34.5. Theorem. Mayer-Vietoris sequence. Let U and V be open subsets in a

manifold M which cover M and admit a subordinated smooth partition of unity.

Then there is an exact sequence

β—

± δ

· · · ’ H k (M ) ’’ H k (U ) • H k (V ) ’ H k (U © V ) ’ H k+1 (M ) ’ · · ·

—

’ ’ ’

It is natural in the triple (M, U, V ). The homomorphisms ±— and β— are algebra

homomorphisms, but δ is not.

Proof. This follows from (34.4) and standard homological algebra.

Since we shall need it later we will now give a detailed description of the connecting

homomorphism δ. Let {fU , fV } be a partition of unity with supp fU ‚ U and

supp fV ‚ V . Let ω ∈ „¦k (U © V ) with dω = 0 so that [ω] ∈ H k (U © V ). Then

(fV .ω, ’fU .ω) ∈ „¦k (U ) • „¦k (V ) is mapped to ω by β, and we have

δ[ω] = [±’1 (d • d)(fV .ω, ’fU .ω)] = [±’1 (dfV § ω, ’dfU § ω)]

= [dfV § ω] = ’[dfU § ω)],

where we have used the following fact: fU + fV = 1 implies that on U © V we have

dfV = ’dfU , thus dfV § ω = ’dfU § ω, and o¬ U © V both are 0.

34.6. Theorem. Let M be a smooth manifold which is smoothly paracompact.

Then the De Rham cohomology of M coincides with the sheaf cohomology of M

with coe¬cients in the constant sheaf R on M .

Proof. Since M is smoothly paracompact it is also paracompact, and thus the

usual theory of sheaf cohomology using the notion of ¬ne sheafs is applicable. For

each k we consider the sheaf „¦k on M which associates to each c∞ -open set U ‚ M

M

the convenient vector space „¦k (U ). Then the following sequence of sheaves

d d

R ’ „¦0 ’ „¦1 ’ . . .

M’ M’

is a resolution of the constant sheaf R by the lemma of Poincar´ (33.20). Since

e

we have smooth partitions of unity on M , each sheaf „¦k is a ¬ne sheaf, so this

M

resolution is acyclic [Godement, 1958], [Hirzebruch, 1962, 2.11.1], and the sequence

of global sections may be used to compute the sheaf cohomology of the constant

sheaf R. But this is the De Rham cohomology.

34.7. Theorem. Let M be a smooth manifold which is smoothly paracompact.

Then the De Rham cohomology of M coincides with the singular cohomology with

coe¬cients in R via a canonical isomorphism which is induced by integration of

p-forms over smooth singular simplices.

k

Proof. Denote by S∞ the sheaf which is generated by the presheaf of singular

k

smooth cochains with real coe¬cients. In more detail: let us put S∞ (U, R) =

∞

R = RC (∆k ,U ) , where σ : ∆k ’ U is any mapping which extends to a smooth

σ

34.7

34.7 34. De Rham cohomology 357

mapping from a neighborhood of the standard k-simplex ∆k ‚ Rk+1 into U , where

U is c∞ -open in M . This de¬nes a presheaf. The associated sheaf is denoted by

k

S∞ . The sequence

— —

0δ 1δ 2

R’ S∞ ’’ S∞ ’’ S∞ ’ . . .

of sheafs is a resolution, because if U is a small open set, say di¬eomorphic to

a radial neighborhood of 0 in the modeling convenient vector space, then U is

—

smoothly contractible to a point. Smooth mappings induce mappings in the S∞ -

—

cohomology, thus H k (S∞ (U, R), ‚) = 0 for k > 0. This implies that the associated

sequence of stalks is exact, so the sequence above is a resolution. A standard

k

argument of sheaf theory shows that each sheaf S∞ is a ¬ne sheaf, so they form an

—

acyclic resolution, and H k (S∞ (M, R), ‚) coincides with the sheaf cohomology with

coe¬cients in the constant sheaf R.

Furthermore, integration of p-forms over smooth singular p-simplices de¬nes a map-

ping of resolutions

w„¦ w„¦ w ···

RRTR „¦0 1 2

R

¢ u

u u

wS wS w ···

0 1 2

S∞ ∞ ∞

which induces an isomorphism from the De Rham cohomology of M to the coho-

mology H — (S∞ (M, R), ‚).

—

Now we consider the resolution

δ— δ—

R ’ S0 ’ S1 ’ S2 ’ . . .

