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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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