The Fiber Integral of Differential Forms

Heinrich Hartmann

April 17, 2010

1 Statements

Proposition 1.1. Let f : M → B be an oriented submersion with boundary ofrelative dimension d. There is a unique morphism of CN modules∫

f

: f!ΩdM/B −→ CN

satisfying (∫f

α

)(x) =

∫f−1(x)

α

for all relative differential forms with proper support α ∈ f!ΩdM/B.Moreover there is a natural morphism of CM modules

Φ : Ωk+dM −→ ΩdM/N ⊗ f

∗ΩkN

satisfyingΦ(α ∧ β)→ α⊗ β (1)

for all α ∈ ΩdM , β ∈ f∗ΩkB where α is the image of α in ΩdM/N . This morphisminduces the a fiber integral ∫

f

: f!Ωk+dM → ΩkN .

Definition 1.2. In the situation of the proposition we call f∗ :=∫f

the pushforward or fiber integral of differential forms.

Theorem 1.3. Let f : (M,∂M)→ B be an oriented submersion with boundaryof relative dimension d. The fiber integral has the following properties

1. (Normalisation) If f : M → pt then

f∗α =∫M

α

1

2. (Fubini-Theorem/Naturality) If g : B → B′ is another submersion, then

g∗ f∗ = (f g)∗

3. (Base Change Formular) If there is a cartesian diagram

M ′g′−−−−→ Myf ′ yf

B′g−−−−→ B.

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where f (and hence f ′) is a submersion and g is any differentiable map.Then

g∗f∗(α) = f ′∗g′∗(α)

for all α ∈ f!(Ωk+dM ).

4. (Projection Formular) For α ∈ f!(ΩdM ) and β ∈ ΩkM )

f∗(α ∧ f∗β) = (f∗α) ∧ β

5. (Stokes Theorem) Recall that we deonte by ∂f the restriction of f to theboundray ∂M . For α ∈ f!(Ωk+d

M ) it is

f∗(dα) = (−1)dd(f∗α) + (f |∂M )∗(α|∂M )

if we use the graded commutator this reads.

[f∗, d] = ∂f

6. If M is a closed manifold and f proper, then f induces a morphism ofcohomology groups

f∗ : HkDR(M)→ Hk−d

DR (B).

If M and B are compact this map gets identified with the Poincare dualof the homology pushforward

f∗ : Hm−k(M)→ Hm−k(N).

2 Concepts

2.1 Orientation

Definition 2.1. An subersion with boundary f : (M,∂M) → B is a differen-tiable map from a manifold with boundary (M,∂M) to a (ordenary) manifoldB, with such that the restrictions

fo : Mo := M \∂M −→ B, and ∂f : ∂M −→ B

of f to the (ordenary) manifolds Mo and ∂M are submersions in the usual sense.

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Proposition 2.2 (Existence of fiber space coordinates). Let f : (M,∂M)→ Bbe a submersion with boundary of relative dimension d. Then there are opencoverings Uii∈I of M and Vii∈I of B, and open embedings zi : Ui → Rm+ :=Rm−1 × R≥0 and yi : Vi → Rb such that f(Ui) ⊂ Vi and the following diagramis commutative

Uizi−−−−→ Rm+yf yp

Vibi−−−−→ Rb.

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where p is the projectionon the first b coordinates.

Theorem 2.3 (Ehresmann [2] [1]). Let f : (M,∂M)→ B be a proper subersionwith boundary. Then for all b ∈ B there is an open neighbourhood U ⊂ B anda diffeomorphism

Φ : (M,∂M)|U −→ (Mb, ∂Mb)× U

with f = Φ pr2

Definition 2.4. Let ξ : E → M be a vector bundle of rank n. The framebundle of E is the principal GLn-bundle

F (ξ) :=∐x∈M

Isom(Rn, Ex)

The orienation bundle Or(E) of E is the fiber bundle associated to F inducedby the GLn-action

GLn −→ Z×, A 7→ det(A)/|det(A)|

An orientation of E is a global section of Or(E).

