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A COMPARISON BETWEEN TWO DE RHAM COMPLEXES IN
DIFFEOLOGY
KATSUHIKO KURIBAYASHI
Abstract. There are two de Rham complexes in diffeology. The original one
is due to Souriau and another one is the singular de Rham complex defined
by a simplicial differential graded algebra. We compare the first de Rham
cohomology groups of the two complexes within the ˇ
Cech–de Rham spectral
sequence in diffeology. In particular, a comparison map enables us to conclude
that the first singular de Rham cohomology for the irrational torus Tθis iso-
morphic to the direct sum of the original one and the group of equivalence
classes of flow bundles over Tθwith connection 1-forms.
1. Introduction
This manuscript is a sequel to [7, Appendix C]. The de Rham complex due to
Souriau [8] is very beneficial in the development of diffeology; see [2, Chapters 6,7,8,
and 9]. In fact, the de Rham calculus is applicable to not only diffeological path
spaces but also more general mapping spaces. While the complex is isomorphic to
the usual de Rham complex if the input diffeological space is a manifold, the de
Rham theorem does not hold in general.
Another complex called the singular de Rham complex is introduced in [7] via
simplicial arguments; see [5] for a cubic de Rham complex. An advantage of the new
complex is that the de Rham theorem holds for every diffeological space. Moreover,
the singular de Rham complex allows us to construct Leray–Serre and Eilenberg–
Moore spectral sequences in the diffeological framework; see [7, Theorems 5.4 and
5.5]. Furthermore, there exists a natural morphism α: Ω(X)→A(X) of differential
graded algebras from the original de Rham complex Ω(X) due to Souriau to the
new one A(X) which induces an isomorphism on the cohomology provided Xis a
manifold; see [5] and [7, Theorem 2.4].
The aim of this short manuscript is to compare the first de Rham cohomology
groups for the complexes A(X) and Ω(X) within the ˇ
Cech–de Rham spectral se-
quence [3] by using the morphism αmentioned above; see Theorem 2.1 for more
details. In particular, by a comparison map, it is shown that the first singular de
Rham cohomology for the irrational torus Tθis isomorphic to the direct sum of the
original one and the group of equivalence classes of flow bundles over Tθwith con-
nection 1-forms; see Corollary 2.2. In consequence, we see that, as an algebra, the
singular de Rham cohomology H∗(A(Tθ)) is isomorphic to the tensor product of the
original de Rham cohomology and the exterior algebra generated by a flow bundle
over Tθ; see Corollary 2.3. Thus, it seems that the singular de Rham cohomology
has K-theoretical information.
2010 Mathematics Subject Classification: 57P99, 55U10, 58A10.
Key words and phrases. Diffeology, ˇ
Cech–de Rham spectral sequence, singular de Rham complex.
Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto,
Nagano 390-8621, Japan e-mail:kuri@math.shinshu-u.ac.jp
1
2 KATSUHIKO KURIBAYASHI
2. Main theorem
We begin by recalling the original de Rham complex. Let (X, DX) be a diffeo-
logical space. For an open set Uof Rn, let DX(U) be the set of plots with Uas the
domain and Λ∗(U) = {h:U−→ ∧∗(⊕n
i=1Rdxi)|his smooth}the usual de Rham
complex of U. Let Open denote the category consisting of open sets of Euclidian
spaces and smooth maps between them. We can regard DX( ) and Λ∗( ) as func-
tors from Openop to Sets the category of sets. A p-form is a natural transformation
from DX( ) to Λ∗( ). Then the de Rham complex Ω∗(X) is the cochain algebra of
p-forms for p≥0; that is, Ω∗(X) is the direct sum of the modules
Ωp(X) :=
Openop
DX
))
Λp
55
ωSets
ωis a natural transformation
with the cochain algebra structure defined by that of Λ∗(U) pointwisely.
