Integration along fibers

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In differential geometry, the integration along fibers of a k-form yields a ( k − m ) {\displaystyle (k-m)} {\displaystyle (k-m)}-form where m is the dimension of the fiber, via "integration". It is also called the fiber integration.

Definition

Let π : E → B {\displaystyle \pi :E\to B} {\displaystyle \pi :E\to B} be a fiber bundle over a manifold with compact oriented fibers. If α {\displaystyle \alpha } {\displaystyle \alpha } 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 {\displaystyle (\pi _{*}\alpha )_{b}(w_{1},\dots ,w_{k-m})=\int _{\pi ^{-1}(b)}\beta } {\displaystyle (\pi _{*}\alpha )_{b}(w_{1},\dots ,w_{k-m})=\int _{\pi ^{-1}(b)}\beta }

where β {\displaystyle \beta } {\displaystyle \beta } is the induced top-form on the fiber π − 1 ( b ) {\displaystyle \pi ^{-1}(b)} {\displaystyle \pi ^{-1}(b)}; i.e., an m {\displaystyle m} {\displaystyle m}-form given by: with w i ~ {\displaystyle {\widetilde {w_{i}}}} {\displaystyle {\widetilde {w_{i}}}} lifts of w i {\displaystyle w_{i}} {\displaystyle w_{i}} to E {\displaystyle E} {\displaystyle E},

β ( v 1 , … , v m ) = α ( v 1 , … , v m , w 1 ~ , … , w k − m ~ ) . {\displaystyle \beta (v_{1},\dots ,v_{m})=\alpha (v_{1},\dots ,v_{m},{\widetilde {w_{1}}},\dots ,{\widetilde {w_{k-m}}}).} {\displaystyle \beta (v_{1},\dots ,v_{m})=\alpha (v_{1},\dots ,v_{m},{\widetilde {w_{1}}},\dots ,{\widetilde {w_{k-m}}}).}

(To see b ↦ ( π ∗ α ) b {\displaystyle b\mapsto (\pi _{*}\alpha )_{b}} {\displaystyle b\mapsto (\pi _{*}\alpha )_{b}} is smooth, work it out in coordinates; cf. an example below.)

Then π ∗ {\displaystyle \pi _{*}} {\displaystyle \pi _{*}} is a linear map Ω k ( E ) → Ω k − m ( B ) {\displaystyle \Omega ^{k}(E)\to \Omega ^{k-m}(B)} {\displaystyle \Omega ^{k}(E)\to \Omega ^{k-m}(B)}. By Stokes' formula, if the fibers have no boundaries(i.e. [ d , ∫ ] = 0 {\displaystyle [d,\int ]=0} {\displaystyle [d,\int ]=0}), the map descends to de Rham cohomology:

π ∗ : H k ⁡ ( E ; R ) → H k − m ⁡ ( B ; R ) . {\displaystyle \pi _{*}:\operatorname {H} ^{k}(E;\mathbb {R} )\to \operatorname {H} ^{k-m}(B;\mathbb {R} ).} {\displaystyle \pi _{*}:\operatorname {H} ^{k}(E;\mathbb {R} )\to \operatorname {H} ^{k-m}(B;\mathbb {R} ).}

This is also called the fiber integration.

Now, suppose π {\displaystyle \pi } {\displaystyle \pi } 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 {\displaystyle 0\to K\to \Omega ^{*}(E){\overset {\pi _{*}}{\to }}\Omega ^{*}(B)\to 0} {\displaystyle 0\to K\to \Omega ^{*}(E){\overset {\pi _{*}}{\to }}\Omega ^{*}(B)\to 0}, K the kernel, which leads to a long exact sequence, dropping the coefficient R {\displaystyle \mathbb {R} } {\displaystyle \mathbb {R} } and using H k ⁡ ( B ) ≃ H k + m ⁡ ( K ) {\displaystyle \operatorname {H} ^{k}(B)\simeq \operatorname {H} ^{k+m}(K)} {\displaystyle \operatorname {H} ^{k}(B)\simeq \operatorname {H} ^{k+m}(K)}:

