On powers of the Euler class for flat circle bundles

Apparently a lost theorem of Thurston states that the cube of the Euler class $e^3\in H^6(BDif{f}_{\omega }^{\delta }(S^1);\mathbb{Q})$ is zero where $Dif{f}_{\omega }^{\delta }(S^1)$ is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita’s theorem that the powers of the Euler class are nonzero in $H^{\ast }(BDif{f}^{\delta }(S^1);\mathbb{Q})$ where $Dif{f}^{\delta }(S^1)$ is the orientation preserving $C^{\mathrm{\infty }}$- diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class $e^k\in H^{\ast }(BDif{f}_{\omega }^{\delta }(S^1);\mathbb{Z})$ in fact are nonzero in cohomology with integer coefficients. We also give a short proof of Morita’s theorem.