We introduce another de Rham complex for a diffeological space. Indeed, it is
a diffeological counterpart of the singular de Rham complex in [1, 10, 11]. Let
An:= {(x0, ..., xn)∈Rn+1 |n
i=0 xi= 1}be the affine space equipped with
the sub-diffeology of Rn+1 and (A∗
DR)•the simplicial cochain algebra defined by
(A∗
DR)n:= Ω∗(An) for each n≥0. For a diffeological space (X, DX), let SD
•(X)
denote the simplicial set defined by
SD
•(X) := {{σ:An→X|σis a C∞-map}}n≥0.
The simplicial set and the simplicial cochain algebra (A∗
DR)•give rise to a cochain
algebra
Sets∆op (SD
•(X),(A∗
DR)•) :=
∆op
SD
•(X)
))
(A∗
DR)•
55
ωSets
ωis a natural transformation
whose cochain algebra structure is defined by that of (A∗
DR)•. In what follows, we
call the complex A(X) := Sets∆op (SD
•(X),(A∗
DR)•) the singular de Rham complex
of X; see also [7, Section 2]. We define a morphism α: Ω(X)→A(X) of cochain
algebras by α(ω)(σ) = σ∗(ω). The result [7, Theorem 2.4] asserts that αis a
quasi-isomorphism if Xis a manifold, a smooth CW-complex or a parametrized
stratifold; see [4, 5] and [6] for a smooth CW-complex and a startifold, respectively.
Moreover, the map αinduces a monomorphism H(α) : H1(Ω(X)) →H1(A(X))
for every diffeological space X; see [7, Proposition 6.9]. Thus one might concern
the difference between the first de Rham cohomology groups. Theorem 2.1 below
which is our main theorem relates the cohomology groups within the ˇ
Cech–de Rham
spectral sequence introduced below.
Let GXbe the generating family of the diffeology DXconsisting of all plots
whose domains are open balls in Euclidian spaces.We assume that GXcontains the
set C∞(R0, X); see [2, 1.76]. Then we define the nebula NXof Xassociated with
GXby
NX:=
φ∈GX{φ} × dom(φ)
with sum diffeology, where dom(φ) denotes the domain of the plot φ. We may
write N(GX) for NXwhen expressing the generating family. It is readily seen that
the evaluation map ev :NX→Xdefined by ev(φ, r) = φ(r) is smooth. The gauge
A COMPARISON BETWEEN TWO DE RHAM COMPLEXES IN DIFFEOLOGY 3
monoid MXis a submonoid of the monoid of endomorphisms on the nebula NX
defined by
MX:= {f∈C∞(NX,NX)|ev ◦f=ev and ♯Supp f < ∞},
where Suppf:= {φ∈ G | f|{φ}×dom(φ)= 1{φ}×dom(φ)}. In what follows, we denote
the monoid MXby Mif the underlying diffeological space is clear form the context.
The original de Rham complex Ω∗(NX) is a left Mop-module whose actions are
defined by f∗induced by endomorphisms f∈ NX. Moreover, the complex Ω∗(NX)
is regarded as a two sided Mop-module for which the right module structure is
trivial. Then we have the Hochschild complex C∗,∗={Cp,q, δ, dΩ}p,q≥0with
Cp,q = HomRMop⊗RM(RMop ⊗(RMop )⊗p⊗RM,Ωq(NX)) ∼
=map(Mp,Ωq(NX)),
where the horizontal map δis the Hochshcild differential and the vertical map dΩ
is induced by the de Rham differential on Ω∗(NX); see [3, Subsection 8]. The hori-
zontal filtration F∗={Fj}j≥0defined by Fj=⊕q≥jC∗,q of the the total complex
Tot C∗,∗gives rise to a first quadrant spectral sequence {ΩE∗,∗
r, dr}converging to
the ˇ
Cech cohomology ˇ
H(X) := HH∗(RM,map(G,R)) with
Ep,q
2∼
=Hq(HHp(RMop,Ω∗(NX)), dΩ),
where HH∗(-) denotes the Hochschild cohomology; see [3, Subsections 9 and 16].