Definition 2.5. Let f : M → B be a submersion of relative dimension d. Anorientation of f (or M over B) is an orientation of the relative tangent bundleTM/B . An orientation of a submersion with boundary f : (M,∂M) → B is anorienation of fo : Mo → B.

Remark 2.6. If f : (M,∂M)→ N is an oriented submersion with boundary, thenthe fiber f−1(x) over x ∈ N has a natural structure of an oriented manifold withboundary.

Recall that an orientation on a manifold with boundary (M,∂M) is definedto be an orienation of the interior Mo and induces a natural orientation of ∂M(defined using outwards pointing vectorfilds), in the same way an orientation off induces an orientation of ∂f .

3 Proofs

Prop. 1.1. The first condition clearly defines∫fα as a function on N . It has

just to be checked that this function is smooth. This is a local statement in

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N . Since moreover supp(α) is proper over M we can assume there is a finitecover of supp(α) admitting fiber space coordinates and a partition of unitywith compact supports on each open subset. The smoothness now follows formstandard results of Analysis in Rn.

To proof the existence of the morphism Φ we consider the insertion map

ins : ΛdTM/B ⊗ Ωk+dM −→ ΩkM .

In the exact sequence (see the appendix)

0→ f∗ΩkB → ΩkMϕ→ f∗Ω1

B ⊗ Ωk−1M → Sym2(f∗Ω1

B)⊗ Ωk−2M → . . .

the arrow ϕ is adjoint to the insertion map f∗TB⊗ΩkM → Ωk−1M . Hence compo-

sition β ins is adjoint to the insertion d+1 vertical vectorfiedls in a k+d formand therefore zero. Thus ins factorizes over f∗ΩkB . And we get the requiredmorphism Φ by adjunction.

Note that Φ satisfies the formular 1, since ∧ is itself adjoint to insertion.The definition of

∫f

is now easy. Application of proper pushforward yieldsa map

f!ΩkM → f!(ΩdM/B ⊗ f∗ΩkB) ∼= f!(ΩdM/B)⊗ ΩkB

which we compose with the above constructed integral. (One needs a proj.formular for f! ... Reference?/Proof?)

Remark 3.1. In the situation above the morphism Φ can be described veryconcretely in coordinates. Let x1, . . . , xd, y1, . . . , yb be fiber space coordinatesfor f : M → B. On the canonical basis Φ takes the values

Φ : dxI ∧ dyJ 7→dxI ⊗ dyJ if I = [d] := 1, . . . , d

0 else

It is not hard to check, that this assignment is compatiple with coordinate trans-formations - and hence can be used as definition. Indeed let x1, . . . , xd, y1, . . . , yb

be another set of fiber space coordinates. We have

dxI ∧ dyJ =∑IJ

D(I , J)dxI ∧ dyJ

where the coefficient D(I , J) is the determinant of the matrix

(∂xI/∂xI 0∂xI/∂yJ ∂yJ/∂yJ

):=

∂xi1 . . . ∂xik ∂yj1 . . . ∂yjl

∂xa1

... ∂xip

∂xaq 0∂xar

∂yb1

... ∂xip

∂ybq∂yjp

∂yjq

∂ybk

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where of course I = i1 < · · · < ik, etc. Note that in general this determinantis not so easy to compute since the building blocks are not neccessarily diagonalmatrices. However in our case the situation is more fortunate. By definition Φtakes all forms dxI ∧ dyJ to zero if not I = [d], but then the first d rows arelineary dependent if not also I = [n]. So Φ takes the transfomed form to

det(∂xI/∂xI)dxI ⊗∑J

det(∂yJ/∂yJ)dyJ

which is precisley the transformation of dxI ⊗ dyJ as claimed.

