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 Mat. Sb., 2008, Volume 199, Number 10, Pages 127–158 (Mi msb4524)

Local formulae for characteristic classes of a principal $\mathrm{GL}_n$-bundle

G. I. Sharygin

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: 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.

DOI: https://doi.org/10.4213/sm4524

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English version:
Sbornik: Mathematics, 2008, 199:10, 1547–1577

Bibliographic databases:

UDC: 515.145.253+512.7
MSC: Primary 55R40; Secondary 55N30, 55U15, 57R20, 58A12
Received: 26.02.2008 and 17.06.2008

Citation: G. I. Sharygin, “Local formulae for characteristic classes of a principal $\mathrm{GL}_n$-bundle”, Mat. Sb., 199:10 (2008), 127–158; Sb. Math., 199:10 (2008), 1547–1577

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb4524
• https://doi.org/10.4213/sm4524
• http://mi.mathnet.ru/eng/msb/v199/i10/p127

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This publication is cited in the following articles:
1. Kubarski J., “Cyclic Cech-Hochschild Bicomplex”, Miskolc Math. Notes, 14:2 (2013), 713–720
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