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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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