Observe that the differential dris of bidegree (1 −r, r). This spectral sequence is
called the ˇ
Cech–de Rham spectral sequence.
The same construction as that of the spectral sequence above is applicable to the
singular de Rham complex A(X). Then replacing the original de Rham complex
Ω(-) with A(-), we have a spectral sequence {AE∗,∗
r, dr}. Since the Poincar´e lemma
holds for the complex A(-), it follows that the target of the spectral sequence
for A(X) is also the ˇ
Cech cohomology ˇ
H(X). Thus the naturality of the map
α:A(X)→Ω(X) gives rise to a commutative diagram of isomorphisms
H1(Ω(X)) ⊕ΩE1,0
3
Θ
∼
=//H1(A(NX)M)⊕AE1,0
3
ˇ
H1(X;R).
edge2
∼
=44
i
i
i
i
i
i
i
i
i
i
i
i
edge2
∼
=
jjUUUUUUUUUUU
In fact, the edge map edge1:= ev∗:H∗(Ω(X)) →ΩE0,∗
2=H1(Ω(NX)M) induced
by the evaluation map ev :X→ NXis an isomorphism; see [3, 6.Proposition].
Moreover, the morphism α: Ω(X)→A(X) of cochain algebras induces a map
H(Tot(α)) between the total complexes which define the spectral sequences above.
Thus the naturality of the map αenables us to obtain a commutative diagram
(2.1)
H∗(Ω(X))
H(α)
ev∗
∼
=//H∗(Ω(NX)M) = ΩE0,∗
2////
f(α)2
ΩE0,∗
∞////
f(α)∞
H∗(Tot C∗,∗)
H∗(Tot(α))
ˇ
H∗(X).
edge2
∼
=
jjT
T
T
T
T
T
edge2
∼
=
ttj
j
j
j
j
j
H∗(A(X)) ev∗
//H∗(A(NX)M) = AE0,∗
2////AE0,∗
∞////H∗(Tot ′C∗,∗)
By the degree reasons, we see that the surjective maps KE0,1
2→KE0,1
∞are isomor-
phisms and KE1,0
3∼
=KE1,0
∞for K= Ω and A. Thus the map H∗(Tot(α)) yields the
homomorphism Θ which fits in the triangle. In consequence, Θ is an isomorphism.
4 KATSUHIKO KURIBAYASHI
In a particular case where a diffeological space Xappears as the base space of a
diffeological fibration; see [2, Chapter 5], we relate H1(Ω(X)) to H1(A(X)) in the
ˇ
Cech–de Rham spectral sequence.
Theorem 2.1. Let Xbe a connected diffeological space which admits a diffeo-
logical fibration of the form F→Mπ
→Xin which Mis a connected manifold
and Fis connected diffeological space with H1(A(F)) = 0. Then the edge map
edge1:= ev∗:Hi(A(X)) →Hi(A(NX)M) = AE0,∗
2(X)is a monomorphism for
i= 1,2. Moreover, the restriction of the map Θmentioned above to the cohomology
H1(Ω(X)) is the composite of the monomorphism H(α) : H1(Ω(X)) →H1(A(X))
and the edge map edge1.
Before describing corollaries, we recall results on principal R-bundles (flow bun-
dles) in [3]. For a diffeological space X, we consider a Hochschild cocycle τ:M→
Ω0(NX) = C∞(NX,R) in Z1,0
δ. Then an M-action Aτon NX×Ris defined by
Aτ(b, s) = (A(b), s +τ(A)(b)). The action gives rise to a principal R-bundle of the
form Yτ:= NX×τR→ NX/M∼
=NX/ev ∼
=Xover X, where Yτis the quotient
space of NX×Rby the M-action. More precisely, the equivalence relation is gen-
erated by the binary relation which the M-action induces. Observe that the second
diffeomorphism is given by the evaluation map ev :NX→X.
Let Fl(X) be the abelian group of equivalence classes of flow bundles. The
sum is given by the quotient of the direct sum of two flow bundles by the anti
diagonal action of R; see [3, Proposition 2]. Then a map ΩE1,0
1→Fl(X) defined by
assigning the equivalence class of the flow bundle Yτ→Xto [τ] is an isomorphism.
