f ∈ Diff
∞
k (R),
respectively. Then the following proposition holds.
Proposition 1.1. — The following γ2− and γ2+ are 2-cocycles of the
group Diff∞1 (R).
γ2− =
(1
2α4 − 3α2α3 + 3α
3
2
)
^
(
α3 −
3
2α
2
2
)
− 12α2^
(
α3 −
3
2α
2
2
)2
,
γ2+ = −α2^
( 1
10α3α5 −
1
8α
2
4 −
3
20α
2
2α5
+ 12α2α3α4 −
4
9α
3
3 +
1
2α
2
2α
2
3 −
3
4α
4
2α3 +
3
8α
6
2
)
+
(1
2α4 − 3α2α3 + 3α
3
2
)
^
( 1
10α5 − α2α4 −
1
3α
2
3 + 4α22α3 − 3α42
)
+
(1
5α5 −
3
2α2α4 − α
2
3 + 6α22α3 −
15
4 α
4
2
)
^
(
− 12α4 + 3α2α3 − 3α
3
2
)
+
( 1
30α6 −
3
10α2α5 −
1
2α3α4 +
3
2α
2
2α4 + 2α2α23 − 5α32α3 +
9
4α
5
2
)
^
(
α3 −
3
2α
2
2
)
.
Our main theorem is the following.
Theorem 1.2. — Let γ2 : H2(Diff∞1 (R);Z) → R2 be the homomor-
phism defined by
γ2(ξ) = (γ2−(ξ), γ2+(ξ)).
Then γ2 is surjective.
The author is extremely grateful to Professor Takashi Tsuboi for many
helpful advices. He suggested that commutators would be useful to con-
struct 2-cycles of groups. The author also would like to thank Professor
Shigeyuki Morita for his teachings. The construction of the above cocycles
was done under his supervision when the author was in the master course.
ANNALES DE L’INSTITUT FOURIER
COHOMOLOGY OF THE GROUP OF C1-FLAT DIFFEOMORPHISMS 79
2. First cohomology of Diff∞k (R)
In this section, we review a result of Fukui [1] and compute 1-cocycles of
H1(Diff∞k (R);R).
Definition 2.1. — For a topological Lie algebra g, we denote by A∗C(g)
the differential graded algebra of all continuous alternating forms on g.
The Gel’fand-Fuks cohomology of g is defined to be the cohomology of the
complex (A∗C(g), d). Here, d is the ordinary differential mapping of cochain
complexes of Lie algebras. We denote the Gel’fand-Fuks cohomology of g
by H∗GF (g).
In the case g = ak1 , the complex A∗C(ak1) is an exterior algebra generated
by δ(k+1), δ(k+2), . . . , where δ(l)’s are the 1-forms on ak1 defined by
δ(l)
(
f(x) d
dx
)
= (−1)lf (l)(0) for f(x) ∈ R[[x]].
Since it is easily seen that dδ(l) = 0 if and only if k + 1 6 l 6 2k + 1, we
obtain the following proposition.
Proposition 2.2. — For k > 1,
H1GF (ak1) ∼= Rk+1.
Moreover, δ(k+1), δ(k+2), . . . , δ(2k+1) generate H1GF (ak1).
In particular, H1GF (a11) is generated by δ′′ and δ′′′.
On the other hand, Fukui proved a proposition about the homology of
groups corresponding to ak1 .
Theorem 2.3 (Fukui[1]). — For k > 1,
H1(Diff∞k (R);Z) ∼= Rk+1.
Theorem 2.3 is obtained from the fact that the group homomorphism
Ψk : Diff∞k (R)→ Ṙk+1
defined by
Ψk(f) :=
(
1
(k + 1)!f
(k+1)(0), 1(k + 2)!f
(k+2)(0), . . . , 1(2k + 1)!f
(2k+1)(0)
)
induces an isomorphism in the first homology. Here, Ṙk+1 means the group
