Abstract
Apparently a lost theorem of Thurston [1] states that the cube of the Euler class [Formula: see text] is zero where [Formula: see text] 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 [Formula: see text] where [Formula: see text] is the orientation preserving [Formula: see text]-diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class [Formula: see text] in fact are nonzero in cohomology with integer coefficients. We also give a short proof of Morita’s theorem [5].
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