Moreover, we see that ΩE1,0
2= Ker{dΩ:ΩE1,0
1→ΩE1,1
1}is isomorphic to Fl•(X)
the subgroup of Fl(X) consisting of equivalence classes of flow bundles over Xwith
connection 1-forms; see also [2, 8.37].
Thanks to the injectivity of the edge map in Theorem 2.1 and a result on flow
bundles due to Iglesias-Zemmour mentioned above, we have
Corollary 2.2. Let Tθbe the irrational torus. Then the map Θin Theorem 2.1
gives rise to an isomorphism Θ : H1(Ω(Tθ)) ⊕Fl•(Tθ)∼
=
→H1(A(Tθ)).
We recall the diffeomorphism ψ:R/(Z+θZ)→Tθdefined by ψ(t) = (0, e2πit)
in [2, Exercise 31, 3)]. Then there exist isomorphisms Ω(Tθ)∼
=Ω(R/(Z+θZ)) ∼
=
(∧∗(R), d ≡0) which are induced by ψand the subduction R→R/(Z+θZ),
respectively; see [2, Exercise 119]. On the other hand, we see that H∗(A(Tθ)) ∼
=
∧(t1, t2) as an algebra, where deg ti= 1; see the proof of Corollary 2.2. Thus the
corollary above yields the following result.
Corollary 2.3. One has H∗(A(Tθ)) ∼
=∧(Θ(t),Θ(ξ)) as an algebra, where t∈
H∗(Ω(Tθ)) ∼
=∧(t)is a generator and ξ∈Fl•(Tθ)∼
=Ris a flow bundle over Tθwith
a connection 1-form, which is a generator of the group Fl•(Tθ).
3. Proofs of Theorem 2.1 and Corollary 2.2
We begin by considering invariant differential forms on nebulae of dfiffeological
spaces.
Lemma 3.1. Let π:Y→Xbe a subduction and GYa generating family of
Y. Then the map π∗:A(NX)→A(NY)induced by πgives rise to a map π∗:
A(NX)MX→A(NY)MY, where the nebula NXis defined by the generating family
π∗GY:= {π◦ϕ|ϕ∈ GY}induced by GY.
A COMPARISON BETWEEN TWO DE RHAM COMPLEXES IN DIFFEOLOGY 5
Proof. For ω∈A∗(NX)MXand η∈MY, we show that η·π∗(ω) = π∗(ω). Let
σ:An→ NYbe an element in SD
n(NX), namely a smooth map from An. Since An
is connected, it follows that the image of σis contained in a component {ϕ}×dom(ϕ)
of NY. We define a smooth map η:NX→ NXby η(π◦ϕ, u)=(π◦ϕ′, η(u))
and by the identity maps in other components, where η(ϕ, u)=(ϕ′, η(u)). Since
ϕ(u) = ev(η(ϕ, u)) = ev(η(ϕ′, η (u))) = ϕ′(η(u)), it follows that ev ◦η=ηand hence
η∈MX. Observe that π◦η◦σ=η◦π◦σ. Thus we see that (η·π∗(ω))(σ) =
π∗(ω)(η◦σ) = ω(π◦ησ) = ω(η◦π◦σ) = (η·ω)(π◦σ) = ω(π◦σ) = π∗(ω)(σ). This
completes the proof. □
Under the assumption in Theorem 2.1, we have a commutative diagram
(3.1) H∗(A(X)) ev∗=edge1//
H∗(A(π))
H∗(A(NX)MX)
π∗
=AE0,∗
2(X)
H∗(A(M)) ev∗
//H∗(A(NM)MM) = AE0,∗
2(M)
H∗(Ω(M))
H(α)∼
=
OO
ev∗
∼
=//H∗(Ω(NM)MM)
f(α)2
OO
=ΩE0,∗
2(M).
