A simplicial formula and bound for the Euler class

LetG be the discrete group of orientation preserving diffeomorphisms of the circle. An explicit simplicial formula on the level of the bar construction is given for the Euler Class of a circle bundle with structure groupG. An upper bound for the Euler Class is obtained which, when the base space of the bundle is a closed orientable surface, reduces to that of J. Wood. An invariant of circle bundles, complexity, is defined which “detects” the upper bound. Partially supported by a grant from the N.S.F.