Remark 3.2. A submersion determnines a principal fiber bundle on M definedby

F (f) :=∐x∈Mϕ : Rd+b ∼=→ TxM |ϕ(Rd × 0n) ⊂ Txf = ker dfx

=∐x∈M

Isom(Rd ⊂ Rd+b, Txf ⊂ TxM)

with structure group

GL(d, b) := invertible block- (d+ b)× (d+ b) -matrices(A 0B C

)

= Aut(Rd ⊂ Rd+b)

All vector bundles occured so far are associated to this principal fiber bundle.So we could have formulated everything in terms of representation theory forGL(d, b).

Theorem 1.3. . The Normalisation is obvious form the definition of f∗.Let us first make the statement of the base change formular precise: Recall

the situation we put ourselfs into:

M ′g′−−−−→ Myf ′ yf

B′g−−−−→ B.

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The fiber integral for f is a morphism of sheaves on B:

f∗ : f!Ωk+dM −→ ΩkB .

We can pull back this morphism to B′ using g and compose with the differentialdg of g, (this is usually deoted by g∗)

g∗f!Ωk+dM

g∗(f∗)−→ g∗ΩkBdg−→ ΩkB′ .

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On the other hand g∗f!Ωk+dM is canonically isomorphic to f ′! g

′∗Ωk+dM . The dif-

ferential dg′ induces a map

f ′! g′∗Ωk+d

M

f!(dg′)−→ f ′! Ω

k+dM ′

which we can compose with the fiber integral f∗. The Base Change formularclaims now, that this two morphisms concide, i.e. the following diagram ofsheaves on B′ commutes:

f ′! g′∗Ω

k+dM

g∗(f∗)−−−−→ g∗ΩkB

f ′! dg′y g∗

yf ′! Ω

k+dM ′

(f ′∗)−−−−→ ΩkB′ .

By definition of the fiber integral the rows are the proper direct image f! ofmorphism on M ′ which we can further decompose... use adjunction propertiesof f !, f∗, f

∗ in the category of ringed spaces...We can also prove this directly in coordinates: Let b′ ∈ B′ set b := g(b′)0.

take α ∈ (g∗f!Ωk+dM )b, that is a form defined on an open subset f−1(U) ⊂ M

where U is an arbitary small neighbourhood of b in B, which has proper supportover B.

By Ehresmanns theorem can assume there is a fiber preserving diffeomor-phism

MUΨ−→Mb × U

and coodrdinates yj : U → R. We now choose a cover of Mb admitting co-ordinates xi on each patch V ⊂ Mb and a partition of unity with compactsupports.

By definition of fiber square M ′ is canonically isomorphic to B′×gM , hencef−1(g−1(U)) is canonically isomorphic to g−1(U)×Mb. After making U smaller,we can again assume there are coordinates y′j on g−1(U).

So we have reduced the statement to the affine situation1. Let

α =∑I,J

αIJ(x, y)dxI ∧ dyJ

be a k + d-form on Rd+ × Rb proper over Rb.

g : Rb′→ Rb

be a differentiable map. Then∑|J|=k

(∫α[d]J(x, y)dx1 ∧ · · · ∧ dxd) ∧ dyJ

...1A little attention has to be paid to the orientation. Either we assume that the chart

transions of Mb are orientation preserving, or we multiply by dxI by Or(∂/∂x1, . . . , ∂/∂xd)before going to the affine case.

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4 Appendix

4.1 Exterior and Symmetric Algebras

We recall some baisc facts about exterior algebras.Let V be a finite dimensional vector space over a field K. Set

T (V ) =⊕k≥0

V ⊗k

be the Tensor algebra over V . As T is functorial in V we get a canonical actionGL(V ) on T (V ) by Algebra Homomorphisms. The action of the subgroupK× ⊂ GL(V ) induces a N0-grading on T (V ) which coincedes with the gradingwe used in the definition (by k ≥ 0).

