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Some problems of integration in fiber bundles. (English. Russian original) Zbl 0790.58003
Math. Notes 51, No. 5, 501-509 (1992); translation from Mat. Zametki 51, No. 5, 114-124 (1992).
Let $$E\to B$$ be a smooth fiber bundle, and consider the iterated integral $$\int_{h'}\{\int_{h(x)} \omega\}$$, where $$\omega \in \Lambda^ k(E)$$ is a closed differential form, $$h'$$ is a homology class of $$B$$ and $$h(x)$$ is a homology class of the fiber $$F_ x$$, depending smoothly on $$x$$. In the authors’ work on constructing global asymptotic expansions of solutions of (partial) differential equations on complex manifolds the question arose of when such an iterated integral can be written as a multiple integral $$\int_{H}\omega$$, where $$H$$ is a homology class of $$E$$. This is a version of Fubini’s theorem, and one cannot always find such a representation when the dimension of the cycle $$h(x)$$ (representing that class) is less than the dimension of the fiber.
In this paper the authors show that $$\int_ H\omega$$ can be understood as an integral over $$H = h' \times h(x)$$, which is an element of the $$E^ 2_{p,q}$$-term of a homology spectral sequence of the bundle, $$E^ 2_{p,q} \cong H_ p(B,{\mathcal L};H_ q(F,Y))$$, where $$(F,Y)$$ is a fiber of a certain fibration of a pair $$(E,X)\to B$$ and $$\mathcal L$$ is a subcomplex of $$B$$. In this context $$\omega$$ is integrated as an element of the $$E^{p,q}_ 2$$-term of a cohomology spectral sequence. The main result is that the integral defines a duality pairing of these two spectral sequences. This does not seem to be a consequence of (generalized) Hodge theory, or of the usual theory of ramified differential forms.
##### MSC:
 58A10 Differential forms in global analysis 58J37 Perturbations of PDEs on manifolds; asymptotics 32C30 Integration on analytic sets and spaces, currents
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##### References:
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