SL2

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". More precisely, let $\pi :E\to B$ be a fiber bundle over a manifold with compact oriented fibers. If $\alpha$ is a k-form on E, then let:

$(\pi _{*}\alpha )_{b}(w_{1},\dots ,w_{k-m})=\int _{\pi ^{-1}(b)}\beta$

where $\beta$ is the induced top-form on the fiber $\pi ^{-1}(b)$; i.e., an $m$-form given by

$\beta (v_{1},\dots ,v_{m})=\alpha ({\widetilde {w_{1}}},\dots ,{\widetilde {w_{k-m}}},v_{1},\dots ,v_{m}),\quad {\widetilde {w_{i}}}{\text{ the lifts of }}w_{i}.$

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

$\pi _{*}$ is then a linear map $\Omega ^{k}(E)\to \Omega ^{k-m}(B)$, which is in fact surjective. By Stokes' formula, if the fibers have no boundaries, the map descends to de Rham cohomology:

$\pi _{*}:\operatorname {H} ^{k}(E)\to \operatorname {H} ^{k-m}(B).$

This is also called the fiber integration. Now, suppose $\pi$ is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence $0\to K\to \Omega ^{*}(E){\overset {\pi _{*}}{\to }}\Omega ^{*}(B)\to 0$, K the kernel, which leads to a long exact sequence, using $\operatorname {H} ^{k}(B)\simeq \operatorname {H} ^{k+m}(K)$:

$\dots \rightarrow \operatorname {H} ^{k}(B){\overset {\delta }{\to }}\operatorname {H} ^{k+m+1}(B){\overset {\pi ^{*}}{\rightarrow }}\operatorname {H} ^{k+m+1}(E){\overset {\pi _{*}}{\rightarrow }}\operatorname {H} ^{k+1}(B)\rightarrow \dots$,

called the Gysin sequence.

Example

Let $\pi :M\times [0,1]\to M$ be an obvious projection. For simplicity, assume $M=\mathbb {R} ^{n}$ with coordinates $x_{j}$ and consider a k-form:

$\alpha =f\,dx_{i_{1}}\wedge \dots \wedge dx_{i_{k}}+g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}}.$

Then, at each point in M,

$\pi _{*}(\alpha )=\pi _{*}(g\,dt\wedge dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}})=\left(\int _{0}^{1}g(\cdot ,t)\,dt\right)\,{dx_{j_{1}}\wedge \dots \wedge dx_{j_{k-1}}}.$

From this the next formula follows easily: if $\alpha$ is any k-form on $M\times I,$

$\pi _{*}(d\alpha )=\alpha _{1}-\alpha _{0}-d\pi _{*}(\alpha )$

where $\alpha _{i}$ is the restriction of $\alpha$ to $M\times \{i\}$. This formula is a special case of Stokes' formula. As an application of this, let $f:M\times [0,1]\to N$ be a smooth map (thought of as a homotopy). Then the composition $h=\pi _{*}\circ f^{*}$ is a homotopy operator:

$d\circ h+h\circ d=f_{1}^{*}-f_{0}^{*}:\Omega ^{k}(N)\to \Omega ^{k}(M),$

which implies $f_{1},f_{0}$ induces the same map on cohomology. For example, let U be an open ball with center at the origin and let $f_{t}:U\to U,x\mapsto tx$. Then $\operatorname {H} ^{k}(U)=\operatorname {H} ^{k}(pt)$, the fact known as the Poincaré lemma.

See also

• Gysin homomorphism
• Transgression map

References

• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004