which is Rk+1 as a set, where the addition is defined by
(a1, a2, . . . , ak+1) + (b1, b2, . . . , bk+1)
= (a1 + b1, a2 + b2, . . . , ak + bk, ak+1 + bk+1 + (k + 1)a1b1).
TOME 62 (2012), FASCICULE 1
80 Tomohiko ISHIDA
Since Ψk is a group homomorphism, αk+1, αk+2, . . . , α2k are 1-cocycles of
Diff∞k (R) with real coefficients. Moreover, if we denote the cochain α̃2k+1
by
α̃2k+1 = α2k+1 −
1
2
(
2k + 1
k
)
α2k+1,
then it is also a 1-cocycle. In particular, α2 and α̃3 = α3 −
3
2α
2
2 are 1-
cocycles of Diff∞1 (R).
Remark 2.4. — The same argument can be applied to the groups G1(1)
and G∞1 (1) instead of Diff
∞
1 (R). Hence Theorem 2.3 also holds for G1(1)
and G∞1 (1). By regarding αl’s as the 1-cochains of G1(1) and G∞1 (1), the
1-cocycles αk+1, αk+2, . . . , α2k and α̃2k+1 of Diff∞1 (R) can be considered
as 1-cocycles of the groups G1(1) and G∞1 (1), respectively.
3. Construction of the 2-cocycles of Diff∞1 (R)
In this section, we recall the definition of the Massey products following
[4], and construct the 2-cocycles γ2± of the group Diff∞1 (R).
Definition 3.1 ([4]). — Let A = (An, d) be a differential graded al-
gebra. For ui ∈ Hpi(A), we set ai a cocycle representative of ui. We de-
fine p(i, j) to be
j∑
r=i
(pr − 1). A collection of cochains A = (a(i, j)) for
1 6 i 6 j 6 k and (i, j) 6= (1, k) is a defining system of {a1, . . . , ak} if
(i) a(i, i) = ai ∈ Api ,
(ii) a(i, j) ∈ Ap(i,j)+1, and
(iii) da(i, j) =
j−1∑
r=i
(−1)deg a(i,r)a(i, r)a(r + 1, j).
Definition 3.2 ([4]). — When a defining system A of {a1, . . . , ak} ex-
ists, we define c(A) ∈ Ap(1,k)+2 by setting
c(A) =
k−1∑
r=1
(−1)deg a(i,r)a(1, r)a(r + 1, k).
Then c(A) is a cocycle and the set
{a cohomology class of c(A);A is a defining system of {a1, . . . , ak}}
depends only on the cohomology classes u1, . . . , uk. We call the elements
of the set the Massey products of {u1, . . . , uk}.
ANNALES DE L’INSTITUT FOURIER
COHOMOLOGY OF THE GROUP OF C1-FLAT DIFFEOMORPHISMS 81
By Goncharova’s theorem [3][7], which gives dimHpGF (ak1) for any p, k >
1, we know
HpGF (a
1
1) ∼= R2 for any p.
Furthermore, the following theorem is known.
Theorem 3.3 (Millionschikov[5]). — For any p > 2, there exist gen-
erators gp−, g
p
+ ∈ H
p
GF (a11) of H
p
GF (a11) ∼= R2, which are described by
the Massey products. In particular, both the triple Massey product of
{δ′′, δ′′′, δ′′′} and the 5-fold Massey product {δ′′, δ′′′, δ′′, δ′′, δ′′′} determine
non-trivial cohomology classes in H2GF (a11), which are linearly independent.
In fact, the defining systems of {δ′′, δ′′′, δ′′′} and {δ′′, δ′′′, δ′′, δ′′, δ′′′} can
be written as
δ
′′ −12δ
(4) ∗
δ′′′ 0
δ′′′
 and