The exterior and symmetric algebras can be constructed out of T (V ) in twoways:

4.1.1 Construction as a quotient

Let I be the two sided ideal of T (V ) generated by the elements

x⊗ y + y ⊗ x |x, y ∈ V

Then Λ(V ) := T (V )/I is the exterior algebra. As I is a graded ideal the gradingof T (V ) induces a natural grading on Λ∗(V ). It is I moreover invariant for theaction of GL(V ), so Λ(V ) carries a natural GL(V ) action, too.

We denote the image of x1 ⊗ . . .⊗ xk in Λ(V ) by x1 ∧ · · · ∧ xk.For the symmetric algebra Symk(V ) we exchange x⊗y+y⊗x with x⊗y−y⊗x

above. The same statements hold mutatis mutandis in this case.We denote the image of x1 ⊗ . . .⊗ xk in Sym(V ) by x1 · · ·xk.We now turn to the second description

4.1.2 Construction as a subspace

There is a natural Sk = AutSET(1, . . . , k) left action on T k(V ) given by

τ : x1 ⊗ . . .⊗ xk 7→ xτ1 ⊗ . . .⊗ xτk

for τ ∈ Sk.Note that if we embed Sk × Sl into Sk+l by letting Sk act on 1, . . . , k and

Sl on k + 1, . . . , l in the obvious way, then the product

T k(V )⊗ T l(V )→ T k+l(V )

is Sk × Sl-equivariant.For a character χ : Sk → Z× define an operator Pχ : T (V )→ T (V ) by

x1 ⊗ . . .⊗ xk →1k!

∑τ∈Sk

χ(τ) τ · (x1 ⊗ . . .⊗ xk).

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One checks easily that Pχ is a projection operator i.e. P 2χ = Pχ. Hence we have

a canonical decomposition:

T k(V ) = Image(Pχ)⊕Kernel(Pχ)

which is invariant for the Sk-action as well as for the GL(V ) action on T k(V )!We say S = P1 is the symmetrisation operator and A = Psign is the anti-

symmetrisation operator. In degree k ≥ 2 we have S A = A S = 0. Wedenote the image of A and S by Ak(V ) resp. Sk(V ).

Proposition 4.1. The compositions

A(V )→ T (V )→ Λ(V )

andS(V )→ T (V )→ Sym(V )

are natural isomorphisms of vectorspaces, GL(V )-representations and induceisomorphisms of Sk-representations in the k-th graded component.

These isomorphisms can be made explicit as follows: Take x1 ∧ · · · ∧ xk ∈Λ(V ) represent it by some (non-anti-symmetric) tensor x1⊗ . . .⊗ xk. Then theanti-symmetrisation A(x1 ⊗ . . . ⊗ xk) lies in A(V ) and is an inverse image forx1 ∧ · · · ∧ xk.

4.1.3 An exact sequence

Let0 −→ S −→ V −→ Q −→ 0 (5)

be an exact sequence of finite dimensional k-vector spaces. The induced seqences

0 −→ Symk(S) −→ Symk(V ) −→ Symk(Q) −→ 0

and0 −→ Λk(S) −→ Λk(V ) −→ Λk(Q) −→ 0

are exact at the right and left, but not in the middle. Instead we have natural(longer) exact sequences

0→ Symk(S)→ Symk(V )→ Symk−1(V )⊗Q→ . . . (6). . .→ Sym2(V )⊗ Λk−2(Q)→ V ⊗ Λk−1(Q)→ Λk(Q)→ 0

0→ Λk(S)→ Λk(V )→ Λk−1(V )⊗Q→ . . . (7). . .→ Λ2(V )⊗ Symk−2(Q)→ V ⊗ Symk−1(Q)→ Symk(Q)→ 0

and the dual versions

0→ Symk(S)→ V ⊗ Symk−1(S)→ Λ2(V )⊗ Symk−2(S)→ . . .

. . .→ Λk−1(V )⊗ S → Λk(V )→ Λk(Q)→ 0

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0→ Λk(S)→ V ⊗ Λk−1(S)→ Sym2(V )⊗ Λk−2(S)→ . . .

. . .→ Symk−1(V )⊗ S → Symk(V )→ Symk(Q)→ 0.