’ ’

of the constant sheaf R, where S k is the usual sheaf induced by the singular contin-

uous cochains. Since M is (even smoothly) paracompact and locally contractible,

this is an acyclic resolution, and the embedding of smooth singular chains into

continuous singular chains de¬nes a mapping of resolutions

wS wS w ···

RRTR S0 1 2

R

¢ u

u u

wS wS w ···

0 1 2

S∞ ∞ ∞

which induces an isomorphism from the singular cohomology of M to the cohomol-

ogy H — (S∞ (M, R), ‚).

—

34.7

358 Chapter VII. Calculus on in¬nite dimensional manifolds 35.2

35. Derivations on Di¬erential Forms

and the Fr¨licher-Nijenhuis Bracket

o

35.1. In this section let M be a smooth manifold. We consider the graded com-

mutative algebra

∞ ∞

∞

k

Lk (T M, M „¦k (M )

C (M ← — R)) =

„¦(M ) = „¦ (M ) = alt

k≥0 k=0 k=’∞

of di¬erential forms on M , see (33.22), where „¦0 (M ) = C ∞ (M, R), and where we

put „¦k (M ) = 0 for k < 0. We denote by Derk „¦(M ) the space of all (graded)

derivations of degree k, i.e., all bounded linear mappings D : „¦(M ) ’ „¦(M ) with

D(„¦l (M )) ‚ „¦k+l (M ) and D(• § ψ) = D(•) § ψ + (’1)kl • § D(ψ) for • ∈ „¦l (M ).

Convention. In general, derivations need not be of local nature. Thus, we consider

each derivation and homomorphism to be a sheaf morphism (compare (32.1) and

the de¬nition of modular 1-forms in (33.2)), or we assume that all manifolds in

question are again smoothly regular. This is justi¬ed by the obvious extension of

(32.4) and (33.3).

Lemma. Then the space Der „¦(M ) = k Derk „¦(M ) is a graded Lie algebra with

the graded commutator [D1 , D2 ] := D1 —¦ D2 ’ (’1)k1 k2 D2 —¦ D1 as bracket. This

means that the bracket is graded anticommutative and satis¬es the graded Jacobi

identity:

[D1 , D2 ] = ’(’1)k1 k2 [D2 , D1 ],

[D1 , [D2 , D3 ]] = [[D1 , D2 ], D3 ] + (’1)k1 k2 [D2 , [D1 , D3 ]]

(so that ad(D1 ) = [D1 , ] is itself a derivation of degree k1 ).

Proof. Plug in the de¬nition of the graded commutator and compute.

In section (33) we have already met some graded derivations: for a vector ¬eld X

on M the derivation iX is of degree ’1, LX is of degree 0, and d is of degree 1. In

(33.18) we already met some some graded commutators like LX = d iX + iX d =

[iX , d].

35.2. A derivation D ∈ Derk „¦(M ) is called algebraic if D | „¦0 (M ) = 0. Then

D(f.ω) = f.D(ω) for f ∈ C ∞ (M, R) and ω ∈ „¦(M ).

If the spaces Lk (Tx M ; R) are all re¬‚exive and have the bornological approximation

alt

property, then an algebraic derivation D induces for each x ∈ M a derivation

Dx ∈ Derk (L— (Tx M ; R)), by a method used in (33.5). It is not clear whether it

alt

su¬ces to assume that just the model spaces of M are all re¬‚exive and have the

bornological approximation property.

35.2

35.2 35. The Fr¨licher-Nijenhuis bracket

o 359

In the sequel, we will consider the space of all vector valued kinematic di¬erential

forms, which we will de¬ne by

C ∞ (M ← Lk (T M ; T M ))

„¦k (M ; T M ) =

„¦(M ; T M ) = alt

k≥0 k≥0

Note that „¦0 (M ; T M ) = X(M ) = C ∞ (M ← T M ). For simplicity™s sake, we will

not treat other kinds of vector valued di¬erential forms.

Theorem. (1) For K ∈ „¦k+1 (M ; T M ) the formula

(iK ω)(X1 , . . . , Xk+l ) =

1

= sign σ .ω(K(Xσ1 , . . . , Xσ(k+1) ), Xσ(k+2) , . . . )

(k+1)! (l’1)!

σ∈Sk+l

(k+1)(k+2)

(’1)i1 +···+ik+1 ’

= ω(K(Xi1 , . . . , Xik+1 ), X1 , . . . , Xi1 , . . . ),

2

i1 <···<ik+1

for f ∈ C ∞ (M, R) = „¦0 (M ),

iK f = 0

for ω ∈ „¦l (M ), Xi ∈ X(M ) de¬nes an algebraic graded derivation iK = i(K) ∈

Derk „¦(M ).