δ′′ −12δ
(4) −15δ
(5) − 130δ
(6) ∗
δ′′′
1
2δ
(4) 1
10δ
(5) 0
δ′′
1
3δ
′′′ 1
10δ
(5)
δ′′ −12δ
(4)
δ′′′

,
respectively.
Proof of Proposition 1.1. — For Diff∞1 (R) we checked that the defining
systems of both of {α2, α̃3, α̃3} and {α2, α̃3, α2, α2, α̃3} also exist. In fact,
they can be written as
α2 β1 ∗α̃3 β2
α̃3
 and

α2 β1 β5 β8 ∗
α̃3 β3 β6 β9
α2 β4 β7
α2 β1
α̃3
 ,
TOME 62 (2012), FASCICULE 1
82 Tomohiko ISHIDA
respectively. Here,
β1 = −
1
2α4 + 3α2α3 − 3α
3
2, β2 =
1
2 α̃
2
3,
β3 =
1
2α4 − 2α2α3 +
3
2α
3
2, β4 =
1
3α3,
β5 = −
1
5α5 +
3
2α2α4 + α
2
3 − 6α22α3 +
15
4 α
4
2,
β6 =
1
10α5 −
1
2α2α4 −
1
3α
2
3 +
3
2α
2
2α3 −
3
4α
4
2,
β7 =
1
10α5 − α2α4 −
1
3α
2
3 + 4α22α3 − 3α42,
β8 = −
1
30α6 +
3
10α2α5 +
1
2α3α4 −
3
2α
2
2α4 − 2α2α23 + 5α32α3 −
9
4α
5
2,
β9 =
1
10α3α5−
1
8α
2
4−
3
20α
2
2α5+
1
2α2α3α4 −
4
9α
3
3 +
1
2α
2
2α
2
3 −
3
4α
4
2α3 +
3
8α
6
2.
Following to the definition of the Massey products, we obtain cocycles
γ2− = −α2^β2 − β1^α̃3,
and
γ2+ = −α2^β9 − β1^β7 − β5^β1 − β8^α̃3,
of Proposition 1.1. 
4. Proof of the main theorem
Throughout this section, for any two diffeomorphisms f and g, the mul-
tiplication fg means that g is applied first.
In this section, we prove the non-triviality of γ2± by constructing un-
countably many 2-cycles ξ±2 ∈ Z[Diff
∞
1 (R)2] such that γ2−(ξ−2 ) 6= 0 and
γ2+(ξ+2 ) 6= 0. Then this proves Theorem 1.2. To construct ξ
±
2 , we use the
following lemma.
Lemma 4.1. — For any k, l > 1(k 6= l) and any f ∈ Diff∞k+l(R), there
exist g ∈ Diff∞k (R) and h ∈ Diff∞l (R) such that f = [g, h]. Here, [g, h]
means ghg−1h−1.
Moreover, for any k, l > 1(k 6= l) it is true that [Diff∞k (R),Diff∞l (R)] =
Diff∞k+l(R). In the case k = l, Fukui proved that [Diff∞k (R), Diff∞k (R)] =
Diff∞2k+1(R) for k > 1 in [1].
ANNALES DE L’INSTITUT FOURIER
COHOMOLOGY OF THE GROUP OF C1-FLAT DIFFEOMORPHISMS 83
Proof. — We may assume that k > l and the ∞-jet of f is written as
f(x) = x+
∞∑
n=k+l+1
anx
n.
If we take h ∈ Diff∞l (R) so that h can be described as
h(x) = x+ xl+1
in a some neighborhood of 0, then the ∞-jet of fh at 0 is written as
fh(x) = x+ xl+1 + ak+l+1xk+l+1 + . . . .
Here, we apply the following theorem of the normal forms of diffeomor-
phisms of (R, 0).
Theorem 4.2 (Takens[6]). — For any l > 1 and ψ ∈ Diff∞l (R), there
exists ϕ ∈ Diff∞0 (R) such that
ϕψϕ−1(x) = x+ δxl+1 + αx2l+1,
in a some neighborhood of 0 for some δ = ±1 and α ∈ R. Here δ and α are
uniquely determined by the (2l + 1)-jet of ψ.
Because of the uniqueness of δ and α, there exists ϕ such that ϕ−1fhϕ =
h in some neighborhood U of 0. By Takens’ construction of ϕ in Theo-
rem 4.2, it is seen that one can choose ϕ to be Cl-flat to the identity at 0.
We denote the composition ϕ−1fhϕ by Φ. If we take h so that both of h
