Vanishing powers of the Euler class

Let H=Homeo+S1 be the discrete group of orientation preserving homeomorphisms of the circle S1, and let G be a subgroup. In this work the Euler Class [e(G)] for discrete G-bundles is studied in order to determine the range of powers for which [e(G)] vanishes. A new invariant is introduced, the orbit class of G, as well as an integer associated to it, its holonomy. The first vanishing power of the Euler Class results from the non-vanishing of the holonomy of the orbit class. The highest non-vanishing power of the Euler Class is a consequence of the vanishing of the holonomy. Applications focus on the Based Mapping Class Groups, Mg. These can be represented as subgroups of H which exhibit a certain degree of transitivity of their actions depending on their genus g. This leads to a vanishing/non-vanishing result for the powers of the Euler Class of the Mgʹs. The vanishing theorem and its application to Mapping Class Groups: the k-th power of the Euler Class [ek(Mg)] is zero for k⩾g, is described in this article. The non-vanishing theorem will appear in a sequel.