⋯ → H k ⁡ ( B ) → δ H k + m + 1 ⁡ ( B ) → π ∗ H k + m + 1 ⁡ ( E ) → π ∗ H k + 1 ⁡ ( B ) → ⋯ {\displaystyle \cdots \rightarrow \operatorname {H} ^{k}(B){\overset {\delta }{\to }}\operatorname {H} ^{k+m+1}(B){\overset {\pi ^{*}}{\rightarrow }}\operatorname {H} ^{k+m+1}(E){\overset {\pi _{*}}{\rightarrow }}\operatorname {H} ^{k+1}(B)\rightarrow \cdots } {\displaystyle \cdots \rightarrow \operatorname {H} ^{k}(B){\overset {\delta }{\to }}\operatorname {H} ^{k+m+1}(B){\overset {\pi ^{*}}{\rightarrow }}\operatorname {H} ^{k+m+1}(E){\overset {\pi _{*}}{\rightarrow }}\operatorname {H} ^{k+1}(B)\rightarrow \cdots },

called the Gysin sequence.

Example

Let π : M × [ 0 , 1 ] → M {\displaystyle \pi :M\times [0,1]\to M} {\displaystyle \pi :M\times [0,1]\to M} be an obvious projection. First assume M = R n {\displaystyle M=\mathbb {R} ^{n}} {\displaystyle M=\mathbb {R} ^{n}} with coordinates x j {\displaystyle x_{j}} {\displaystyle 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 . {\displaystyle \alpha =f\,dx_{i_{1}}\wedge \dots \wedge dx_{i_{k}}+g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}}.} {\displaystyle \alpha =f\,dx_{i_{1}}\wedge \dots \wedge dx_{i_{k}}+g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{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 . {\displaystyle \pi _{*}(\alpha )=\pi _{*}(g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}})=\left(\int _{0}^{1}g(\cdot ,t)\,dt\right)\,{dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}}}.} {\displaystyle \pi _{*}(\alpha )=\pi _{*}(g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}})=\left(\int _{0}^{1}g(\cdot ,t)\,dt\right)\,{dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}}}.}[1]

From this local calculation, the next formula follows easily (see Poincaré_lemma#Direct_proof): if α {\displaystyle \alpha } {\displaystyle \alpha } is any k-form on M × [ 0 , 1 ] , {\displaystyle M\times [0,1],} {\displaystyle M\times [0,1],}

π ∗ ( d α ) = α 1 − α 0 − d π ∗ ( α ) {\displaystyle \pi _{*}(d\alpha )=\alpha _{1}-\alpha _{0}-d\pi _{*}(\alpha )} {\displaystyle \pi _{*}(d\alpha )=\alpha _{1}-\alpha _{0}-d\pi _{*}(\alpha )}

where α i {\displaystyle \alpha _{i}} {\displaystyle \alpha _{i}} is the restriction of α {\displaystyle \alpha } {\displaystyle \alpha } to M × { i } {\displaystyle M\times \{i\}} {\displaystyle M\times \{i\}}.

As an application of this formula, let f : M × [ 0 , 1 ] → N {\displaystyle f:M\times [0,1]\to N} {\displaystyle f:M\times [0,1]\to N} be a smooth map (thought of as a homotopy). Then the composition h = π ∗ ∘ f ∗ {\displaystyle h=\pi _{*}\circ f^{*}} {\displaystyle h=\pi _{*}\circ f^{*}} is a homotopy operator (also called a chain homotopy):

d ∘ h + h ∘ d = f 1 ∗ − f 0 ∗ : Ω k ( N ) → Ω k ( M ) , {\displaystyle d\circ h+h\circ d=f_{1}^{*}-f_{0}^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M),} {\displaystyle d\circ h+h\circ d=f_{1}^{*}-f_{0}^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M),}

