(FlX )— LX ω

= t

X— X— X—

d d

ds |0 (Fls ) (Flt ) ω

dt (Flt ) ω =

LX (FlX )— ω.

= t

We may commute ds |0 with the bounded linear mapping (FlX )— from the space

d

t

of di¬erential forms on U to that of forms on V , where V is open in U such that

FlX (V ) ‚ U for all r ∈ [0, t]. We may ¬nd such open U and V because the

r

c -topology on R — M is the product of the c∞ -topologies, by corollary (4.15).

∞

33.20. Lemma of Poincar´. Let ω ∈ C ∞ (U, Lk+1 (D0 E; F )) be a closed form

±

e alt

i.e., dω = 0, where U is a star-shaped c∞ -open subset of a convenient vector space

E, with values in a convenient vector space F . Here D± may be any of T , D, D(k) ,

etc.

Then ω is exact, i.e. ω = d• where

1

tk ω(tx)(x, v1 , . . . , vk )dt

•(x)(v1 , . . . , vk ) =

0

33.20

33.21 33. Di¬erential forms 351

is a di¬erential form • ∈ C ∞ (U, Lk (D0 E, F )).

±

alt

Proof. We consider µ : R — E ’ E, given by µ(t, x) = µt (x) = tx. Let I ∈ X(E)

be the vector ¬eld I(x) = x, then µ(et , x) = FlI (x). So for (x, t) in a neighborhood

t

k, alt

of U — (0, 1], in HomC ∞ (U,R) (C (U ← D U ), C ∞ (U, R)) we have

∞ ±

d—

(FlI t )— ω = 1 (FlI t )— LI ω by

d

dt µt ω = (33.19)

log log

dt t

1— —

1

= t µt (iI dω + diI ω) = t dµt iI ω.

For X1 , . . . , Xk ∈ D0 E we may compute

( 1 µ— iI ω)x (X1 , . . . , Xk ) = 1 (iI ω)tx (Dx µt .X1 , . . . , Dx µt .Xk )

tt t

= 1 ωtx (tx, Dx µt .X1 , . . . , Dx µt .Xk ) = ωtx (x, Dx µt .X1 , . . . , Dx µt .Xk ).

t

Since Tx (µt ) = t. IdE and D(1) µt = µ—— = t. IdE we can make the last com-

t

putation more explicit if all Xi ∈ E or E . So if k ≥ 0, the k-form 1 µ— iI ω

tt

is de¬ned and smooth in (t, x) for all t ∈ [0, 1] and describes a smooth curve in

C ∞ (U, Lk (D0 E, F )). Clearly, µ— ω = ω and µ— ω = 0, thus

±

1 0

alt

1

µ— ω µ— ω d—

’

ω= = dt µt ωdt

1 0

0

1 1

d( 1 µ— iI ω)dt 1—

= =d t µt iI ωdt = d•.

tt

0 0

Remark. We were unable to prove the Lemma of Poincar´ for modular forms

e

which are given by module homomorphisms, because µ— ω does not make sense in

t

a di¬erentiable way for t = 0. One may ask whether a closed modular di¬eren-

tial form ω ∈ Homk,∞ (M,R) (C ∞ (M ← D± M ), C ∞ (M, R)) already has to be in

alt

C

C ∞ (Lk (D± M, M — R)).

alt

33.21. Review of operations on di¬erential forms.

f—

LX

Space d

C ∞ (M ← Λ— (D M )) “ “ +

C ∞ (M ← Λ— (T M )) “ “ +

Λ— ∞ (M,R) HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)) “ “ di¬

C

Λ— ∞ (M,R) C ∞ (M ← T M ) “ “ +

C

C ∞ (L— (DM, M — R)) ¬‚ow “ +

alt

—, alt

HomC ∞ (M,R) (C (M ← DM ), C ∞ (M, R))

