Geodesic
Local formulae for characteristic
Sbornik. Mathematics, Tome 199 (2008) no. 10, pp. 1547-1577 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $P$ be a principal $\mathrm{GL}_n$-bundle over a smooth compact manifold $X$ given by a finite atlas $\mathscr U=\{U_\alpha\}$ with transition functions $g_{\alpha\beta}$. A method is described for constructing the cocycles corresponding to the Chern classes of the bundle $P$ in the Čech complex with coefficients in the sheaf of de Rham forms on the manifold associated with the atlas $\mathscr U$. It is proved that for every rational characteristic class $c$ of the bundle $P$ there exists a cocycle in the aforementioned complex depending only on the gluing functions and corresponding to the class $c$ under the canonical identification of the cohomologies of the complex and the de Rham cohomologies of the manifold $X$ (a simple algorithm is given that enables one to calculate this cocycle explicitly). One of the key ideas leading to these results is the idea of using the notion of a twisting cochain for constructing the cocycles. Bibliography: 14 titles. 
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G. I. Sharygin. Local formulae for characteristic. Sbornik. Mathematics, Tome 199 (2008) no. 10, pp. 1547-1577. http://geodesic.mathdoc.fr/item/SM_2008_199_10_a6/

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