In differential geometry, the integration along fibers of a k-form yields a ( k − m ) -form where m is the dimension of the fiber, via "integration".
Let π : E → B be a fiber bundle over a manifold with compact oriented fibers. If α is a k-form on E, then for tangent vectors wi's at b, let
( π ∗ α ) b ( w 1 , … , w k − m ) = ∫ π − 1 ( b ) β where β is the induced top-form on the fiber π − 1 ( b ) ; i.e., an m -form given by: with w i ~ the lifts of w i to E,
β ( v 1 , … , v m ) = α ( v 1 , … , v m , w 1 ~ , … , w k − m ~ ) . (To see b ↦ ( π ∗ α ) b is smooth, work it out in coordinates; cf. an example below.)
Then π ∗ is a linear map Ω k ( E ) → Ω k − m ( B ) . By Stokes' formula, if the fibers have no boundaries, the map descends to de Rham cohomology:
π ∗ : H k ( E ; R ) → H k − m ( B ; R ) . This is also called the fiber integration.
Now, suppose π is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence 0 → K → Ω ∗ ( E ) → π ∗ Ω ∗ ( B ) → 0 , K the kernel, which leads to a long exact sequence, dropping the coefficient R and using H k ( B ) ≃ H k + m ( K ) :
⋯ → H k ( B ) → δ H k + m + 1 ( B ) → π ∗ H k + m + 1 ( E ) → π ∗ H k + 1 ( B ) → ⋯ ,
called the Gysin sequence.
Let π : M × [ 0 , 1 ] → M be an obvious projection. First assume M = R n with coordinates x j and consider a k-form:
α = f d x i 1 ∧ ⋯ ∧ d x i k + g d t ∧ d x j 1 ∧ ⋯ ∧ d x j k − 1 . Then, at each point in M,
π ∗ ( α ) = π ∗ ( g d t ∧ d x j 1 ∧ ⋯ ∧ d x j k − 1 ) = ( ∫ 0 1 g ( ⋅ , t ) d t ) d x j 1 ∧ ⋯ ∧ d x j k − 1 . From this local calculation, the next formula follows easily: if α is any k-form on M × I ,
π ∗ ( d α ) = α 1 − α 0 − d π ∗ ( α ) where α i is the restriction of α to M × { i } .
As an application of this formula, let f : M × [ 0 , 1 ] → N be a smooth map (thought of as a homotopy). Then the composition h = π ∗ ∘ f ∗ is a homotopy operator:
d ∘ h + h ∘ d = f 1 ∗ − f 0 ∗ : Ω k ( N ) → Ω k ( M ) , which implies f 1 , f 0 induces the same map on cohomology, the fact known as the homotopy invariance of de Rham cohomology. As a corollary, for example, let U be an open ball in Rn with center at the origin and let f t : U → U , x ↦ t x . Then H k ( U ; R ) = H k ( p t ; R ) , the fact known as the Poincaré lemma.
Given a vector bundle π : E → B over a manifold, we say a differential form α on E has vertical-compact support if the restriction α | π − 1 ( b ) has compact support for each b in B. We written Ω v c ∗ ( E ) for the vector space of differential forms on E with vertical-compact support. If E is oriented as a vector bundle, exactly as before, we can define the integration along the fiber:
π ∗ : Ω v c ∗ ( E ) → Ω ∗ ( B ) . The following is known as the projection formula. We make Ω v c ∗ ( E ) a right Ω ∗ ( B ) -module by setting α ⋅ β = α ∧ π ∗ β .
Proof: 1. Since the assertion is local, we can assume π is trivial: i.e., π : E = B × R n → B is a projection. Let t j be the coordinates on the fiber. If α = g d t 1 ∧ ⋯ ∧ d t n ∧ π ∗ η , then, since π ∗ is a ring homomorphism,
π ∗ ( α ∧ π ∗ β ) = ( ∫ R n g ( ⋅ , t 1 , … , t n ) d t 1 … d t n ) η ∧ β = π ∗ ( α ) ∧ β . Similarly, both sides are zero if α does not contain dt. The proof of 2. is similar. ◻