∞

+ + di¬

C ∞ (L— (D[1,∞) M, M — R)) ¬‚ow “ +

alt

—, alt

HomC ∞ (M,R) (C (D[1,∞) M ), C ∞ (M, R))

∞

+ + di¬

C ∞ (M ← L— (D(1) M, M — R)) ¬‚ow “ +

alt

—, alt

HomC ∞ (M,R) (C (M ← D(1) M ), C ∞ (M, R))

∞

¬‚ow ? di¬

C ∞ (M ← L— (T M, M — R)) + + +

alt

—, alt

HomC ∞ (M,R) (X(M ), C ∞ (M, R)) + + di¬

33.21

352 Chapter VII. Calculus on in¬nite dimensional manifolds 33.22

In this table a ˜“™ means that the space is not invariant under the operation on

top of the column, a ˜+™ means that it is invariant, ˜di¬™ means that it is invariant

under f — only for di¬eomorphisms f , and ˜¬‚ow™ means that it is invariant under

LX for all kinematic vector ¬elds X which admit local ¬‚ows.

33.22. Remark. From the table (33.21) we see that for many purposes only one

space of di¬erential forms is fully suited. We will denote from now on by

„¦k (M ) := C ∞ (M ← Lk (T M, M — R))

alt

the space of di¬erential forms, for a smooth manifold M . By (30.1) it carries the

structure of a convenient vector space induced by the closed embedding

C ∞ (U± , Lk (E, R))

„¦k (M ) ’ alt

±

s ’ pr2 —¦ ψ± —¦ (s|U± ),

where (U± , u± : U± ’ E) is a smooth atlas for the manifold M , and where ψ± :=

Lk (T u’1 , R)) is the induced vector bundle chart.

±

alt

Similarly, we denote by

„¦k (M, V ) := C ∞ (M ← Lk (T M, M — V ))

alt

the space of di¬erential forms with values in a convenient vector space V , and by

„¦k (M ; E) := C ∞ (M ← Lk (T M, E))

alt

the space of di¬erential forms with values in a vector bundle p : E ’ M .

Lemma. The space „¦k (M ) is isomorphic as convenient vector space to the closed

linear subspace of C ∞ (T M —M . . . —M T M, R) consisting of all ¬berwise k-linear

alternating smooth functions in the vector bundle structure T M • · · · • T M from

(29.5).

Proof. By (27.17), the space C ∞ (T M —M . . . —M T M, R) carries the initial struc-

ture with respect to the closed linear embedding

C ∞ (T M —M . . . —M T M, R) ’ C ∞ (u± (U± ) — E — . . . — E, R),

±

and C ∞ (u± (U± )—E—. . .—E, R) contains an isomorphic copy of C ∞ (U± , Lk (E, R))

alt

as closed linear subspace by cartesian closedness.

Corollary. All the important mappings are smooth:

d : „¦k (M ) ’ „¦k+1 (M )

i : X(M ) — „¦k (M ) ’ „¦k’1 (M )

L : X(M ) — „¦k (M ) ’ „¦k (M )

f — : „¦k (M ) ’ „¦k (N )

33.22

34.1 34. De Rham cohomology 353

where f : N ’ M is a smooth mapping. The last mappings is even smooth consid-

ered as mapping (f, ω) ’ f — ω, C ∞ (N, M ) — „¦k (M ) ’ „¦k (N ).

Recall once more the formulas for ω ∈ „¦k (M ) and Xi ∈ X(M ), from (33.12.3),

(33.10), (33.17) :

k

(’1)i Xi (ω(X0 , . . . , Xi , . . . , Xk ))+

(dω)(x)(X0 , . . . , Xk ) =

i=0

(’1)i+j ω([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ),

+

i<j

(iX •)(X1 , . . . , Xk’1 ) = •(X, X1 , . . . , Xk’1 ),

k

(LX ω)(X1 , . . . , Xk ) = X(ω(X1 , . . . , Xk )) ’ ω(X1 , . . . , [X, Xi ], . . . , Xk ).

i=1

Proof. For d we use the local formula (33.12.1), smoothness of i is obvious, and

for the Lie derivative we may use formula (33.18.6). The pullback mapping f — is

induced from T f — . . . — T f .

