Local formulae for characteristic

G. I. Sharygin

Sbornik. Mathematics, Tome 199 (2008) no. 10, pp. 1547-1577

Résumé

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.