User:TakuyaMurata/Integration in fiber

In differential geometry, the integration in fiber 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 vector bundle over a manifold with compact fibers. If $$\alpha$$ is a k-form on E, then let:
 * $$(\pi_* \beta)_b(w_1, \dots, w_{k-m}) = \int_{\pi^{-1}(b)} \beta$$

where $$\beta$$ is a form on a fiber; i.e., a $$m$$-form given by
 * $$\beta(v_1, \dots, v_m) = \alpha(\widetilde{w_1}, \dots, \widetilde{w_{k-m}}, v_1, \dots, v_m).$$

$$\pi_*$$ is then a linear map from $$\Omega^k(E) \to \Omega^{k-m}(B)$$. Since it clearly commutes with d, the map descends to de Rham cohomology:
 * $$\pi_*: H^k(E) \to H^{k-m}(B).$$