34. De Rham Cohomology

Section (33) provides us with several graded commutative di¬erential algebras con-

sisting of various kinds of di¬erential forms for which we can de¬ne De Rham

cohomology, namely all those from the list (33.21) which have + in the d-column.

But among these only C ∞ (L— (T M, M — R)) behaves functorially for all smooth

alt

mappings; the others are only functors over categories of manifolds where the mor-

phisms are just the local di¬eomorphisms. So we treat here cohomology only for

these di¬erential forms.

34.1. De Rham cohomology. Recall that for a smooth manifold M we have

denoted

„¦k (M ) := C ∞ (Lk (T M, M — R)).

alt

We now consider the graded algebra „¦(M ) = k≥0 „¦k (M ) of all di¬erential forms

on M . Then the space Z(M ) := M ) := {ω ∈ „¦(M ) : dω = 0} of closed forms is

a graded subalgebra of „¦ (i. e. it is a subalgebra, and „¦k (M ) © Z(M ) = Z k (M )),

and the space B(M ) := {d• : • ∈ „¦(M )} of exact forms is a graded ideal in

Z(M ). This follows directly from d2 = 0 and the derivation property d(• § ψ) =

d• § ψ + (’1)deg • • § dψ of the exterior derivative.

De¬nition. The algebra

{ω ∈ „¦(M ) : dω = 0}

Z(M )

H — (M ) := =

{d• : • ∈ „¦(M )}

B(M )

34.1

354 Chapter VII. Calculus on in¬nite dimensional manifolds 34.2

is called the De Rham cohomology algebra of the manifold M . It is graded by

ker(d : „¦k (M ) ’ „¦k+1 (M ))

— k

H (M ) = H (M ) = .

im(d : „¦k’1 (M ) ’ „¦k (M ))

k≥0 k≥0

If f : M ’ N is a smooth mapping between manifolds then f — : „¦(N ) ’ „¦(M )

is a homomorphism of graded algebras by (33.9), which satis¬es d —¦ f — = f — —¦ d

by (33.15). Thus, f — induces an algebra homomorphism which we also call f — :

H — (N ) ’ H — (M ). Obviously, each H k is a contravariant functor from the category

of smooth manifolds and smooth mappings into the category of real vector spaces.

34.2. Lemma. Let f , g : M ’ N be smooth mappings between manifolds which

are C ∞ -homotopic, i.e., there exists h ∈ C ∞ (R — M, N ) with h(0, x) = f (x) and

h(1, x) = g(x). Then f and g induce the same mapping in cohomology f — = g — :

H — (N ) ’ H — (M ).

Remark. f , g ∈ C ∞ (M, N ) are called homotopic if there exists a continuous

mapping h : [0, 1] — M ’ N with h(0, x) = f (x) and h(1, x) = g(x). For ¬nite

dimensional manifolds this apparently looser relation in fact coincides with the

relation of C ∞ -homotopy. We sketch a proof of this statement: let • : R ’ [0, 1]

be a smooth function with •(t) = 0 for t ¤ 1/4, •(t) = 1 for t ≥ 3/4, and •

¯ ¯

monotone in between. Then consider h : R — M ’ N , given by h(t, x) = h(•(t), x).

¯ ˜

Now we may approximate h by smooth functions h : R — M ’ N without changing

it on (’∞, 1/8) — M where it equals f , and on (7/8, ∞) — M , where it equals g.

This is done chartwise by convolution with a smooth function with small support

on Rm . See [Br¨cker, J¨nich, 1973] for a careful presentation of the approximation.

o a

It is an open problem to extend this to some in¬nite dimensional manifolds.