First of all we have to construct natural morphisms

Symk(V )⊗ Λl(Q)→ Symk−1(V )⊗ Λl+1(Q)

Λk(V )⊗ Syml(Q)→ Λk−1(V )⊗ Syml+1(Q)

andSymk(S)⊗ Λl(V )→ Symk−1(S)⊗ Λl+1(V )

Λk(S)⊗ Syml(V )→ Λk−1(S)⊗ Syml+1(V )

To cunstruct the first morphism we compose the tensored product of the naturalembedings

Symk(V )→ T k(V ), Λl(Q)→ T k(Q)

with the canonical map

T k(V )⊗ T l(Q)→ T k−1(V )⊗ T l+1(Q)

mapping the last V variable to its image in Q. Now we use the quotient desc-tription of Λ and Sym to end up with the required morphism. The remainingmorphism are defined similary.

The vanishing of the composition

Symk+1(V )⊗ Λl−1(Q)→ Symk(V )⊗ Λl(Q)→ Symk−1(V )⊗ Λl+1(Q)

and the exactnes can be checked in coordinates - it holds true basically becauseS A = A S = 0 and the ideals are generated in degree 2.

4.2 Multiplication Formular

Let f : M → N be a differentiable map of smooth manifolds with localcoordinates xi, i = 1 . . .m and yi, i = 1, . . . , n respectively. For two Indexsets I ⊂ [m] := 1, . . . ,m, J ⊂ [n] of cardinality k. we define the matrix∂yJ

∂xI:= (∂y

jf∂xi )j∈J,i∈I where we order the indices in ascending order.

Lemma 4.2. With this notation we have:

dyJ = dyj1 ∧ · · · ∧ dyjk =∑

I⊂[m],|I|=k

det(∂yJ

∂xI)dxI

where of course j1 < · · · < jk.

Proof. We have

dyJ =∑

i1=1..k

· · ·∑

ik=1..k

∂yj1

∂xi1dxi1 ∧ · · · ∧ ∂y

jk

∂xikdxik

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if in this sum ir = is for some r 6= s then the corresponding summand is zero.Hence the set i1, . . . , ik ⊂ [m] can be assumed to have cardinality k. There isnow an unique permutation π ∈ Sk such that iπ1 < · · · < iπk. We further notethat

dxi1 ∧ · · · ∧ dxik = sign(π)dxiπ1 ∧ · · · ∧ dxiπk .

Combining this with the previews formular yields:

dyJ =∑

I⊂[m],|I|=k

∑π∈Sk

sign(π)∂yj1

∂xiπ1. . .

∂yjk

∂xiπkdxI =

∑I⊂[m],|I|=k

det(∂yJ

∂xI)dxI

where now i = i1, . . . , ik with i1 < · · · < ik.

4.3 Localised Manifolds (obsolete)

Definition 4.3. A localised manifold M = (CM , AM ) is a manifold CM togetherwith a subset AM ⊂ CM . We call CM the craig and AM the core of M .

A morphism of localised manifolds f : (CM , AM )→ (CN , AN ) is an equica-lence class of diffeneriable mappings

Cf : D(Cf )→ N with f(AM ) ⊂ AN

where D(Cf ) is an open neighbourhood of AM in CM . Two maps Cf and Cg areequivalent iff there is an open neighbourhood V of A contained in D(Cf )∩D(Cg)and

f |V = g|V .

If f : M → M ′ and g : M ′ → M ′′ are morphisms of localised manifolds, wedefine the composition by composing representatives 2. So we get a category oflocalised Manifolds which we will denote by LDIFF.

The dimension M is the dimension of CM , which is preserved under isomor-phisms.

Question: Is the functor LDIFF → LkRSp which maps (M,A) to i−1(CM )where i : M → A fully faithful?What about HomLDIFF(Rn, 0)→ (Rn, 0))? It works?!If not, is it the restriction to mfds. with boundary f.f. ?