which implies f 1 , f 0 {\displaystyle f_{1},f_{0}} {\displaystyle f_{1},f_{0}} induce 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 {\displaystyle f_{t}:U\to U,x\mapsto tx} {\displaystyle f_{t}:U\to U,x\mapsto tx}. Then H k ⁡ ( U ; R ) = H k ⁡ ( p t ; R ) {\displaystyle \operatorname {H} ^{k}(U;\mathbb {R} )=\operatorname {H} ^{k}(pt;\mathbb {R} )} {\displaystyle \operatorname {H} ^{k}(U;\mathbb {R} )=\operatorname {H} ^{k}(pt;\mathbb {R} )}, the fact known as the Poincaré lemma.

Projection formula

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 ) {\displaystyle \alpha |_{\pi ^{-1}(b)}} {\displaystyle \alpha |_{\pi ^{-1}(b)}} has compact support for each b in B. We write Ω v c ∗ ( E ) {\displaystyle \Omega _{vc}^{*}(E)} {\displaystyle \Omega _{vc}^{*}(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 ) . {\displaystyle \pi _{*}:\Omega _{vc}^{*}(E)\to \Omega ^{*}(B).} {\displaystyle \pi _{*}:\Omega _{vc}^{*}(E)\to \Omega ^{*}(B).}

The following is known as the projection formula.[2] We make Ω v c ∗ ( E ) {\displaystyle \Omega _{vc}^{*}(E)} {\displaystyle \Omega _{vc}^{*}(E)} a right Ω ∗ ( B ) {\displaystyle \Omega ^{*}(B)} {\displaystyle \Omega ^{*}(B)}-module by setting α ⋅ β = α ∧ π ∗ β {\displaystyle \alpha \cdot \beta =\alpha \wedge \pi ^{*}\beta } {\displaystyle \alpha \cdot \beta =\alpha \wedge \pi ^{*}\beta }.

Proposition—Let π : E → B {\displaystyle \pi :E\to B} {\displaystyle \pi :E\to B} be an oriented vector bundle over a manifold and π ∗ {\displaystyle \pi _{*}} {\displaystyle \pi _{*}} the integration along the fiber. Then

  1. π ∗ {\displaystyle \pi _{*}} {\displaystyle \pi _{*}} is Ω ∗ ( B ) {\displaystyle \Omega ^{*}(B)} {\displaystyle \Omega ^{*}(B)}-linear; i.e., for any form β on B and any form α on E with vertical-compact support,
    π ∗ ( α ∧ π ∗ β ) = π ∗ α ∧ β . {\displaystyle \pi _{*}(\alpha \wedge \pi ^{*}\beta )=\pi _{*}\alpha \wedge \beta .} {\displaystyle \pi _{*}(\alpha \wedge \pi ^{*}\beta )=\pi _{*}\alpha \wedge \beta .}
  2. If B is oriented as a manifold, then for any form α on E with vertical compact support and any form β on B with compact support,
    ∫ E α ∧ π ∗ β = ∫ B π ∗ α ∧ β {\displaystyle \int _{E}\alpha \wedge \pi ^{*}\beta =\int _{B}\pi _{*}\alpha \wedge \beta } {\displaystyle \int _{E}\alpha \wedge \pi ^{*}\beta =\int _{B}\pi _{*}\alpha \wedge \beta }.