Here, in constructing the spectral sequences, we use the generating family GMof
Mconsisting of all plots whose domains are open balls in Euclidian spaces.
(I) The injectivity of H∗(A(π))): Let {LS E∗,∗
r, dr}be the Leray–Serre spectral
sequence {LS E∗,∗
r, dr}for the bundle F→Mπ
→Xin [7, Theorem 5.4]. By as-
sumption, the first cohomology H1(A(F)) is trivial. This yields that the edge ho-
momorphism Hi(A(X)) →LS Ei,0
2∼
=LS Ei,0
∞→Hi(A(M)) is injective for i= 1,2.
The edge homomorphism is nothing but the map H∗(A(π)). This follows from one
of the properties of the Leray–Serre spectral sequence.
(II) The injectivity of f(α)2: Recall the commutative diagram (2.1). By the degree
reasons, we see that the elements in ΩE0,1
2are non-exact. Since Mis a manifold, it
follows from the argument in [3, Section 20] that ΩE1,0
2is trivial and then each ele-
ment in ΩE0,2
2is also non-exact. This yields that the upper-left hand side surjective
map in (2.1) is bijective. It turn out that the map f(α)2is injective for ∗= 1,2.
Proof of Theorem 2.1. Consider the commutative diagram (3.1). The injectivity of
the maps described in (I) and (II) implies the result. The latter half of the assertion
follows from the commutativity of the left square in the diagram (2.1). □
Before proving Corollary 2.2, we recall a result on the ˇ
Cech cohomology of a dif-
feological torus. Let TKbe a diffeological torus, namely a quotient Rn/K endowed
with the quotient diffeology, where Kis a discrete subgroup of Rn.
Proposition 3.2. ([3, Corollary]) One has an isomorphism ˇ
H∗(TK,R)∼
=H∗(K;R).
Here H∗(K;R)denotes the ordinary cohomology of group K.
Proof of Corollary 2.2. Let Tθbe the irrational torus. By definition, Tθis the
diffeological space T2/Sθendowed with the quotient diffeology, where Sθis the
subgroup {(e2πit, e2πiθt )∈T2|t∈R}which is diffeomorphic to Ras a Lie group.
Then we have a principal R-bundel of the form R→T2→Tθwhich is a diffeological
fibration. Then the Leray-Seere spectral sequence enables us to conclude that
6 KATSUHIKO KURIBAYASHI
H∗(A(Tθ)) ∼
=H∗(A(T2)) ∼
=H∗(Ω(T2)) ∼
=∧(t1, t2), where deg ti= 1. In particular,
H1(A(Tθ)) ∼
=R⊕R.
Moreover, by virtue of Theorem 2.1, we see that the map edge1:H1(A(Tθ)) →
AE0,1
2is a monomorphism. Since Tθis isomorphic to a diffeological torus of the form
R/(Z+θZ) ; see [2, Exercise 31, 3)], it follows from Proposition 3.2 that ˇ
H∗(Tθ,R)∼
=
H∗(Z+θZ;R)∼
=H∗(Z⊕Z;R). This yields that AE0,1
2⊕AE1,0
3∼
=ˇ
H1(Tθ,R)∼
=R⊕R.
The injectivity of the edge map above implies that AE1,0
3(Tθ) = 0 and hence the
map Θ induces an isomorphism H1(Ω(Tθ)) ⊕ΩE1,0
3
∼
=
→H1(A(Tθ)). It follows from
[3, Section 19] that ΩE1,0
2∼
=Fl•(Tθ). Furthermore, we have H2(Ω(Tθ)) = 0; see [2,
Exercise 119]. It turns out that ΩE1,0
2∼
=ΩE1,0
3. We have the result. □
4. From the second singular de Rham cohomology to the ˇ
Cech
cohomology
We define the edge map edge : Hi(A(X)) →ˇ
Hi(X) by the composite of the
maps in the lower sequence in (2.1). For degree reasons, we see that each element
in AE0,1
2the E2-term of the ˇ
Cech-de Rham spectral sequence is non exact. Then,
the map edge : H1(A(X)) →ˇ
H1(X) is injective under the same assumption as in
Theorem 2.1. In order to consider the edge map in degree 2, we generalize Lemma
3.1 introducing a generating family of a multi-set. Let π:Y→Xbe a subduction
and GYa generating family of Y. We define Gmulti
Xby the multi-set ϕ∈GY{π◦ϕ}.