The lemma of Poincar´ (33.20) is an immediate consequence of this result.

e

Proof. For ω ∈ „¦k (M ) we have h— ω ∈ „¦k (R — M ). We consider the insertion

operator inst : M ’ R — M , given by inst (x) = (t, x). For • ∈ „¦k (R — M ) we then

have a smooth curve t ’ ins— • in „¦k (M ).

t

Consider the integral operator I0 : „¦k (R — M ) ’ „¦k (M ) given by I0 (•) :=

1 1

1

ins— • dt. Let T := ‚t ∈ C ∞ (R — M ← T (R — M )) be the unit vector ¬eld

‚

t

0

in direction R.

We have inst+s = FlT —¦ inss for s, t ∈ R, so

t

— —

— T—

T

‚ ‚ ‚

‚t 0 (Flt —¦ inss ) • = ‚t 0 inss (Flt ) •

‚s inss • =

ins— ‚t 0 (FlT )— • = (inss )— LT •

‚

= by (33.19).

s t

We have used that (inss )— : „¦k (R — M ) ’ „¦k (M ) is linear and continuous, and so

34.2

34.4 34. De Rham cohomology 355

one may di¬erentiate through it by the chain rule. Then we have in turn

1 1

ins— • dt d ins— • dt

1

d I0 •=d =

t t

0 0

1

ins— d• dt = I0 d •

1

= by (33.15).

t

0

1 1

(ins— ins— )• —

ins— LT • dt

‚

’ = ‚t inst • dt =

1 0 t

0 0

1 1

I0 LT •

= = I0 (d iT + iT d)• by (33.18.6).

¯

Now we de¬ne the homotopy operator h := I0 —¦ iT —¦ h— : „¦k (M ) ’ „¦k’1 (M ). Then

1

we get

g — ’ f — = (h —¦ ins1 )— ’ (h —¦ ins0 )— = (ins— ’ ins— ) —¦ h—

1 0

¯¯

= (d —¦ I 1 —¦ iT + I 1 —¦ iT —¦ d) —¦ h— = d —¦ h ’ h —¦ d,

0 0

which implies the desired result since for ω ∈ „¦k (M ) with dω = 0 we have g — ω ’

¯ ¯ ¯

f — ω = dhω ’ hdω = dhω.

34.3. Lemma. If a manifold is decomposed into a disjoint union M = M± of

±

open submanifolds, then H k (M ) = ± H k (M± ) for all k.

Proof. „¦k (M ) is isomorphic to ± „¦k (M± ) via • ’ (•|M± )± . This isomorphism

commutes with the exterior derivative d and induces the result.

34.4. The setting for the Mayer-Vietoris Sequence. Let M be a smooth

manifold, let U , V ‚ M be open subsets which cover M and admit a subordinated

smooth partition of unity {fU , fV } with supp(fU ) ‚ U and supp(fV ) ‚ V . We

consider the following embeddings:

‘ 99

e GA U ©V

”

‘ jU

j V

U

! heiu

eV

e

i U V

M.

Lemma. In this situation, the sequence

β

±

0 ’ „¦(M ) ’ „¦(U ) • „¦(V ) ’ „¦(U © V ) ’ 0

’ ’

is exact, where ±(ω) := (i— ω, i— ω) and β(•, ψ) = jU • ’ jV ψ. We also have

— —

U V

(d • d) —¦ ± = ± —¦ d and d —¦ β = β —¦ (d • d).

Proof. We have to show that ± is injective, ker β = im ±, and that β is surjective.

The ¬rst two assertions are obvious. For • ∈ „¦(U ©V ) we consider fV • ∈ „¦(U ©V ).

Note that supp(fV •) is closed in the closed subset supp(fV )©U of U and contained

in the open subset U © V of U , so we may extend fV • by 0 to a smooth form

•U ∈ „¦(U ). Likewise, we extend ’fU • by 0 to •V ∈ „¦(V ). Then we have

β(•U , •V ) = (fU + fV )• = •.

34.4

356 Chapter VII. Calculus on in¬nite dimensional manifolds 34.7