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Integration of simplicial forms and Deligne cohomology. (English) Zbl 1101.14024

The purpose of the paper is to construct an integration map the fiber for the Deligne cohomology, and study its basic properties. In fact, two approaches to the integration along the fiber are given by using two models of Deligne cohomology. The first one is a simplicial model introduced by the first author and F. W. Kamber [Commun. Math. Phys. 253, 253–282 (2005; Zbl 1079.57023)]. That model considers the geometric realization \(| N{\mathcal U}| \) of a nerve of a good open covering \(\mathcal U\) of a manifold \(Z\), and then a class in the Deligne cohomology \(H_{\mathcal D}^{l+1}(Z,{\mathbb Z})\) is represented by a simplicial form \(\omega\in\Omega^l(| N{\mathcal U}| )\). For a fiber bundle \(\pi:Y\to Z\) with compact, oriented \(n\)-dimensional fibers and suitable open coverings \(\mathcal V\) and \(\mathcal U\) of \(Y\) and \(Z\), the authors define the integration along the fibers as a map \[ \int_{[Y/Z]}:\Omega^{*+n}(| N{\mathcal V}| )\to\Omega^*(| N{\mathcal U}| ) \] which satisfies the Stokes type formula \[ \int_{[Y/Z]}d\omega=\int_{[\partial Y/Z]}\omega+(-1)^n\,d\int_{[Y/Z]}\omega\;; \] in particular, when \(\partial Y=\emptyset\), it induces a map \(\pi_!:H_{\mathcal D}^{*+n}(Y,{\mathbb Z})\to H_{\mathcal D}^*(Z,{\mathbb Z})\) independent of the all choices and compatible with the usual integration along the fibers. The proof of these properties leads to the second approach to the integration along the fibers, which uses a more combinatorial model of Deligne cohomology: each cohomology class is represented by a simplicial form in the triangulated nerve associated to a triangulation of the bundle. The authors also describe the product in Deligne cohomology via a product \(\tilde\wedge\) on the first cochain model, which is well adapted to the integration along the fibers: \[ \left(\int_{[Y/Z]}\omega_1\right)\tilde\wedge\omega_2 =\int_{[Y/Z]}(\omega_1\tilde\wedge\pi^*\omega_2)\;. \] Among all approaches to this subject in the literature, this one has the advantages of involving local data, and being defined for general fiber bundles (not only for products).

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
55N35 Other homology theories in algebraic topology
58A10 Differential forms in global analysis

Citations:

Zbl 1079.57023
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