Proof: 1. Since the assertion is local, we can assume π is trivial: i.e., π : E = B × R n → B {\displaystyle \pi :E=B\times \mathbb {R} ^{n}\to B} {\displaystyle \pi :E=B\times \mathbb {R} ^{n}\to B} is a projection. Let t j {\displaystyle t_{j}} {\displaystyle t_{j}} be the coordinates on the fiber. If α = g d t 1 ∧ ⋯ ∧ d t n ∧ π ∗ η {\displaystyle \alpha =g\,dt_{1}\wedge \cdots \wedge dt_{n}\wedge \pi ^{*}\eta } {\displaystyle \alpha =g\,dt_{1}\wedge \cdots \wedge dt_{n}\wedge \pi ^{*}\eta }, then, since π ∗ {\displaystyle \pi ^{*}} {\displaystyle \pi ^{*}} 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 . {\displaystyle \pi _{*}(\alpha \wedge \pi ^{*}\beta )=\left(\int _{\mathbb {R} ^{n}}g(\cdot ,t_{1},\dots ,t_{n})dt_{1}\dots dt_{n}\right)\eta \wedge \beta =\pi _{*}(\alpha )\wedge \beta .} {\displaystyle \pi _{*}(\alpha \wedge \pi ^{*}\beta )=\left(\int _{\mathbb {R} ^{n}}g(\cdot ,t_{1},\dots ,t_{n})dt_{1}\dots dt_{n}\right)\eta \wedge \beta =\pi _{*}(\alpha )\wedge \beta .}

Similarly, both sides are zero if α does not contain dt. The proof of 2. is similar. ◻ {\displaystyle \square } {\displaystyle \square }

See also

Notes

  1. If α = g d t ∧ d x j 1 ∧ ⋯ ∧ d x j k − 1 {\displaystyle \alpha =g\,dt\wedge dx_{j_{1}}\wedge \cdots \wedge dx_{j_{k-1}}} {\displaystyle \alpha =g\,dt\wedge dx_{j_{1}}\wedge \cdots \wedge dx_{j_{k-1}}}, then, at a point b of M, identifying ∂ x j {\displaystyle \partial _{x_{j}}} {\displaystyle \partial _{x_{j}}}'s with their lifts, we have:
    β ( ∂ t ) = α ( ∂ t , ∂ x j 1 , … , ∂ x j k − 1 ) = g ( b , t ) {\displaystyle \beta (\partial _{t})=\alpha (\partial _{t},\partial _{x_{j_{1}}},\dots ,\partial _{x_{j_{k-1}}})=g(b,t)} {\displaystyle \beta (\partial _{t})=\alpha (\partial _{t},\partial _{x_{j_{1}}},\dots ,\partial _{x_{j_{k-1}}})=g(b,t)}
    and so
    π ∗ ( α ) b ( ∂ x j 1 , … , ∂ x j k − 1 ) = ∫ [ 0 , 1 ] β = ∫ 0 1 g ( b , t ) d t . {\displaystyle \pi _{*}(\alpha )_{b}(\partial _{x_{j_{1}}},\dots ,\partial _{x_{j_{k-1}}})=\int _{[0,1]}\beta =\int _{0}^{1}g(b,t)\,dt.} {\displaystyle \pi _{*}(\alpha )_{b}(\partial _{x_{j_{1}}},\dots ,\partial _{x_{j_{k-1}}})=\int _{[0,1]}\beta =\int _{0}^{1}g(b,t)\,dt.}
    Hence, π ∗ ( α ) b = ( ∫ 0 1 g ( b , t ) d t ) d x j 1 ∧ ⋯ ∧ d x j k − 1 . {\displaystyle \pi _{*}(\alpha )_{b}=\left(\int _{0}^{1}g(b,t)\,dt\right)dx_{j_{1}}\wedge \cdots \wedge dx_{j_{k-1}}.} {\displaystyle \pi _{*}(\alpha )_{b}=\left(\int _{0}^{1}g(b,t)\,dt\right)dx_{j_{1}}\wedge \cdots \wedge dx_{j_{k-1}}.} By the same computation, π ∗ ( α ) = 0 {\displaystyle \pi _{*}(\alpha )=0} {\displaystyle \pi _{*}(\alpha )=0} if dt does not appear in α.
  2. Bott & Tu 1982, Proposition 6.15.; note they use a different definition than the one here, resulting in change in sign.

References