and Φ have no fixed points except for 0, then Φ is conjugate to h. In the
case l is odd and x < 0, there exists an integer nx > 0 such that Φn(x) is
in U for any n > nx and we define ϕ̃(x) = Φ−nxhnx(x). Otherwise, for any
x there exists an integer nx > 0 such that Φ−n(x) is in U for any n > nx
and we define ϕ̃(x) = Φnxh−nx(x). Then ϕ̃−1Φϕ̃ coincides with h. If we set
g = ϕϕ̃, then g is contained in Diff∞k (R) and Lemma 4.1 is proved. 
Proof of Theorem 1.2. — If f, g ∈ Diff∞1 (R) and the ∞-jet of them are
written as
f(x) = x+
∞∑
n=2
anx
n, g(x) = x+
∞∑
n=2
bnx
n,
then
γ2−(f, g) = 36(b3 − b22)(2a4 − 6a2a3 + 4a32 − a2b3 + a2b22).
Thus if fi ∈ Diff∞i (R) for i = 1, 2, 3, 4, then γ2−(f2, f3) = γ2−(f1, f4) =
γ2−(f4, f1) = 0. On the other hand, if both of the coefficient of x4 in the jet of
f3 and the coefficient of x3 in the jet of f2 are non-zero, then γ2−(f3, f2) 6= 0.
Therefore, we assume f3 ∈ Diff∞3 (R) r Diff∞4 (R) and f2 ∈ Diff∞2 (R) r
TOME 62 (2012), FASCICULE 1
84 Tomohiko ISHIDA
Diff∞3 (R). By Lemma 4.1, we can choose f1 ∈ Diff∞1 (R) and f4 ∈ Diff∞4 (R)
such that [f2, f3] = [f1, f4]. If we set
ξ−2 = (f3, f2)− (f2, f3) + ([f2, f3], f3f2)
− (f4, f1) + (f1, f4)− ([f1, f4], f4f1) ∈ Z[Diff∞1 (R)2],
then ξ−2 is a cycle and
γ2−(ξ−2 ) = γ
−
2 ((f3, f2)) =
1
2α4(f3)α3(f2) 6= 0.
Therefore, the non-triviality of γ2− is proved.
The non-triviality of γ2+ can be proved similarly. Let g3 ∈ Diff∞3 (R) r
Diff∞4 (R) and g4 ∈ Diff∞4 (R) r Diff∞5 (R). If we choose g1 ∈ Diff∞1 (R) and
g6 ∈ Diff∞6 (R) such that [g3, g4] = [g1, g6] and set
ξ+2 = (g4, g3)− (g3, g4) + ([g3, g4], g4g3)
− (g6, g1) + (g1, g6)− ([g1, g6], g6g1) ∈ Z[Diff∞1 (R)2],
then ξ+2 is a cycle and
γ2+(ξ+2 ) = γ
+
2 ((g4, g3)− (g3, g4)) = −
1
20 α5(g4)α4(g3) 6= 0.
Consequently, γ2± are non-trivial cohomology classes in H2(Diff∞1 (R);R).
Furthermore, clearly γ2−(ξ+2 ) = 0 and γ2−(ξ
−
2 ) can take any real value by
changing f2 or f3. Similarly, γ2+(ξ+2 ) also can take any value. This concludes
the proof of Theorem 1.2. 
Moreover, the following corollary holds.
Corollary 4.3. — For any g > 2, there exist uncountably many iso-
morphism classes of flat R-bundles on genus g surface Σg, such that the
images of their holonomy homomorphisms
π1(Σg)→ Diff∞(R)
are contained in Diff∞1 (R).
Remark 4.4. — The same argument in §3, and §4 can be applied to the
groups G1(1) and G∞1 (1) instead of Diff
∞
1 (R). Therefore, we can regard γ2±
as the 2-cochains of G1(1) and G∞1 (1), and Theorem 1.2 also holds for G1(1)
and G∞1 (1), respectively.
On the other hand, for any group G and commuting f, g ∈ G, the chain
(f, g)− (g, f) is the simplest 2-cycle of G. However, if we regard γ2± as the
2-cocycles of G∞1 (1), then it is seen that γ2±((f, g) − (g, f)) = 0 for any
commuting f, g ∈ G∞1 (1). Hence the following is true.
ANNALES DE L’INSTITUT FOURIER