Yes, then we can skip all this, and define mf. with boundary as lk-ringedspaces.

Remark 4.4. If AM ⊂ CM is open, then (CM , AM ) ∼= (AM , AM ) as localisedmanifolds.

Definition 4.5. Let M = (CM , AM ) be a localised manifold, an open sub-manifold of M is the localised manifold (U,AM ∩ U) given by an open subset

2The set of definition D(Cf ) can be made smaller to ensure the latter morphism is definedon the image

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U ⊂ CM of the craig. We have an obvious morphism of localised manifolds(U,AM ∩ U)→M .

An open cover of M is a collection Uii∈I of open submanifolds such thatAUi cover AM .

Remark 4.6. We have a localisation functor

L : DIFF→ LDIFF, M 7→ (M,M), f 7→ f

which is full and faithful. And a restriction functor

A : LDIFF→ TOP, (CM , AM ) 7→ AM , f 7→ Af = f |AM : AM → AN .

which is neither faithful nor full.

Definition 4.7. A morphism f : M → N of localied manifolds is called submer-sion/immersion if there exists an representative Cf which has the correspondigproperty.

f is called injective/surjective/proper/open/closed, if Af : AM → AN hasthe correspondig property.

We call f an open embeding, if f is open and induces an isomorphism oflocalised manifolds M → (N, Image(Af )).

Definition 4.8. A manifold with bounday is a localised manifold M locallyisomorphic to Rn+ := (Rn,Rn−1 × R≥0), i.e. there is an open cover Uii∈I ofM and open embedings ϕi : Ui → Rn+ of localised manifolds.

Let (CM , AM ) be a manifold with boundary. We denote the topologicalboundary of AM ⊂ CM by ∂M . As AM is closed in CM we have ∂M ⊂ AMand ∂M is hence preserved under isomorphisms of localised manifolds. Theboundary ∂M carries a natural structure of an (ordenary) submanifold of CM .Moreover Mo := AM \ ∂M is a (orendenary) manifold, too.

We also use the notation (M,∂M) for a manifold with boundary, whichmeans M = (CM , AM ) is a manifold with boundary and ∂M ⊂ AM its bound-ary.

Definition 4.9. A submersion of manifolds with boundary is a submersion ofa localised manifolds f : M → B, where B = (B,B) is an (ordenary) manifoldand M = (CM , AM ) a manifold with bounday, such that ∂f , the restriction off to ∂M , is still submersive.

The condition that ∂f is submersive is not automatic. Let for example beD ⊂ R2 be the closed disc, then

pr1 : (R2,D)→ (R,R)

is clearly a submersion of localised manifolds but not submersive on ∂D.

Remark 4.10. In the situation of the definition, let U ⊂ M be an open subset.Then the restriction of π to U,M ∩ U is again a submersion with boundary.

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Proposition 4.11. Let M → B be a submersion of manifolds with boundary,dim(M) = m,dim(B) = b. Then locally on M there are fiber space coordinates.This means there are covers Uii∈I of M and Vii∈I of B, and open embedingszi : Ui → Rm+ and bi : Vi → Rb such that f(Ui) ⊂ Vi and the following diagramis commutative

Uizi−−−−→ Rm+yf yp

Vibi−−−−→ Rb.

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Here p is the projection to the first b corrdinates.

Definition 4.12. A sheaf on a localised manifold M is a sheaf on AM . (Is thisequivalent to sheafs on the Site of open covers?)

The structure sheaf of M is i−1(CCM )The tangent sheaf of M is i−1(TCM ) where i : AM → CM is the inclusion

and TCM the Tangent sheaf of CM .The relative tangent sheaf of a morphism f : M → N of localised manifolds

is...

References

[1] C. Ehresmann. Sur les espaces fibres differentiables. CR Acad. Sci. Paris,224:1611–1612, 1947.

[2] J.A. Wolf. Differentiable fibre spaces and mappings compatible with Rie-mannian metrics. Michigan Math. J, 11(1):65–70, 1964.

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