Torus bundles and the group cohomology of \(GL(N,\mathbb{Z})\) (Q1383647)

The authors study certain characteristic classes of flat complex vector bundles, which are known in algebraic \(K\)-theory as Borel regulator classes. They prove that if \(N\) is odd and if the structure group of a rank-\(N\) vector bundle is contained in \(GL(N, \mathbb{Z})\), then the Borel class of degree \(2N-1\) is zero. In the proof the characteristic class is written explicitly as an exact form on the base of the bundle.