Proposition 4.1. Under the same assumption as in Theorem 2.1, the edge map
H2(A(X)) →ˇ
H2(X)is injective, where ˇ
H2(X)is the ˇ
Cech cohomology associated
with Gmulti
X.
Remark 4.2.In the proof of [3, Proposition in §5], we need the condition (*) for a
generating family GXthat for any plot P:U→Xand each r∈U, there exists a
plot q:B→Yin GXsuch that q=P|B. To this end, we have chosen the generating
family GYconsisting of all plots whose domains are open balls in Euclidian spaces.
Let Gmulti
Xbe the generating multi-family associated with GY. Then Gmulti
Xalso
satisfies the condition (*) above. We observe that the inclusion π∗GY→ Gmulti
X
induces a diffeomorphism N(π∗GY)/ev∼
=
→ N (Gmulti
X)/evbetween nebulae and hence
the evaluation map gives rise to a diffeomorphism N(Gmulti
X)/ev∼
=
→X; see [2, 1.76].
With the notation Remark 4.2, for a map in the monoid MY, we define η(π◦
ϕ, r) = (π◦ψ, η (r)), where η(ϕ, r)=(ψ, η(r)). Then we have a morphism π′:
MY→MXof monoids defined by π′(η) = η. Moreover, we define
π:Cp,q
X:= map(Mp
X, Kq(NX)) →map(Mp
Y, Kq(NY)) =: Cp,q
Y
for K= Ω and Aby π(φ)(η1, .., ηp) = π∗(φ(η1, ..., ηp)). The straightforward calcu-
lation shows that πis compatible with the differentials dΩ,dAand the Hochschild
differential δ. Thus we have
Proposition 4.3. The map πinduces a morphism of spectral sequences {f(π)r}:
{KE∗,∗
r(X), dr} → {KE∗,∗
r(Y), dr}for K= Ω and A.
We are ready to prove the main result in this section.
Proof of Proposition 4.1. Suppose that there exists a non-zero element xin the
kernel of the map edge : H2(A(X)) →ˇ
H2(X). We recall the commutative dia-
gram (3.1). For the map πin the right hand side, we see that π∗=f(˜π)0,∗
2. This
A COMPARISON BETWEEN TWO DE RHAM COMPLEXES IN DIFFEOLOGY 7
follows from the construction the morphism {f(π)r}of the spectral sequence for
the singular de Rham complex in Proposition 4.3. The arguments in (I) and (II)
before the proof of Theorem 2.1 enable us to deduce that ev∗(x)∈AE0,2
2(X)
and f(π)2(ev∗(x)) ∈AE0,2
2(M) are non-zero elements. Since xis in the ker-
nel, it follows that ev∗(x) is a d2-exact element. The naturality of f(π)2im-
plies that f(π)2(ev∗(x)) is also d2-exact. Then, the commutativity of the dia-
gram (2.1) obtained by replacing Xwith Mimplies that the non-zero element
(ev∗◦H(α)−1◦H∗(A(π)))(x) in ΩE0,2
2(M) is d2-exact. For degree reasons, we
see that d1,0
2is nontrivial and then so is ΩE1,0
2. On the other hand, since Mis
a manifold, it follows that ΩE1,0
1(M)∼
=FL(M) = 0. In fact, the fibre Rof a
flow bundle is contractible and then the bundle admits a smooth global section;
see [9, 6.7 Theorem] for differentiable approximation of a section. Thus, we have
ΩE1,0
2= Ker{Ωd:ΩE1,0
1(M)→ΩE1,1
1(M)}= 0, which is a contradiction. This
completes the proof. □
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