On powers of the Euler class for flat circle bundles

Apparently a lost theorem of Thurston [1] states that the cube of the Euler class e3 H6(BDiff Í(S1); âš) is zero where DiffÍ(S1) is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem [5] that the powers of the Euler class are nonzero in Hâ-(BDiff(S1); âš) where Diff(S1) is the orientation preserving C∞-diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class ek Hâ-(BDiff Í(S1);) in fact are nonzero in cohomology with integer coefficients. We also give a short proof of Morita's theorem [5]. ;copy 2018 World Scientific Publishing Company.