]§ on „¦—+1 (M ; T M ) for K ∈ „¦k+1 (M ; T M ), L ∈

(2) We de¬ne a bracket [ ,

„¦l+1 (M ; T M ) by

[K, L]§ := iK L ’ (’1)kl iL K,

where iK (L) is given by the same formula as in (1). This de¬nes a graded Lie

algebra structure with the grading as indicated, and we have i([K, L]§ ) = [iK , iL ] ∈

Der „¦(M ). Thus, i : „¦—+1 (M ; T M ) ’ Der— „¦(M ) is a homomorphism of graded

Lie algebras, which is injective under the assumptions of (35.1).

The concomitant [ , ]§ is called the algebraic bracket or the Nijenhuis-Richardson

bracket, compare [Nijenhuis, Richardson, 1967].

Proof. (1) We know that iX is a derivation of degree ’1 for a vector ¬eld X ∈

X(M ) = „¦0 (M ; T M ) by (33.11). By direct evaluation, one gets

(3) [iX , iK ] = i(iX K).

Using this and induction on the sum of the degrees of K ∈ „¦k (M ; T M ), • ∈ „¦(M ),

and ψ ∈ „¦(M ), one can then show that

iX iK (• § ψ) = iX (iK • § ψ + (’1)k deg• • § iK ψ)

holds, which implies that iK is a derivation of degree k.

35.2

360 Chapter VII. Calculus on in¬nite dimensional manifolds 35.4

(2) By induction on the sum of k = deg K ’ 1, l = deg L ’ 1, and p = deg •, and

by (3) we have

[iX , [iK , iL ]]• = [[iX , iK ], iL ]• + (’1)k [iK , [iX , iL ]]•

= [i(iX K), iL ]• + (’1)k [iK , i(iX L)]•

= i i(iX K)L ’ (’1)(k’1)l iL iX K •

+ (’1)k i iK iX L ’ (’1)k(l’1) i(iX L)K •

= i iX iK L ’ (’1)kl iX iL K • = i (iX [K, L]§ ) •,

iX [iK , iL ]• = [iX , [iK , iL ]]• + (’1)k+l [iK , iL ]iX •

= i (iX [K, L]§ ) • + (’1)k+l i([K, L]§ )iX •

= i (iX [K, L]§ ) • ’ i(iX [K, L]§ )• + iX i([K, L]§ )•

= iX i([K, L]§ )•.

This implies i([K, L]§ ) = [iK , iL ] since the iX for X ∈ T M separate points, in both

cases of the convention (35.1). From iK df = df —¦ K it follows that the mapping

i : „¦(M ; T M ) ’ Der(„¦(M )) is injective, so („¦—+1 (M ; T M ), [ , ]§ ) is a graded

Lie algebra.

35.3. The exterior derivative d is an element of Der1 „¦(M ). In view of the formula

LX = [iX , d] = iX d + d iX for vector ¬elds X (see (33.18.6)), we de¬ne for K ∈

„¦k (M ; T M ) the Lie derivative LK = L(K) ∈ Derk „¦(M ) by

LK := [iK , d] = iK d ’ (’1)k’1 d iK .

Since the 1-forms df for all local functions on M separate points on each Tx M , the

mapping L : „¦(M ; T M ) ’ Der „¦(M ) is injective, because LK f = iK df = df —¦ K

for f ∈ C ∞ (M, R).

From (35.2.1) it follows that i(IdT M )ω = kω for ω ∈ „¦k (M ). Hence, L(IdT M )ω =

i(IdT M )dω ’ d i(IdT M )ω = (k + 1)dω ’ kdω = dω, and thus L(IdT M ) = d.

35.4. Proposition. For K ∈ „¦k (M ; T M ) and ω ∈ „¦l (M ) the Lie derivative of ω

along K is given by the following formula, where the Xi are (local) vector ¬elds on

M.

(LK ω)(X1 , . . . , Xk+l ) =

1

sign σ LK(Xσ1 ,...,Xσk ) (ω(Xσ(k+1) , . . . , Xσ(k+l) ))

= k! l!

σ

+ (’1)k 1

sign σ ω([Xσ1 , K(Xσ2 , . . . , Xσ(k+1) )], Xσ(k+2) , . . . )

k! (l’1)!

σ

1

’ sign σ ω(K([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(k+2) , . . . )

(k’1)! (l’1)! 2!

σ

Proof. Consider LK ω = [iK , d]ω = iK dω ’ (’1)k’1 diK ω, and plug into this the

de¬nitions (35.2.1), second version, and (33.12.3). After computing some signs the

expression above follows.

35.4

35.5 35. The Fr¨licher-Nijenhuis bracket

o 361

35.5. De¬nition and theorem. For K ∈ „¦k (M ; T M ) and L ∈ „¦l (M ; T M ) we

de¬ne the Fr¨licher-Nijenhuis bracket [K, L] by the following formula, where the Xi

o

are vector ¬elds on M .