COHOMOLOGY OF THE GROUP OF C1-FLAT DIFFEOMORPHISMS 85
Proposition 4.5. — For any group homomorphism ρ :π1(T 2)→G∞1 (1),
ρ∗γ2± = 0.
BIBLIOGRAPHY
[1] K. Fukui, “Homologies of the group Diff∞(Rn, 0) and its subgroups”, J. Math.
Kyoto Univ. 20 (1980), no. 3, p. 475-487.
[2] I. M. Gel′fand & D. B. Fuks, “Cohomologies of the Lie algebra of formal vector
fields”, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), p. 322-337.
[3] L. V. Goncharova, “The cohomologies of Lie algebras of formal vector fields on the
line”, Funct. Anal. and Appl. 7 (1973), p. 91-97, 194-203.
[4] D. Kraines, “Massey higher products”, Trans. Amer. Math. Soc. 124 (1966), p. 431-
449.
[5] D. Millionschikov, “Algebra of formal vector fields on the line and Buchstaber’s
conjecture”, Funct. Anal. Appl. 43 (2009), p. 264-278.
[6] F. Takens, “Normal forms for certain singularities of vectorfields”, Ann. Inst.
Fourier (Grenoble) 23 (1973), no. 2, p. 163-195, Colloque International sur l’Analyse
et la Topologie Différentielle (Colloques Internationaux du Centre National de la
Recherche Scientifique, Strasbourg, 1972).
[7] F. V. Weinstein, “Filtering bases: a tool to compute cohomologies of abstract sub-
algebras of the Witt algebra”, in Unconventional Lie algebras, Adv. Soviet Math.,
vol. 17, Amer. Math. Soc., Providence, RI, 1993, p. 155-216.
Manuscrit reçu le 6 janvier 2010,
accepté le 15 octobre 2011.
Tomohiko ISHIDA
The University of Tokyo
Graduate School of Mathematical Sciences
Komaba, Meguro-ku ,Tokyo 153-8914 (Japan)
ishidat@ms.u-tokyo.ac.jp
TOME 62 (2012), FASCICULE 1


Second cohomology classes of the group of $C^1$-flat diffeomorphisms

Greither, Cornelius

Johnston, Henri

Non-existence and splitting theorems for normal integral bases

Pasquale, Angela

Sundari, Maddala

Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

Kotschick, D.

Three-manifolds and Kähler groups

Benoist, Olivier

Degrés d’homogénéité de l’ensemble des intersections complètes singulières

Tilouine, J.

Formes compagnons et complexe BGG dual pour $GSp_4$

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Erratum: “Weil representation and beta-extensions” by Corinne Blondel, volume 62.4 (2012), pp. 1319-1366.

Surjectivity of Siegel $\Phi$ -operator for square free level and small weight

Integral models for moduli spaces of $G$ -torsors

de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities

Second cohomology classes of the group of $C^1$ -flat diffeomorphisms

Non-existence and splitting theorems for normal integral bases

Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

Three-manifolds and Kähler groups

Degrés d’homogénéité de l’ensemble des intersections complètes singulières

Formes compagnons et complexe BGG dual pour $GSp_4$

3,503

3,732