The Euler Class in homological algebra

Let Gτ be the topological group of orientation preserving homeomorphisms of the circle, and Gδ the same group with the discrete topology. Motivated by the classical problem of reducing a circle bundle with structure group Gτ to a totally disconnected subgroup K⊂Gδ, and more currently, applications to mapping class groups, we analyze, in a homological algebra setting, the role played by the Topological and Discrete Euler Classes. In particular we describe the Discrete Euler Class of G, and any of its subgroups K, explicitly as a group extension. We apply our constructions to show that the values of the Discrete Euler Class are bounded on any space, and we state triviality and non-triviality conditions for its powers in the based mapping class groups.