(1) [K, L](X1 , . . . , Xk+l ) =

1

= sign σ [K(Xσ1 , . . . , Xσk ), L(Xσ(k+1) , . . . , Xσ(k+l) )]

k! l!

σ

+ (’1)k 1

sign σ L([Xσ1 , K(Xσ2 , . . . , Xσ(k+1) )], Xσ(k+2) , . . . )

k! (l’1)!

σ

1

’ sign σ L(K([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(k+2) , . . . )

(k’1)! (l’1)! 2!

σ

’ (’1)kl+l 1

sign σ K([Xσ1 , L(Xσ2 , . . . , Xσ(l+1) )], Xσ(l+2) , . . . )

(k’1)! l!

σ

1

’ sign σ K(L([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(l+2) , . . . ) .

(k’1)! (l’1)! 2!

σ

Then [K, L] ∈ „¦k+l (M ; T M ), and we have

[L(K), L(L)] = L([K, L]) ∈ Der „¦(M ).

dim M

Therefore, the space „¦(M ; T M ) = k=0 „¦k (M ; T M ) with its usual grading is a

graded Lie algebra for the Fr¨licher-Nijenhuis bracket. So we have

o

[K, L] = ’(’1)kl [L, K]

[K1 , [K2 , K3 ]] = [[K1 , K2 ], K3 ] + (’1)k1 k2 [K2 , [K1 , K3 ]]

IdT M ∈ „¦1 (M ; T M ) is in the center, i.e., [K, IdT M ] = 0 for all K.

For vector ¬elds the Fr¨licher-Nijenhuis bracket coincides with the Lie bracket. The

o

—

mapping L : „¦ (M ; T M ) ’ Der— „¦(M ) is an injective homomorphism of graded

Lie algebras.

Proof. We ¬rst show that [K, L] ∈ „¦k+l (M ; T M ). By convention (35.1), this

is a local question in M , thus we may assume that M is a c∞ -open subset of a

convenient vector space E, that Xi : M ’ E, that K : M ’ Lk (E; E), and that

alt

l

L : M ’ Lalt (E; E). Then each expression in the formula is a kinematic vector

¬eld, and for such ¬elds Y1 , Y2 the Lie bracket is given by [Y1 , Y2 ] = dY2 .Y1 ’dY1 .Y2 ,

as shown in the beginning of the proof of (32.8). If we rewrite the formula in this

way, all terms containing the derivative of one Xi cancel, and the following local

expression for [K, L] remains, which is obviously an element of „¦k+l (M ; T M ).

[K, L](X1 , . . . , Xk+l ) =

1

= sign σ (dL.K(Xσ1 , . . . , Xσk ))(Xσ(k+1) , . . . )

k! l!

σ

’ (dK.L(Xσ(k+1) , . . . ))(Xσ1 , . . . , Xσk )

+ l L((dK.Xσ(k+1) )(Xσ1 , . . . , Xσk ), Xσ(k+2) , . . . )

’ k K((dL.Xσ1 )(Xσ(k+1) , . . . , Xσ(k+l) ), Xσ2 , . . . , Xσk ) .

35.5

362 Chapter VII. Calculus on in¬nite dimensional manifolds 35.5

Next we show that L([K, L]) = [LK , LL ] holds, by the following purely algebraic

method, which is adapted from [Dubois-Violette, Michor, 1997]. The Chevalley

coboundary operator for the adjoint representation of the Lie algebra X(M ) is

given by [Koszul, 1950], see also [Cartan, Eilenberg, 1956]

1

‚K(X1 , . . . , Xk+1 ) = sign σ [Xσ1 , K(Xσ2 , . . . , Xσ(k+1) )]

k!

σ

1

’ sign σ K([Xσ1 , Xσ2 ], Xσ3 , . . . , Xσ(k+1) ),

(k’1)! 2!

σ

(’1)i [Xi , K(X0 , . . . , Xi , . . . , Xk )]

‚K(X0 , . . . , Xk ) =

0¤i¤k

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

+

0¤i<j¤k

and it is well known that ‚‚ = 0. The following computation and close relatives

will appear several times in the remainder of this proof, so we include it once.

(i‚K ω)(X1 , . . . , Xk+l ) =

1

= sign(σ)ω(‚K(Xσ1 , . . . ), Xσ(k+2) , . . . )

(k+1)! (l’1)!

σ∈Sk+l

k+1

(’1)i’1 ω([Xσi , K(Xσ1 , . . . , Xσi , . . . )], Xσ(k+2) , . . . )

1